ast802 advanced topics of theoretical cosmologyjyoo/class/2019/ast802.pdf · 2019. 9. 12. ·...

AST802 Advanced Topics of Theoretical Cosmology

Jaiyul Yoo

Center for Theoretical Astrophysics and Cosmology, Institute for Computational SciencePhysics Institute, University of Zurich, Winterthurerstrasse 190, CH-8057, Zurich, Switzerland

e-mail: [email protected]

September 12, 2019

The purpose of this intensive block course is to introduce active research fields and provide essentialingredients and tools for research in theoretical cosmology. The course will focus on large-scalestructure probes of inflationary cosmology. The prerequisite for this course is AST513 TheoreticalCosmology with good understanding of general relativity.

• Two-hour long lectures (10am to noon), every day for two weeks (from September 2nd to 13th)• Lecture room: Y36J33 at UZH, ECTS 4 point, course number 4332, no final exam• Online registration for UZH students, ETH students should directly register at UZH• Audit is also welcome, but send me email before the class

Contents

1 Spherical Collapse Model in Cosmology 11.1 Spherical Collapse Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Dark Matter Halo Mass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Halo Mass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Halo Mass Functions in Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.4 Astrophysical Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 From Probability Functional to Correlation Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.1 Density Probability Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.2 Mildly Non-Gaussian Halo Mass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.3 Peak-Background Split and Statistics of Thresholded Region . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Beyond the Spherical Model: Peak Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.1 Basic Idea of Peak Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.2 Multivariate Gaussian Joint Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4.3 Gaussian Peaks Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.4 Conditional Probability for Ellipticity and Prolateness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Newtonian Perturbation Theory and Galaxy Bias 132.1 Standard Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Basic Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Recurrence Relation and Third Order Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.3 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1.4 Unified Treatment of the Standard Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Galaxy Bias Primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.1 Summary of Spherical Collapse Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Renormalized Galaxy Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2.3 Gravitational Tidal Tensor Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.2.4 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Lagrangian Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.1 Basic Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.3.2 Zel’dovich Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3.3 Resummation in LPT: One-Loop Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.3.4 Zel’dovich Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.5 Galaxy Bias in the Lagrangian Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.3.6 Perturbation Theory and Nonlocal Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3.7 Examples of Bias Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3.8 Multi-Point Propagator for Matter and Biased Tracers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.9 The Recursion Relation in Lagrangian Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Redshift-Space Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.4.1 Large-Scale Velocity Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4.2 Gaussian Streaming Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.4.3 Complete Treatment of Redshift-space Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4.4 Improved Model of the Redshift-Space Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.5 Summary of Other Phenomenoligical Models of FoG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Effective Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.1 Basic Formalsim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.5.2 Effective Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.5.3 One-Loop Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Nonlinear Relativistic Dynamics: ADM Formalism and its Cosmological Applications 383.1 Arnowitt-Deser-Misner (ADM) Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1.2 Hamiltonian Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1.3 FRW Metric and Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.1.4 Relations to ADM Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2 Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.1 General Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.2 Tetrad Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.3 Frame Choice for Energy-Momentum Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.4 Normal Frame and Its Relation to Energy Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3 Fully Nonlinear Einstein Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.1 ADM Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.2 Second-Order ADM Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3.3 Decomposed Perturbation Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.3.4 Closed Form Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Standard Inflationary Models 474.1 Standard Inflationary Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.1.1 Scalar Field Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.2 Background Relation and Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.3 de-Sitter Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.4 Slow-Roll Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.5 Linear-Order Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.6 Quadratic Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.1.7 Quantum Fluctuations in Quadratic Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.1.8 In-In Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.1.9 Quantum Fluctuations in Cubic Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.1.10 Lyth Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Primordial Non-Gaussianity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3 Miscellaneous Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.4 δN -Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Applications of the Effective Field Theory 615.1 Basics of EFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.1.1 Simple Toy Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2 Effective Field Theory of Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2.1 Weinberg 2008: Effective Field Theory for Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.2.2 Goldstone Action in Inflationary Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2.3 Cheung, Creminelli, Fitzpatrick, Kaplan, Senatore 2008: Effective Field Theory of Inflation . . . . . . . . . 64

6 Modification of Gravity and Dark Energy Models 656.1 Dark Energy Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2 Brans-Dicke Scalar-Tensor Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.3 Scalar-Tensor Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 666.4 Effective Field Theory of Quintessence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.5 Weinberg Approach to Cosmic Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 676.6 Effective Field Theory of Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.7 A Unifying Description of Dark Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7 CMB Temperature Anisotropies 717.1 Collisionless Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.1.1 Geodesic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717.1.2 Collisionless Boltzmann Equation for Massless Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 727.1.3 Convention for Multipole Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.1.4 Multipole Expansion of the Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737.1.5 Massless Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

ii

7.2 Collisions of the Baryon-Photon Fluid: Thompson Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.2.1 Collisional Boltzmann Equation for Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.2.2 Collisional Boltzmann Equation for Baryons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.3 Initial Conditions for the Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.4 Observed CMB Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

7.4.1 Acoustic Oscillation: Tight-Coupling Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.4.2 Diffusion Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787.4.3 Free Streaming: Line-of-Sight Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797.4.4 CMB Power Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

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1 Spherical Collapse Model in Cosmology

1.1 Spherical Collapse ModelA simple spherical collapse model was developed long time ago to serve as a toy model for dark matter halo formation (see Peebles(1980) for details). The idea is that a slightly overdense region in a flat universe evolves as if the region were a closed universe,such that it expands almost together with the background universe but eventually turns around and collapses. The overdense regiondescribed by the closed universe would collapse to a singularity, but in reality it virializes and stops contracting. By using theanalytical solutions for the two universes, we can readily derive many useful relations about the evolution of such overdense regions.

Einstein-de Sitter Universe

A flat homogeneous universe dominated by pressureless matter is called the Einstein-de Sitter Universe:

H2 =8πG

3ρm , ρm ∝

1

a3. (1.1)

This simple model is indeed a good approximation to the late Universe, before dark energy starts to dominate the energy budget. Theevolution equations are

a =

(t

t0

)2/3

=

(η

η0

)2

,t

t0=

(η

η0

)3

, η0 = 3t0 , (1.2)

H =2

3t, H =

2

η, ρm =

1

6πGt2, r = η0 − η =

2

H0

(1− 1√

1 + z

), (1.3)

where the reference point t0 satisfies a(t0) = 1, but it can be any time t0 ∈ (0,∞). At a given epoch t0, one can define a mass scale

M :=4π

3ρ0 =

H20

2G=

2

9Gt20, H0 =

2

3t0, (1.4)

Closed Homogeneous Universe

An analytic solution can be derived for a closed universe with again pressureless matter. The evolution equations for a closed universeare

a

at=

1− cos θ

2, t =

ttπ

(θ − sin θ) =a2t (θ − sin θ)

2√K

, dη =at√Kdθ , (1.5)

H2 =8πG

3ρm −

K

a2=K

a2

(ata− 1

), (1.6)

where we used tilde to distinguish quantities in the closed universe from the flat universe and the maximum expansion (or turn-aroundat) is reached at θ = π (Ht = 0). The density parameters are related to the curvature K of the universe as

Ωm − 1 = −Ωk = − K

a2H20

, K =8πG

3

ρ0

at=H2

0

at=

2GM

at=π2a2

t

4t2t. (1.7)

Spherical Collapse Model

Matching the density equal at some early time, say t0 (i.e., δ0 = 0), the time evolution of the overdense region can be derived in anon-perturbative way as

1 + δ =ρmρm

=(aa

)3

=9

2

(θ − sin θ)2

(1− cos θ)3, (1.8)

where we used

a3 =

(t

t0

)2

=

(ttπt0

)2

(θ − sin θ)2, a3 =

(at2

)3

(1− cos θ)3

=2

9

(ttπt0

)2

(1− cos θ)3. (1.9)

AST802 Advanced Theoretical Cosmology JAIYUL YOO

Therefore, the density contrast δt at its maximum expansion

1 + δt =9π2

16' 5.6 , (1.10)

is about a few, while the density contrast δv at its virialization

1 + δv = 18π2 ' 177.7 , (1.11)

is a few hundreds, under the assumption that the overdensity region virialized at the half of its maximum expansion. Note that theuniverse further expands and the background density is reduced by factor 4, until it collapses at tv = 2tt (or θ = 2π).

Finally, expanding the expressions to the linear order,

a =1

361/3

(ttπt0

)2/3

θ2 + · · · , δ =3

20θ2 + · · · , (1.12)

the density contrast linearly extrapolated to today and its value at virialization are then derived as

δL =D

Diδi =

a

aiδi =

3

10

(9

2

)1/3

(θ − sin θ)2/3 , δv ' 1.686 . (1.13)

Assuming the non-perturbative expression is valid for |δ| 1, we have

δ = δL +17

21δ2L +

341

567δ3L +

55805

130977δ4L + · · · , δL = δ − 17

21δ2 +

2815

3969δ3 − 590725

916839δ4 + · · · . (1.14)

Biased Tracer

For any biased tracer δX , the Eulerian and the Lagrangian bias parameters can be written in a series

δX =

∞∑n=1

bnn!δn , δLX =

∞∑n=1

bLnn!δnL , (1.15)

where the superscript L represents quantities in the Lagrangian space. If the number density of the objects X is conserved

ρ d3x = ρ d3q , ρX d3x = ρLX d3q , ∴ 1 + δX = (1 + δ)(1 + δLX) , (1.16)

the bias parameters are related as

b1 = bL1 + 1 , b2 = bL2 +8

21bL1 , b3 = bL3 −

13

7bL2 −

796

1323bL1 , b4 = bL4 −

40

7bL3 +

7220

1323bL2 +

476320

305613bL1 . (1.17)

This simple relation owes to the fact that the spherical collapse model is local in both Eulerian and Lagrangian spaces.

1.2 Dark Matter Halo Mass Function

1.2.1 Basic IdeaGiven the simple spherical collapse model, we would like to associate the collapsed region with some virialized objects like darkmatter halos.

1.2.2 Halo Mass FunctionGiven the simple spherical collapse model, we would like to associate the collapsed region with some virialized objects like massivegalaxy clusters or dark matter halos. Of our main interest is then the number density of such objects in a mass range M ∼M + dM ,and this is called the mass function.

A simple model called, the excursion set approach, was developed: One starts with a smoothing scale R and its associatedmass M . The density fluctuation δR after smoothing with R is very small (δR = 0, if R =∞), and this region has never reached thecritical density threshold δc in its entire history. This implies that there is no virialized object associated with such mass. One thendecreases the smoothing scale (or mass), and looks for the collapsed probability: Some overdense regions have at some point in the

2


past reached the critical density, while some underdense regions have not. Therefore, the total fraction Fc of collapse can be obtainedby using the survival probability Ps of a given scale, and it is related to the mass function as

Fc = 1−∫ δc

−∞dδ Ps =

∫ ∞M

dMdn

dM

M

ρm, ∴

dn

dM=ρmM

(−∂Fc∂M

)≡ ρmM

f(ν)d ln ν

dM, (1.18)

where it is assumed that the mass function only depends on mass and we defined the multiplicity function f through the relation

ν ≡ δc(z)

σ(M),

∫ ∞0

dν

νf = 1 . (1.19)

The task of obtaining the mass function boils down to computing the survival probability and expressing it in terms of themultiplicity function. The way to find the survival probability at a given mass scale M is to derive the evolution of the densityfluctuation as we decrease the smoothing scale R. The reason is that the region may have already collapsed at a larger mass scale orsmoothing scale, and this contribution should be removed in computing the survival probability at a lower mass scale. The survivalprobability at n-th step depends on the entire history of the trajectory (non-Markovian process) as

Ps(δn, σn)dδn = dδn

∫ δc

−∞dδn−1 · · ·

∫ δc

−∞dδ1 Ps(δ1, · · · δn, σ1, · · · , σn) , (1.20)

it is notoriously difficult to solve, even numerically. However, once we assume that the fluctuations are independent at each smoothingand are Gaussian distributed (true only in Fourier space at linear order), the trajectory only depends on the previous step (Markovianprocess) and the survival probability becomes

Ps(δn, σn) =

∫ δc

−∞dδn−1Pt(δn, σn|δn−1, σn−1) Ps(δn−1, σn−1) , (1.21)

where the transition probability Pt is nothing but a conditional probability. With the boundary condition Ps = 0 at δ = δc, thesolution is (derived by Chandrasekhar for other purposes)

Ps =1√2πσ

exp

(− δ2

2σ2

)− 1√

2πσexp

[− (2δc − δ)2

2σ2

]. (1.22)

The survival probability for its simplest case is described by a Gaussian distribution, but the second term reflects that there existequally likely trajectories around the threshold that have reached the threshold in the past. The collapsed fraction is

Fc = 1− 1

2erf

(νc√

2

)− 1

2erf

(νc√

2

)= erfc

(νc√

2

), (1.23)

and the multiplicity function is

f(ν) =

√2

πν e−ν

2/2 . (1.24)

Of course, this model relies on many approximations, and it is not accurate. However, it provides physical intuitions, connectingthe complicated formation of galaxy clusters and the dynamical evolution of the matter density fluctuations. In general, numericalN -body simulations are run, and dark matter halos are identified by using some algorithm such as the friends-of-friends method orits variants to derive the mass function from the simulations.

1.2.3 Halo Mass Functions in LiteratureIn general, numerical N -body simulations are run, and dark matter halos are identified by using some algorithm such as the friends-of-friends method or its variants. These halo mass functions differ from the simple analytic formula we derived for the Gaussianrandom field. However, the functional form is relativel resilient, such that an introduction of a few nuisance parameters to the massfunction can provide a good fit to the simulation results. A lot of variants exist in literature, but the standard and simplest cases areone by Sheth and Tormen (1999)

fST = A

√2a

π

[1 +

(σ2

aδ2c

)p]δcσ

exp

(− aδ

2c

2σ2

), A = 0.3222 , a = 0.707 , p = 0.3 , (1.25)

and one by Jenkins et al. (2001)fVirgo = 0.315 exp

(−∣∣lnσ−1 + 0.61

∣∣3.8) . (1.26)

They all have a limited range of validity.

3


1.2.4 Astrophysical Applicationshalo merger trees, semi-analytic models for galaxy formation, void distributions, and so on.

1.3 From Probability Functional to Correlation FunctionAssuming that the density proability functional is known, we derive the one-point statistics and its spatial correlation function. Thosefor the cumulative distributions are also derived. In this case, we focus on one population of halos or samples, rather than multiplepopulations considered in the mass function approach.

1.3.1 Density Probability FunctionalOne-Point Statistics

Given a probability functional P [δ(x)] defined in all space x, a one-point pdf (i.e., the probability to have δ1 at a given point x1) is

P (δ1) =

∫dδ(x) P [δ(x)] δD[δ(x1)− δ1] =

∫dδ(x) P [δ(x)]

∫dJ1

2πeiJ1(δ(x1)−δ1) ≡

∫dJ1

2πe−iJ1δ1Z(J1) , (1.27)

where the pdf is the Fourier counterpart of the generating function (and the cumulant generating function)

Z(J1) ≡∫dδ(x) P [δ(x)] eiJ1δ(x1) =

⟨eiJ1δ(x1)

⟩=

∞∑n=0

(iJ1)n

n!〈δn(x1)〉 = eω , ω(J1) ≡ lnZ =

∞∑n=0

(iJ1)n

n!〈δn(x1)〉c .

(1.28)Note the difference in the ensemble average in Z and ω, and we used the cumulant expansion theorem

⟨eiX⟩

=

∞∑n=0

in

n!〈Xn〉 = exp

[⟨eiX⟩c

]= exp

[ ∞∑n=1

in

n!〈Xn〉c

], (1.29)

⟨eiδ⟩

= e−σ2/2 = 1− 1

2σ2 +

1

2!

(σ4

4

)− 1

3!

(σ6

8

)· · · = 1− 1

2

⟨δ2⟩

+1

4!

⟨δ4⟩− 1

6!

⟨δ6⟩

+ · · · . (1.30)

For the Gaussian distribution, we have the usual

ωG = −1

2J2σ2 , PG(δ1) =

∫dJ1

2πe−iJ1δ1− 1

2J21σ

2

=1√

2π σe−δ

21/2σ

2

. (1.31)

For a smoothed density field δR(x1), we have

iJ1δ(x1)→ iJ1δR(x1) = iJ1

∫d3y1 δ(y1)WR(x1 − y1) . (1.32)

N -Point Statistics

Similarly, two-point pdf of δ1 at x1 and δ2 at x2 is

P (δ1, δ2) =

∫dδ(x)P [δ(x)] δD[δ(x1)−δ1] δD[δ(x2)−δ2] =

∫dJ1

2πe−iJ1δ1

∫dJ2

2πe−iJ2δ1Z(J1, J2) , Z =

⟨eiJ1δ(x1)+iJ2δ(x2)

⟩.

(1.33)Generalizing the above practice to∞-point pdf, we have

P [δ(x)] =

∞∏i=1

∫dJi2π

eiJiδiZ , Z =

∫dδ(x) P [δ(x)] e

∑∞i iJiδ(xi) =

⟨e∫d3x iJ(x)δ(x)

⟩, ω =

⟨e∫d3x iJ(x)δ(x)

⟩c, (1.34)

and from the generating function we “generate” cumulants

〈δ(x1) · · · δ(xn)〉c ≡ ξ(n)c (x1, · · · ,xn) =

1

inδn lnZ

δJ(x1) · · · δJ(xn)

∣∣∣∣J=0

. (1.35)

4


Further Generalization

In fact, this can be further generalized by considering not only the density field δ(x), but also other fields Aµ(x) with µ = 1, · · · , Nsuch as ηi(x) = ∂iδ(x), ζij(x) = ∂ijδ(x). For example, the peak model pdf requires N = 10 (1: δ, 3: ηi, 6: ζij) all at the samespatial point (such that N ×M -number of fields for M different spatial points). Another example is N = ∞ for the density field.In general, the generating function can be constructed for N -number of Aµ by using the cumulants as (here, N is the total numberincluding different quantities and spatial points)

M(n)µ1,··· ,µn ≡ 〈Aµ1 · · ·Aµn〉c =

1

inδn lnZ

δJµ1 · · · δJµn

∣∣∣∣J=0

, (1.36)

lnZ[J] =

∞∑n=1

in

n!

N∑µ1=1

· · ·N∑

µn=1

M(n)µ1,··· ,µnJµ1 · · · Jµn , Z = e−J·MT ·J exp

[ ∞∑n=3

in

n!

N∑µ1,··· ,µn

M(n)µ1,··· ,µnJµ1 · · · Jµn

],

where we isolated the Gaussian part. Finally, using the trick Jµ → i∂/∂Aµ we have the one-point pdf with N -Aµ

P (A) =

N∏i=1

[∫ ∞−∞

dJi(2π)

]e−iJ·A Z(J) = exp

[ ∞∑n=3

(−1)n

n!

N∑µ1,··· ,µn

M(n)µ1,··· ,µn

∂n

∂Aµ1· · · ∂Aµn

]N∏i=1

[∫ ∞−∞

dJi(2π)

]e−J·MT ·J−iJ·A

= exp

[ ∞∑n=3

(−1)n

n!

N∑µ1,··· ,µn

M(n)µ1,··· ,µn

∂n

∂Aµ1 · · · ∂Aµn

]PG(A) . (1.37)

Another useful generalization for a one-point pdf is that two probability distribution functions and their generating functions arerelated as

Pa(δ) = exp

[ ∞∑n=1

κan − κbnn!

(− d

dδ

)n]Pb(δ) , Za(J) = exp

[ ∞∑n=1

κan − κbnn!

(iJ)n

]Zb(J) , (1.38)

where the cumulant is κn = 〈δn〉c.The cumulative N -point probability distributions are from Eq. (1.37)

P1(> ν) =1√2π

∫ ∞ν

dy exp

[ ∞∑N=3

(−1)N⟨δN⟩c

σNN !

dN

dyN

]e−y

2/2 , (1.39)

P2(> ν, r) =1

2π

∫ ∞ν

dy1

∫ ∞ν

dy2 exp

[ ∞∑N=2

N∑m=0

(−1)Nw

(N,m)s (r)

m!(N −m)!

∂N

∂ym1 ∂yN−m2

]e−

12 (y2

1+y22) ,

where

w(N,m)s (r) ≡

w

(2,m)s = ξ

(2,m)s (r)/σ2

0s (m = 1)

w(2,m)s = 0 (m = 0 or 2)

w(N,m)s = ξ

(N,m)s (r)/σN0s (N > 2)

, ξ(N,m)s (r) ≡

⟨δs(x1) · · · δs(x1)︸︷︷︸

m times

δs(x2) · · · δs(x2)︸︷︷︸N−m times

⟩c. (1.40)

Again, keeping only the linear order term in Eq. (1.39), i.e., exp[· · · ] = 1 + · · · , we have

P1(> ν) ≈ 1

2erfc

(ν√2

)+

∞∑N=3

1√2π

w(N,0)s

N !HN−1(ν) e−ν

2/2 , (1.41)

P2(> ν, r) ≈[

1

2erfc

(ν√2

)]2

+

√1

2πerfc

(ν√2

) ∞∑N=3

w(N,0)s

N !HN−1(ν)e−ν

2/2

+1

2π

∞∑N=2

N−1∑m=1

w(N,m)s

m!(N −m)!Hm−1(ν)HN−m−1(ν)e−ν

2

,

where Hm(x) is the Hermite polynomials

5


1.3.2 Mildly Non-Gaussian Halo Mass FunctionGiven the non-Gaussian PDF, the non-Gaussian mass function can be derived as

dn

dM= −2

ρmM

dP (> νR)

dM= 2

ρmM

dν

dM

1√2π

exp

[ ∞∑n=3

Snσn

n!σ2

(− d

dν

)n]e−ν

2/2

−2ρmM

∫ ∞ν

dx1√2π

d

dM

exp

[ ∞∑n=3

Snσn

n!σ2

(− d

dx

)n]e−x

2/2

,

where the reduced cumulant is Sn = κn/κn−12 , i.e., κn = (Snσ

n/σ2)σn. Note the reduced cumulant follows the convention inliterature, but it is dimensionful. One needs to solve the above equation for the non-Gaussian mass function. Various non-Gaussianmass functions differ in truncation of the expansion above (see below). For example, Matarrese et al. (2000) truncated at somecumulant order. LoVerde et al. (2008) truncated by using the Edgeworth expansion. LoVerde and Smith (2011) truncated by usingthe log of Edgeworth.

Only keeping the linear order term exp[x] = 1 + x + · · · and using Eq. (1.37), the one-point non-Gaussian PDF is by (oftenknown as Gram-Chalier A Series)

Pa(δ) = exp

[ ∞∑n=3

κnn!

(− d

dδ

)n]1√2πσ

e−δ2/2σ2

=1√2πσ

e−δ2/2σ2

[1 +

S3σ

6H3 +

S4σ2

24H4 + · · ·

]. (1.42)

However, since it only accounts for the linear order term in the exponential, this series is not positive-definite (hence Pa is not aproper PDF) and often diverges.

Following Lucchin & Matarrese (1988) and manipulating the variables, we obtain

ν ≡ δ

σ, J ≡ z − iν

σ, −iJδ+ω(J) =

(−1

2J2σ2 − iJδ

)+

(ω +

1

2J2σ2

)= −1

2z2− 1

2ν2+

1

σ2

∞∑n=3

Snn!σn(ν+iz)n , (1.43)

and using the contour integration, we have the one-point pdf (known as the Edgeworth expansion)

P (δ) = e−12ν

2

∫ ∞−∞

dz

2πσexp

[−1

2z2 +

1

σ2

∞∑n=3

Snn!σn(ν + iz)n

](1.44)

=e−

12ν

2

√2πσ

[1 +

S3σ

6H3(ν) + σ2

(S4

24H4(ν) +

S23

72H6(ν)

)+ σ3

(S5

120H5(ν) +

S3S4

144H7(ν) +

S33

1296H9(ν)

)+ · · ·

],

where the probabilists’ Hermite polynomial is

HL(x) =1√2π

∫ ∞−∞

dz e−12 z

2

(x+ iz)L , (1.45)

but one can also derive from Eq. (1.37) by going to higher order terms in the exponential exp[x] = 1 + x+ x2/2 + · · · .

1.3.3 Peak-Background Split and Statistics of Thresholded RegionHere we exclusively focus on the statistics of the density distribution above a given threshold.

One-Point Statistics

When the underlying smoothed density field (Rs) obeys the Gaussian statistics, the probability P1 of exceeding the threshold ν andthe effect of adding a long-wavelength (background) perturbation δl of characteristic wavelength Rl Rs to the small scale densityfield (peak) δs are

P1(> ν) =1√2π

∫ ∞ν

dx e−x2/2 =

1

2erfc

(ν√2

), P1(> ν, δl) = P1

(> ν − δl

σs

). (1.46)

We define the peak-background split cumulative bias factors cN as the fractional change of P1 with δl via

cN ≡1

P1(> ν)

dN P1(> ν, δl)

d δNl=

(− 1

σs

)N1

P1(> ν)

dN[P1(> ν)

]dνN

=

√2

π

[erfc

(ν√2

)]−1e−ν

2/2

σNsHN−1 (ν) , (1.47)

6


and in the high peak limit ν →∞HN → νN , cN ≈ νHN−1(ν)/σNs ≈ νN/σNs . (1.48)

Here, HN is the statistician’s Hermite polynomial defined by [HphysN (x) = 2N/2HN

(√2x)]

HN (x) ≡ (−1)Nex2/2 d

N

dxN

(e−x

2/2), H0 = 1 , H1 = x , H2 = x2−1 , H3 = x3−3x , H4 = x4−6x2 + 3 . (1.49)

For differential mass function, we have (vs. cumulative)

nh(M) = −2ρmM

d

dMP1 (> ν) = 2

ρ

M2

νe−ν2/2

√2π

∣∣∣∣ d lnσsd lnM

∣∣∣∣ , P1(> ν) =1

2ρm

∫ ∞M

dM ′M ′nh(M ′) . (1.50)

where the factor of 2 is introduced to account for the fact that regions with δ < δc may be embedded in regions with δ > δc on scale> Rs (clouds-in-clouds). Using the definition of the cumulative bias, we have

cN =

[∫ ∞M

dM ′M ′nh(M ′)

]−1 ∫ ∞M

dM ′M ′[dN

dδNlnh(M ′)

],

[dN

dδNlnh(M ′)

]≡ bN (M ′)nh(M ′) , (1.51)

where the PBS bias is

bN (M) =1

νe−ν2/2

(− 1

σs

)NdN

dνN

(νe−ν

2/2)

=1

ν

HN+1(ν)

σNM. (1.52)

It is only in the high-peak limit (ν 1) that the mass-weighted cumulative bias cN and the bias bN (M) asymptote to the samevalues.

Two-Point Statistics

For a Gaussian, we have (see Eq. 1.39)

ξ>ν(r) =P2(r)

P 21

− 1 =2

π

[erfc

(ν√2

)]−2 ∫ ∞ν

dy1

∫ ∞ν

dy2 exp

[ξs(r)

σ2s

∂2

∂y1∂y2

]e−

12 (y2

1+y22) − 1 , (1.53)

and the double integration is

∫ ∞ν

dy1

∫ ∞ν

dy2

∞∑N=0

1

N !

[ξs(r)

σ2s

]N (∂2

∂y1∂y2

)Ne−

12 (y2

1+y22) =

∞∑N=0

1

N !

[ξs(r)

σ2s

]N [∫ ∞ν

dy

(∂

∂y

)Ne−y

2/2

]2

, (1.54)

where the N = 0 term is simply the erfc function for the Gaussian piece and the integration is∫ ∞ν

dy

(∂

∂y

)Ne−y

2/2 = −(∂

∂y

)N−1

e−y2/2 = (−1)Ne−y

2/2HN−1(y) . (1.55)

Therefore, the correlation function is

ξ>ν(r) =2

π

[erfc

(ν√2

)]−2 ∞∑N=1

[ξs(r)

]NN !σ2N

s

[HN−1(ν)

]2e−ν

2

=

∞∑N=1

c2NN !

[ξs(r)

]N= c21ξs(r) +

1

2c22ξ

2s (r) + · · · . (1.56)

Since the local bias expansion can be written as

δ>ν(x) =

∞∑N=1

cNN !

[δs(x)

]N= c1δs +

1

2c2δ

2s + · · · , (1.57)

we see that the coefficient cN is different from the cN appearing in the correlation function: the coefficient c2N includes not only c2N ,but also terms such as cN cN+2mσ

2ms for all positive integers m ≥ N/2. This clearly shows that the bias parameters cN from the

peak-background split are to be seen as “renormalized” bias parameters, which take all the higher order moments into account.

7


Non-Gaussian Two-Point Statistics

Similarly, the correlation function in the non-Gaussian case is obtained by using Eq. (1.41) and keeping the linear order only

ξ>ν(r) =2

π

[erfc

(ν√2

)]−2 ∞∑N=2

N−1∑m=1

w(N,m)s

m!(N −m)!Hm−1(ν)HN−m−1(ν)e−ν

2

=

∞∑N=2

N−1∑m=1

cmcN−mm!(N −m)!

ξ(N,m)s (r) . (1.58)

Now we compute the power spectrum. For simplicity and without loss of generality, we will assume that a single non-GaussianN -point function (N ≥ 3) dominates. We then have

ξ>ν(r) = c21ξs(r) +

N−1∑m=1

cmcN−mm!(N −m)!

ξ(N,m)s (r) , P>ν(k) = c21Ps(k) +

N−1∑m=1

cmcN−mm!(N −m)!

ξ(N,m)s (k) , (1.59)

where for m = 1 (and m = N − 1) we have

ξ(N,1)s (k) = Ms(k)

N−2∏i=1

(∫d3ki(2π)3

Ms(ki)

)Ms(q) ξ

(N)Φ (k,k1, . . . ,kN−2,q;X) , (1.60)

ξ(N,2)s (k) =

N−2∏i=1

(∫d3ki(2π)3

Ms(ki)

)Ms(|k− k1|)Ms(q) ξ

(N)Φ (k− k1,k1, . . . ,kN−2,q;X) , (1.61)

where q = −k1 − · · · − kN−2 − k, and X is a set of variables characterizing the primordial N -point function such as fNL, gnl

depending on the details of the model of non-Gaussianity. On large scales, ξ(N,m)s terms (m = 2 · · ·N − 2) all add white-noise

contributions to the power spectrum of thresholded regions, and only the terms with m = 1, N − 1 contribute to the scale-dependentbias. Therefore, the power spectrum is

P>ν(k) ' c21Ps(k) + 2c1cN−1

(N − 1)!ξ(N,1)s (k) =

[c21 + 2

4

(N − 1)!c1cN−1σ

2sM−1

s (k)F (N)s (k,X)

]Ps(k) , (1.62)

where the shape factor is

F (N)s (k,X) ≡ M−1

s (k)

4σ2sPφ(k)

ξ(N,1)s (k) =

1

4σ2s Pφ(k)

N−2∏i=1

∫d3ki(2π)3

Ms(ki)

Ms(q) ξ

(N)Φ (k1, · · · ,kN−2, q, kz;X) . (1.63)

Noting that P>ν = (c21 +2c1∆c1)Ps to leading order in the non-Gaussian corrections, we can identify the scale-dependent correctionto the linear bias, and for the fNL-case we have

∆c1(k) =4cN−1

(N − 1)!σ2s

F (N)s (k)

Ms(k), P>ν(k) =

[c21 + 4fNLc1c2σ

2sM−1

s (k)]Ps(k)

ν1= b21

[1 + 4fNL

δcMs(k)

]Ps(k) . (1.64)

Note that in general the scale-dependent correction is cN−1, not the usual (b1 − 1).

1.4 Beyond the Spherical Model: Peak Models

1.4.1 Basic Idea of Peak ModelsGoing beyond the spherical collapse model, we consider the density peak positions for the sites for halo formation. Compared to thesimplest version of the spherical collapse models, not only the density threshold, but also its derivative is considered to describe thehalo formation process. In the framework of this peak model, we derive the statistics of such peaks including the number density, theshape parameters (deviation from sphericity), and the correlation. For the Gaussian probability distribution, analytic calculations arepossible, and much of the work was done in Bardeen, Bond, Kaiser, and Szalay (1986).

1.4.2 Multivariate Gaussian Joint Probability DistributionThe notational convention is

F (x, t) = δm(x, t) , ηi(r) = ∇iF (r) , ζij(r) = ∇i∇jF (r) , Npk(ν) = 〈Npk(r, ν)〉 . (1.65)

8


The joint Gaussian probability distribution of F , ηi, ζij at a given one-point r is described by its correlations

〈FF 〉 = σ20 , 〈ηiηj〉 =

σ21

3δij , 〈Fζij〉 = −σ

21

3δij , 〈ζijζkl〉 =

σ22

15(δijδkl+δikδjl+δilδjk) , 〈Fηi〉 = 〈ηiζjk〉 = 0 , (1.66)

where σ2j =

∫d ln k ∆2

k k2j and [σ2

j ] = L2j dimensionful. Note that if one consider a PDF for density F only, one just needs σ20 (it

is one-point, not spatial two-point correlation). Because of the symmetry of ζij , only six components are independent: A = 1 − 6,referring to the ij = 11, 22, 33, 23, 13, 12 components. The covariance matrix M has dimension 10.

M =

σ20 0 0 0 −σ2

1/3 −σ21/3 −σ2

1/3 0 0 00 σ2

1/3 0 0 0 0 0 0 0 00 0 σ2

1/3 0 0 0 0 0 0 00 0 0 σ2

1/3 0 0 0 0 0 0−σ2

1/3 0 0 0 σ22/5 σ2

2/15 σ22/15 0 0 0

−σ21/3 0 0 0 σ2

2/15 σ22/5 σ2

2/15 0 0 0−σ2

1/3 0 0 0 σ22/15 σ2

2/15 σ22/5 0 0 0

0 0 0 0 0 0 0 σ22/15 0 0

0 0 0 0 0 0 0 0 σ22/15 0

0 0 0 0 0 0 0 0 0 σ22/15

. (1.67)

To diagonalize the remaining four dimensions, we introduce a new set of variables ζA, A = 1, 2, 3 → x, y, z, where

σ2x = −∇2F = −(ζ1 + ζ2 + ζ3) , σ2y = −(ζ1 − ζ3)/2 , σ2z = −(ζ1 − 2 ζ2 + ζ3)/2 , (1.68)

ζ1 = −σ2

3(x+ 3y + z) , ζ2 = −σ2

3(x− 2z) , ζ3 = −σ2

3(x− 3y + z) . (1.69)

Further manipulation gives

(F

σ0,ζ1σ2,ζ2σ2,ζ3σ2

)→

1 −γ/3 −γ/3 −γ/3−γ/3 1/5 1/15 1/15−γ/3 1/15 1/5 1/15−γ/3 1/15 1/15 1/5

, (ν, x, y, z)→

1 γ 0 0γ 1 0 00 0 1/15 00 0 0 1/15

, (1.70)

where γ is dimensionless and

⟨ν2⟩

= 1 ,⟨x2⟩

= 1 , 〈xν〉 = γ ,⟨y2⟩

=1

15,⟨z2⟩

=1

5, γ ≡

⟨k2⟩

〈k4〉1/2=

σ21

σ2σ0. (1.71)

Note⟨k2⟩

= σ21/σ

20 = −3ξ′′(0)/ξ(0). So, if Pm(k) ∼ δD(k − k0), then γ = 1, and if ∆2

k 'constant, then γ 1.For multivariate Gaussians (y1, · · · , yn), the joint Gaussian probability is

P (y1, · · · , yn) dy1 · · · dyn =e−Q

[(2π)n det M]1/2

dy1 · · · dyn , (1.72)

where Q = 12 (y − 〈y〉)T M−1 (y − 〈y〉) and the covariance matrix M =

⟨(y − 〈y〉) (y − 〈y〉)T

⟩. Therefore, for the Gaussian

variables ν, η, ζA (A = 1− 6), the joint probability is

P (F, ηi, ζA) dF d3η d6ζ =e−Q

[(2π)10 det M]1/2

dF d3η d6ζ , (1.73)

where M is the covariance matrix and

2Q = ν2 +(x− x∗)2

(1− γ2)+ 15y2 + 5z2 +

3 η · ησ2

1

+

6∑A=4

15 ζ2A

σ22

, x∗ ≡ γν . (1.74)

Now we choose the principal axes of ζij and let the eigenvalues λi, where i = 1, 2, 3:

ζ = −

λ1 0 00 λ2 00 0 λ3

, (1.75)

9


where ζA = −λA for A = 1, 2, 3 (note x, y, z 6= 0) and ζA = 0 for A = 4, 5, 6 (no off-diagonal terms). The volume element in thesix-dimensional space of symmetric real matrices is

d6ζA =

6∏A=1

dζA = |(λ1 − λ2)(λ2 − λ3)(λ1 − λ3)|dλ1 dλ2 dλ3dΩS3

6

= 2 σ32 |y (y2 − z2)| 2

3σ3

2 dx dy dzdΩS3

6=

2

9σ6

2 |y (y2 − z2)| dx dy dz dΩS3 , (1.76)

dλ1 dλ2 dλ3 = σ32 |J−1

R | dx dy dz =2

3σ3

2 dx dy dz , (1.77)

y (y2 − z2) = − 1

2σ22

(λ1 − λ2)(λ2 − λ3)(λ1 − λ3) . (1.78)

Therefore, the joint probability is

P (F, ηi, ζA) dF d3η d6ζ =e−Q

[(2π)10 det M]1/2

(σ0dν) d3η

(2

9σ6

2 |y (y2 − z2)| dx dy dz dΩS3

), (1.79)

and since Q is independent of the Euler angles, integrating over the angles gives 2π2/3! and the joint probability is

P (ν,η, x, y, z) dν d3η dx dy dz =e−Q

[(2π)10 det M]1/2

σ02

9σ6

2 |y (y2 − z2)| 2π2

3!dν d3η dx dy dz

=(15)5/2

32π3

σ30

σ31(1− γ2)1/2

|2y (y2 − z2)| e−Qdν dx dy dz d3η

σ30

, (1.80)

where the prefactor is N and the determinant is

det M =

(σ2

1

3

)3(σ2

2

15

)31− γ33152

σ20σ

62 . (1.81)

1.4.3 Gaussian Peaks ModelsGiven λ1 ≥ λ2 ≥ λ3 and peaks are at extremum η(rp) = 0, the density fluctuation and its derivative near peaks are

F (r) ' F (rp) +1

2

∑ij

ζij(rp)(r− rp)i(r− rp)j , ηi(r) '∑j

ζij(rp)(r− rp)j . (1.82)

Therefore, the full density field for the maxima of height between ν0 and ν0 + dν (differential number density of peaks) is

Npk(r, ν0) dν =∑i

δD(r− rp,i) = δD[(ζ−1)η] θ(λ1) θ(λ2) θ(λ3) δD(ν − ν0) dν (1.83)

= |det ζ| δD(η) θ(λ1) θ(λ2) θ(λ3)δD(ν − ν0) dν .

Similarly, the phase-space distribution function is fpk(r,v0) = δD(v − v0) npk(r). Unfortunately, the probability function ofnpk = Npk(> ν) is analytically intractable. However, the average Npk = 〈Npk(r)〉 can be obtained by integrating Eq. (1.80) as

Npk(ν, x, y, z) dν dx dy dz = 〈Npk〉 dν dx dy dz =

∫d3η Npk P (ν,η, x, y, z) dν dx dy dz

=55/231/2

(2π)3

(σ2

σ1

)31

(1− γ2)1/2e−QF (x, y, z) χ dν dx dy dz , (1.84)

where

F (x, y, z) =27

2

λ1λ2λ3(λ1 − λ2)(λ2 − λ3)(λ1 − λ3)

σ62

= (x− 2z)[(x+ z)2 − (3y)2

]y(y2 − z2) , (1.85)

Q =ν2

2+

(x− x∗)2

2(1− γ2)+

5

2(3y2 + z2) , (1.86)

10


and χ = 1 if the constraints in the x, y, z domain are satisfied (χ = 0 otherwise). Further integration of Eq. (1.84) yields

Npk(ν, x) dν dx =e−ν

2/2

(2π)2R3∗f(x)

exp[−(x− x∗)2/2(1− γ2)

][2π(1− γ2)]1/2

dν dx , (1.87)

with

f(x) =3255/2

√2π

[∫ x/4

0

dy e−(15/2)y2

∫ y

−ydz F (x, y, z) e−(5/2)z2

+

∫ x/2

x/4

dy e−(15/2)y2

∫ y

3y−xdz F (x, y, z) e−(5/2)z2

](1.88)

= (x3 − 3x)

erf

[(5

2

)1/2

x

]+ erf

[(5

2

)1/2x

2

]/2 +

(2

5π

)1/2 [(31x2

4+

8

5

)e−5x2/8 +

(x2

2− 8

5

)e−5x2/2

].

The asymptotic limits of this function is

f(x)→ 3555/2

7 211√

2πx8

(1− 5x2

8

)for x→ 0 , f(x)→ x3 − 3x for x→∞ . (1.89)

Final integration to get the average number density Npk(ν) dν needs to be done numerically over x as

Npk(ν) dν =1

(2π)2R3∗e−ν

2/2

∫ ∞0

dx f(x)exp

[−(x− x∗)2/2(1− γ2)

][2π(1− γ2)]1/2

, (1.90)

where R∗ ≡√

3 (σ1/σ2) and x∗ = γν. The integral is denoted as G(γ, x∗) and its fitting formula is given. The cumulative numberdensity of peaks higher than height ν is

npk(ν) =

∫ ∞ν

dν′ Npk(ν′) , npk(−∞) =29− 6

√6

53/22(2π)2R3∗

= 0.016R−3∗ , (1.91)

and the high peak limit ν →∞ is

Npk dν →[⟨k2⟩/3]3/2

(2π)2(ν3 − 3ν) e−ν

2/2 dν , n(ν)→[⟨k2⟩/3]3/2

(2π)2(ν2 − 1) e−ν

2/2 ,⟨k2⟩≡ σ2

1

σ20

= 3(γ/R∗)2 . (1.92)

For a Gaussian filtering, the power spectrum is P (k,Rf ) ∼ kn e−(kRf )2

, and other quantities are

σ21(Rf )

σ20(Rf )

=n+ 3

2R2f

,σ2

2(Rf )

σ20(Rf )

=(n+ 5)(n+ 3)

4R4f

, γ2 =n+ 3

n+ 5, R∗ =

(6

n+ 5

)1/2

Rf , Npkdν ∝ 1/R3∗ ∝ 1/R3

f . (1.93)

1.4.4 Conditional Probability for Ellipticity and ProlatenessThe conditional probability for x = −∇2F/σ2 is independent of e and p and is

P (x|ν) dx =Npk(ν, x) dx

Npk(ν)=

exp[−(x− x∗)2/2(1− γ2)]

[2π(1− γ2)]1/2f(x) dx

G(γ, x∗). (1.94)

For high peaks (large x∗), it is more likely that the peaks are located at large x. The conditional probability for y, z given ν, x issimply

P (y, z|ν, x) dy dz =Npk(ν, x, y, z) dν dx dy dz

Npk(ν, x) dν dx=

3255/2

√2π

F (x, y, z) χ

f(x)exp

[−5

2(3y2 + z2)

], (1.95)

and independent of ν. We characterize the asymetry as ellipticity and prolateness

e =λ1 − λ3

2∑i λi

=y

x, p =

λ1 − 2λ2 + λ3

2∑i λi

=z

x. (1.96)

Thus, e (0 ≤ e ≤ 1/2) is a measure of the ellipticity of the distribution in the 1-3 plane, and p determines the degress of oblateness(0 ≤ p ≤ e) or prolateness (0 ≥ p ≥ −e) of the triaxial ellipsoid. Oblate spheroids (football) have p = e, and prolate spheroids(disk) have p = −e. The characteristic function χ = 1 if 0 ≤ e ≤ 1/4 and −e ≤ p ≤ e or 1/4 ≤ e ≤ 1/2 and −(1 − 3e) ≤ p ≤ e(zero, otherwise). (e, p) is confined within (0, 0), (1/4,−1/4), (1/2, 1/2).

11


Therefore, the conditional probability for ellipticity and prolateness is

Pep(e, p|ν, x) de dp = Pep(e, p|x) de dp =3255/2

√2π

x8

f(x)e−(5/2)x2(3e2+p2) W (e, p) de dp , (1.97)

where

W (e, p) =F (x, y = ex, z = px) χ

x8= e (e2 − p2)(1− 2p)[(1 + p)2 − 9 e2] χ . (1.98)

Notice that the most likely value of p quickly goes to zero. High ν peaks are neither oblate nor prolate, but they are definitely triaxiallyasymmetric, since λ2 ' (λ1 + λ3)/2. Indeed, in the large x limit, e, p are small, and we can apprxoimate the PDF by a Gaussian:

Pep(e, p) ' Pep(em, pm) exp

[− (e− em)2

2σ2e

− (p− pm)2

2σ2p

], (1.99)

whereem =

1√5x[1 + 6/(5x2)]1/2

, pm =6

5x4[1 + 6/(5x2)]2, σe =

em√6, σp =

em√3. (1.100)

For high peaks, x is large and thus em ' pm ' 0 (more spherically symmetric).

12

2 Newtonian Perturbation Theory and Galaxy BiasIn this chapter we study the Eulerian and the Lagrangian perturbation theories in Newtonian dynamics and their connection tomodeling the galaxy (or halo) distribution.

2.1 Standard Perturbation TheoryIn Newtonian dynamics, fully nonlinear equation presureless fluid can be written down:

δ +1

a∇ · v = −1

a∇ · (vδ) , ∇ · v +H∇ · v +

3H2

2aΩmδ = −1

a∇ · [(v · ∇) v] , ∇2φ = 4πGρ . (2.1)

The Euler equation can be split into one for divergence and one for vorticity. The vorticity vector ∇ × v decays at the linear order.At nonlinear level, if no anisotropic pressure and no initial vorticity, the vorticity vanishes at all orders. However, in reality, theanisotropic pressure arises from shell crossing, generating vorticity on small scales, even in the absence of the initial vorticity. Thismodifies the SPT equation, such that there exist additional source terms for two kernels.

2.1.1 Basic FormalismWe consider multi-component fluids in the presence of isotropic pressure. In case of n-fluids with the mass densities %i, the pressurespi, the velocities vi (i = 1, 2, . . . n), and the gravitational potential Φ, we have

%i +∇ · (%ivi) = 0 , vi + vi · ∇vi = − 1

%i∇pi −∇Φ , ∇2Φ = 4πG

n∑j=1

%j . (2.2)

Assuming the presence of spatially homogeneous and isotropic but temporally dynamic background, we introduce fully nonlinearperturbations as

%i = %i + δ%i, pi = pi + δpi, vi = Hr + ui, Φ = Φ + δΦ, (2.3)

where H ≡ a/a, and a(t) is a cosmic scale factor. We move to the comoving coordinate x where

r ≡ a(t)x, ∇ = ∇r =1

a∇x,

∂

∂t=

∂

∂t

∣∣∣r

=∂

∂t

∣∣∣x

+

(∂

∂t

∣∣∣rx

)· ∇x =

∂

∂t

∣∣∣x−Hx · ∇x. (2.4)

In the following we neglect the subindex x. To the background order we have

%i + 3H%i = 0,a

a= −4πG

3

∑j

%j , H2 =8πG

3

∑j

%j +2E

a2, (2.5)

where E is an integration constant which can be interpreted as the specific total energy in Newton’s gravity; in Einstein’s gravity wehave 2E = −Kc2 where K can be normalized to be the sign of spatial curvature. To the perturbed order we have

δi +1

a∇ · ui = −1

a∇ · (δiui) , ui +Hui +

1

aui · ∇ui = − 1

a%i

∇δpi1 + δi

− 1

a∇δΦ ,

1

a2∇2δΦ = 4πG

∑j

%jδj . (2.6)

By introducing the expansion θi and the rotation −→ω i of each component as

θi ≡ −1

a∇ · ui, −→ω i ≡

1

a∇× ui, (2.7)

we derive

θi + 2Hθi + 4πG∑j

%jδj =1

a2∇ · (ui · ∇ui) +

1

a2%i∇ ·(∇δpi1 + δi

), (2.8)

−→ω i + 2H−→ω i = − 1

a2∇× (ui · ∇ui) +

1

a2%i

(∇δi)×∇δpi(1 + δi)2

. (2.9)


By introducing decomposition of perturbed velocity into the potential- and transverse parts as

ui ≡ −∇Ui + u(v)i , ∇ · u(v)

i ≡ 0; θi =∆

aUi,

−→ω i =1

a∇× u

(v)i , (2.10)

instead of Eq. (2.9) we have

u(v)i +Hu

(v)i = −1

a

[ui · ∇ui +

1

%i

∇δpi1 + δi

−∇∆−1∇ ·(

ui · ∇ui +1

%i

∇δpi1 + δi

)]. (2.11)

Combining equations above, we can derive

δi + 2Hδi − 4πG∑j

%jδj = − 1

a2[a∇ · (δiui)]· +

1

a2∇ · (ui · ∇ui) +

1

a2%i∇ ·(∇δpi1 + δi

). (2.12)

These equations are valid to fully nonlinear order. Notice that for vanishing pressure these equations have only quadratic ordernonlinearity in perturbations.

2.1.2 Recurrence Relation and Third Order SolutionBy assuming the separability of the time and the spatial dependences, the standard perturbation theory (SPT) takes a perturbativeapproach to the nonlinear solution:

δN (t,k) ≡∞∑n=1

Dn(t)

[n∏i

∫d3qi(2π)3

δ(qi)

](2π)3δD(k− q12···n)F (s)

n (q1, · · · ,qn) ≡∞∑n=1

Dn(t)δ(n)(k) , (2.13)

θN (t,k)

Hf1≡

∞∑n=1

Dn(t)

[n∏i

∫d3qi(2π)3

δ(qi)

](2π)3δD(k− q12···n)G(s)

n (q1, · · · ,qn) ≡∞∑n=1

Dnθ(n)(k) , (2.14)

where q12···n ≡ q1 + · · ·+qn, δ(n)(k) and θ(n)(k) are time-independent n-th order perturbations, F (s)n andG(s)

n are the SPT kernelssymmetrized over its arguments. The (dimensionless) Newtonian linear-order growth factorD(t) is normalized to unity at some earlyepoch t0 when the nonlinearities are ignored, and the initial linear density perturbation is set up in terms of which the perturbativeexpansion is given:

δN (t0,k) ≡ δ(1)1 (t0,k) ≡ δ(k) , D(t) ≡ D1(t)

D1(t0), D + 2HD − 4πGρmD = 0 . (2.15)

With these decompositions in the Fourier space, the LHS of the Newtonian dynamical equations become

δN + θN = Hf1

∞∑n=1

Dn(nδ(n) − θ(n)

), θN + 2HθN − 4πGρmδN = H2f2

1

∑ Dn

2

[(1 + 2n)θ(n) − 3δ(n)

], (2.16)

where we adopted the usual assumption Ωm = f1 = 1 in SPT and utilized the relation between the growth factor and the growth rateD = HDf1. The RHS of the Newtonian dynamical equations are the convolution in the Fourier space:[

−1

a∇ · (δNvN )

](k) =

∫d3Q1

(2π)3

∫d3Q2

(2π)3(2π)3δD(k−Q12)α12θN (Q1, t)δN (Q2, t) ≡ Hf1

∞∑n=1

DnAn(k) ,(2.17)

1

a2∇ · [(vN · ∇)vN ]

(k) =

∫d3Q1

(2π)3

∫d3Q2

(2π)3(2π)3δD(k−Q12)β12θN (Q1, t)θN (Q2, t) ≡ H2f2

1

∞∑n=1

DnBn(k) ,(2.18)

where the vertex functions are defined as

α12 ≡ α(Q1,Q2) ≡ 1 +Q1 ·Q2

Q21

, β12 ≡ β(Q1,Q2) ≡ |Q1 +Q2|2Q1 ·Q2

2Q21Q

22

, (2.19)

and the n-th order perturbation kernels An(k) and Bn(k) are

An(k) =

[n∏i

∫d3qi(2π)3

δ(qi)

](2π)3δD(k− q12···n)

n−1∑i=1

α12Gi(q1, · · · ,qi)Fn−i(qi+1, · · · ,qn) , (2.20)

Bn(k) =

[n∏i

∫d3qi(2π)3

δ(qi)

](2π)3δD(k− q12···n)

n−1∑i=1

β12Gi(q1, · · · ,qi)Gn−i(qi+1, · · · ,qn) , (2.21)

14


withQ1 = q1···i andQ1 +Q2 = k.Therefore, the two Newtonian dynamical equations become algebraic equations with the time-dependence removed:

nδ(n) − θ(n) = An , (1 + 2n)θ(n) − 3δ(n) = 2Bn , (2.22)

and the well-known recurrence formulas for the solutions are

δ(n) =(1 + 2n)An + 2Bn

(2n+ 3)(n− 1), θ(n) =

3An + 2nBn(2n+ 3)(n− 1)

, (2.23)

and similarly so for the SPT kernels

Fn =

n−1∑i=1

Gi(2n+ 3)(n− 1)

[(1 + 2n)α12Fn−i + 2β12 Gn−i] , Gn =

n−1∑i=1

Gi(2n+ 3)(n− 1)

[3α12Fn−i + 2nβ12Gn−i] .

(2.24)Up to the third order in perturbations, with F1 = G1 = 1 these SPT kernels are explicitly

F2 =5

7+

2

7

(q1 · q2)2

q21q

22

+q1 · q2

2q1q2

(q1

q2+q2

q1

), G2 =

3

7+

4

7

(q1 · q2)2

q21q

22

+q1 · q2

2q1q2

(q1

q2+q2

q1

), (2.25)

F3 =2k2

54

[q1 · q23

q21q

223

G2(q2,q3) + cycl.

]+

7

54k ·[

q12

q212

G2(q1,q2) + cycl.

]+

7

54k ·[

q1

q21

F2(q2,q3) + cycl.

], (2.26)

G3 =k2

9

[q1 · q23

q21q

223

G2(q2,q3) + cycl.

]+

1

18k ·[

q12

q212

G2(q1,q2) + cycl.

]+

1

18k ·[

q1

q21

F2(q2,q3) + cycl.

]. (2.27)

Using the recurrence relations, the SPT kernels Fn ∼ Gn ∝ k2 for n > 1 in the limit k → 0, with the individual momentum qi heldfinite. This originates from the momentum conservation of the nonlinear evolution.• Compute the one-loop power spectrum

2.1.3 Asymptotic BehaviorThe leading-order terms are

P22 =

∫d ln q

q3PL(q)

2π2

∫ 1

−1dµ PL(|k− q|)

[3r + 7µ− 10rµ2

14 r(1 + r2 − 2rµ)

]2, lim

q→0F2(q, k − q) = lim

q→kF2(q, k − q) , (2.28)

2P13 = 6PL(k)

∫d3q

(2π)3PL(q)F3(q,−q,k) = 6PL(k)

∫d ln q

q3PL(q)

2π2

1

24

[6k6 − 79k4q2 + 50k2q4 − 21q6

63k2q4+

(q2 − k2)3(7q2 + 2k2)

42k3q5ln

∣∣∣∣k + q

k − q

∣∣∣∣] ,where the solid angle integration of F3 is performed over q. The asymptotic behaviors are

F2 →

kµ2q →∞ if k →∞ (q → 0)(3−5µ2)k2

7q2 → 0 if k → 0 (q →∞), F3 →

− 61k2

1890q2 →∞ if k →∞ (q → 0)

− k2

18q2 → 0 if k → 0 (q →∞), (2.29)

where the angle-averaged F3 is the square bracket above with 1/24. The one-loop terms asymptotically approach, and the leadingcorrection after cancellation of two terms is

limk→∞

P22 = limk→∞

−2P13 =k2P (k)

6π2

∫ ∞0

dq P (q) , limk→∞

(P22 + 2P13) ∝ 0 + P (k)

∫ ∞0

dq q2P (q) ∝ P (k)×∞ , (2.30)

where the learding term k2P (k) is cancelled. For a scale-free power spectrum PL ∝ kn, P22 diverges with n ≥ 1/2 at UV andn ≤ −1 at IR, while P13 diverges with n ≥ −1 at UV and n ≤ −1 at IR. The sum diverges with n > −3 at UV and n ≤ −3 at IR.

Peebles Argument

Given the matter number density (or galaxies), the nonlinear correction can only give rise to k2 in the large scale limit:

ng(x) =

N∑i

δD(x− xi) , ∆ng(k) =

∫d3x [ng(x)− ng(x; t0)] eik·x =

N∑i

eik·xi − eik·xinii =

N∑i

eik·xinii[eik·∆xi − 1

]=k→0

=

N∑i

eik·xinii

N∑j

(1− 1) + ik ·∆xj − (k ·∆xj)2 + · · ·

∝ k2ninig (k) , PNL(k) = Plin(k) + k2Plin(k) + · · · .

The first term vanishes due to mass conservation, and the second term vanishes due to momentum conservation. Therefore, thecorrection should fall as k2 (k2Plin in the power spectrum), compared to the initial.

15


2.1.4 Unified Treatment of the Standard Perturbation TheoryThe master equation can be rephrased as[δab

∂

∂η+ Ωab(η)

]Φb(k; t) = ΛabΦb(k; t) =

∫d3k1 d

3k2

(2π)3δD(k− k1 − k2) γabc(k1,k2) Φb(k1; t) Φc(k2; t), η ≡ lnD(t) ,

(2.31)where Φa = (δm,−θ/f)→ (δm, δm) in the linear regime, θ = ∇ · v/H, the time-dependent matrix and the vertex funtion are

Ωab(η) =

0 −1

− 3

2f2Ωm(η)

3

2f2Ωm(η)− 1

, γabc(k1,k2) =

12

1 + k2·k1

|k2|2

; (a, b, c) = (1, 1, 2)

12

1 + k1·k2

|k1|2

; (a, b, c) = (1, 2, 1)

(k1·k2)|k1+k2|22|k1|2|k2|2 ; (a, b, c) = (2, 2, 2)

0 ; otherwise

. (2.32)

The formal solution can be obtained as

Φa(k; η) = gab(η, η0)ub δ0(k) +

∫ η

η0

dη′gab(η, η′)

∫d3k1 d

3k2

(2π)3δD(k− k1 − k2) γbcd(k1,k2)Φc(k1; η′)Φd(k2; η′) . (2.33)

From the master equation (2.31), the power spectrum is the integral of bispectrum, and so on. This hierarchy arises from thenonlinearity, and it must be truncated to solve it self-consistently.

With the definition of the nonlinear propogator, the linear propagator satisfies

δD(k− k′)Gab(|k|, η, η′) ≡⟨δΦa(k; η)

δΦb(k′; η′)

⟩, 0 =

[δab

∂

∂η+ Ωab(η)

]gbc(η, η

′) = Λabgbc , gab(η, η) = δab , (2.34)

and the power spectrum can be expressed as

Pab(k; η) = Gac(k|η, η0)Gbd(k|η, η0)ucudPlin(k; η0) + P(MC)ab (k; η, η0) , Gab(k|η, η′) = gab(η, η

′) +G(MC)ab (k; η, η′) , (2.35)

where ua = (1, 1) represents the growing mode solution and the mode-coupling terms are defined by the above equation. SeeEq. (2.133).

2.2 Galaxy Bias Primer

2.2.1 Summary of Spherical Collapse ModelFull extension of Kaiser (1984) to N -point without approximation (thresholded sample), they have

1 + ξ(N)ν =

∞∑m=0

wm1212

m12!

wm1313

m13!· · ·Am1Am2 · · ·AmN , A0 = 1 , An =

2xHn−1(x)2−n/2√πxex2erfc(x)

, (2.36)

where x = ν/√

2, w(r) = ξ(r)/σ2, andm =

∑k,l

mkl , mkl = 0 if k ≥ l . (2.37)

In the limit limν→∞An = νn (thresholded sample becomes a peak sample), we have

1 + ξ(N)ν =

∞∑m=0

ν2wm1212

m12!

ν2wm1313

m13!· · · = exp

[ν2(w12 + w13 + · · · )

], (2.38)

corresponding to the Politzer and Wise (1984) equation. For the two-point correlations, we have

ξν =

∞∑m=1

wm

m!A2m → ν2w . (2.39)

The thresholded correlation vanishes, whenever the matter correlation vanishes.

16


2.2.2 Renormalized Galaxy BiasAssuming the galaxy number density is a function of smoothed density field, one can perform a naive expansion:

ng(x) = Fg[δ(x); x] , δg(x) = c0 + c1δ(x) +c22δ2(x) + . . . , (2.40)

where δ is a smoothed matter density with some filter R, cn are the bias parameters. Additional dependence on x indicates thatthere exists some stochasticity on small scales (this scatter is equivalent to the dependence of ng on the small-scale fluctuations δsin the given region), but for Gaussian case this stochasticity does not matter on large scales. When two-point correlation function iscomputed, the correlation function depends on the zero-lag correlators (e.g., σ2 and hence the dependence on the smoothing scaleR),signaling the break-down of the local expansion.

This is to be renormalized, absorbing the zero-lag terms into the bare bias parameters. It is observed that when δg and δm areplotted, the bare bias parameters determined from the scatter plot change as a function of smoothing radius R. But on large scales,the correlation function should be independent of R. Physically, the renormalized bias parameters quantify the response of the meanabundance of tracers to a change in the background matter density ρ of the Universe

bN =ρN

ng

∂N ng∂ρN

, ξg(r) =

∞∑N=1

b2NN !

[ξ(r)]N, (2.41)

where the latter holds for a Gaussian density. Note that renormalization removes the zero-lag matter correlators from the expressionfor the tracer correlation function at all orders, and that the same bias parameters bN describe both the tracer auto- and the cross-correlation with matter.

Basics

The PBS argument can be summarized as follows: if the description of the clustering solely through their dependence on δ issufficient, the expected abundance of tracers in a region with smoothed overdensity δ = D is sufficiently well approximated by themean abundance 〈ng〉 in a fictitious Universe with modified background density ρ′ = ρ(1 + D). The advantage of this approach isthat we only need 〈ng〉, not Fg:

〈ng〉 |D = 〈Fg[0]〉∞∑n=0

cnn!〈(δ +D)n〉 , bN ≡

1

〈ng〉∂N 〈ng[D]〉∂DN

∣∣∣D=0

=ρN

〈ng〉∂N 〈ng〉∂ρN

. (2.42)

Starting with the assumption of ng above and assuming that the small-scale fluctuations are not correlated, we have

ng(x) =

∞∑n=0

1

n!F (n)g [δ = 0; x] [δ(x)]n , 〈F (n)

g [0; x] [δ(x)]n〉 = 〈F (n)g [0; x]〉〈[δ(x)]n〉 , (2.43)

and then the mean is

〈ng(x)〉 =∑n

1

n!

⟨F (n)g [0; x]

⟩〈δn〉 = 〈Fg[0]〉

(1 +

c22σ2L +

c36〈δ3L〉+ . . .

), cn ≡

1

〈Fg[0]〉

⟨F (n)g [0]

⟩. (2.44)

The renormalized bias parameter can be derived by using the PBS argument in Eq. (2.42)

bN =1

N

∞∑n=N

cnn!

n!

(n−N)!〈δn−N 〉 , N =

∞∑n=0

cnn!〈δnL〉 . (2.45)

Using the binomial expansion

〈δn(1)δ(2)〉n∑

N=0

(n

N

)〈δn−N 〉〈δN (1)δ(2)〉c , 〈δn1 δm2 〉 =

n∑k=0

m∑l=0

(n

k

)〈δk1 〉

(m

l

)〈δl2〉〈δn−k1 δm−l2 〉nzl , (2.46)

the two-point correlation function is derived as

ξg(r) =

∑∞n,m=0

cncmn!m! 〈δ

n(1)δm(2)〉∑∞n,m=0

cncmn!m! 〈δn〉〈δm〉

− 1 = · · · =∞∑

N,M=1

bNN !

bMM !〈δN (1)δM (2)〉nzl , (2.47)

ξgm(r) =

∑∞n=1

cnn! 〈δ

n(1)δ(2)〉∑∞n=0

cnn! 〈δn〉

=1

N

∞∑n=1

cnn!

n∑N=0

(n

N

)〈δn−N 〉〈δN (1)δ(2)〉c =

∞∑N=1

bNN !〈δNL (1)δL(2)〉c , N ≡

∞∑n=0

cnn!〈δn〉 ,

17


where nzl: no zero-lag correlator.• Example 1 — Universal mass function:

bN =(−1)N

〈ng〉∂N 〈ng〉∂δNc

=(−1)N

σN1

f(νc)

dNf(νc)

dνNc. (2.48)

• Example 2 — Thresholded sample:

〈ng〉 = P1(νc) =1√2π

∫ ∞νc

dxe−x2/2 =

1

2erfc

(νc√

2

), ξg(r) =

P2(νc; r)

[P1(νc)]2−1 =

2

π

[erfc

(νc√

2

)]−2 ∞∑N=1

[ξ(r)

]NN !σ2N

[HN−1(νc)

]2e−ν

2c ,

(2.49)which are consistent with the renormalized bias parameters.

Curvature bias

Now, consider additional dependence on the coarse-grained Laplacian of the density field, i.e. the curvature, and we follow the sameprocedure:

ng(x) = Fg[δ(x);∇2δ(x); x] , c∇2δ ≡1

〈Fg[0]〉

⟨∂Fg

∂(∇2δ)

∣∣∣δ=0,∇2δ=0

⟩, (2.50)

ξg(r) = c21 〈δ(1)δ(2)〉+ 2c1c∇2δ

⟨δ(1)∇2δ(2)

⟩+O(∇4ξ) = c21

[ξ(r) + 2R2∇2ξ(r)

]+ 2c1c∇2δ∇2ξ(r) + · · · , (2.51)

which is again phrased in terms of disconnected matter correlators and R-dependent bare bias parameters. We need to introduce aR-independent PBS bias parameter for∇2δ as before. We would like a transformation where the Laplacian of the density perturbationshifts by a constant:

∇2δα(x) = ∇2δ(x) +α

l2−→ δα(x) = δ(x) +

α

6l2

(x2 + ~A · x + C

)−→ δα(x) = δ(x) +

α

6l2x2 , (2.52)

where α is a dimensionless small parameter, we have added a constant length scale l (which will disappear in bN ), and we usedsymmetry argument to remove A and C. We can now defined a (renormalized) PBS bias parameter through

b∇2δ =l2

〈ng〉∂ 〈ng[0;α]〉

∂α

∣∣∣α=0

. (2.53)

non-Gaussianity

In a traditional way for computing the primordial non-Gaussianity, we have

ξg(r) = b21ξL(r) + b1b2⟨δ(1)δ2(2)

⟩+O(δ4) ,

⟨δ(1)δ2(2)

⟩= 4fNLσ

2ξφδ(r) , (2.54)

where the appearance of σ2 indicates that the description of the tracer density as a function of the matter density δ alone is insufficienteven on large scales in the non-Gaussian case. Instead, we need to include a dependence of the tracer density on the amplitudeof small-scale fluctuations. This dependence is present regardless of the nature of the initial conditions; however, only in the non-Gaussian case are there large-scale modulations of the small-scale fluctuations, due to mode coupling, whereas in the Gaussian casewe were able to neglect the small-scale fluctuations in the large-scale description. For simplicity, we will parametrize the dependencethrough the variance of the density field on a single scale R∗ < R < r.

We first define the small-scale density field as the local fluctuations around the coarse-grained field δ:

δs(x) ≡ δ∗(x)−δ(x) =

∫d3y[W∗(x−y)−WR(x−y)]δ(y) =

∫d3k

(2π)3Ws(k)δ(k)eik·x , Ws(k) = W∗(k)−WR(k) . (2.55)

We quantify the dependence of the tracer abundance on the amplitude of small-scale fluctuations through

y∗(x) ≡ 1

2

(δ2s(x)

σ2s

− 1

), σ2

s ≡⟨δ2s

⟩=

∫d3k

(2π)3|Ws(k)|2P (k) , (2.56)

In the Gaussian case, ξs(r) → 0 for r RL, so that the small-scale density field and y∗ have no large-scale correlations. We nowgeneralize to explicitly include the dependence on y∗1

ng(x) = Fg[δ(x), y∗(x); x] , 〈ng〉 = 〈Fg[0]〉∑n,m

cnmn!m!

〈δnym∗ 〉 , cnm ≡1

〈Fg[0]〉

⟨∂n+mFg∂δn∂ym∗

∣∣∣δ=0,y∗=0

⟩. (2.57)

1Although this approach here is formally similar to the bivariate local expansion in δ and φ adopted in Giannantonio and Porciani (2010), there is somewhat of aconceptual difference. The effect of non-Gaussianity, and the fact that it derives from a potential φ, only enter through the expressions for the correlators between δand y∗ here. The nature of non-Gaussianity thus decouples from the description of the tracers (which only know about the matter density field) in this approach.

18


Then the two-point correlation function is

ξg(r) =1

N 2

∞∑n,m,n′,m′=0

cnmcn′m′

n!m!n′!m′!

⟨δn(1)ym∗ (1)δn

′(2)ym

′

∗ (2)⟩− 1 , N ≡

∞∑n,m=0

cnmn!m!

〈δnym∗ 〉 , (2.58)

and to the lowest order

ξg(r) =1

N 2

[ (c210 + c10c30σ

2)ξ(r) +

c220

2ξ(r)2 +

(2c10c01 + c01c30σ

2 + 2c10c20σ2)

2fNLξφδ(r) + 2c11c202fNLξφδ(r)ξ(r)].

(2.59)We would like to introduce a physically motivated bias parameter which quantifies the response of the tracer number density to a

change in the amplitude of small-scale fluctuations, without making reference to any coarse-graining on the scaleR. The simplest wayto parametrize such a dependence is to rescale all perturbations by a factor of 1+ε from their fiducial value, where ε is an infinitesimalparameter. Note that this means that the scaled cumulants 〈δn∗ 〉c /σn∗ are invariant, whereas the primordial non-Gaussianity parameterfNL ∼ BΦ/P

2Φ, if non-zero, scales as (1 + ε)−1 under this transformation. Specifically, under this rescaling δ and y∗ transform as

δ(x)→ (1 + ε)δ(x) , y∗(x)→ y∗(x) +

(ε+

ε2

2

)δ2s(x)

σ2s

, (2.60)

where the parameter σ2s in the definition of y∗ is just a constant normalization, and does not change under the ε-transformation. This

is in analogy to keeping ρ fixed in the D-transformation. We can then define a set of bivariate PBS bias parameters bNM as

bNM ≡1

〈ng〉D=0,ε=0

∂N+M 〈ng〉D,ε∂DN∂εM

∣∣∣∣D=0,ε=0

. (2.61)

2.2.3 Gravitational Tidal Tensor BiasThe traceless tidal tensor and its simplest scalar are

sij = ∂i∂jΦ−1

3δKijδ , s2 = sijsij , s2(k) =

∫d3q

(2π)3δqδk−qS2(q,k− q) , S2(k1,k2) =

(k1 · k2)2

k21k

22

− 1

3, (2.62)

where Φ ∝ αχ is a normalized potential. Interestingly, the gravitational instability generates the tidal contribution. By using the SPT,we have

δ(2)(k) =

∫d3q

(2π)3δqδk−qF2(q,k− q) , F2 =

17

21+

1

2

k1 · k2

k1k2

(k1

k2+k2

k1

)+

2

7

[(k1 · k2

k1k2

)2

− 1

3

], (2.63)

where we have re-arranged the usual F2, representing the growth, the shift, and the anisotropy. Note that

vχ = −Hfkδ , v = −

vχ,ik

= HfΨ , Ψ =ikik2

δ , (2.64)

where Ψ is the usual Lagrangian displacement vector. Therefore, the shift term is −Ψ · ∇δ and the anisotropy term is (2/7)s2 inconfiguration space.

2.2.4 MiscellaneousStochasticity

The stochasticity can be defined as

r(R) =〈δm(R)δX(R)〉σm(R)σX(R)

, r(k) =PmX(k)√Pm(k)PX(k)

, (2.65)

and r(k) = 1 to the linear order, while r(R) can be scale-dependent to the linear order, if b1(k) is scale-dependent. To the lowestorder contribution to the stochasticity, we have

1− r(k) =1

4[b1(k)]2PL(k)

∫d3k′

(2π)3[b2(k′,k − k′)]2 PL(k′)PL(|k − k′|) → (bL2 )2

4[b1(k)]2PL(k)

∫d3k′

(2π)3[PL(k′)]

2, (2.66)

where the latter is the large-scale limit (b2 → bL2 ) with constant Lagrangian bias. Nonzero bL2 can generate stochasticity.

19


Deterministic (nonlocal) Bias Arguments

The peak-background split method is necessary because the halo approach is based on a statistical nature of extended Press-Schechtermass function. In such an approach, the local mass function is obtained by averaging over small-scale fluctuations, while large-scalefluctuations are considered as background modulation field, which leads spatial fluctuations of number density of halos. Comparingthe fluctuations of the halo number density field and those of mass, the halo bias is analytically derived. However, the biasing canbe seen as a deterministic process at a most fundamental level, in which any statistical information is not required. One can thinkof getting a halo catalog in numerical simulations to understand the situation. Just one realization of the initial condition determin-istically gives subsequent nonlinear evolutions and formation sites of halos. When only leading growing modes are considered in aperturbation theory, any structure in the universe is deterministically related to the linear density field. The biasing relation shouldnot require statistical information of the field. Any statistical quantities, such as the short-mode power spectrum in the method ofpeak-background split, are not expected to appear at the most fundamental level.

Halo Exclusion

Due to the halo exclusion, the correlation function ξd becomes −1 below the virial radius (d for discrete object of finite size, c forunderlying continuous field). Based on this observation:∫ ∞

0

d3x ξdh(r)j0(kr) = −∫ R

0

d3x j0(kr) +

∫ ∞R

d3x ξch(r)j0(kr) , (2.67)

we re-arrange the integral by using ξch = ξch,NL∫ ∞R

d3x ξch(r)j0(kr) =

∫ ∞R

d3x(ξch,NL − ξch,lin

)j0(kr)−

∫ R

0

d3x ξch,linj0(kr) +

∫ ∞0

d3x ξch,linj0(kr) . (2.68)

Therefore, the halo power spectrum is

P dh (k) = −VWR(k) +

∫ ∞R


)j0(kr)−

∫ R

0

d3x ξch,linj0(kr) +1

nh+ P ch,lin(k) . (2.69)

where we defined the window function and V is the exclusion volume

VWR(k) ≡∫ R

0

d3x j0(kr) . (2.70)

Therefore, the shot noise in the k = 0 limit is

P dh (k → 0) =1

nh− V −

∫ R

0

d3x ξch,lin +

∫ ∞R


). (2.71)

The exclusion contributes negative power to the shot-noise, while the nonlinear effect contributes positive power. Also, due to thescale-dependence in the k = 0 limit, they isolate the linear part within R.

2.3 Lagrangian Perturbation Theory

2.3.1 Basic IdeaThe Lagrangian Perturbation Theory attempts to provide description of the matter and the galaxy distributions today by modelingthose at the very early time and tracing their motion to today. So the critical quantity in this approach is the displacement field Ψ thatrelates the initial (Lagrangian) position q to the final (Eulerian) position x:

x(q, t) = q + Ψ(q, t) . (2.72)

Rather than modeling the density and the velocity in the Standard Perturbation Theory, the Lagrangian Perturbation Theory modelsthe evolution of the displacement field. Therefore, when completely expanded at each order, they both agree, but in general the LPTexpressions correspond to the SPT expression with non-trivial ressumation of different perturbation orders.

At the linear order (Zel’dovich approximation), we have

Ψ(1) = ik

k2δ(1)m (t,k) , v = HfΨ(1) . (2.73)

20


In general, the displacement field in Fourier space is generally represented as

Ψ(n)(p) =iDn

n!

∫d3p1

(2π)3· · · d

3pn(2π)3

(2π)3δD

n∑j=1

pj − p

L(n)(p1, . . . ,pn)δ0(p1) · · · δ0(pn) , (2.74)

and we have

L(1)(p1) =k

k2, L(2)(p1,p2) =

3

7

k

k2

[1−

(p1 · p2

p1p2

)2]. (2.75)

2.3.2 Zel’dovich ApproximationThe Zel’dovich approximation is basically the linear order LPT, in which the displacement field is computed only at the linear orderin perturbations. The density of objects X is related to its Lagrangian quantities as

(1 + δX)d3x = (1 + δq)d3q , δ(x) =

1 + δqJ− 1 =

∫d3q (1 + δq)δ

D [x− q −Ψ(q)]− 1 , (2.76)

where the last equation can be verified by integrating over d3x, and the Jacobian matrix and the determinant are (derivatives are w.r.t.qi)

Jij =∂xi

∂qj= δij + Ψi,j , (J−1)ij ' δij −Ψi,j + · · · , ∇x =

∂

∂xi=∂qj

∂xi∂

∂qj=(J−1

)ji

∂

∂qj,

∂

∂qi= Jji

∂

∂xj, (2.77)

and the determinant is

J = detJij = exp [Tr (ln δij + Ψi,j)] = exp

[Tr

(∑n=1

(−1)n−1(Ψi,j)n

n

)](2.78)

= 1 + Tr

(∑n=1

(−1)n−1(Ψi,j)n

n

)+ Tr

(∑n=1

(−1)n−1(Ψi,j)n

n

)2

+ · · · = 1 + Tr (Ψi,j) +1

2

[Tr2 (Ψi,j)− Tr (Ψi,j)

2]

+O(3)

= 1 +∇ ·Ψ + (∂xΨx∂yΨy + · · −∂xΨy∂yΨx − ··) · · · .

The initial density field is often assumed to be δq = 0. For a Newtonian pressureless self-gravitating fluid embedded in an expandinguniverse, the displacement field is governed by the equation of motion,

r = −∇rφ , r = a x , ∇r =1

a∇x , Φ = φ+ φ , ∇2

xΦ = 4πGρma2(1 + δ) , ∇2

xφ = 4πGρma2δ =

3

2H2Ωm(z)

[1

J− 1

],

x + 2Hx = − 1

a2∇xφ , Ψ + 2HΨ = − 1

a2∇xφ (q + Ψ) → J∇x · Ψ + 2JH∇x · Ψ =

3

2H2Ωm(J − 1) ,

x′′ +Hx′ = −∇xφ , Ψ′′ +HΨ′ = −∇xφ (q + Ψ) → J∇x ·Ψ′′ + JH∇x ·Ψ′ =3

2H2Ωm(J − 1) , (2.79)

where the gradient∇x needs to be expanded and we removed the constant motion (through Jean’s swindle) in r:

a

a= −4πG

3ρm , ∇x ·

(a

ax

)= −4πGρm = − 1

a2∇2

xφ , v = ax = aΨ , ∇x × v = 0 . (2.80)

Assuming that Ψ(n) ∝ Dn, we derive2

D = HDf , Ψ(n) = nHfΨ(n) , Ψ(n) = H2f2n

[H

H2f+

f

Hf2+ n

]Ψ(n) ,

H

H2f+

f

Hf2+

2

f=

3

2

Ωmf2− 1 ,

∴ ∇x · Ψ + 2H∇x · Ψ =∑n

∇x ·Ψ(n)H2f2n

[H

H2f+

f

Hf2+ n+

2

f

]=∑n

∇x ·Ψ(n)H2f2n

[n− 1 +

3

2

Ωmf2

]

=3

2H2Ωm

(1− 1

J

)→ ∇x ·Ψ(n)n

[n− 1 +

3

2

Ωmf2

]=

3

2

Ωmf2

(1− 1

J

)(n)

, (2.81)

2Sometimes, it is assumed Ψ(n) ∝ Dn. Then, we have D1 = D, D2 = 3D2/7, f2 = d lnD2/d ln a = 2f1, and so on.

21


where we assumed a flat Λ but presureless medium. As in the SPT, as long as Ωm(z)/f2 ' 1 the time-dependence of the displacementfield can be separated. The peculiar velocity needs some care due to coordinates.

v(q, t) = ax = aΨ(q, t) , v(x, t) = aΨ(x−Ψ, t) = aΨi − aΨi,jΨj + · · · , (2.82)

and v(q, t) is the velocity assigned to particles that are initially placed in a grid at q and then displaced to a position x in theZel’dovich or 2LPT simulations.

In the original Zel’dovich paper, strange notations are used:

Ψ ≡ D(z)p(q) ≡ −D(z)∇Φ , p = −∇Φ = − ikk2

δ(1)m (t,k)

D(z), Φ = −δ

(1)m (t,k)

k2D(z)= −δ

(1)(q)

k2. (2.83)

Initial Condition in Numerical Simulations

In numerical simulations the initial density distribution is set up at the initial redshift zi ' 50. By using P (k, zi) one generatesδEul.(x, zi) = δEul.(q, zi) + O(2) and use δEul.(q, zi) to generate the displacement field at each grid. Finally, particles at each gridis then displaced by using Ψ.

2.3.3 Resummation in LPT: One-Loop Power SpectrumFollowing Matsubara (2008), the polyspectra of the displacement field are defined as⟨

Ψi1(p1) · · · ΨiN (pN )⟩

c= (2π)3δD(p1 + · · ·+ pN )(−i)N−2Ci1···iN (p1, . . . ,pN ) , Ψi(p) =

∫d3q e−ip·qΨi(q) , (2.84)

where i = x, y, z, representing a vector (including scalar) quantity. The relation p1 + · · · + pN = 0 is always satisfied because ofthe translational invariance. The factors (−i)N−2 in the RHS are there to ensure that the polyspectra Ci1··· are real numbers:

Ψi(−p) = Ψ∗i (p) → Ci1···iN (−p1, · · · ,−pN ) = (−1)NC∗i1···iN (p1, · · · ,pN ) , (2.85)

Note that this relation holds in general. For the displacement field, we have another condition:⟨Ψi1(p1) · · · ΨiN (pN )

⟩∗c

= (−1)N⟨

Ψi1(p1) · · · ΨiN (pN )⟩

c, (2.86)

which together guarantees that the polyspectra are real. For N = 2, Cij(p) = Cij(p,−p), for simplicity. The power spectrum isthen

P (k) =

∫d3q e−ik·q

⟨e−ik·[Ψ(q1)−Ψ(q2)]

⟩− 1

= exp

[−2

∞∑n=1

ki1 · · · ki2n(2n)!

A(2n)i1···i2n

]∫d3q e−ik·q

exp

[ ∞∑N=2

ki1 · · · kiNN !

B(N)i1···iN (q)

]− 1

, (2.87)

where we have⟨[k · (Ψq1

−Ψq2)]N⟩

c= [1 + (−1)N ]

⟨[k ·Ψ(0)]N

⟩c

+

N−1∑j=1

(−1)N−j(Nj

)⟨(k ·Ψq1

)j(k ·Ψq2)N−j

⟩c, (2.88)

A(2n)i1···i2n =

∫d3p1

(2π)3· · · d

3p2n

(2π)3(2π)3δD(p1 + · · ·+ p2n) Ci1···i2n(p1, . . . ,p2n) , (2.89)

B(N)i1···iN (q) =

N−1∑j=1

(−1)j−1

(Nj

)∫d3p1

(2π)3· · · d

3pN(2π)3

(2π)3δD(p1 + · · ·+ pN ) ei(p1+···+pj)·q Ci1···iN (p1, . . . ,pN ) .

The calculation is done at the one-loop power spectrum: all we need are

Cij(p,−p) = C(11)ij + C

(22)ij + C

(13)ij + C

(31)ij , Cijk(p1,p2,p3) = C

(112)ijk + C

(121)ijk + C

(211)ijk , A

(2)ij =

∫d3p

(2π)3Cij(p) ,

B(2)ij = 2

∫d3p

(2π)3eip·qCij(p) , B

(3)ijk = 3

∫d3p1

(2π)3

∫d3p2

(2π)3

[eip1·q − ei(p1+p2)·q

]Cijk(p1,p2,−p1 − p2) , (2.90)

22


and hence the power spectrum is

P (k) = exp[−kikjA(2)

ij

] ∫d3q e−ik·q

[kikj

2B

(2)ij +

kikjkk6

B(3)ijk +

kikjkkkl8

B(2)ij B

(2)kl

](2.91)

= exp[−kikjA(2)

ij

] [kikjCij(k,−k) + kikjkk

∫d3p

(2π)3Cijk(k,−p,p− k) +

kikjkkkl2

∫d3p

(2π)3Cij(p)Cij(k,−p)

].

With this, the resulting one-loop power spectrum is

P (k) = exp

[− k2

6π2

∫dpPL(p)

] [PL(k) + P 1-loop

SPT (k) +k2

6π2PL(k)

∫dpPL(p)

], k−2

NL ≡1

6π2

∫dpPL(p) . (2.92)

When expanded, it is identical to the SPT at one-loop, but due to the resummation, it differs from SPT. The LPT formula breaks downat k ' kNL due to the exponential damping.

2.3.4 Zel’dovich Power SpectrumThe Zel’dovich power spectrum is one-loop LPT power spectrum, in a sense that the displacement field is computed only at the linearorder. However, the quantities are not expanded, but kept in the exponential, such that it is a nonlinear analytic solution under theassumption that the displacement field is only linear.

Defining the 1-D rms displacement, we have

σ2Ψ,1D =

1

3

⟨|Ψ|2

⟩=

1

3

∫d3k

(2π)3

Pm(k)

k2=

1

6π2

∫ ∞0

dk P (k) = I0(0) , Il(q) ≡1

3

∫ ∞0

dp

2π2Pm(p)jl(pq) . (2.93)

To the linear order, we have

P (k) = exp[−kikjA(2)

ij

] ∫d3q e−ik·q

exp

[kikj

2B

(2)ij (q)

]− 1

, A

(2)ij = 〈Ψi(0)Ψj(0)〉 , kikjA

(2)ij = k2σ2

Ψ , (2.94)

where we omitted 1-D in the notation. Assuming that q//z and k in x-z plane (µk = q · k, µp = p · q), the term in the exponentialcan be written as

kikjpipj =

(kp√

1− µ2k

√1− µ2

p cosφp + kpµkµp

)2

, (2.95)

kikj2

B(2)ij (q) =

∫ ∞0

p2dp

2π2

∫dµp2

∫ 2π

0

dφp2π

eipqµpkikjpipj

2p4Pm(p) (2.96)

=

∫ ∞0

p2dp

2π2

Pm(p)

p4

∫dµp2

eipqµpk2p2

[1

2(1− µ2

k)(1− µ2p) + µ2

kµ2p

]= k2(1− µ2

k)[I0(q) + I2(q)] + k2µ2k[I0(q)− 2I2(q)] = −1

2k2(1− µ2

k)σ2⊥(q)− 1

2k2µ2

kσ2‖(q) + k2σ2

Ψ ,

where the cosφp term averages out, the cos2 φp term yields 1/2, q = q1−q2 is the Lagrangian space position in configuration space,and

σ2‖ =

⟨[Ψ‖(q1)−Ψ‖(q2)

]2⟩= 2σ2

Ψ − 2I0(q) + 4I2(q) , (2.97)

σ2⊥ =

⟨[Ψ⊥(q1)−Ψ⊥(q2)]

2⟩

= 2σ2Ψ − 2I0(q)− 2I2(q) , lim

q→0σ2⊥(q) = lim

q→0σ2‖(q) = 0 . (2.98)

2.3.5 Galaxy Bias in the Lagrangian FrameThe local Lagrangian (deterministic) bias factor is introduced as

〈F 〉 = 1 , ρLobj(q) = ρobj F [δR(q)] → 1 + δL = F [δR(q)] =

∫dλ

2πeiδRλ F (λ) , (2.99)

and then the power spectrum is then

P (k) =

∫d3q e−ik·q

[⟨e−ikz [Ψz(q2)−Ψz(q1)]

(1 + δLq1

) (1 + δLq2

)⟩− 1]

=

∫d3q e−ik·q

[∫dλ1

2π

dλ2

2πF (λ1)F (λ2)

⟨ei[λ1δR(q1)+λ2δR(q2)]−ik·[Ψ(q1)−Ψ(q2)]

⟩− 1

]. (2.100)

23


Using the cumulant theorem, we have

⟨ei[λ1δR(q1)+λ2δR(q2)]−ik·[Ψ(q1)−Ψ(q2)]

⟩= exp

∑n1+n2+m1+m2≥1

in1+n2+m1+m2

n1!n2!m1!m2!λ1n1λ2

n2Bn1n2m1m2

(k, q)

, (2.101)

where the multinomial theorem is used and using the translational invariance and the parity symmetry Bn1n2m1m2

= Bn2n1m2m1

, we have

Bn1n2m1m2

(k, q) = (−1)m1 〈[δR(q1)]n1 [δR(q2)]n2(k ·Ψq1)m1(k ·Ψq2

)m2〉c = (−1)m1+m2Bn2n1m2m1

(k,−q) = (−1)m1+m2Bn1n2m1m2

(k,−q) .(2.102)

For a Gaussian (initial) random field, we have a few cases, where we can solve

Bn1n200 (k, q) =

ξR(|q|) , n1 = n2 = 1,

σ2R , (n1 = 2, n2 = 0) or (n1 = 0, n2 = 2),

0 , otherwise,(2.103)

and for n1 = n2 = 0 (matter part) we have

A2m(k) ≡⟨[k ·Ψ(0)]2m

⟩c, Bm1m2

(k, q) ≡ (−1)m1 〈[k ·Ψ(q1)]m1 [k ·Ψ(q2)]m2〉c , (2.104)

B00m1m2

(k, q) =

A2m(k) , (m1 = 2m,m2 = 0) or (m1 = 0,m2 = 2m),

Bm1m2(k, q) , m1 ≥ 1 and m2 ≥ 1,

0 , otherwise .(2.105)

Therefore, the resulting power spectrum is in full generality

Pobj(k) = exp

[2

∞∑m=1

(−1)m

(2m)!A2m(k)

]∫d3q e−ik·q exp

∞∑m1,m2≥1

im1+m2

m1!m2!Bm1m2

(k, q)

×∫ ∞−∞

dλ1

2π

dλ2

2πF (λ1)F (λ2) e−

12 [(λ1σR)2+(λ2σR)2] exp

−λ1λ2ξR(|q|) +

∞∑n1 + n2 ≥ 1m1 +m2 ≥ 1

in1+n2+m1+m2

n1!n1!m1!m2!λ1n1λ2

n2Bn1n2m1m2

(k, q)

−(2π)3δ3

D(k) . (2.106)

With one-loop corrections, the real-space power spectrum can be obtained by expanding the above big square bracket as

Pobj(k) = exp[− (k/kNL)

2]

(1 + 〈F ′〉)2PL(k) +

9

98Q1(k) +

3

7Q2(k) +

1

2Q3(k) + 〈F ′〉

[6

7Q5(k) + 2Q7(k)

]+〈F ′′〉

[3

7Q8(k) +Q9(k)

]+ 〈F ′〉2 [Q9(k) +Q11(k)] + 2〈F ′〉〈F ′′〉Q12(k) +

1

2〈F ′′〉2Q13(k)

+6

7(1 + 〈F ′〉)2

[R1(k) +R2(k)]− 8

21(1 + 〈F ′〉)R1(k)

, (2.107)

where the Lagrangian linear bias factors are

bLn =

⟨∂nδL∂δnR

⟩=

∫ ∞−∞

dλ

2πF (λ)e−λ

2σ2R/2(iλ)n =

1√2π σR

∫ ∞−∞

dδe−δ2/2σR

2 dnF

dδn≡⟨F (n)

⟩. (2.108)

At the linear order, we have the Eulerian bias b1 = 1 + 〈F ′〉, derived without assuming spherical collapse.• compute the scale-dependent coefficients

2.3.6 Perturbation Theory and Nonlocal BiasConsidering the translational invariance (given density fields, coordinates can be freely chosen, i.e., origin and so on), the galaxyfluctuation field can be written in terms of non-local bias (deterministic) functions as

δX(x) =

∞∑n=1

1

n!

∫d3x1 · · · d3xn bn(x− x1, . . . ,x− xn)δm(x1) · · · δm(xn) , (2.109)

24


and its Fourier transform is

δX(k) =

∞∑n=1

1

n!

∫d3k1

(2π)3· · · d

3kn(2π)3

(2π)3δ3D(k1···n−k)× bn(k1, . . . ,kn) δm(k1) · · · δm(kn) , k1···n ≡ k1 + · · ·+kn , (2.110)

where the bias functional is (and the renormalized bias functional)

bn(k1, . . . ,kn) = (2π)3n

∫d3k′

(2π)3

δnδX(k′)

δδm(k1) · · · δδm(kn)

∣∣∣∣δL=0

, cn(k1, . . . ,kn) = (2π)3n

∫d3k′

(2π)3

⟨δnδX(k′)

δδL(k1) · · · δδm(kn)

⟩,

(2.111)and should be rotationally invariant, i.e., b1(k), b2(k1, k2, k12) and so on. Furthermore, the mass density contrast δm(x) is also anonlocal and nonlinear functional of a linear density field δL(x):

δm(k) =

∞∑n=1

1

n!

∫d3k1

(2π)3· · · d

3kn(2π)3

(2π)3δ3D(k1···n − k)Fn(k1, . . . ,kn)δL(k1) · · · δL(kn) , (2.112)

where the SPT kernels are the usual

F1(k) = 1 , F2(k1,k2) =10

7+

(k1

k2+k1

k2

)k1 · k2

k1k2+

4

7

(k1 · k2

k1k2

)2

. (2.113)

Combining the above, the galaxy fluctuation field is

δX(k) =

∞∑n=1

1

n!

∫d3k1

(2π)3· · · d

3kn(2π)3

(2π)3δ3D(k1···n − k)Kn(k1, . . . ,kn)δL(k1) · · · δL(kn) , (2.114)

where we have K1(k) = b1(k), K2(k1,k2) = b1(k)F2(k1,k2) + b2(k1,k2), and

K3(k1,k2,k3) = b1(k)F3(k1,k2,k3) + [b2(k1,k23)F2(k2,k3) + cyc.] + b3(k1,k2,k3) . (2.115)

Similarly, we express the galaxy fluctuation field in the Lagrangian space at initial time as in Eq. (2.109), and the galaxy fluctuationis moved via Lagrangian picture: x = q + Ψ and (1 + δX)d3x = (1 + δLX)d3q as

δX(k) =

∫d3qe−ik·q

[1 + δL

X(q)]e−ik·Ψ(q) − (2π)3δ3

D(k) =

∞∑n+m≥1

(−i)m

n!m!

∫d3k1

(2π)3· · · d

3kn(2π)3

d3k′1(2π)3

· · · d3k′m

(2π)3

×(2π)3δ3D(k1···n + k′1···m − k)bLn(k1, . . . ,kn)δL(k1) · · · δL(kn)[k · Ψ(k′1)] · · · [k · Ψ(k′m)] . (2.116)

Combining the above, we derive the relation (and the relation between the Eulerian and the Lagrangian bias functionals)

K1(k) = k ·L1(k) + bL1 (k) , (2.117)K2(k1,k2) = k ·L2(k1,k2) + [k ·L1(k1)][k ·L1(k2)] + bL1 (k1)[k ·L1(k2)] + bL1 (k2)[k ·L1(k1)] + bL2 (k1,k2) ,

K3(k1,k2,k3) = k ·L3(k1,k2,k3) + [k ·L1(k1)][k ·L2(k2,k3)] + cyc.+ [k ·L1(k1)][k ·L1(k2)][k ·L1(k3)]

+bL1 (k1)[k ·L2(k2,k3)] + cyc.

+bL1 (k1)[k ·L1(k2)][k ·L1(k3)] + cyc.

+bL2 (k1,k2)[k ·L1(k3)] + cyc.

+ bL3 (k1,k2,k3) .

2.3.7 Examples of Bias Models• Local Lagrangian bias: with 〈FX(δR)〉 = 0

δLX = FX (δR) , bLn(k1, . . . ,kn) = F

(n)X (0) , cLn(k1, . . . ,kn) =

⟨F

(n)X (δR)

⟩= (−1)n

∫ ∞−∞

dδR P(n)(δR)FX(δR) , (2.118)

where we assume the smoothing kernel is unity W (kR) = 1, valid in the large scale limit. The thresholded sample is

FX(δR) = AΘ(δR − δt)− 1 , A = 〈Θ(δR − δt)〉−1=

[∫ ∞δt

dδR P(δR)

]−1

,⟨F

(n)X (δR)

⟩= (−1)nA

∫ ∞δt

dδR P(n)(δR) .

(2.119)Halo model bias factors are spherically symmetric, and hence they are local in both Lagrangian and Eulerian.

25


• Spherical collapse model: The number density of halos of mass M , identified at redshift z, in a region of Lagrangian radius R0 inwhich the linear overdensity extrapolated to the present time is δ0, is given by

n(M, z|δ0, R0)dM =ρ

MfMF(ν′) d ln ν′ , ν′ =

δc(z)− δ0[σ2(M)− σ2

0 ]1/2

, σ0 = σ(M0) , M0 =4π

3ρR3

0 . (2.120)

The halo of mass M is collapsed at z, while M0 is assumed uncollapsed at z = 0, and thus we always have δc(z) > δ0. Theconditional number density represents the biasing for the Lagrangian number density of halos. The smoothed density contrast δ0 ofmass modulates the number of halos, and the number density becomes unconditional at R0 → ∞ (δ0 → 0, σ0 → 0). The densitycontrast of halos in Lagrangian space is given by

d ln ν′

d ln ν=

σ2(M)

[σ2(M)− σ20 ], δL

h =n(M, z|δ0, R0)

n(M, z)− 1 =

σ2(M)

σ2(M)− σ20

fMF(ν′)

fMF(ν)− 1 → FX(δR0) , δR → D(z)δ0 . (2.121)

To evaluate the bias functions, the derivatives F (n)X need to be derived, and we consider a limit of the peak-background split σ2(M)

σ20 for simplicity. In this limit, we have

F(n)X (δR) ' 1

Dn(z)

(∂

∂δ0

)nδLh =

(−1

D(z)σ(M)

)nf

(n)MF(ν′)

fMF(ν), F

(n)X (0) '

⟨F

(n)X (δR)

⟩= bLn(k1, . . . ,kn) = cLn(k1, . . . ,kn) .

(2.122)Using the PS, we have

bLn(M) = F(n)X (0) =

νn−1Hn+1(ν)

δcn , Hn(ν) = eν

2/2

(− d

dν

)ne−ν

2/2 , Hn+1(ν) = νHn(ν)− nHn−1(ν) . (2.123)

• Halo model example: Assuming a universal mass function (i.e., only depends on ν), we have

n(x,M) = −2ρ0

M

∂

∂MΘ[δM (x)− δc] , 〈Θ(δM (x)− δc)〉 = P (M, δc) =

1

2F (ν) , F (ν) =

∫ ∞ν

fMF(ν)

νdν , (2.124)

where for the original PS Θ is a step function, but it can be modified for other MF. Also, note that for the PS, there is no statisticalnature at a given point: it is either collapsed or not. Using

δnn(x,M)

δδL(k1) · · · δδL(kn)= −e

i(k1+···+kn)·x

(2π)3n

2ρ0

M

∂

∂M

[Θ(n)(δM − δc)W (k1R) · · ·W (knR)

], Θ(n)(x) = dnΘ(x)/dxn , (2.125)

δM (x) =

∫d3k

(2π)3eik·xW (kR)δL(k) ,

δδM (x)

δδL(k1)=eik1·x

(2π)3W (k1R) ,

δn(x,M)

δδL(k1)= −2ρ0

M

∂

∂M

[Θ(1)(δM − δc)

eik1·x

(2π)3W (k1R)

],

(2.126)and noting that δL

h (x) = n(x,M)/n(M)− 1 we have two equivalent expressions

cLn(k1, . . . ,kn) =

(−1)n∂

∂M

[∂nP (M, δc)

∂δcn W (k1R) · · ·W (knR)

]∂P (M, δc)

∂M

=An(M)

δcn W (k1R) · · ·W (knR) +

An−1(M)σMn

δcn

d

d lnσM

[W (k1R) · · ·W (knR)

σMn

]= bLn(M)W (k1R) · · ·W (knR) +

An−1(M)

δcn

d

d lnσM[W (k1R) · · ·W (knR)] ,

An(M) ≡n∑j=0

n!

j!δcj bLj (M) , An = nAn−1 + δc

nbLn ,

where A0 = 1, A1 = 1 + δcbL1 (M), A2 = 2 + 2δcb

L1 (M) + δ2

c bL2 (M), and note that cLn → bLn in the limit k → 0 (there are a few

steps in the above calculations and cLn for n ≤ 2 are explicitly computed).

26


•Multivariate Lagrangian bias: for multiple variables χα,

χα(q) =

∫d3q′Uα(q − q′)δL(q′), δL

X = FX (χ1, χ2, . . .) , χi = φ → U = −Ga2ρm

|q − q′|, (2.127)

where the last one is an example of additional variable.

bLn(k1, . . . ,kn) =∑

α1,...,αn

∂nFX

∂χα1· · · ∂χαn

∣∣∣∣χα=0

Uα1(k1) · · ·Uαn(kn) , cLn(k1, . . . ,kn) =

∑α1,...,αn

⟨∂nFX

∂χα1· · · ∂χαn

⟩Uα1

(k1) · · ·Uαn(kn) .

(2.128)The peaks formalism is a ten-dimensional multi-variate case:

npk = θ (δR/σR − ν) δ3D (∇δR) |det (∇∇δR)| θ (λ3) , (χα) = (δR,∇δR,∇∇δR) , (Uα) = [W (kR), ikiW (kR),−kikjW (kR)] ,

(2.129)and only renormalized bias functionals are defined for peaks:

cL1 (k) =(A1 +B1k

2), cL2 (k1,k2) =

[A2 +B2

(k2

1 + k22

)+ C2k1 · k2 +D2k

21k

22 + E2 (k1 · k2)

2], (2.130)

where the exact coefficients are calculated by Vincent.

2.3.8 Multi-Point Propagator for Matter and Biased TracersThe original RPT propagator was identified as one-point propagator, and it was extended to multi-point propagators. Here we consideronly one-component (density, not velocity) multi-point propagator for matter and biased object:⟨

δnδm(k)

δδL(k1) · · · δδL(kn)

⟩= (2π)3−3nδ3

D(k− k1···n)Γ (n)m (k1, . . . ,kn) , Γ (n)

m = n!2−n(Dinit

D

)nΓ

(n)1a1···anua1

· · ·uan , (2.131)

where the latter shows the relation to the original multi-point propagator. Furthermore, the renormalized bias cn is the multi-pointpopagators of biasing in Lagrangian space. Thus, we have in terms of the SPT kernels,

Γ (n)m (k1, . . . ,kn) =

∞∑m=0

1

m!

∫d3k′1(2π)3

· · · d3k′m

(2π)3Fn+m(k1, . . . ,kn,k

′1, . . . ,k

′m) 〈δL(k′1) · · · δL(k′m)〉 , (2.132)

and the resulting power spectrum is

PX(k) =

∞∑n=0

1

n!

∫d3k1

(2π)3· · · d

3kn(2π)3

(2π)3δ3D(k − k1···n)

∣∣∣Γ (n)X (k1, . . . ,kn)

∣∣∣2 PL(k1) · · ·PL(kn) (2.133)

→ G2(k, z)Plin(k) + Pmc(k, z) ,

where we can replace F by K for the multi-point propagator for biased objects. The mode-coupling term must arise from the multi-point propagator with n > 1, and the one-point nonlinear propagator G = Γ. Note that as is above, the one-point (or any multi-point)nonlinear propagator can be formally expressed in terms of the SPT kernels, but it cannot be solved analytically.

2.3.9 The Recursion Relation in Lagrangian Perturbation TheoryThe recursion relation in SPT is well known, but LPT has no such, because even for the irrotational fluid, the LPT displacement fieldhas transverse component, starting at third order, which makes things complicated. However, the recursion relation for LPT kernelcan be derived by comparing to SPT at each order (they should be same when the density field is literally expanded).

The density in the Eulerian space is written in terms of LPT displacement as

δ(k, t) =

∫d3q e−ik·q

[e−ik·Ψ − 1

]=

∞∑n=1

Dn(t)δ(n)(k) , δ(n)(k) = SPT(n) , (2.134)

and this matching condition gives

F(s)1 (p1) = k · S(1)

L⊕T (p1) , F(s)2 (p1,p2) = k · S(2)

L⊕T (p1,p2) +1

2k · S(1)

L⊕T (p1) k · S(1)L⊕T (p2) , S

(n)L⊕T ≡ S

(n)L + S

(n)T .

F(s)3 (p1,p2,p3) = k · S(3)

L⊕T (p1,p2,p3) +1

6k · S(1)

L⊕T (p1) k · S(1)L⊕T (p2) k · S(1)

L⊕T (p3) +1

3

k · S(1)

L⊕T (p1) k · S(2)L⊕T (p2,p3) + two perms.

,

27


where k = p1,··· ,n and SL and ST are LPT displacement kernels for longitudinal and transverse parts, respectively. Noting that

S(1)T = S

(2)T = 0 , S

(n)L =

p1···n

p21···n

S(n)L , k · S(n)

L⊕T (p1, · · · ,pn) = k · S(n)L (p1, · · · ,pn) = S

(n)L , (2.135)

one can derive the n-th longitudinal kernel from the lower order kernels and the n-th order SPT kernel, where S(n)L is the n-th order

scalar function.The transverse kernel is a bit tricky to derive, while it is not much needed in most cases. Nevertheless, the irrotational condition

gives rise to the relation:

0 = ∇x × v → εijkd

dηΨk,j − εijkΨl,j

d

dηΨl,k = Ψi,n εnjk

(Ψl,j

d

dηΨl,k −

d

dηΨk,j

), (2.136)

where the time derivative is w.r.t dη = dt/a2 and the partial derivatives are with respect to the Lagrangian coordinates. This conditioncan be expressed perturbatively as

C(n)i ≡

∑p+q=n

εijk

(d

dηΨ

(n)k,j −Ψ

(p)l,j

d

dηΨ

(q)l,k

), ∴ C

(n)i = −

∑p+q=n

Ψ(p)i,mC

(q)m , (2.137)

and the solution is C(n)i = 0. Therefore, the nth order solution of the transverse kernel satisfies

εijkd

dη

(Ψ

(n)T

)k,j

=∑

p+q=n

εijk

(Ψ

(p)L + Ψ

(p)T

)l,j

d

dη

(Ψ

(q)L + Ψ

(q)T

)l,k

. (2.138)

Furthermore, by denoting that the time evolution of the nth order displacement is∝ η−2n (≡ D), we can separate the time evolution ofthe nth order displacement from its longitudinal and transverse part: Ψ

(n)L (η,q) ≡ L(n)(q) η−2n, and Ψ

(n)T (η,q) ≡ T(n)(q) η−2n.

Then, we can evaluate the temporal derivatives and obtain

εijkT(n)k,j =

1

2

∑0<p<n

n− 2p

nεijk (L + T)

(p)l,j (L + T)

(n−p)l,k . (2.139)

• derive the solution

2.4 Redshift-Space Power SpectrumThe distance of objects in cosmology is estimated by measuring the redshift of such objects, and the redshift is affected not only bythe cosmological expansion, but also by the peculiar motion of objects. The power spectrum we measure is therefore affected bypeculiar motion, and it is called the redshift-space distortion. To separate quantities, those without redshift-space distortion are oftencalled real-space quantities.

The real-space position x is mapped in observation as the redshift-space position s as

s = x+z · vaH

z , v = ax , (2.140)

where we assumed that the line-of-sight direction is aligned with z-axis. The redshift-space distortion can be readily incorporatedinto the Lagrangian picture:

Ψs = Ψ +z · ΨH

z , Ψ(n) = nHfΨ(n) , Ψs(n) = Ψ(n) + nf(z ·Ψ(n)) z → Ls(n)i = [δij + nfzizj ]L

(n)j , (2.141)

and the resulting power spectrum at the one-loop level is

Ps(k) = e−k2[1+f(f+2)µ2]σ2

A

[(1 + fµ2)2PL(k) + P 1-loop

sSPT (k) + (1 + fµ2)2[1 + f(f + 2)µ2]k2PL(k)σ2A

], (2.142)

whereH2f2σ2A = σ2

1D is the one-dimensional velocity dispersion.

28


2.4.1 Large-Scale Velocity CorrelationThe velocity field is very difficult to measure in observation, but the velocity correlation is one of the basic statistics in cosmology.The velocity correlation function is

Ψij(r) = 〈vi(x)vj(x+ r)〉 =

∫d3k

(2π)3eik·r H2f2P (k)

kikjk4

= Ψ⊥(r)δij +[Ψ‖(r)−Ψ⊥(r)

]δizδjz , (2.143)

where z-direction is the radial direction and

Ψ⊥ = Ψxx = Ψyy =

∫dk

2π2H2f2P (k)

j1(kr)

kr, Ψ‖ = Ψzz =

∫dk

2π2H2f2P (k)

[j0(kr)− 2j1(kr)

kr

]=

d

dr[rΨ⊥(r)] .

(2.144)Finally, we have

σ23D =

∫d3k

(2π)3

H2f2P (k)

k2=

∫dk

2π2H2f2P (k) , σ2

1D =1

3σ2

3D , 〈v(x) · v(x+ r)〉 = Ψ‖(r) + 2Ψ⊥(r) . (2.145)

Often in the redshift-space distortion literature the velocity is scaled with the conformal Hubble parameters v = v/H andθ = θ/H, and the velocity power spectrum often mean Pθ. The velocity vector is

v = −1

kv,α = i

k

k2a δ =

Hfk2

δ,α = iHf kk2

δ = Hf∇∇−2δ , θ ≡ −∇ · v = Hfδ = −kv , Pθ = H2f2P (k) . (2.146)

2.4.2 Gaussian Streaming ModelFisher (1995): A popular streaming model (Peebles) is the convolution of the real-space correlation function with the relative velocityprobability distribution:

ξ(rσ, rπ) =

∫dy ξ(r) Pv

[rπ − y −

y

rv12(r)

], r2 = y2 + r2

σ , H ≡ 1 , (2.147)

where y/r is the cosine angle along the line-of-sight direction and rπ is the redshift-space distance. The PDF is often assumed to bean isotropic Gaussian. However, this streaming model is known to fail to recover the Kaiser formula (Hamilton) in the linear regime.In fact, the PDF is not independent: Consider a vector η = (δ, δ′, vz, v

′z), where primes mean quantities at x′. By using the number

conservation, we have

1 + ξ(rσ, rπ) =

∫d4η dy(1 + δ)(1 + δ′)Pη(η) δD(rπ − y − v′z + vz) , (2.148)

where the PDF is a Gaussian

Pη =1

(2π)2√

detCexp

(−1

2η†C−1η

),

⟨viv′j

⟩= Ψ⊥(r)δij +

[Ψ‖(r)−Ψ⊥(r)

]rirj , (2.149)

the mean relative velocity above is a number weighted as 〈v12(r)〉 = 〈(v′ − v)(1 + δ)(1 + δ′)〉 ≡ v12(r)r, and

Ψ⊥(r) =β2

2π2

∫dk P (k)

j1(kr)

kr, Ψ‖(r) =

β2

2π2

∫dk P (k)

[j0(kr)− 2j1(kr)

kr

], σ2

v =β2

3 2π2

∫dk P (k) . (2.150)

Using Pηd4η = Pη′d4η′, we change the variable η to η′ as

η′ =

δ+δ−V+

V−

≡

δ′+δΓ+(r)δ′−δΓ−(r)v′+vσ+(r)v′−vσ−(r)

, Γ2±(r) = 2 [ξ(0)± ξ(r)] , σ2

±(r) = 2

[σ2v ±

(yr

)2

Ψ‖(r)±(rσr

)2

Ψ⊥(r)

], (2.151)

and the covariance matrix is then

C′ =⟨η′η′†⟩ =

1 0 0 κ1

0 1 −κ2 00 −κ2 1 0κ1 0 0 1

, C′−1

=

1

1−κ21

0 0 −κ1

1−κ21

0 11−κ2

2

κ2

1−κ22

0

0 κ2

1−κ22

11−κ2

20

−κ1

1−κ21

0 0 11−κ2

1

, (2.152)

29


where

κ1 =y

r

v12(r)

Γ+σ−, κ2 =

y

r

v12(r)

Γ−σ+. (2.153)

Therefore, the PDF in terms of new variable is

Pη′ =1

(2π)2(1− κ21)1/2(1− κ2

2)1/2exp

−1

2

[V 2

+ + V 2− +

(σ+ − κ1V−)2

1− κ21

+(σ− − κ2V+)2

1− κ22

], (2.154)

and the PDF becomes two independent Gaussians when κ1 1 and κ2 1. Now noting that (1 + δ)(1 + δ′) = 1 + Γ+δ+ +(Γ2

+δ2+ − Γ2

−δ2−)/4, we have

1+ξ(rσ, rπ) =

∫dy√

2πσ−(r)exp

[−1

2

(rπ − y)2

σ2−(r)

]×

1 +1

4(Γ2

+ − Γ2−) + Γ+κ1

rπ − yσ−(r)

− 1

4κ2

1Γ2+

[1− (rπ − y)2

σ2−(r)

], (2.155)

where d4η′ are integrated out. Finally, once the PDF is expanded to the linear order (lengthy), then it recovers the linear theory. Onemissing point is that this derivation is based on Gaussianity, while the Kaiser formula is just linear, independent of its Gaussianity.

2.4.3 Complete Treatment of Redshift-space DistortionScoccimarro (2004): Starting from the basic relation, we have the conservation relation (1 + δz)d

3s = (1 + δ)d3r and δz(s) =J−1

[1 + δ(r)

]− 1 with the Jacobian J = 1 +∇z∆ ,

(2π)3δD(k) + δz(k) =

∫d3r [1 + δ(x)] e−ik·s , s ≡ r + ∆ , ∆ = V/H , (2.156)

δz(k) =

∫d3s (1 + δ − J) e−ik·s =

∫d3r

δ(r)− ∇zV (r)

H(z)

ei(kµ V/H+k·r) , (2.157)

where two expressions are quite common and largely equivalent. Since with x2 = x1 + r, we have

(2π)3δD(k) + Pz(k) =

∫d3r e−ik·r

⟨e−ikz [∆(x2)−∆(x1)] (1 + δx1) (1 + δx2)

⟩, (2.158)

Pz(k) =

∫d3r e−ik·r

⟨e−ikz [∆(x2)−∆(x1)]

(δ − ∇zVH(z)

)x1

(δ − ∇zVH(z)

)x2

⟩,

and by Fourier transforming back we have

1 + ξz(s) =

∫dkz2π

∫d3r δD(sx − x)δD(sy − y)

⟨eikz(sz−z−[∆x2

−∆x1]) (1 + δx1) (1 + δx2)

⟩, (2.159)

ξz(s) =

∫dkz2π

∫d3r δD(sx − x)δD(sy − y)

⟨eikz(sz−z−[∆x2

−∆x1])(δ − ∇zVH(z)

)x1

(δ − ∇zVH(z)

)x2

⟩.

We further define useful quantities:

1 + ξz(s) =

∫dkz2π

∫dz eikz(sz−z) [1 + ξ(r)]M =

∫dkz2π

∫dz eikz(sz−z)Z =

∫dz [1 + ξ(r)]P , (2.160)

where the pairwise velocity generating functionM, the pairwise velocity probability distribution function P ,

Z ≡ [1 + ξ(r)]M(kz, r) ≡⟨e−ikz [∆x2

−∆x1] (1 + δx1) (1 + δx2)

⟩, P(sz − z, r) =

∫dkz2π

eikz(sz−z)M(kz, r) . (2.161)

The pairwise velocity moments are defined as

v12(r) ≡(∂M∂kz

)kz=0

, σ212(r) ≡

(∂2M∂k2

z

)kz=0

. (2.162)

In summary, we have

ξz(s) =

∫dkz dz

2πeikz(sz−z) [Z − 1] , Pz(k) =

∫d3r e−ik·r [Z − 1] , (2.163)

and for a Gaussian distribution Z can be exactly solved.

30


2.4.4 Improved Model of the Redshift-Space DistortionThe exact redshift-space power spectrum is

Pz(k) =

∫d3x e−ik·x

⟨e−ikµ f∆uz δ(r) + f∇zuz(r) δ(r′) + f∇zuz(r′)

⟩, uz = − vz

Hf= −iµk

δkk, ∇zuz = µ2

kδk .

(2.164)Note that no dynamical information for velocity and density fields, i.e., Euler equation and/or continuity equation, is invoked inderiving this equation. The above equation is in the form of

Pz(k, µ) =

∫d3x e−ik·x

⟨ej1A1A2A3

⟩, j1 = −i kµf , A1 = uz(r)−uz(r′) , A2(r) = δ+f∇zuz , A3(r′) = δ+f∇zuz .

(2.165)Now consider an arbitrary vector j = (j1, j2, j3), taking the derivative twice with respect to j2 and j3, and setting j2 = j3 = 0, wederive

〈ej·A〉 = exp〈ej·A〉c

→

⟨ej1A1A2

⟩=⟨ej1A1A2

⟩c

exp⟨ej1A1

⟩c, (2.166)

〈ej1A1A2A3〉 = exp〈ej1A1〉c

[〈ej1A1A2A3〉c + 〈ej1A1A2〉c〈ej1A1A3〉c

], (2.167)

and therefore, the redshift-space power spectrum is

Pz(k, µ) =

∫d3x e−ik·x exp

〈ej1A1〉c

[〈ej1A1A2A3〉c + 〈ej1A1A2〉c〈ej1A1A3〉c

]. (2.168)

By comparing to the phenomenolgoical models that incorporate the FoG effect, we deduce that in those models 1) the ensembleaverage product is zero by setting j1 = 0, while keeping the exponential prefactor

〈ej1A1A2〉c〈ej1A1A3〉c ' 〈A2〉c〈A3〉c = 0 , (2.169)

2) the exponential prefactor becomes a Gaussian

A1 = uz(r)− uz(r′) , j1 = −ikµf , exp〈ej1A1〉c

= exp

[∑n

jn1n!

]〈An1 〉c → e−(kµfσv)2

→ 〈An1 〉c ' 2σ2v , n = 2 ,

(2.170)i.e., the spatial correlation is ignored and other higher momements are ignored. Based on this observation, they argue that the Gaussiandamping term is pretty accurate (in the sense, they get resummed), so we keep the second approximation, but make a perturbativecorrection to the first approximation. Up to the second order in j1, we have

〈ej1A1A2A3〉c + 〈ej1A1A2〉c〈ej1A1A3〉c ' 〈A2A3〉+ j1〈A1A2A3〉c + j21

1

2〈A2

1A2A3〉c + 〈A1A2〉c〈A1A3〉c

+O(j31), (2.171)

and the term 〈A21A2A3〉c turns out to be higher order (not clear, doesn’t matter). Therefore, we have with σv fitting parameter

Pz(k, µ) = e−(kµfσv)2 [Pδδ(k)− 2fµ2Pδθ(k) + f2µ4Pθθ(k) +A(k, µ) +B(k, µ)

], (2.172)

where θ = −∇ · v/(Hf), two additional corrections are

A(k, µ) = j1

∫d3x e−ik·x〈A1A2A3〉c = (kµ f)

∫d3q

(2π)3

qzq2Bσ(q,k− q,−k)−Bσ(q,k,−k− q) ,

B(k, µ) = j21

∫d3x e−ik·x〈A1A2〉c〈A1A3〉c = (kµf)2

∫d3q

(2π)3F (q)F (k− q) ,

F (q) =qzq2

[Pδθ(q) + f

q2z

q2Pθθ(q)

],⟨θk1

(δk2

+ fµ22θk2

) (δk3

+ fµ23θk3

)⟩= (2π)3δD(k123)Bσ(k1,k2,k3) .

The bispectrum Bσ is also computed to the 1-loop level, and the extension of the above with linear bias is simple.

2.4.5 Summary of Other Phenomenoligical Models of FoGSome of the popoular models are Eq. (2.172) above, and

Pz(k, µ) = DFoG(kµfσv) P (k, µ) , P =

(1 + fµ2)2Pδ(k) linear ,Pδδ(k) + 2fµ2Pδθ(k) + f2µ4 Pθθ(k) nonlinear ,

(2.173)

31


where D(x) = exp[−x2] for Gaussian, D = 1/(1 + x2) for Lorentzian, and some variation D = 1/(1 + x2/2)2 (Cole et al., 1995).The nonlinear model is Scoccimarro (2004).

• LPT at the one-loop level in Eq. (2.142) is

Ps(k) = e−k2[1+f(f+2)µ2]σ2

A

[(1 + fµ2)2PL(k) + P 1-loop

sSPT (k) + (1 + fµ2)2[1 + f(f + 2)µ2]k2PL(k)σ2A

], (2.174)

whereH2f2σ2A = σ2

1D is the one-dimensional velocity dispersion, but it is often taken as a free parameter.

2.5 Effective Field Theory

2.5.1 Basic FormalsimLet fn(x,p) be the single particle phase space density defined such that fn(x,p) d3x d3p is the probability of particle n occupyingan infinitesimal phase space element. For a point particle, the phase space density is

fn(x,p) = δ3D(x− xn) δ3

D(p−mavn) (2.175)

(where both x and p are co-moving). By summing over n, we define the total phase space density f , mass density ρ, momentumdensity πi, kinetic-tensor σij as

f(x,p) =∑n

δ3D(x− xn) δ3

D(p−mavn) , ρ(x) = ma−3

∫d3p f(x,p) =

∑n

ma−3 δ3D(x− xn) ,

πi(x) = a−4

∫d3p pif(x,p) =

∑n

ma−3 vin δ3D(x− xn) , σij(x) = m−1a−5

∫d3p pi pjf(x,p) =

∑n

ma−3 vin vjn δ

3D(x− xn) ,

The Newtonian potential is sensitive to an infrared quadratic divergence in an infinite homogeneous universe. To isolate this diver-gence we introduce an exponential infrared regulator with cutoff µ (a mass term) and will take the µ→ 0 limit whenever it is allowed.The Newtonian potential φ is

φn(x) = −Ga2

∫d3x′

ρn(x′)

|x− x′|e−µ|x−x′| = − mG

a|x− xn|e−µ|x−xn| , (2.176)

φ(x) = −Ga2

∫d3x′

ρ(x′)− ρb|x− x′|

e−µ|x−x′| =∑n

φn +4πGa2ρb

µ2, µ2

∑n

φn → −4πGa2ρb as µ→ 0 . (2.177)

The k-space version of the Newtonian potential is

φn(k) = − 4πmG

a(k2 + µ2)e−ik·xn , φ(k) =

∑n

φn(k) +4πGa2ρb

µ2(2π)3δ3

D(k) , ∇2φ = 4πGa2(ρ(x, t)− ρb(t)) , (2.178)

where the final term evidently subtracts out the zero-mode. The collisionless Boltzmann equation becomes in the Newtonian limit:(pµ

∂

∂xµ+ Γµαβp

αpβ∂

∂pµ

)fn = 0 → 0 =

Df

Dt=∂f

∂t+

p

ma2· ∂f∂x−m

∑n 6=n

∂φn∂x· ∂fn∂p

, (2.179)

where the final term now involves a double summation over n and n.Let us define the following Gaussian smoothing function

WΛ(x) ≡(

Λ√2π

)3

exp

(−Λ2x2

2

), WΛ(k) = exp

(− k2

2 Λ2

),

∫d3xW (x) = 1 . (2.180)

Of course our results will not depend on the choice of smoothing function, but the Gaussian is chosen for convenience. For certainobservables O(x), we will define the smoothed value by the convolution

Ol(x) = [O]Λ(x) ≡∫d3x′WΛ(x− x′)O(x′) . (2.181)

32


The smoothed versions of the phase space density fl, mass density ρl, momentum density πil , stress-tensor σijl , derivative of Newto-nian potential ∂iφl are

fl(x,p) =∑n

WΛ(x− xn) δ3D(p−mavn) , ρl(x) =

∑n

ma−3WΛ(x− xn) , (2.182)

πil(x) =∑n

ma−3 vinWΛ(x− xn) , σijl (x) =∑n

ma−3 vin vjnWΛ(x− xn) , (2.183)

where the l-subscript indicates that these only depend on the long modes. Similarly, the smoothed version of the Newtonian potentialφl is

φl,n(x) = − mG

a|x− xn|Erf[

Λ|x− xn|√2

]e−µ|x−xn| , φl(x) =

∑n

φl,n +4πGa2ρb

µ2. (2.184)

We now write down the smoothed version of the Boltzmann equation by multiplying it by WΛ and integrating over space

0 =

[Df

Dt

]Λ

=∂fl∂t

+p

ma2· ∂fl∂x−m

∑n 6=n

∫d3x′WΛ(x− x′)

∂φn∂x′

(x′) · ∂fn∂p

(x′,p) , (2.185)

where we applied the smoothing kernel and integrated (integration by part is needed for the spatial derivative). The zeroth momentgives the continuity equation:

ρl + 3Hρl +1

a∂i(ρl v

il) = 0 , vil(x) ≡ πil(x)

ρl(x)=

∑n v

inWΛ(x− xn)∑

nWΛ(x− xn). (2.186)

The first moment gives the Euler equation:

vil +Hvil +1

avjl ∂jv

il +

1

a∂iφl = − 1

a ρl∂j[τ ij]Λ≡ −J il , [τ ij ]Λ ≡ κijl + Φijl , (2.187)

where we defined the source term, which is explicitly

a ρb Jil = ∂j(σ

ijl −ρlv

ilvjl ) +

∑n 6=n

[ρn ∂iφn]Λ− ρl∂iφl , [ρn ∂iφn]Λ =m2G(xin − xin)

a4|xn − xn|3(1 +µ|xn−xn|)e−µ|xn−xn|WΛ(x−xn) ,

(2.188)which requires one to perform a double summation over n and n (which can be computationally expensive). Furthermore, we splitthe effective-stress tensor into two, where κijl is a type of kinetic dispersion and Φijl is a type of gravitational dispersion:

κijl = σijl − ρlvilvjl , Φijl = −

wkkl δij − 2wijl8πGa2

+∂kφl∂kφlδ

ij − 2∂iφl∂jφl8πGa2

, (2.189)

where the former describes the dispersion around the mean velocity and the latter describes the higher gravitational multipoles, andwe have defined

wijl (x) =


[∂i′φ(x′) ∂j′φ(x′)−

∑n

∂i′φn(x′) ∂j′φn(x′)]. (2.190)

Note that we have subtracted out the self term in wijl , and used∇2φ = 4πGa2(ρ− ρb) and∇2φl = 4πGa2(ρl− ρb) to express Φl interms of φ and φl. In the limit in which there are no short modes, it is simple to see from the definition of κl and Φl that they vanishin this limit.

2.5.2 Effective Stress TensorThe derivation of the effective stress-tensor is as follows. We define the short modes to be

σijs ≡ m−1a−5

∫d3p (pi − pil(x))(pj − pjl (x))f(x,p) =

∑n

m

a3(vin − vil(xn))(vjn − v

jl (xn)) δ3

D(x− xn) ,

φs,n ≡ φn − φl,n , ∂iφs =∑n

∂iφs,n , wijs ≡ ∂iφs ∂jφs −∑n

∂iφs,n ∂jφs,n , pil(x) ≡ mavil(x) ,

σijl =[σijs]Λ

+[ρmv

ilvjl

]Λ

+[vil(π

j − ρmvjl ) + vjl (πi − ρmvil)

]Λ, σijs 6= σij − σijl . (2.191)

33


Therefore, we have

κijl '[σijs]Λ

+ρl∂kv

il∂kv

jl

Λ2+O

(1

Λ4

),

Φijl ' − [wkks ]Λδij − 2[wijs ]Λ

8πGa2+∂m∂kφl∂m∂kφlδ

ij − 2∂m∂iφl∂m∂jφl8πGa2Λ2

+O(

1

Λ4

), (2.192)

where some terms are ignored and expanded (not really derived). Now, we re-arrange the effective stress-tensor as

[τ ij]Λ

= κijl + Φijl ≡[τ ijs]Λ

+[τ ij]∂2

,[τ ijs]Λ

=[σijs]Λ− [wkks ]Λδ

ij − 2[wijs ]Λ8πGa2

, (2.193)

[τ ij]∂2

=ρl∂kv

il∂kv

jl

Λ2+∂m∂kφl∂m∂kφlδ

ij − 2∂m∂iφl∂m∂jφl8πGa2Λ2

+O(

1

Λ4

), (2.194)

and by taking the derivative ∂j this leading piece becomes

∂j [τijs ]Λ = ∂j

[σijs]Λ

+ [ρs∂iφs]Λ , [ρs∂iφs]Λ =∑n 6=n

ma−3∂iφs,n(xn)WΛ(x− xn)− [ρl∂iφs]Λ , (2.195)

where we have [ρl∂iφs]Λ = − 12Λ2 ρl∂i∂

2φl + · · · and

∑n 6=n

ma−3∂iφs,n(xn)WΛ(x− xn) =∑n 6=n

m2G

a4

(xn − xn)i

|xn − xn|3

(Erfc

[Λ|xn − xn|√

2

]+

4π|xn − xn|Λ2

WΛ(xn − xn)

)WΛ(x− xn) .

(2.196)It is of some interest to compute the trace of the stress-tensor, so-called mechanical pressure. This includes the gravitational piece

Φiil = − wkkl8πGa2

+∂kφl∂kφl8πGa2

, (2.197)

where the first term is approximately given by

− wkkl8πGa2

≈ 1

2


[δρ(x′)φ(x′)−

∑n

δρn(x′)φn(x′)]

(∇φ)2 = −φ∇2φ+1

2∇2(φ2)

= −1

2

∑n 6=n

Gm2

a4|xn − xn|e−µ|xn−xn|WΛ(x− xn) +

1

2

∑n

4πGmρbaµ2

WΛ(x− xn) ,

and we dropped all terms that are suppressed by the ratio of low k-modes to high k-modes. Therefore, the trace of the stress-tensor isroughly

[τ ]Λ ≈∫d3x′WΛ(x− x′)

[ρ(x′)

(vs(x

′)2 +1

2φs(x

′))− 1

2

∑n

ρs,n(x′)φn(x′)]. (2.198)

The background pressure has the zero mode contribution

pb =1

3〈[τ ]Λ〉 , (2.199)

where we have ignored a correction from the bulk viscosity. There are also stochastic fluctuations to the pressure. Now, sincethe density field ρ(x) can be arbitrarily large for dense objects on small scales, it suggests that each of the contributions to therenormalized pressure, both the kinetic and the gravitational, can be quite large. However, for virialized structures, these two termscancel each other. Hence the only significant contribution to the integral comes from modes of order k ∼ knl which have yet tovirialize.

The effective stress-tensor [τ ij ]Λ in the Euler equation (first moment) comes from smoothing over the short modes (and thesecond-moment) and therefore is sourced by the short modes. When it comes to n-point correlation functions, coupling betweenlong modes is connected to the non-linear terms in the continuity and Euler equations, while coupling between long and shortmodes is connected to the stress-tensor, which generates non-zero correlation functions 〈[τ ij ]Λ δl〉 and 〈[τ ij ]Λ vkl 〉. In order to makefurther progress, we write the stress-tensor as an expansion in terms of the long fields, whose coefficients are determined by various

34


correlation functions. This will involves a type of pressure perturbation term ∝ δij δl, a shear viscosity term ∝ ∂jvil + ∂ivjl −23δij ∂kv

kl , and a bulk viscosity term ∝ δij ∂kvkl . Demanding rotational symmetry, we write a type of effective field theory expansion

for the stress-tensor as

[τ ij ]Λ = ρb

[c2s δ

ij(γ−1 + δl)−c2bvHa

δij ∂kvkl −

3

4

c2svHa

(∂jvil + ∂ivjl −

2

3δij ∂kv

kl

)]+ ∆τ ij , (2.200)

where γ would correspond to the ratio of specific heats in an ordinary fluid (e.g., γ = 5/3 for an ideal monatonic gas) but here it justparameterizes the background pressure term, cs is a sound speed, and csv , cbv are viscosity coefficients with units of speed. Notethat cs, csv, cbv are allowed to depend on time, but not space. Our fluid coefficients are related to the conventional fluid quantities:background pressure pb, pressure perturbation δp, shear viscosity η, and bulk viscosity ζ by

pb =c2sρbγ

, δp = c2sρbδl , η =3ρbc

2sv

4H, ζ =

ρbc2bv

H. (2.201)

In addition to those, there is an entire tower of higher order corrections carrying the appropriate rotational symmetry, guaranteedto exist by the principles of effective field theory. These will be parametrically suppressed at low wave number k compared to thenon-linear wavenumber knl. Here ∆τ ij represents stochastic fluctuations due to fluctuations in the short modes, with 〈∆τ ij〉 = 0.

For later convenience, let’s introduce various quantities:

a J il =1

ρb∂j[τ ij]Λ

= c2s ∂iδl +3

4c2sv ∂jΘ

jil +

(c2sv4

+ c2bv

)∂iΘl , Θl ≡ −

∂kvkl

Ha, Θki

l ≡ −∂kv

il

Ha, (2.202)

a2Akil ≡ 1

ρb∂k∂j

[τ ij]Λ

= a∂kJil = c2s ∂k∂iδl +

3

4c2sv ∂k∂jΘ

jil +

(c2sv4

+ c2bv

)∂k∂iΘl , (2.203)

a2Al ≡1

ρb∂i∂j

[τ ij]Λ

= a∂iJil = c2s ∂

2δl + (c2sv + c2bv)∂2Θl , a2Bl ≡

1

ρb

(∂i∂j −

δij

3∂2

)[τ ij]Λ

= c2sv ∂2Θl .

With these definitions, the EFT parameters are

c2s =PAΘ(x) ∂2PδΘ(x)− PAδ(x) ∂2PΘ Θ(x)

(∂2PδΘ(x))2/a2 − ∂2Pδ δ(x) ∂2PΘ Θ(x)/a2, c2v =

PAδ(x) ∂2PδΘ(x)− PAΘ(x) ∂2Pδ δ(x)

(∂2PδΘ(x))2/a2 − ∂2Pδ δ(x) ∂2PΘ Θ(x)/a2,

c2sv =4

3

PAki Θki(x)− PAΘ(x)

∂2PΘki Θki(x)/a2 − ∂2PΘ Θ(x)/a2=

PBΘ(x)

∂2PΘ Θ(x)/a2, c2v ≡ c2sv + c2bv , PAB(x) = 〈A(x + x′)B(x′)〉 ,

where we defined various correlation function PAB(x). Similarly, in Fourier space, we have

c2s =PAΘ(k)PδΘ(k)− PAδ(k)PΘ Θ(k)

−k2PδΘ(k)2/a2 + k2Pδ δ(k)PΘ Θ(k)/a2, c2v =

PAδ(k)PδΘ(k)− PAΘ(k)Pδ δ(k)

−k2PδΘ(k)2/a2 + k2Pδ δ(k)PΘ Θ(k)/a2,

c2sv =4

3

PAki Θki(k)− PAΘ(k)

−k2PΘki Θki(k)/a2 + k2PΘ Θ(k)/a2=

PBΘ(k)

−k2PΘ Θ(k)/a2.

Let us now compare the relative size of the terms that appear in the Euler equation. We use the Hubble friction term Hvil as thequantity to compare to.

Pressure,ViscosityHubble Friction

∼c2s,v k δl/a

Hvl∼ c2s,v

(k

Ha

)2

∼c2s,v

10−5c2δl ∼ δl ,

non-linear VelocityHubble Friction

∼ k v2l /a

Hvl∼ 10−5

(ck

Ha

)2

∼ δl . (2.204)

2.5.3 One-Loop Power SpectrumIn the absence of vorticity and ignoring stochastic fluctuation ∆τ ij , we can solve the continuity equation and the Euler equation

vil +Hvil + vjl ∂jvil +

1

a∂iφl = −1

ac2s ∂iδl +

3c2sv4Ha2

∂2vil +4c2bv + c2sv

4Ha2∂i∂jv

jl −∆J i , ∆J i ≡ ρ−1

b ∂j∆τij/a , (2.205)

35


using the standard procedure:

dδldτ

+ θl = −∫

d3k′

(2π)3α(k,k′)δl(k− k′)θl(k

′) , α(k,k′) ≡ k · k′

(k′)2, β(k,k′) ≡ k2 k′ · (k− k′)

2|k′|2|k− k′|2,

dθldτ

+Hθl +3

2H2Ωmδl = −

∫d3k′

(2π)3β(k,k′)θl(k− k′)θl(k

′) + c2sk2δl −

c2vk2

Hθl . (2.206)

The time dependence is removed by assuming the EdS-like behavior, but we have to consider the additional EFT terms, which areapproximated as

c2s(a,Λ) = a c2s,0(Λ) , c2v(a,Λ) = a c2v,0(Λ) , Cs,0(k) ≡c2s,0k

2

H20

, Cv,0(k) ≡c2v,0k

2

H20

, (2.207)

where the Λ dependence is implied.For n > 1 we find the following set of relationships between the fields at different orders

An(k) = n δn(k)− θn(k) , Bn(k) = 3δn(k)− (2n+ 1)θn(k)− 2Cs,0(k)δn−2(k)− 2Cv,0(k)θn−2(k) ,

An(k) =

∫d3k1

(2π)3

∫d3k2 δ

3D(k1 + k2 − k)α(k,k1)

n−1∑m=1

θm(k1)δn−m(k2) ,

Bn(k) = −∫

d3k1

(2π)3

∫d3k2 δ

3D(k1 + k2 − k)2β(k,k1)

n−1∑m=1

θm(k1)θn−m(k2) ,

and then

δn(k) =1

(2n+ 3)(n− 1)

[(2n+ 1)An(k)−Bn(k)− 2Cs,0(k)δn−2(k)− 2Cv,0(k)θn−2(k)

]=

n∑j=1

∫d3q1

(2π)3. . .

∫d3qj δ

3D(q1 + · · ·+ qj − k)Fn,j(q1, . . . ,qj) δ1(q1) . . . δ1(qj) , (2.208)

θn(k) =1

(2n+ 3)(n− 1)

[3An(k)− nBn(k)− 2nCs,0(k)δn−2(k)− 2nCv,0(k)θn−2(k)

]=

n∑j=1

∫d3q1

(2π)3. . .

∫d3qj δ

3D(q1 + · · ·+ qj − k)Gn,j(q1, . . . ,qj) δ1(q1) . . . δ1(qj) , (2.209)

where the kernels are

Fn,n(q1, . . . ,qn) =

n−1∑m=1

Gm,m(q1, . . . ,qm)

(2n+ 3)(n− 1)

[(2n+ 1)α(k,k1)Fn−m,n−m(qm+1, . . . ,qn) + 2β(k1,k2)Gn−m,n−m(qm+1, . . . ,qn)

],

Gn,n(q1, . . . ,qn) =

n−1∑m=1

Gm,m(q1, . . . ,qm)

(2n+ 3)(n− 1)

[3α(k,k1)Fn−m,n−m(qm+1, . . . ,qn) + 2nβ(k1,k2)Gn−m,n−m(qm+1, . . . ,qn)

],

and also Fn,1 = Gn,1 = 0 for n even (Gn,1 = nFn,1) and

Fn,1(k) =

n∏m=3,5,...

−2(Cs(k) + (m− 2)Cv(k))

(2m+ 3)(m− 1), for n odd (2.210)

Gn,1(k) = n

n∏m=3,5,...

−2(Cs(k) + (m− 2)Cv,0(k))

(2m+ 3)(m− 1), for n odd . (2.211)

At the one-loop order, we have the usual SPT kernels F (s)2,2 , G(s)

2,2 (F2,1 = G2,1 = 0), and

F3,3(q1,q2,q3) =1

18

[7α(k,q1)F2,2(q2,q3) + 2β(q1,q2 + q3)G2,2(q2,q3) + (7α(k,q1 + q2) + 2β(q1 + q2,q3)G2,2(q1,q2)

],

G3,3(q1,q2,q3) =1

18

[3α(k,q1)F2,2(q2,q3) + 6β(q1,q2 + q3)G2,2(q2,q3) + (3α(k,q1 + q2) + 6β(q1 + q2,q3)G2,2(q1,q2)

],

F3,1(k) = −1

9(Cs,0(k) + Cv,0(k)) , G3,1(k) = −

1

3(Cs,0(k) + Cv,0(k)) . (2.212)

36


The final result for P13 and P22 have two contributions: the contributions from IR modes and the contribution from UV modes,which we write in the following obvious notation

P13(k) = P13,IR(k,Λ) + P13,UV (k,Λ) , P22(k) = P22,IR(k,Λ) + P22,UV (k,Λ) . (2.213)

The IR contribution is the usual SPT but with the cutoff

P13,IR(k,Λ) = 3P11(k)

∫ Λ d3q

(2π)3F

(s)3,3 (q,−q,k)P11(q)

=1

504

k3

4π2P11(k)

∫ Λ/k

0

dr P11(k r)

(12

r2− 158 + 100r2 − 42r4 +

3

r3(r2 − 1)3(7r2 + 2) ln

∣∣∣∣1 + r

1− r

∣∣∣∣) ,

P22,IR(k,Λ) = 2

∫ Λ d3q

(2π)3

[F

(s)2,2 (q,k− q)

]2P11(q)P11(|k− q|)

=1

98

k3

4π2

∫ Λ/k

0

dr

∫ 1

−1

dxP11(k r)P11(k√

1 + r2 − 2rx)(3r + 7x− 10rx2)2

(1 + r2 − 2rx)2,

and the UV contributions are

P13,UV (k,Λ) = F3,1(k,Λ)P11(k) = −(c2s,0(Λ) + c2v,0(Λ))k2

9H20

P11(k) , P22,UV (k,Λ) ≡ ∆P22(k,Λ) , (2.214)

where P13 is set by the (Λ dependent) sound speed and viscosity, and ∆P22 is set by the stochastic fluctuations (ignored later). Inorder to extract the Λ dependence of P13(k,Λ) we take the large r limit inside the integrand (k Λ1 < Λ)

P13,IR(k,Λ) = P13,IR(k,Λ1)− 488

5

1

504

k2

4π2P11(k)

∫ Λ

Λ1

dq P11(q) , (2.215)

and this Λ-dependence should be cancelled by the UV piece with the same k-dependence as

c2s,0(Λ) + c2v,0(Λ) =

(488

5

1

504

9H20

4π2

∫Λ

dq P11(q)

)+ c2s,0(∞) + c2v,0(∞) (2.216)

The constant contributions are determined by explicit matching to numerical simulations. This structure is sort of expected as follows.For the cutoff in the perturbative regime (Λ . knl), the Λ dependence of the fluid parameters is adequately described by the lineartheory. This allows us to estimate the value of the fluid parameters c2s and c2v and their time dependence as a function of the linearpower spectrum. The sound speed is roughly given by the velocity dispersion, so in linear theory we estimate the sound speed byan integral over the velocity dispersion of the short modes. The linear theory is not applicable at very high k, so we shall include aconstant correction as follows

c2s(a,Λ) = α

∫Λ

d ln q ∆2v(q) + c2s(a,∞) , (2.217)

where ∆2v is the velocity dispersion, α is an O(1) constant of proportionality (which is fixed as above), since the sound speed arises

from integrating out the short modes. In the Λ→∞ limit, which we shall eventually take once we cancel the Λ dependence, we findthat c2s(Λ) is non-zero (due to the UV dependence), which we account for with the c2s(∞) constant correction.

For viscous fluids there is a famous dimensionless number which captures its tendency for laminar or turbulent flow; the Reynoldsnumber. The Reynolds number is defined as

Re ≡ρ vL

η∼ HvL

c2sv∼ H2a2

c2svk2δ . 10 (2.218)

where η is shear viscosity, ρ is density, v is a characteristic velocity, and L is a characteristic length scale. The Reynolds number isnot very large, and the system is therefore not turbulent. Furthermore, if we were to estimate the viscosity by Hubble friction, then wewould have Re ∼ δ and so the Reynolds number would be even smaller in the linear or weakly non-linear regime. For cosmologicalparameters ρb ∼ 3×10−30 [g/cm3],H = 70 [km/s/Mpc], and if we take a plausible value for the shear viscosity of c2sv ∼ 2×10−7c2,then the viscosity coefficient is found to be η ∼ 20 Pa s which is perhaps surprisingly not too far from unity in SI units. (For instance,it is somewhat similar to the viscosity of some everyday items, such as chocolate syrup.)

37

3 Nonlinear Relativistic Dynamics: ADM For-malism and its Cosmological Applications

3.1 Arnowitt-Deser-Misner (ADM) Formalism

3.1.1 BasicsThe ADM (Arnowitt et al., 1962) equations are based on splitting the spacetime into the spatial and the temporal parts using a normalvector field na (and in fact another time-like vector ta = (1, 0, 0, 0) defined by the coordinate system, i.e., coordinate observer). It isa fully nonlinear description. The metric is written as

ds2 = −N2dt2 + hαβ(dxα +Nαdt)(dxβ +Nβdt) = (NαNα −N2)dt2 + 2Nαdxαdt+ hαβdx

αdxβ , (3.1)

and

g00 ≡ −N2 +NαNα, g0α ≡ Nα, gαβ ≡ hαβ , g00 = − 1

N2, g0α =

Nα

N2, gαβ = hαβ − NαNβ

N2, (3.2)

where Nα is based on hαβ as the metric and hαβ is an inverse metric of hαβ . The time flow ta is split into its normal and spatial partswith respect to the hypersurface Σt of constant t: the lapse function N = −gabtanb = (na∇at)−1 and the shift vector Nα = pαb t

b,where pab = gab + nanb is the projection tensor. hαβ = gabp

aαp

bβ is in fact the induced 3-metric on the hypersurface.

• Notation convention.— In the ADM formalism, the spacetime coordinate is simply (t, xα). When applied to the FLRW universe,the zeroth time coordinate can be the cosmic time dt = adη, but in that approach one needs conversion to compare with quantitiesderived in the usual perturbation analysis, where 0-th component is the conformal time. Another approach is to put dt = adηin Eq. (3.1), but the metric components (e.g., g00) carry not only the ADM variables, but also the expansion factor a. The otherapproach we adopt here is that the zeroth time component in the ADM formalism is simply considered as the conformal time η, inwhich easy comparison can be made. Last, one has to be careful in lowering and raising indicies in the ADM formalism, as they areall based on hαβ = a2(gαβ + 2Cαβ). Note, however, it is quite often the case the 0-th component is a proper time.

The normal vector na is introduced as:

n0 ≡ −N, nα ≡ 0, n0 =1

N, nα = − 1

NNα . (3.3)

where the normal observer is literally normal to the η-constant hypersurface dxa = (0, dxα) and 0 = nadxa. The fluid quantities are

defined as: (the energy density, the momentum flux, the stress tensor, its trace and traceless part)

E ≡ nanbT ab = N2T 00, Jα ≡ −nbT bα = NT 0α, Sαβ ≡ Tαβ , S ≡ hαβSαβ , Sαβ ≡ Sαβ −

1

3hαβS, (3.4)

where Jα and Sαβ are based on hαβ . The ADM formulation is based on the normal-frame vector ua = na , and there exists onlyone observer (normal observer), which is defined in relation to metric, completely independent of fluid components. Hence in themulti-component situation, e.g., E =

∑iE(i) = nanb

∑i T

ab(i).

Connection and CurvatureThe extrinsic curvature (of 3-geometry in 4-D spacetime) is introduced as

Kαβ ≡1

2N

(Nα:β +Nβ:α − hαβ,0

)= −nα;β = −NΓ0

αβ , Kαβ ≡1

2N

(Nα:β +Nβ:α + hαβ,0

), K ≡ hαβKαβ , Kαβ ≡ Kαβ −

1

3hαβK,

(3.5)where Kαβ is based on hαβ . A colon ‘:’ denotes a covariant derivative based on hαβ . The extrinsic curvature is K = −3H + κ to allorders. The connections become:

Γ000 =

1

N

(N,0 +N,αN

α −KαβNαNβ

), Γ0

0α =1

N

(N,α −KαβN

β), Γ0

αβ = − 1

NKαβ ,

Γα00 =1

NNα

(−N,0 −N,βNβ +KβγN

βNγ)

+NN ,α +Nα,0 − 2NKαβNβ +Nα:βNβ ,

Γα0β = − 1

NN,βN

α −NKαβ +Nα

:β +1

NNαNγKβγ , Γαβγ = Γ

(h)αβγ +

1

NNαKβγ , (3.6)


where Γ(h)αβγ is the connection based on hαβ as the metric, Γ

(h)αβγ ≡

12h

αδ (hβδ,γ + hδγ,β − hβγ,δ). The intrinsic curvatures (of3-geometry) are based on hαβ as the metric:

R(h)α

βγδ ≡ Γ(h)α

βδ,γ − Γ(h)α

βγ,δ + Γ(h)ε

βδΓ(h)α

γε − Γ(h)ε

βγΓ(h)α

δε , R(h)αβ ≡ R

(h)γαγβ , R(h) ≡ hαβR(h)

αβ ,

R(h)αβ ≡ R

(h)αβ −

1

3hαβR

(h) , R = R(h) +KαβKαβ +K2 +2

N(−K,0 +K,αN

α −N :αα) , (3.7)

where the last is related to the Gauss-Codazzi equation. They are fully nonlinear equations. In FRW background, the intrinsiccurvatures are R(3)

αβ = 2Kg(3)αβ and R(3) = 6K, and the extrinsic curvatures are Kαβ = −Hg(3)

αβ and K = −3H , where K = 0,±1is the normalized spatial curvature. These are curvatures of 3D space.•When proper time coordinate (with carat below) is used for 0-th component instead of conformal time:

N = aN , Nα = aNα , Nα = aNα , hαβ = hαβ , Kαβ = Kαβ ,∂

∂t=

1

a

∂

∂τ, (3.8)

and the Christoffel symbols are related as

Γηηη = H+ aΓttt , Γηηα = Γttα , Γηαβ =1

aΓtαβ , Γαηη = a2Γαtt , Γαηβ = aΓαtβ , (3.9)

and of course the expressions for spatial quantities like R(h) and 4D quantities R remain unchanged. Note that when written in termsof ADM quantities, their functional form remains unchanged regardless whether the time coordinate is dt or dη.

3.1.2 Hamiltonian ApproachThe Legendre transformation relates the Lagrangian L(q, q, t) to the Hamiltonian H(q, p, t) = qp − L with canonical momentump = ∂L/∂q, i.e.,

∂H

∂p= q ,

∂H

∂q= −∂L

∂q= − d

dt

(∂L

∂q

)= −p , ∂H

∂t= −∂L

∂t,

d

dt

(∂L

∂q

)− ∂L

∂q= 0 , (3.10)

where we used the Euler-Lagrange equation. The Hamilton-Jacobi theory is based on the canonical transformation from the currentphase space (q, p, t) to new phase space (Q,P, t), but in particular the constant initial condition (Q,P ) = (q0, p0) at t0, i.e., newcoordinates and momenta are constants of the motion. The generating function S(q, P ) of this canonical transformation satisfies

p =∂S

∂q, Q =

∂S

∂P, K = H

[q,∂S

∂q, t

]+∂S

∂t= 0 , (3.11)

where K is the Kamiltonian. The typical example of the generating function is S = qP , which corresponds to the identity trans-formation. The generating function with (Q,P ) = (q0, p0) is identical to the action (note that action is invariant under translationS → S + α).

In GR, a Lagrangian formulation is spacetime covariant: An action is specified on a spacetime manifold. However, a Hamiltonianformulation requires a breakup of spacetime into space and time. The first step is to choose a time-like vector ta and its hypersurfaceΣt. In E&M, one can choose to take the four vector Aa = (φ,Aα) as the configuration field q. However, since the Lagrangian doesnot depend on φ, the conjugate momentum vanishes πφ = 0, indicating that φ is not a dynamical variable. Indeed, this is related tothe gauge arbitrariness in E&M. So, one can takeAα as the configuration field and φ will act as a Lagrange multiplier, which providesa constraint equation.

The ADM-Hamiltonian formalism considers N and Nα as two Lagrange multipliers and the spatial metric hαβ as dynamicvariables. The Einstein-Hilbert Lagrangian is

√−g

16πGR =

N√h

16πG

[R(h) +KαβK

αβ −K2],

√−g = N

√h , (3.12)

and the canonical conjugate momentum is then

παβ =δL

δ(hαβ,0)=

√h

16πG

[hαβK −Kαβ

],

δKαβ

δ(hγδ,0)= − 1

2Nδγαδ

δβ . (3.13)

39


However, the Lagrangian does not depend on the time derivatives of N and Nα, indicating that they are not dynamic variables. TheHamiltonian is then

H = παβhαβ − L = −√hNR(h) +

N√h

(παβπαβ −

1

2π2

)+ 2παβNβ:α (3.14)

=√h

[N

(− R(h)

16πG+ 16πG

παβπαβh

− 16πG

2hπ2

)− 2Nβ

(παβ√h

):α

+ 2

(Nβπ

αβ

√h

):α

],

where the last term becomes the boundary term. Therefore, the constraints of two Lagrange multipliers (Euler-Lagrange equation)are two round parenthesis set zero:

∂H∂N

=∂H∂Nα

= 0 → H ≡√h

(− R(h)

16πG+ 16πG

παβπαβh

− 16πG

2hπ2

)= 0 , Hi ≡ −2παβ :α = 0 . (3.15)

The dynamical equations are

hαβ =δH

δπαβ=

2N√h

(παβ −

1

2πhαβ

)+ 2N(α:β) , (3.16)

παβ = − δH

δhαβ= −N

√h

(Rαβ − 1

2Rhαβ

)+

1

2Nhαβ√h

(πγδπγδ −

1

2π2

)− 2N√

h

(παγπβγ −

1

2ππαβ

)−√h(N ,α:β − hαβN :γ

,γ

)+(παβNγ

):γ−Nα

:γπγβ −Nβ

:γπγα . (3.17)

3.1.3 FRW Metric and ConnectionWe use the following convention for the metric variables:

g00 ≡ −a2 (1 + 2A) , g0α ≡ −a2Bα, gαβ ≡ a2(g

(3)αβ + 2Cαβ

), (3.18)

where A, Bα and Cαβ are perturbed order variables and are assumed to be based on g(3)αβ as the metric. To the second-order, we can

write the perturbation variables explicitly as:

A ≡ A(1) +A(2), Bα ≡ B(1)α +B(2)

α , Cαβ ≡ C(1)αβ + C

(2)αβ . (3.19)

The inverse metric expanded to the second-order in perturbation variables is (note gacgcb = δab holds to all orders):

g00 =1

a2

(−1 + 2A− 4A2 +BαB

α), g0α =

1

a2

(−Bα + 2ABα + 2BβC

αβ), gαβ =

1

a2

(g(3)αβ − 2Cαβ −BαBβ + 4Cαγ C

βγ).

(3.20)The components of the frame four-vector ua are introduced as:

u0 ≡ 1

a

(1−A+

3

2A2 +

1

2V αVα − V αBα

), uα ≡ 1

aV α , u0 = −a

(1 +A− 1

2A2 +

1

2V αVα

),

uα = a(Vα −Bα +ABα + 2V βCαβ

)≡ a vα ≡ a(−v,α + v(v)

α ) , (3.21)

where V α is based on g(3)αβ . The connections are:

Γ000 =

a′

a+A′ − 2AA′ −A,αBα +Bα

(Bα′ +

a′

aBα), Γ0

0α = A,α −a′

aBα − 2AA,α + 2

a′

aABα −BβCβ′α +BβB[β|α] , (3.22)

Γα00 = A|α −Bα′ −a′

aBα +A′Bα − 2A,βC

αβ + 2Cαβ

(Bβ′ +

a′

aBβ),

Γ0αβ =

a′

ag(3)αβ − 2

a′

ag(3)αβA+B(α|β) + C′αβ + 2

a′

aCαβ +

a′

ag(3)αβ

(4A2 −BγBγ

)− 2A

(B(α|β) + C′αβ + 2

a′

aCαβ

)−Bγ

(2Cγ

(α|β) − C|γ

αβ

),

Γα0β =a′

aδαβ +

1

2

(B|αβ −Bα|β

)+ Cα′β +Bα

(A,β −

a′

aBβ

)+ 2Cαγ

(B[γ|β] − C′γβ

),

Γαβγ = Γ(3)α

βγ +a′

ag(3)βγB

α + 2Cα(β|γ) − C|α

βγ − 2Cαδ

(2Cδ(β|γ) − C

|δβγ

)− 2

a′

ag(3)γβ

(ABα +BδCαδ

)+Bα

(B(β|γ) + C′βγ + 2

a′

aCβγ

),

where a vertical bar indicates a covariant derivative based on g(3)αβ . An index 0 indicates the conformal time η, and a prime indicates

a time derivative with respect to η.

40


3.1.4 Relations to ADM QuantitiesThe normal-frame vector na has a property nα ≡ 0. Thus we have

n0 ≡ 1

a

(1−A+

3

2A2 − 1

2BαBα

), nα ≡ 1

a

(Bα −ABα − 2BβCαβ

), n0 = −a

(1 +A− 1

2A2 +

1

2BαBα

), nα = 0.

(3.23)Therefore, the normal frame condition can be derived by imposing Vα −Bα +ABα + 2V βCαβ = 0 . Using Eqs. (3.2) and (3.3) theADM metric variables become:

N = a

(1 +A−

1

2A2 +

1

2BαBα

), Nα ≡ −a2Bα , Nα = −Bα+2BβCαβ , hαβ ≡ gαβ , hαβ =

1

a2

(g(3)αβ − 2Cαβ + 4Cαγ C

βγ). (3.24)

The connection becomes

Γ(h)γ

αβ = Γ(3)γ

αβ +(g(3)γδ − 2Cγδ

) (Cδα|β + Cδβ|α − Cαβ|δ

). (3.25)

The extrinsic curvature in Eq. (3.5) gives:

Kαβ = −a[(Hg(3)αβ +B(α|β) + C′αβ + 2HCαβ

)(1−A) +

1

2Hg(3)αβ

(3A2 −BγBγ

)−Bγ

(2Cγ

(α|β) − C|γ

αβ

)],

Kαβ = −

1

a

[(Hδαβ +B(β|γ)g

αγ + Cα′β

)(1−A) +

1

2Hδαβ (3A2 −BγBγ)−Bγ

(2Cγ

(δ|β)gαδ − Cα|γβ

)− 2Cαγ

(B(γ|β) + C′γβ

)],

Kαβ = −1

a3

[(Hgαβ +B(α|β) + C′αβ − 2HCαβ

)(1−A) +

1

2Hgαβ

(3A2 −BγBγ

)+ 4HCαγ Cβγ −Bγ

(2Cγ(α|β) − Cαβ|γ

)−4Cγ(αC

′β)γ − Cαγ

(Bβ |γ +B

|βγ

)− Cβγ

(Bα |γ +B

|αγ

)],

K = −1

a

[(3H+Bα|α + Cα′α

)(1−A) +

3

2H(3A2 −BαBα

)−Bβ

(2Cαβ|α − C

αα|β

)− 2Cαβ

(C′αβ +Bα|β

)]

≡ −3H + κ = −3H +

[3(−ϕ+Hα)−

∆

a2χ

]+O(2) ,

Kαβ = −a(

B(α|β) + C′αβ

)(1−A)−Bγ

(2Cγ

(α|β) − C|γ

αβ

)−

2

3Cαβ

(Bγ|γ + Cγ′γ

)−

1

3g(3)αβ

[(Bγ|γ + Cγ′γ

)(1−A)−Bγ

(2Cδγ|δ − C

δδ|γ

)− 2Cγδ

(C′γδ +Bγ|δ

)]

−→No VT

− (1− α)χ,α|β + 2χ,(αϕ,β) −1

3g(3)αβ [− (1− α) ∆χ+ 2χ,γϕ,γ ] −→

χ→00 , (3.26)

where Kαβ = σαβ of the normal observer and vanishes if χ = 0 (ignoring vector and tensor). To the background and linear order,we have

R = 6

(2H2 + H +

K

a2

), δR = 2

[−κ− 4Hκ+

(k2

a2− 3H

)α+ 2

k2 − 3K

a2ϕ

]. (3.27)

3.2 Energy-Momentum TensorGiven the gravitational action and the matter Lagrangian, the energy momentum tensor is derived as

Tab = gabLm − 2δLmδgab

, T ab = gabLm + 2δLmδgab

,√−g T ab = 2

δ(√−gLm)

δgab. (3.28)

3.2.1 General DecompositionThe energy-momentum tensor is decomposed into fluid quantities based on the four-vector field ua as (the most general decomposi-tion)

Tab ≡ µuaub + p (gab + uaub) + qaub + qbua + πab, (3.29)

whereuaqa = 0 = uaπab, πab = πba, πaa = 0. (3.30)

41


The variables µ, p, qa and πab are the energy density, the isotropic pressure (including the entropic one), the energy flux and theanisotropic pressure based on ua-frame, respectively (i.e., they are measured by an observer with ua). We have

µ ≡ Tabuaub, p ≡ 1

3Tabh

ab, qa ≡ −Tcduchda, πab ≡ Tcdhcahdb − phab, (3.31)

where qa represents the “spatial” energy flux measured by the observer or in the observer rest frame. To the perturbed order wedecompose the fluid quantities as:

µ ≡ µ+ δµ, p ≡ p+ δp, qα ≡ aQα, παβ ≡ a2Παβ , c2s ≡p

µ, δp ≡ c2sδµ+ e , e ≡ pΓ , Γ =

δp

p− δρ

ρ, (3.32)

where Qα and Παβ are based on g(3)αβ . We have Πα

α − 2CαβΠαβ = 0 , which follows from πaa = 0 or Sαα = 0. At the backgroundlevel, we have µ = µ and p = p and zeros for the other fluid quantities (i.e., they are perturbations). For adiabatic perturbations(e = 0), c2s = δp/δµ = p/µ.

Note that the energy (total energy density) µ and so on in the energy-momentum tensor depends on which observer ua measures,i.e., energy and momentum are frame-dependent quantities. It is better defined as above (e.g., µ = Tabu

aub), rather than defined insome cases, in terms of components of the four momentum vector.

3.2.2 Tetrad ApproachGiven an observer ua (timelike −1 = uaua), one can define a local Lorentz frame (where the metric is Minkowski) by constructingthree spacelike orthonormal vectors [ei]

a. For example, one can construct three rectangular basis vectors [ex]a, [ey]a, [ez]a and of

course [et]a = ua = −[et]a, where the component index a represents that their components are written in a FRW coordinate. The

orthonormality condition and the spacelike normalization is

δij = gab[ei]a[ej ]

b , 1 = [ei]a[ei]a , (3.33)

where the tetrad index i should not be summed over. The tetrad index can be raised and lowered by using the Minkowski metric ηab,but there is no difference for spacelike vectors in the Minkowski metric.

The fluid quantities measured by the observer are

µ ≡ Tabuaub = Tab[et]a[et]

b , p ≡ 1

3Tabh

ab =1

3Tab

3∑i=1

[ei]a[ei]

b , hab ≡ gab + uaub =

3∑i=1

[ei]a[ei]

b . (3.34)

The energy flux vector has four components in a FRW coordinate and three components in the observer rest frame, and their compo-nents are related as

qa ≡ −Tcduchda = −Tcduc3∑i=1

[ei]a[ei]d , [qi] = [ei]

aqa = −Tcduc[ei]d . (3.35)

Similarly, the anisotropic tensor can be derived as

πab ≡ Tcdhcahdb − phab, [πij ] = πab[ei]a[ej ]

b = Tab[ei]a[ej ]

b − 1

3p δij . (3.36)

To the linear order, the tetrad basis vectors are

[ei]a =

1

a

[Vi −Bi, δαi (1− ϕ)− γ |α,i

], [ei]a = a

[−Vi, g(3)

αi (1 + ϕ) + γ,i|α

]. (3.37)

3.2.3 Frame Choice for Energy-Momentum TensorA frame choice is not a gauge choice; coordinates xa remains unchanged by any frame choice. Therefore, components of the energy-momentum tensor are not affected by our frame choice, i.e., each component of the energy-momentum tensor is identical in any frame.However, since fluid quantities (µ, p, π, etc) are measured by a specific observer, they are different in different frames.

To the linear-order we notice that δµ, δp, Παβ are independent of the frame choice, andQα+(µ+p)(Vα−Bα) is a frame-invariantcombination (because they are independent of our choice of Qα and Vα). However, to the second-order we no longer have such aluxury. As the fluid quantities are defined based on the frame vector as in eq. (3.31) the values of δµ, δp and Παβ are dependent onthe frame (depending on observers).

The reason we have a freedom to choose a frame (which is a decision about Qα and Vα) is because we have 10 independentinformations in Tab which can be allocated to the energy density µ (one), the pressure p (one), the anisotropic stress Παβ (five,because it is tracefree). The remaining (three) informations can be assigned to either the velocity Vα (three) or the flux Qα (three); orsome combinations of Vα and Qα with total three informations.

42


3.2.4 Normal Frame and Its Relation to Energy FrameThe ADM fluid quantities in Eq. (3.4) correspond to the fluid quantities based on the normal-frame vector as

E ≡ nanbT ab = N2T 00 = µ+ δµ, Sαβ ≡ Tαβ , Jα ≡ −nbT bα = NT 0α = qα ≡ aQα, Jα =

1

a

(Qα − 2CαβQβ

), (3.38)

S ≡ hαβSαβ = 3(p+ δp), Sαβ ≡ Sαβ −1

3hαβS = παβ ≡ a2Παβ , Sαβ = Πα

β − 2CαγΠβγ , Sαβ =1

a2

(Παβ − 4Cγ(αΠβ)

γ

),

From Eqs. (3.30) and (3.32) we have

qα ≡ aQα, q0 = −aQαBα, παβ ≡ a2Παβ , π0α = −a2ΠαβBβ , π00 = 0, Πα

α − 2CαβΠαβ = 0 . (3.39)

Note that the fluid quantities are measured by the observer ua = na, hence uα = nα = 0. The normal-frame quantities (indicated bya superscript N ) are related to the energy-frame quantities (by a superscript E) as QEα ≡ 0

QEα ≡ 0 , δµN = δµE + (µ+ p)(V Eα −Bα

) (V Eα −Bα

), δpN = δpE +

1

3(µ+ p)

(V Eα −Bα

) (V Eα −Bα

),

QNα = (µ+ p)(V Eα −Bα

)+ (µ+ p)

(ABα + 2V EβCαβ

)+(δµE + δpE

) (V Eα −Bα

)+(V Eβ −Bβ

)ΠEαβ ,

ΠNαβ = ΠE

αβ + (µ+ p)(V Eα −Bα

) (V Eβ −Bβ

)− 1

3g

(3)αβ (µ+ p)

(V Eγ −Bγ

) (V Eγ −Bγ

),

δµE = δµN − 1

µ+ pQNαQNα , δpE = δpN − 1

3

1

µ+ pQNαQNα , ΠE

αβ = ΠNαβ −

1

µ+ p

(QNα Q

Nβ −

1

3g

(3)αβQ

NγQNγ

),

(µ+ p)(V Eα −Bα

)= QNα − 2QNβCαβ −

δµN + δpN

µ+ pQNα −QNβ

ΠNαβ

µ+ p− (µ+ p)

(ABα + 2BβCαβ

). (3.40)

3.3 Fully Nonlinear Einstein Equations

3.3.1 ADM EquationsSpacetime is unified in general relativity, but to follow the evolution of a system at some initial time, we need to undo the unificationand split space and time, i.e., need to cast the Einstein’s equation into Cauchy (initial-value) problem — construct initial data consis-tent with the constraint equations, then solve the evolution (dynamical) equations. A simple analogy to E&M is that the Maxwell’sequation is composed of two constraint equations (no time evolution)

CE ≡ ∇ ·E− 4πρ = 0 , CB ≡ ∇ ·B = 0 , (3.41)

and two dynamical equations∂

∂tE = ∇×B− 4πj ,

∂

∂tB = −∇×E . (3.42)

Writing B = ∇×A, the constraint equation CB = 0 is identically satisfied, and the remaining equations become

CB = 0 ,∂

∂tCE = 0 ,

∂

∂tE = −∇2A +∇(∇ ·A)− 4πj ,

∂

∂tA = −E−∇φ . (3.43)

A complete set of the ADM equations is the following Bardeen (1980). The Einstein equation (Gab ∝ Tab) is split: 00-part and0α-part, involving non-dynamical quantities N and Nα (the Lagrangian is independent of their time derivatives). These give twoconstraint equations that relate the energy-momentum to the extrinsic and the intrinsic geometry. αβ-part involves dynamics of hαβ :Two propagation equations (i.e., time derivatives, how they evolve); Finally, there exist two usual conservation equations from theenergy-momentum tensor. Energy constraint and momentum constraint equations:

R(h) = KαβKαβ −2

3K2 + 16πGE + 2Λ , Kβ

α:β −2

3K,α = 8πGJα . (3.44)

Trace and tracefree ADM propagation equations:

K,0N−1 −K,αN

αN−1 +N :ααN−1 − KαβKαβ −

1

3K2 − 4πG (E + S) + Λ = 0 , (3.45)

Kαβ,0N

−1 − Kαβ:γN

γN−1 + KβγNα:γN−1 − Kα

γNγ

:βN−1 = KKα

β −(N :α

β −1

3δαβN

:γγ

)N−1 + R

(h)αβ − 8πGSαβ .

43


Energy and momentum conservation equations:

E,0N−1 − E,αNαN−1 −K

(E +

1

3S

)− SαβKαβ +N−2

(N2Jα

):α

= 0 ,

Jα,0 − Jα:βNβ − JβNβ

:α −KJαN + EN,α +NSβα:β + SβαN,β = 0 . (3.46)

In the multi-component system, two conservation equations hold separately for each component, and interaction terms should beconsidered if there is any between components.

3.3.2 Second-Order ADM EquationsThe basic set of the ADM equations is derived with fluid quantities based on the normal-frame. By using Eq. (3.40) we can recoverthe equations with fluid quantities based on the energy frame. The fluid quantities are based on the energy frame here!The definition of δK:

K + 3H + δK − 3HA+ Cαα +1

aBα|α ≡ n0 , (3.47)

where K ≡ K + δK and K is read from Eq. (3.26).Energy constraint equation:

16πGµ+ 2Λ− 6H2 − 1

a2R(3) + 16πGδµ+ 4HδK − 1

a2

(2C

β|αα β − 2C

α|βα β −

2

3R(3)Cαα

)≡ n1. (3.48)

Momentum constraint equation:[Cβα +

1

2a

(Bβ|α +B |β

α

)]|β− 1

3

(Cγγ +

1

aBγ|γ

),α

+2

3δK,α + 8πGa(µ+ p)(−v,α + v(v)

α ) ≡ n2α . (3.49)

Trace of the ADM propagation equation:

−[3H + 3H2 + 4πG (µ+ 3p)− Λ

]+ δK + 2HδK − 4πG (δµ+ 3δp) +

(3H +

∆

a2

)A ≡ n3. (3.50)

Tracefree ADM propagation equation:[Cαβ +

1

2a

(Bα|β +B

|αβ

)]·+ 3H

[Cαβ +

1

2a

(Bα|β +B

|αβ

)]−

1

a2A|αβ −

1

3δαβ

[(Cγγ +

1

aBγ|γ

)·+ 3H

(Cγγ +

1

aBγ|γ

)−

1

a2A|γγ

]+

1

a2

[Cαγ |βγ + C

γ|αβ γ − C

α|γβ γ − C

γ|αγ β −

2

3R(3)Cαβ −

1

3δαβ

(2C

δ|γγ δ − 2C

γ|δγ δ −

2

3R(3)Cγγ

)]− 8πGΠαβ ≡ n

α4β . (3.51)

Energy conservation equation:

[µ+ 3H (µ+ p)] + δµ+ 3H (δµ+ δp)− (µ+ p) (δK − 3HA) +1

a(µ+ p)

[V α −Bα +ABα + 2V βCαβ

]|α ≡ n5. (3.52)

Momentum conservation equation:

1

a4

[a4 (µ+ p)

(Vα −Bα +ABα + 2V βCαβ

)]·+

1

a(µ+ p)A,α +

1

a

(δp,α + Πβ

α|β

)≡ n6α. (3.53)

Energy conservation equation for the i-th component:[µ(i) + 3H

(µ(i) + p(i)

)+

1

aI(i)0

]+ δµ(i) + 3H

(δµ(i) + δp(i)

)−(µ(i) + p(i)

)(δK − 3HA)

+1

a

(µ(i) + p(i)

) [V α(i) −B

α +ABα + 2V β(i)Cαβ

]|α

+1

aδI(i)0 ≡ n(i)5. (3.54)

Momentum conservation equation for the i-th component:

1

a4

[a4(µ(i) + p(i)

) (V(i)α −Bα +ABα + 2V β(i)Cαβ

)]·+

1

a

(µ(i) + p(i)

)A,α +

1

a

(δp(i),α + Π β

(i)α|β − δI(i)α)≡ n(i)6α.(3.55)

44


3.3.3 Decomposed Perturbation EquationsThe scalar-type perturbation: (note that ni are quadratic terms and hence zero to the linear order)

definition of κ : κ− 3Hα+ 3ϕ+∆

a2χ = n0, χ ≡ aβ + aγ′ , (3.56)

ADM energy constraint G00 : 4πGδµ+Hκ+

∆ + 3K

a2ϕ =

1

4n1, (3.57)

ADM momentum constraint G0α : κ+

∆ + 3K

a2χ− 12πG(µ+ p)av =

3

2∆−1∇αn2α ≡ n2, (3.58)

Raychaudhuri equation Gαα −G00 : κ+ 2Hκ− 4πG (δµ+ 3δp) +

(3H +

∆

a2

)α = n3, (3.59)

trace free of ADM Gαβ −1

3δαβG

γγ : χ+Hχ− ϕ− α− 8πGΠ =

3

2a2 (∆ + 3K)

−1∆−1∇α∇βn β

4α ≡ n4, (3.60)

T b(i)0;b = I(i)0 : δµ(i) + 3H(δµ(i) + δp(i)

)−(µ(i) + p(i)

)(κ− 3Hα+

1

a∆v(i)

)+

1

aδI(i)0 = n5(i), (3.61)

T b(i)α;b = I(i)α :[a4(µ(i) + p(i))v(i)]

·

a4(µ(i) + p(i))− 1

aα− 1

a(µ(i) + p(i))

(δp(i) +

2

3

∆ + 3K

a2Π(i) − δI(i)

)= −

∆−1∇αn6(i)α

µ(i) + p(i)≡ n6(i),

T b0;b = 0 : δµ+ 3H (δµ+ δp)− (µ+ p)

(κ− 3Hα+

1

a∆v

)= n5, (3.62)

T bα;b = 0 :[a4(µ+ p)v]·

a4(µ+ p)− 1

aα− 1

a(µ+ p)

(δp+

2

3

∆ + 3K

a2Π

)= − 1

µ+ p∆−1∇αn6α ≡ n6, (3.63)

δφ+ 3Hδφ− ∆

a2δφ+ V,φφδφ− φ (κ+ α)−

(2φ+ 3Hφ

)α = nφ − φn0, (3.64)

δφ(i) + 3Hδφ(i) −∆

a2δφ(i) +

∑k

V,φ(i)φ(k)δφ(k) − φ(i) (κ+ α)−

(2φ(i) + 3Hφ(i)

)α = nφ(i)

− φ(i)n0. (3.65)

The vector-type perturbation:

∆ + 2K

2a2Ψ(v)α + 8πG(µ+ p)v(v)

α =1

a

(n2α −∇α∆−1∇βn2β

)≡ n(v)

2α , (3.66)

Ψ(v)α + 2HΨ(v)

α − 8πGΠ(v)α = 2a (∆ + 2K)

−1(∇βn β

4α −∇α∆−1∇γ∇βn β4γ

)≡ n(v)

4α , (3.67)

[a4(µ+ p)v(v)α ]·

a4(µ+ p)+

∆ + 2K

2a2

Π(v)α

µ+ p=

1

µ+ p

(n6α −∇α∆−1∇βn6β

)≡ n(v)

6α , (3.68)

[a4(µ(i) + p(i))v(v)(i)α]·

a4(µ(i) + p(i))+

∆ + 2K

2a2

Π(v)(i)α

µ(i) + p(i)− 1

a

δI(v)(i)α

µ(i) + p(i)=

1

µ(i) + p(i)

(n6(i)α −∇α∆−1∇βn6(i)β

)≡ n(v)

6(i)α. (3.69)

The tensor-type perturbation

C(t)αβ + 3HC

(t)αβ −

∆− 2K

a2C

(t)αβ − 8πGΠ

(t)αβ = n4αβ −

3

2

(∇α∇β −

1

3g

(3)αβ∆

)(∆ + 3K)

−1∆−1∇γ∇δn δ

4γ

−2∇(α (∆ + 2K)−1 (∇γn4β)γ −∇β)∆

−1∇γ∇δn δ4γ

)≡ n(t)

4αβ . (3.70)

In order to derive Eqs. (3.60), (3.67), (3.70), it is convenient to show

1

a2

(∇α∇β −

1

3g(3)αβ∆

)(χ+Hχ− ϕ− α− 8πGΠ)+

1

a3

(a2Ψ

(v)(α|β)

)·−8πG

1

aΠ

(v)(α|β)+C

(t)αβ+3HC

(t)αβ−

∆− 2K

a2C

(t)αβ−8πGΠ

(t)αβ = n4αβ , (3.71)

which follows from Eq. (3.51). In our perturbative approach, the second-order perturbations are sourced by the quadratic combinationsof all three-types of linear-order terms.

45


3.3.4 Closed Form EquationsSimilarly, the right-hand side is quadratic and vanishes to the linear order. We have equations that closely resemble the Newtonianhydrodynamics:

Eqs. (3.57), (3.58) :∆ + 3K

a2ϕχ + 4πGδµv =

∆ + 3K

a2ϕ(q)χ + 4πGδµ(q)

v +1

4N1 −HN (s)

2 , (3.72)

Eqs. (3.58), (3.62), (3.63) : δµv + 3Hδµv −∆ + 3K

a2[a(µ+ p)vχ + 2HΠ] = δµ(q)

v + 3Hδµ(q)v −

∆ + 3K

a2a(µ+ p)v(q)

χ

+N5 + (µ+ p)(N

(s)2 + 3aHN

(s)6

), (3.73)

Eq. (3.60) : ϕχ + αχ + 8πGΠ = ϕ(q)χ + α(q)

χ −N(s)4 , or ϕχ + αχ + 8πGΠχ = −N (s)

4χ , (3.74)

Eqs. (3.60), (3.63) : vχ +Hvχ −1

a

(αχ +

δpvµ+ p

+2

3

∆ + 3K

a2

Π

µ+ p

)= v(q)

χ +Hv(q)χ −

1

a

(α(q)χ +

δp(q)v

µ+ p

)+N

(s)6 , (3.75)

Eqs. (3.58), (3.60) : ϕχ +Hϕχ + 4πG(µ+ p)avχ + 8πGHΠ = ϕ(q)χ +Hϕ(q)

χ + 4πG(µ+ p)av(q)χ +

1

3

(N0 −N (s)

2

)−HN (s)

4 .

Using the closed form equations, we can show that

Φ ≡ ϕv −K/a2

4πG(µ+ p)ϕχ =

H2

4πG(µ+ p)a

[ aH

(ϕχ − ϕ(q)

χ

)]·+ 2H2 Π

µ+ p+ Φ(q) +NΦ, (3.76)

Φ =Hc2s∆

4πG(µ+ p)a2

(ϕχ − ϕ(q)

χ

)− H

µ+ p

(e+

2

3

∆

a2Π

)+ Φ(q) +NΦ, (3.77)

Φ(q) ≡ ϕ(q)v −

K/a2

4πG(µ+ p)ϕ(q)χ , NΦ ≡

H2

4πG(µ+ p)

[N

(s)4 +

1

3H

(N

(s)2 −N0

)],

NΦ ≡ 1

3

(1− K/a2

4πG(µ+ p)

)(N0 −N (s)

2

)− Hc2s

4πG(µ+ p)

(1

4N1 −HN (s)

2

)+

K/a2

4πG(µ+ p)HN

(s)4 − aHN (s)

6 .

Combining Eqs. (3.76) and (3.77) we can derive

H2c2s(µ+ p)a3

[(µ+ p)a3

H2c2sΦ

]·− c2s

∆

a2Φ =

Hcsa3√µ+ p

[v′′ −

(z′′

z+ c2s∆

)v

]=

H2c2s(µ+ p)a3

(µ+ p)a3

H2c2s

[− H

µ+ p

(e+

2

3

∆

a2Π

)+ Φ(q) +NΦ

]·− c2s

∆

a2

(2H2 Π

µ+ p+ Φ(q) +NΦ

), (3.78)

µ+ p

H

[H2

(µ+ p)a

( aHϕχ

)·]·− c2s

∆

a2ϕχ =

√µ+ p

a2

[u′′ −

((1/z)′′

1/z+ c2s∆

)u

](3.79)

=4πG(µ+ p)

H

[− H

µ+ p

(e+

2

3

∆

a2Π

)− 2

(H2 Π

µ+ p

)·+NΦ − NΦ

]+µ+ p

H

[H2

(µ+ p)a

( aHϕ(q)χ

)·]·− c2s

∆

a2ϕ(q)χ ,

where we defined a few variables

v ≡ zΦ, u ≡ 1

z

a

Hϕχ, csz ≡

a√µ+ p

H≡ z. (3.80)

46

4 Standard Inflationary Models

4.1 Standard Inflationary ModelsStandard single field inflationary models provide a mechanism for the inflationary expansion (horizon problem) and the perturbationgeneration (initial condition) by a single scalar field, called inflaton. The scalar field Lagrangian has the canonical kinetic term, butvarious single field models differ in the scalar field potential, according to which the inflaton rolls over. In most cases, the slow-rollcondition is adopted, such that the scalar field dynamics is insensitive to the details of the scalar field potential.

The outcome of the standard model predictions is as follows: The curvature fluctuations are scale-invariant (ns ' 1) and highlyGaussian. The tensor fluctuations are also scale-invariant, but its amplitude is very small compared to the scalar fluctuations. Therunning of the indices is very small. Recent observations confirm these predictions and constrain the parameters with high precision.However, beyond these basic features/constraints, we do not have a solid model for inflation. Note that the energy scale of inflationis beyond the validity of the standard model physics, and most inflationary models have many theoretical issues, when quantumcorrections are considered.

4.1.1 Scalar Field ActionIn addition to the Einstein-Hilbert action for gravity, we consider the action for a scalar field with canonical kinetic term and thepotential V :

S =

∫ √−g d4x

[c4

16πGR− 1

2∂µφ ∂

µφ− V (φ)

], (4.1)

where the kinetic term in the Minkowski spacetime reduces to the standard form

− 1

2ηµν∂µφ ∂νφ =

1

2

[(∂tφ)

2 − (∇φ)2]. (4.2)

The Euler-Lagrange equation yields the equation of motion for the scalar field

φ− V,φ = 0 , := gµν∇µ∇ν , (4.3)

and the energy-momentum tensor is

Tµν = gµνLφ − 2δLφδgµν

= φ,µφ,ν −1

2gµν φ,ρφ

,ρ − V gµν . (4.4)

It is often in literature that the Planck unit is adopted, and there exist two different conventions:

M2pl :=

1

8πG, m2

pl :=1

G. (4.5)

4.1.2 Background Relation and Evolution EquationsIn the background, the non-vanishing fluid quantities for a scalar field are the energy density and the pressure

ρφ =1

2φ2 + V (φ) , pφ =

1

2φ2 − V (φ) , (4.6)

and the equation of motion becomes

φ+ 3Hφ+ V,φ = 0 , φ′′ + 2Hφ′ + a2V,φ = 0 . (4.7)

The Friedmann equation for a scalar field is

H2 =ρφ

3M2pl

, H = −ρφ + pφ2M2

pl

= − φ2

2M2pl

, H2 + H =a

a=

1

3M2pl

(V − φ2

), (4.8)

where we assumed a flat universe and no cosmological constant. If the potential energy of the scalar field is the dominant energycomponent of the Universe or the kinetic energy is smaller than the potential energy (slow-roll), the expansion of the Universe isaccelerating a > 0. Various inflationary models with slow-roll condition state that the potential is sufficiently flat, such that V (φ) isnearly constant during the inflationary period and φ slowly evolves (rolls over V ).


4.1.3 de-Sitter SpacetimeThe de-Sitter universe is a highly symmetric spacetime, defined as a background FRW universe with no matter and constant Hubbleparameter. A constant Hubble parameter leads to an exponential expansion, and we parametrize the de-Sitter solution as

H2 :=Λ

3, a(t) = eHt = − 1

Hη, a = (0,∞) , t = (−∞,∞) , η = (−∞, 0) , (4.9)

where the scale factor is normalized at t = 0. The slow-roll parameter for the de-Sitter spacetime is

ε := − H

H2=

d

dt

(1

H

)= 0 . (4.10)

4.1.4 Slow-Roll ParametersIn general, inflationary models slightly deviate from the de-Sitter phase (ε 6= 0), and its deviation is captured by the slow-rollparameter:

ε =d

dt

(1

H

)= − H

H2, H = −H2ε ,

a

a= H2 + H = H2(1− ε) , (3− ε)H2 =

V

M2pl

. (4.11)

To solve the horizon problem, we know that the comoving horizon has to decrease in time

0 >d

dt

(1

H

)= − a

a2H2= −1− ε

a. (4.12)

The background evolution of a scalar field can be re-phrased in terms of the slow-roll parameters as

ε =1

2

φ2

H2M2pl

=3

2(1 + w) , φ2 = ρφ + pφ . (4.13)

If we ignore the second derivative of the field (φ ' 0) in the equation of motion,

3Hφ ' −V,φ , ρφ + pφ '(V,φ3H

)2

, (4.14)

the slow-roll parameters are then further related to the slow-roll parameters defined in terms of the derivatives of the potential alsoused below)

εV :=M2

pl

2

(V,φV

)2

' ε , ηV := M2pl

(V,φφV

)' ε+ η , ξV :=

M4plV,φV,φφφ

V 2, (4.15)

where we used the second slow-roll parameter

η := − φ

Hφ. (4.16)

In fact, one can show the exact relation

ε = εV

(1− 4

3εV +

2

3ηV

). (4.17)

In literature, different convention for slow-roll parameters are often used, in particular, in terms of Hubble flow:

ε1 := ε , ε2 :=1

H

d ln ε

dt= 2(ε− η) , εi+1 :=

1

H

d ln εidt

. (4.18)

Furthermore, the inflation has to last for some time, such that the modes we measure in CMB have to expand at least by 40−60e-folds. So it is convenient to define the number of e-folding for a given mode as the number of e-folds the mode k expanded fromthe horizon crossing until the end of inflation,1

N(φk) := lnaend

a(φk)=

∫ tend

tk

H dt , k = aH , (4.19)

where tk is the time the k-mode crosses the horizon. Using the e-folding number, we can express the slow-roll parameters as

dN = Hdt = d ln a , ε = −d lnH

dN, εi+1 =

d ln εidN

. (4.20)1The end of inflation is a bit ill-defined, as we do not have a concrete model. However, in terms of N we can safely use the condition that the slow-roll parameter

becomes order unity ε ' 1.

48


4.1.5 Linear-Order EvolutionGiven the energy momentum tensor, we can derive the fluid quantities for a scalar field:

δρφ = φδφ− φ2α+ V,φδφ = δρv − 3Hφ δφ , δρv := δρ− ρ′v , (4.21)

δpφ = φδφ− φ2α− V,φδφ = δρv − 3c2sHφ δφ , vφ =δφ

φ′, (4.22)

e := δp− c2s δρ = (1− c2s)δρv , πφαβ = qφα = 0 , (4.23)

where we used the following relation and the sound speed is defined as

ρφ = φ(φ+ V,φ) = −3Hφ2 , pφ = φ(φ− V,φ) = φ(2φ+ 3Hφ) , c2s :=pφρφ

= −1− 2φ

3Hφ. (4.24)

Therefore, the comoving gauge corresponds to the uniform field gauge for the single-field models:

ϕv = ϕ−Hv = ϕ−H δφ

φ= ϕδφ . (4.25)

The equation of motion for a scalar field is then

δφ+ 3Hδφ+

(V,φφ +

k2

a2

)δφ = φ(α+ κ) + (2φ+ 3Hφ)α . (4.26)

Using the Einstein equations, we derive the governing equation for Mukhanov variable Φ (which is the comoving-gauge curvature)

Φ := ϕv −K/a2

4πG(ρ+ p)ϕχ =

H2

4πG(ρ+ p)a

( aHϕχ

)·+

2H2Π

ρ+ p, (4.27)

Φ = − H

4πG(ρ+ p)

k2c2sa2

ϕχ −H

ρ+ p

(e− 2k2

3a2Π

)≡ − Hc2A

4πG(ρ+ p)

k2

a2ϕχ , (4.28)

where the derivation is fully general and we defined the physical sound speed cA for inflaton

c2A := c2s + 4πGa2

k2

e

ϕχ≡ 1 . (4.29)

It is clear that the comoving-gauge curvature is conserved on super horizon scales.

4.1.6 Quadratic ActionThe Lagrangian for a scalar field and its equation of motion are

S =

∫ √−g d4x

[c4

16πGR− 1

2gab∂

aφ∂bφ− V (φ)

], φ−V,φ = 0 , Tab = φ,aφ,b−

1

2φ,cφ

,cgab−V gab , πab = 0 . (4.30)

Note that = gab∂a∂b changes sign depending on the metric signature. In terms of the ADM formalism, the Lagrangian is

S =1

2

∫d4x√h

[NR(h) − 2NV +

EijEij − E2

N+

(φ−N i∂iφ)2

N−Nhij∂iφ∂jφ

], (4.31)

where Kij = −Eij/N and Mpl = 1. Considering N and Ni as Lagrange multipliers and hij and φ as dynamic variables, theconstraint (Euler-Lagrange) equation for the Lagrange multipliers are (ADM energy and momentum constraints)

R(3)−2V − 1

N2(EijE

ij−E2)− 1

N2(φ−N i∂iφ)2−hij∂iφ∂jφ = 0 , ∇i

[1

N(Eij − δijE)

]− 1

N

(φ−N j∂jφ

)∂iφ = 0 , (4.32)

where we used

EijEij−E2 = [hikhjl−hijhkl]EijEkl , 0 = ∂α

δLδ(∂αN)

− δLδN

= 0− δLδN

, 0 = ∂αδL

δ(∂αN i)− δLδN i

. (4.33)

So far the derivation is exact and non-perturbative for a single field.

49


Unitary Gauge (Uniform Field) Action

It is apparent that the uniform-field gauge choice δφ = 0 would greatly simplify the equation. In this gauge choice, the actionbecomes

S =1

2

∫d4x√h

[NR(h) − 6Nρ2 +

EijEij − E2

N+

(N +

1

N

)φ2

], 3ρ2 =

1

2φ2 + V , (4.34)

where we used the background EoM for potential V (φ) = V (φ) = H2(3 − ε) and omitted the bar for simplicity. By setting themetric perturations

N ≡ 1 +N1 , N i ≡ ∂iψ +N iT , ∂iN

iT = 0 , N1 → α , ψ → −χ , N i

T → −aΨα , ζ → ϕδφ , (4.35)

we can expand the Lagrangian up to any orders. However, in principle, we only need first-order terms in N and N i for quadratic andcubic actions.2 Putting this back to the constraint equations, we can derive the linear-order relation between perturbations:

N1 =ζ

H, N i

T = 0 , ψ = − ζ

a2H+ χ , χ ≡ ∆−1(εζ) , ε =

φ2

2ρ2. (4.36)

The exact scalar contributions to R(h) and Eij are

R(h) = −2e−2ρ−2ζ(2∆ζ + ζ,iζ

,i), Eij = e2ρ+2ζ

[(ρ+ ζ − ζ,kψ,k)δij − ψ,ij

]. (4.37)

Before we move on, we consider conserved quantities in all orders on horizon scales, i.e., all powers of fields or metric, butlinear-order in spatial derivatives (gradient expansion). Noting that δN and Ni are zeroth order in spatial derivatives (N ≡ 1 + δN ),the Hamiltonian constraint above yields at the linear order in spatial derivative

2V δN = 2H(

3ζ −∇iN i), S(1) =

∫d4x√h(2V−2V δN) =

∫d4x e3ρ+3ζ

(−6ρ2 + φ2 − 6ζ ρ

)= −2

∫d4x

d

dt

(e3ρ+3ζ ρ

),

(4.38)where the uniform field gauge is assumed. Therefore, ζ and γ = detγij are constant in all orders on superhorizon scales.

4.1.7 Quantum Fluctuations in Quadratic ActionThe background relation describes the inflationary expansion, and the equation of motion describes the evolution of the perturbationsaround the mean. Here we derive their statistical properties.

Scalar Fluctuations

After some integrations by parts of the quadratic action, the second-order scalar action in the comoving gauge is (Mpl = 1)

S(2) =1

2

∫dt d3x a3 φ

2

H2

[ζ2 − 1

a2(∇ζ)2

]=

1

2

∫dτ d3x

[(v′)2 − (∇v)2 +

z′′

zv2

], v ≡ zζ , z2 ≡ a2 φ

2

H2= 2a2ε , (4.39)

where v is the canonically-normalized Mukhanov-Sasaki variable. The Lagrangian appears now like a simple harmonic oscillatorwith time-dependent mass term. In order to quantize the field, we first consider

π =δLδv′

= v′ , H = πv′ − L =1

2

[(v′)2 + (∇v)2 +m2v2

], m2(τ) ≡ −z

′′

z−→z→0

a′′

a=

2

τ2, (4.40)

where the conformal time is used. The Mukhanov-Sasaki variable is decomposed as

v(τ,x) =

∫d3k

(2π)3vk(τ) eik·x ,

(2 +m2

)v = 0 → v′′k + w2

kvk = 0 , w2k = k2 +m2 , (4.41)

where the effective frequency wk corresponds to its energy. The subhorizon solution is oscillating, and the super-horizon solution isvk ∝ z ∝ 1/τ (hence ζk = vk/z is constant, the decaying mode is vk ∝ τ2, and τ = [−∞, 0]). With two solutions for vk(τ), wenow have

vk(τ) ≡ a−k vk(τ)+a+−k v

∗k(τ) , v(τ,x) =

∫d3k

(2π)3

[a−k vk(τ) + a+

−k v∗k(τ)

]eik·x =

∫d3k

(2π)3

[a−k vk(τ)eik·x + a+

k v∗k(τ)e−ik·x],

(4.42)2The action will be expanded in terms of Lagrange multipliers as L(N,N i, · · · ) = L0 + δL

δNN + · · · , such that up to third order in N or N i terms will be

multiplied by the zero-th order or linear-order terms in the derivative, which vanish due to the equation of motion. Therefore, we only need the first-order terms in Nor N i for quadratic and cubic actions.

50


where a+k = (a−k )∗ for reality of v(τ,x). Now, by imposing the canonical quantization relation (~ = 1)

[v(τ,x), π(τ,y)] = iδ(x− y) , [v(τ,x), v(τ,y)] = [π(τ,x), π(τ,y)] = 0 , (4.43)

we promote the classical field to operators and derive the following:

[a−k , a+k′ ] = (2π)3δ(k−k′) , [a−k , a

−k′ ] = [a+

k , a+k′ ] = 0 , a−k |0〉 = 0 , |nk〉 =

√2Ekn!

[(a+k )n]|0〉 , W [vk, v

∗k] ≡ v′kv∗k−vkv′∗k = −i ,

(4.44)where

√2E is put to make it Lorentz invariant, the Wronskian is normalized to give the usual commutation relation for ak, and the

vacuum satisfies 〈0|0〉 = 1.3 In terms of the mode function, the Hamiltonian in Minkowski space is

H =

∫d3x H =

1

2

∫d3k

[a−k a

−−kF

∗k + a+

k a+−kFk + (2a+

k a−k + δD(0))Ek

], Ek ≡ |v′k|2 + k2|vk|2 , Fk ≡ v′2k + k2v2

k ,

(4.47)and therefore the vacuum energy is minimized at⟨

0|H|0⟩

=δ(0)

4

∫d3k Ek , Ek = k ≡ wk , Fk = 0 → vk(τ) =

1√2k

e−ikτ , H =

∫d3k

[a+

k a−k +

1

2δ(0)

]. (4.48)

Similarly, we can minimize the vacuum expectation at each time, as the mode function evolves in time:

vk(τ) =1√

2wk(τ)e−iwk(τ)τ , lim

τ→−∞vk(τ) =

1√2k

e−ikτ , (4.49)

where the Bunch-Davis vacuum is chosen, as the modes were initially deep inside the horizon before inflation. For de Sitter universe,we have exact solution:

vk(τ) =e−ikτ√

2k

(1− i

kτ

), lim

kτ→0vk(τ) =

1

i√

2

1

k3/2τ, 〈0|vkvk′ |0〉 = (2π)3δ(k+k′)|vk|2 , lim

kτ→0k3|vk|2 =

1

2τ2=a2H2

2.

(4.50)For the slow-roll inflation, we have (Stewart), with ν ≡ 1+ε+ε2

1−ε + 12 '

32 + ε+ 1

2ε2,

z′

z= aH

(1 +

1

2ε2

), aH ≈ −

1

τ(1 + ε) ,

z′′

z= (aH)2

(2− ε+

3

2ε2 −

1

2εε2 +

1

4ε22 + ε2ε3

)=

1

τ2

(ν2 −

1

4

)'

1

τ2

[2 + 3

(ε+

1

2ε2

)],

(4.51)where ≈ is exact, if ε is constant, and ' represents the first-order in slow-roll expansion. The exact solution can be derived in terms

of Hankel function

vk(τ) =√−τ[αH(1)

ν (−kτ) + βH(2)ν (−kτ)

]−→

BD vac.

√π

2(−τ)1/2H(1)

ν (−kτ) , β = 0 , α =

√π

2, (4.52)

and the power spectrum in the superhorizon limit is

z ' τ1/2−ν , Pζ =Pvz2

=H2

a2φ2Pv =

1

2a2εPv , ∆2

ζ =k3

2π2Pζ = 22ν−3 Γ2(ν)

Γ2(3/2)(1−ε)2ν−1

(H2

8π2ε

), ns−1 = 3−2ν = −2ε−ε2 ,

(4.53)where we assumed that the slow-roll parameters are constant. Useful relations and the case in de Sitter universe

Pv =(aH)2

2k3, ∆2

ζ =

(H2

8π2ε

), lim

kτ→−∞H(1,2)ν (−kτ) =

√2

π

1√−kτ

e±ikτe±iπ2 (ν+ 1

2 ) , limkτ→0

H(1)ν (−kτ) =

i

πΓ(ν)

(−kτ

2

)−ν.

(4.54)3The non-uniqueness of physical vacuum arises because the mode function vk(τ) is not chosen. Consider a function uk(τ) = αkvk(τ) +βkv

∗k(τ) and construct

operators b±k with uk and u∗k for v(τ,x). Then we can derive the Bogolyubov transformation

a−k = α∗k b−k + βk b

+−k , a+k = αk b

+k + β∗k b

−−k , |αk|

2 − |βk|2 = 1 , (4.45)

which means the vacuum defined by ak is not the vacuum with respect to bk. The normalization is due to the Wronskian normalization. In QFT (Peskin & Schroder),one starts with time-independent operators for SHO, construct H , then construct Heisenberg time-dependent operators

[H, a±] = ±wa± , i∂

∂tO = [O, H] , eiHta±e−iHt = a±e±iEt , (4.46)

and so on, instead of starting with mode functions vk(τ).

51


Tensor Fluctuations: Gravity Waves

The quadratic action for tensor is

S =M2

pl

8

∫dτd3x a2

[(h′ij)

2 − (∇hij)2]

=∑s

∫dτd3k

a2

4M2

pl

[(hs′k )2 − k2(hsk)2

], vsk ≡

a

2Mplh

sk , hij = 2Cij , (4.55)

and we have the tensor power spectrum (sum of each polarization mode)

Pv =(aH)2

2k3, PT = 2Ph = 2

(2

aMpl

)2

Pv =4

k3

H2

M2pl

, r =∆2t

∆2s

= 16ε =8

M2pl

φ2

H2. (4.56)

The large-scale evolution of gravity waves is also simple. The equation of motion is in the absence of anisotropic pressure

0 = c(t) + 3Hc(t) +k2

a2c(t) =

1

a3

(a3c(t)

)·+k2

a2c(t) = v′′ +

(k2 − a′′

a

)v , hij → 2Cij → 2c(t)Q

(t)ij , v = ac(t) , (4.57)

and the exact solution in a universe with K = Λ = π(t) = 0 and w =constant is (x ≡ k|τ |)

c(t) = A1Jν(x)

xν+A2

Yν(x)

xν−→x1

A1

2νΓ(ν + 1)−A22ν

Γ(ν)

πx2ν, c(t) =

mde2A1

j1(x)

x+

√2

πA2

y1(x)

x, ν ≡ 3(1− w)

2(1 + 3w). (4.58)

4.1.8 In-In FormalismIn general, S-matrix is described by states in the far-past and far-future, both of which are described by non-interacting free fieldin Minkowski space. However, in cosmlogy, it is only at very early time (hence deep inside the horizon) that can be described byMinkowski, corresponding to Bunch-Davies vacuum. For the very late time, we want it to be around the horizon crossing. Thein-in formalism to compute cosmological correlations is developed by Schwinger-Keldysh (J. S. Schwinger, J. Math. Phys. 2, 407,1961) with pioneering work (E. Calzetta and B. L. Hu, PRD 35, 495, 1987 and R. D. Jordan, PRD33, 444, 1986) and recent work byMaldacena (2003) and Weinberg (2005).

The interacting vacuum at any time |Ω(τ)〉 can be expanded in terms of free-field (Bunch-Davies) states |n〉 and be evolved fromτ1 to τ2 by using the time-evolution operator U(τ2, τ1) as

|Ω(τ)〉 =∑n

|n〉〈n|Ω(τ)〉 , |Ω(τ2)〉 = U(τ2, τ1)|Ω(τ1)〉 = |0〉〈0|Ω(τ1)〉+∑n≥1

eiEn(τ2−τ1)|n〉〈n|Ω(τ1)〉 . (4.59)

By evolving it to the distant past τ2 = −∞(1− iε), we project out all excited states by the Bunch-Davies vacuum and by evolving itback, we can relate the interacting vacuum at a given time τ to the Bunch-Davies vacuum as

limτ2≡ limτ2→−∞(1−iε)

, limτ2|Ω(τ2)〉 = lim

τ2U(τ2, τ1)|Ω(τ1)〉 = |0〉〈0|Ω(τ1)〉 , U(τ2, τ1) = T exp

[−i∫ τ2

τ1

dτ ′ Hint(τ′)

],

|in〉 ≡ |Ω(τ)〉 = limτ2U(τ, τ2)U(τ2, τ)|Ω(τ)〉 = lim

τ2U(τ, τ2)|0〉〈0|Ω(τ)〉 = lim

τ2T exp

[−i∫ τ

τ2

dτ ′ Hint(τ′)

]|0〉〈0|Ω(τ)〉 ,

where T is the time-ordering operator. Therefore, the expectation value of a product of operators W (τ) is then

〈W (τ)〉 ≡ 〈in|W (τ)|in〉〈in|in〉

=

⟨0

∣∣∣∣(Te−i ∫ τ−∞+ Hintdτ′)†

W (τ)(Te−i

∫ τ−∞+ Hintdτ

′)∣∣∣∣ 0⟩=

⟨0∣∣∣(T e−i ∫ τ−∞− Hintdτ

′)W (τ)

(Te−i

∫ τ−∞+ Hintdτ

′)∣∣∣ 0⟩ , (4.60)

QH(t) = U†QSU ,d

dtQH = −i[QH , H] , QI(t) = U†0Q

SU0 ,d

dtQI = −i[QI , H0] , U0 ≡ exp

[−i∫ t

ti

dt′H0(t′)

],

where T is the anti-time-ordering operator and∞± =∞(1∓ iε). For example, we are interested in W (τ) = R1(τ)R2(τ)R3(τ).The inflationary action is expanded perturbatively to give

S = S0[φ, gab] + S2[R2] + S3[R3] + · · · , H = H0 +Hint , Hint =∑i

fi(ε, η, · · · )R3I(τ) + · · · , (4.61)

52


where S0 defines the backgroundH and slow-roll parameters, the quadratic action S2 defines the Gaussian curvature (RI in the inter-action picture), and the cubic and higher-order actions define the interation Hamiltoninan. So for the quadratic Gaussian caclulations,we have Hint = 0, and hence no in-in formalism is required, but it is for cubic calculations. Expanding the expectation value in termsof the interaction Hamiltonian, we have at each order

〈W (τ)〉(0)= 〈0|W (τ)|0〉 , 〈W (τ)〉(1)

= 2Re

[−i∫ τ

−∞+

dτ ′ 〈0|W (τ)Hint(τ′)|0〉

], (4.62)

〈W (τ)〉(2)= −2Re

[∫ τ

−∞+

dτ ′∫ τ ′

−∞+

dτ ′′ 〈0|W (τ)Hint(τ′)Hint(τ

′′)|0〉

]+

∫ τ

−∞+

dτ ′∫ τ ′

−∞+

dτ ′′ 〈0|Hint(τ′)W (τ)Hint(τ

′′)|0〉 ,

and for the bispectrum calculation the leading order term is 〈W (τ)〉(1).

4.1.9 Quantum Fluctuations in Cubic Action• Overview: Approximately, the comoving-gauge curvature is related as

ζ = N ′δφ+1

2N ′′δφ2+· · · ,

⟨ζ3⟩

= (N ′)3⟨δφ3⟩+N ′′(N ′)2

2

⟨δφ2δφ · δφ

⟩+· · · , ζ = ζ1+

3

5fNLζ

21 +· · · , fNL =

5N ′′

6(N ′)2,

(4.63)such that the leading-order bispectrum comes from two terms: the field bispectrum and the nonlinear interaction. If the field ispurely Gaussian φ = φ1, the comoving-gauge curvature can be written in a local form, such that the nonlinear interaction term in thebispectrum is called local type. However, both terms are important. As in the quadratic computation, it is in fact easier to compute inthe uniform curvature gauge δφϕ.

The action is again expanded up to third order, which is rather straightforward. However, performing lots of integration byparts and using equation of motion (background and linear-order) isolates proper terms that are suppressed by the slow-roll. Thiscalculation is by no means straightforward. The cubic action is

S(3) = M2pl

∫d4x

−ε(2δ+ε)e3ρζ2ζ+ε(2δ+3ε)eρζ∂kζ∂

kζ−2ε2e3ρζ∂kζ∂k(∂−2ζ)−1

2e3ρε3

[ζ2ζ−ζ∂k∂l(∂−2ζ)∂k∂l(∂−2ζ)

],

(4.64)where the last term is negligible (slightly different from Maldacena). The cubic actions for the mixture of gravitons are omitted. Inthe cubic action above, there exist additional quadratic terms that are multiplied by the linear-order EoM. We removed it in the actionabove, because by using the field-redefinition the real cubic action is transformed as

S(3) =

(above

)+

(EoM

)f(ζ) , ζ ≡ ζn + f(ζn) → S(2)(ζ) + S(3)(ζ) = S(2)(ζn) +

(above

)(ζn) , (4.65)

where f is a quadratic function. The quadratic part that is multiplied by the linear-order EoM in the cubic action is cancelled by thecubic term of ζn from the quadratic action.4

Since we are computing the expectation values of quantum fluctuations, we have to account for the vacuum state in interactingtheory.5 In the interaction picture, operators evolve with the free field action, while states evolve with the interaction action:

d

dt|0(t)〉 = −iHI(t)|0(t)〉 , |0(τ)〉 = U(τ,−∞)|0〉 = T exp

[−i∫ τ

−∞dτ ′HI(τ

′)

]|0〉 , |0〉 = |0(−∞)〉 , (4.66)

where we used the conformal time. Since the canonically normalized field is related by a classical factor (non-quantum), the leading-

4The linear-order EoM is non-vanishing, because the theory is now different, i.e., it vanishes for ζ in a free-field theory, but now we consider an interacting field ζ,which includes the cubic action. In fact, the cubic action is simpler if we use the uniform curvature gauge δφϕ (Maldacena indeed derived in two gauges). It turnsout that the quadratic function f is almost (not exactly) the quadratic part of the gauge transformation: ζ = ζn + f(ζn) → ϕδφ = −Hδφϕ/φ + ϕ

(q)δφ , where

ζn ≡ −Hδφ(1,2)ϕ /φ. The cancellation of the quadratic part due to the quadratic action is well described in Seery and Lidsey (2005).5This is the reason we cannot use just the classical expectation. It turns out we won’t need this quantum calculations in other areas but inflation. In the quadratic

calculations, the Mukhanov-Sasaki variable v was split in terms of classical (time-dependent) variables and (time-independent) quantum mode functions. However, inthis interacting theory, the vacuum states as well as mode functions evolve in time. At the leading-order, we have six points (three from external fields and three frominteracting Hamiltonian), but two get annihilated by vacuum, such that it is of order four-point in fields, i.e., the leading correction to bispectrum.

53


order contribution is

⟨0(t)|ζ3|0(t)

⟩=

(H

aφ

)3 ⟨0(t)|φ3

c |0(t)⟩

=

(H

aφ

)3⟨0(t)

∣∣∣∣(Te−i ∫ τ−∞ dτ ′HI(τ ′))†φ3c

(Te−i

∫ τ−∞ dτ ′HI(τ ′)

)∣∣∣∣ 0(t)

⟩' −i

(H

aφ

)3 ∫ τ

−∞dτ ′ 〈0 |[φc(x1)φc(x2)φc(x3), HI(τ

′)]| 0〉+ · · · φc =aφ

Hζ ,

' − iH3

M4pl

(−τ)3

(2ε)3/2

∫ τ

−∞dτ ′ 〈0 |[φc(x1)φc(x2)φc(x3), HI(τ

′)]| 0〉+ · · · , a ' − 1

Hτ, (4.67)

where we adopted the de Sitter limit.For the three-point correlation function of ζ, we need to 1) compute that of ζn with the interaction Hamiltonian in the leading

order, as it is the non-vanishing leading term, and 2) compute the three-point correlation functions of two ζn and one f(ζn):

⟨0(t)|ζ3(t)|0(t)

⟩=⟨0(t)|ζ3

n(t)|0(t)⟩

+ 3⟨0(t)|ζ2

n(t)f(ζn)(t)|0(t)⟩' −i

∫ t

−∞dt′⟨0|[ζ3

n(t), HI(t′)]|0

⟩+ 3

⟨0|ζ2

n(t)f(ζn)(t)|0⟩,

(4.68)where the leading order is unity in the last term, as f(ζn) is already quadratic (next leading-order is one withHI , which is seven-pointcorrelation and vanishes). Noting that the free-field Hamiltonian and the interaction Hamiltonian are6

H0(t) =

∫d3x a3ε

[ζ2 +

1

a2(∇ζ)2

], HI(t) = −

∫d3x

[O1(t, ~x) +O2(t, ~x) +O3(t, ~x) + · · ·

], (4.69)

O1 =1

4e3ρ 1

M2pl

φ4

ρ4ζ2nζn , O2 =

1

4eρ

1

M2pl

φ4

ρ4ζn∂kζn∂

kζn , O3 = −e3ρ φ2

ρ2ζn∂kζn∂

kχn = −1

2e3ρ 1

M2pl

φ4

ρ4ζn∂kζn∂

k(∂−2ζn

),

the three-point correlation function of ζ is then

〈0(t)|ζ(t, ~x)ζ(t, ~y)ζ(t, ~z)|0(t)〉 =

∫~k1,~x

∫~k2,~y

∫~k3,~z

D(~k1 + ~k2 + ~k3)1

32 ε2

H4

M4pl

1

k31k

32k

33

Ak1,k2,k3(t) , (4.70)

and the momentum dependence evaluated at the horizon crossing is

Ak1,k2,k3≡ A1 +A2 +A3 +Af ∝ ε , Af =

[2φ

φρ+

1

M2pl

φ2

ρ2

](k31 + k32 + k33

)+ · · · , Kt ≡ |~k1|+ |~k2|+ |~k3| , (4.71)

A1 +A2 +A3 = −1

2

1

M2pl

φ2

ρ2

(k31 + k32 + k33

)+

1

2

1

M2pl

φ2

ρ2(k21k2 + k1k

22 + k21k3 + k1k

23 + k22k3 + k2k

23) + 4

1

M2pl

φ2

ρ2k21k

22 + k21k

23 + k22k

23

Kt+ · · · ,

where Ai are contributions from each operator Oi in the interaction Hamiltonian and Af is the contribution from the four-pointζ2nf(ζn).

The bispectrum in the equilateral configuration is

〈0|ζkζkζk|0〉 =1

32ε2

H4

M4pl

1

k6

[6φ

φρ+

17

2

1

M2pl

φ2

ρ2+ · · ·

]=

1

32ε2

H4

M4pl

1

k6[17ε+ 6δ + · · · ] , (4.72)

and the bispectrum in the squeezed limit is

limK→0〈0|ζKζkζ−k−K|0〉 =

1

8ε2

H4

M4pl

1

K3k3

[φ

φρ+

1

M2pl

φ2

ρ2

]+ · · · = 1

8ε2

H4

M4pl

1

K3k3(2ε+ δ) + · · · (4.73)

= −〈0|ζKζ∗K |0〉kd

dk〈0|ζkζ∗k |0〉 , 〈0|ζkζ∗k |0〉 =

1

4ε

H2

M2pl

1

k3(−kη)−4ε−2δ + · · · ,

where all quantities including the vacuum state are at the same time. Previous work by Matarrese and/or Riotto violates the consis-tency relation, as they did not consider the change of vacuum.

6Note that the quadratic action is the free-field action H0 without interaction.

54


Tensor Modes

The bispectra for the mixture of scalars and gravitons are

⟨γsk1

ζk2ζk3

⟩= D(k123)

1

16ε

H4

M4pl

εsijki2kj3

k31k

32k

33

4

(−Kt +

k1k2 + k2k3 + k3k1

Kt+k1k2k3

Kt

), (4.74)

⟨ζk1γ

s2k2γs3k3

⟩= D(k123)

1

8

H4

M4pl

εs2ij εijs3

k31k

32k

33

[−1

4k3

1 +1

2k1(k2

2 + k23) +

4k22k

23

Kt

], (4.75)

⟨γs1k1

γs2k2γs3k3

⟩= D(k123)

1

8

H4

M4pl

−4εs1ii′εs2jj′ε

s3ll′t

ijlti′j′l′

k31k

32k

33

(−Kt +

k1k2 + k2k3 + k3k1

Kt+k1k2k3

Kt

), (4.76)

where tijl = kl1δij + ki2δ

jl + kj3δil. Similar consistency relations are available.

4.1.10 Lyth BoundGiven the definition of the e-folds, we can further manipulate it by using the inflaton as a time clock:

N(φk) =

∫ φend

φk

dφH

φ=

∫ φend

φk

dφ

Mpl

√2ε

, r = 16ε =8

M2pl

(dφ

dN

)2

, (4.77)

and this relation further implies that the excursion of the inflaton field is related to the tensor-to-scalar ratio as

∆φkMpl

'∫ Ncmb

Nend

dN

√r

8, (4.78)

where ε(φend) ≡ 1. To solve the horizon problem, the mode k should have expanded at least 40−60 in e-folds. So, this consistencyrelation (Lyth, 1997) implies that an inflationary field variation of the order of the Planck mass is needed to produce r > 0.01. Fromthe theoretical point of view, this sets the upper bound on the amplitude of gravitational waves. Indeed, the standard inflationarymodel predictions are very small.

Note that the uncertainty in e-folds N is due to our ignorance in the reheating era: After the inflationary period ends, the inflatonfield decays into other particles and reheats the Universe. This period is expected to be described by a matter-dominated era, as theinflaton oscillates around the minimum of the potential, effectively acting as a matter. However, we know very little about this period.

The current observational constraint is

As ' 2.2× 10−9 , ns ' 0.96 , ε ' 0.01 . (4.79)

indicating the energy scale of the inflation is

AT =2V

3π2M4pl

= 16εAs , H2 =V

3M2pl

= ε(2× 1014 GeV

)2. (4.80)

4.2 Primordial Non-GaussianityAny deviation from the Gaussian distribution would provide tight constraints on the inflationary models beyond the single fieldmodels.

Definition

It is non-Gaussianity generated at the horizon crossing, not due to the nonlinearities after horizon crossing. The higher-order statisticsare defined by the ensemble average of the connected spectra (also careful with ζ = (5/3)Φ). Considering the translational invariance,the deviation from the Gaussianity can be written as

Bζ(k1, k2, k3) =6

5fNL [Pζ(k1)Pζ(k2) + Pζ(k2)Pζ(k3) + Pζ(k3)Pζ(k1)] , (4.81)

Tζ(k1, k2, k3, k4) =54

25gNL [Pζ(k1)Pζ(k2)Pζ(k3) + 3 perms.] + τNL [Pζ(k1)Pζ(k2)Pζ(|k1 + k3|) + 11 perms.] ,

55


and in fact these are the most general parametrization for constant fNL, gNL, τNL, which lead to the usual local parametrization inconfiguration space. In terms of Φ, we have

BΦ(k1, k2, k3) = 2fNL [PΦ(k1)PΦ(k2) + PΦ(k2)PΦ(k3) + PΦ(k3)PΦ(k1)] , (4.82)

TΦ(k1, k2, k3, k4) = 6gNL [PΦ(k1)PΦ(k2)PΦ(k3) + 3 perms.] +25

9τNL [PΦ(k1)PΦ(k2)PΦ(|k1 + k3|) + 11 perms.] .

By imposing the Cauchy-Schwarz inequality, Suyama and Yamaguchi (2008) find that there exists a consistency relation τNL ≥(65fNL

)2, where the equality holds for a single field. It turns out that the inequality holds very generally for all inflationary models,

i.e., it is true by definition in the limit of infinite volume (hence it can be broken in practice, and it is not easy how to interpret theviolation). Two interesting configurations that maximizes the trispectrum: The squeezed limit is

Tζ =

(2τNL +

54

25gNL

)Pζ(k4) [Pζ(k1)Pζ(k2) + Pζ(k1)Pζ(k3) + Pζ(k2)Pζ(k3)] , k4 k1, k2, k3 , (4.83)

and the folded kite limit isTζ = 4τNLPζ(k12)Pζ(k1)Pζ(k3) , k12 k1 ' k2 , k3 ' k4 , (4.84)

(the usual configuration for the contribution to the covariance matrix).

Shapes of Non-Gaussianity

Following Fergusson & Shellard 2008, it will be convenient to define the shape function of non-Gaussianity as

S(k1, k2, k3) ≡ N(k1k2k3)2BR(k1, k2, k3) , S local(k1, k2, k3) ∝ K3

K111, Sequil(k1, k2, k3) ∝ k1k2k3

K111. (4.85)

where N is an appropriate normalization factor and

Kp =∑i

(ki)p with K = K1 , Kpq =

1

∆pq

∑i 6=j

(ki)p(kj)

q , Kpqr =1

∆pqr

∑i 6=j 6=l

(ki)p(kj)

q(kl)q , kip = Kp − 2(ki)

p with ki = ki1 ,

(4.86)where ∆pq = 1 + δpq and ∆pqr = ∆pq(∆qr + δpr) (no summation).

• For a single-field slow-roll inflation case, the shape function is

SSR(k1, k2, k3) ∝ (ε− 2η)K3

K111+ ε

(K12 + 8

K22

K

)≈ (4ε− 2η)S local(k1, k2, k3) +

5

3εSequil(k1, k2, k3) , (4.87)

where the normalization is such that S local(k, k, k) = Sequil(k, k, k).

• Non-standard vacuum: If inflation started in an excited state rather than in the Bunch-Davies vacuum, the remnant non-Gaussiansignal is peaked at folded triangles (k1 = 2k2 = 2k3) with a shape function (Holman & Trolley 2008)

S folded(k1, k2, k3) ∝ 1

K111(K12 −K3) + 4

K2

(k1k2k3)2. (4.88)

• Derivative interactions: Using the EFT description to the cubic order, the non-Gaussian signal is peaked at equlateral triangles

f equilNL = − 35

108

(1

c2s− 1

)+

5

81

(1

c2s− 1− 2Λ

), Λ ≡

X2P,XX + 23X

3P,XXX

XP,X + 2X2P,XX. (4.89)

In the case of DBI inflation, we have

PDBI(X,φ) = −f−1(φ)√

1− 2f(φ)X + f−1(φ)− V (φ) , fDBINL = − 35

108

(1

c2s− 1

), (4.90)

SDBI(k1, k2, k3) ∝ 1

K111K2(K5 + 2K14 − 3K23 + 2K113 − 8K122) . (4.91)

The DBI action is special in that its form is protected against radiative corrections by a higher-dimensional boost symmetry.

56


4.3 Miscellaneous ModelsMulti-field models, non-canonical kinetic terms, modified gravity, Higgs inflationary models.

4.4 δN -FormalismA convenient way to compute the comoving gauge curvature ζ = ϕv on super-horizon scales at some late time t2 in RDE frominflationary scenarios, including multi-fields. The quantity δN becomes ζ in certain choices of conditions, and it is Taylor expandedin terms of scalar fields at t1 around the horizon crossing with its coefficients that can be computed in terms of background.

With spatial C-gauge condition, we consider a normal observer nα = 0 in a given coordinate, moving along an integral curveγ(τ) parametrized by proper time τ . The e-folding N12 the observer measures is defined as

H ≡ 1

3θ = H

(1−A+

ϕ

H+

∆

3a2Hχ

), dτ = (1+A)dt , N12 =

∫γ

dτ H = N12+ϕ(x, t2)−ϕ(x, t1) , δN ≡ N12−N12 ,

(4.92)where we ignored the Laplacian term on super-horizon scales and performed linear-order calculations (no spatial drift along thecurve). For the sake of computational convenience, we choose a uniform-curvature gauge ϕ(t1) = 0 at t1 and a comoving-gaugeζ = ϕv(t2) at t2.7 Therefore, we have

N12 = N12[φA(x, t1); t2] , δN = ζ(x, t2) =∑n

1

n!NA1A2...Anδφ

A1δφA2 · · · δφAn , NA ≡dN12

dφA

∣∣∣∣t1

=dN

dt1

dt1φ

= −H˙φ

∣∣∣∣t1

,

(4.93)where implicit summation over scalar fields Ai are assumed and the field perturbations δφA are at the horizon crossing t1. Note thatonly the sum

∑ ˙φAδφA = 0 vanishes at t2 due to the gauge condition, not the individual field perturbations. Since we compute ζon super-horizon scales in late time when ζ is conserved, the spatial depdendence x or time dependence t2 is of little importance.Possible dependence of N12 on φA at t1 is ignored by assuming the attactor solution in inflationary models (i.e., by the time it crossesthe horizon, it collapses to the attractor, in which δφ = F [δφ]).

A more rigorous derivation is present in Gong et al. 2011. Choosing the spatial C-gauge condition, one considers the localexpansion and its integral:

hij ≡ a2(e2C)

ij= a2e2ϕ

(e2CT

)ij, h = Det hij = a6(1+ϕ) , θ = na;a = −K =

1

2Nhijhij,0 −

1

NN i

:i '1

2N

d

dtlnh ,

N12 =

∫γ

dτ H =

∫Ndt

1

6N

d

dtln a6(1+ϕ) = N12 + ϕ(t2)− ϕ(t1) , (4.94)

where hij is the ADM spatial metric and we ignored the shift on super-horizon scales. This shows that the formula is valid to allorders inperturbations, accouting for vector and tensor.

• Q&A: i) confusing to have two gauge choices: in two different gauge choices the coordinates are different, such that t1 and t2are different for two coordinates, but we do not specify them, such that it does not matter. The integral is fairly generic, such thatδN = ζ(t2) is a fine statement. The fact that δN is just a function of δφ(t1) is a powerful tool. ii) slow-roll approximation: notnecessary, but attractor argument is needed. iii) ζ is time-varying in multifield scenarios: implicit assumption is that t2 is late enoughthat all the isocurvature components in multi-field scenarios collapse and ζ becomes time-independent again. iv) some use the uniformdensity gauge at t2: ϕδ = ϕ + δµ/3(µ + p), as opposed to the comoving gauge ϕv = ϕ − Hv. The comoving gauge is the rightchoice.

Comoving Curvature Perturbation

At leading order the primordial power spectra depends purely on ζ1, and in the slow-roll limit

ζ1 = NAϕA1 , Pζ(k) = NANBC

AB(k) , s.r. : Pζ(k) = NANAP (k) . (4.95)

To leading order in the field perturbations, the 3-point function of the curvature perturbations depends on ζ1, and the bispectrum isthus

ζ2 = NAϕA2 +NABϕ

A1 ϕ

B1 , 〈ζk1ζk2ζk3〉 = NANBNC〈ϕAk1

ϕBk2ϕCk3〉+1

2NA1A2NBNC

[〈(ϕA1 ∗ ϕA2

)k1ϕBk2

ϕCk3〉+ (2 perms)

],

7The uniform curvature gauge is often called the flat gauge ϕ(= ζ) = 0, and its time-slicing is therefore denoted as Σf . However, it confuses with initial i andfinal f . So we use t1,2 for the initial and final times.

57


where ′∗′ denotes the convolution, defined by(ϕA ∗ ϕB

)k

=∫

d3k′

(2π)3 ϕAk−k′ϕ

Bk′ . Hence the bispectrum of the curvature perturbation

is

Bζ(k1, k2, k3) = NANBNCBABC(k1, k2, k3) +NANBCND

[CAC(k1)CBD(k2) + CAC(k2)CBD(k3) + CAC(k3)CBD(k1)

](4.96)

In the slow-roll limit we can write the bispectrum as (one has to compute the intrinsic bispectrum BABC of fields)

Bζ(k1, k2, k3) = 4π4

∑i k

3i∏

i k3i

P2ζ

(−1

4M2plNCN

C

F(k1, k2, k3)∑i k

3i

+NABN

ANB

(NCNC)2

), F(k1, k2, k3) =

∑i

k3i−∑i 6=j

kik2j−8

∑i>j k

2i k

2j

k1 + k2 + k3.

(4.97)The four-point function of the curvature perturbation at leading order will depend on ζ1, ζ2, and

ζ3 = NAϕA3 +NAB

(ϕA1 ϕ

B2 + ϕA2 ϕ

B1

)+NABCϕ

A1 ϕ

B1 ϕ

C1 . (4.98)

The four-point function at leading order is (sixth order in the field perturbations)

〈ζk1ζk2

ζk3ζk4〉c = NANBNCND〈ϕAk1

ϕBk2ϕCk3

ϕDk4〉c +

1

2NA1A2

NBNCND

[〈(ϕA1 ∗ ϕA2

)k1ϕBk2

ϕCk3ϕDk4〉+ (3 perms)

]+

1

4NA1A2

NB1B2NCND

[〈(ϕA1 ∗ ϕA2

)k1

(ϕB1 ∗ ϕB2

)k2ϕCk3

ϕDk4〉+ (5 perms)

]+

1

3!NA1A2A3

NBNCND

[〈(ϕA1 ∗ ϕA2 ∗ ϕA3

)k1ϕBk2

ϕCk3ϕDk4〉+ (3 perms)

]. (4.99)

In the end, we find the connected part of the trispectrum of the curvature perturbation is

〈ζk1ζk2

ζk3ζk4〉c ≡ Tζ(k1,k2,k3,k4)(2π)3δD(k1 + k2 + k3 + k4) , (4.100)

where

Tζ(k1,k2,k3,k4) = NANBNCNDTABCD(k1,k2,k3,k4) +NA1A2

NBNCND[CA1B(k1)BA2BC(k12, k3, k4) + (11 perms)

]+NA1A2

NB1B2NCND

[CA2B2(k13)CA1C(k3)CB1D(k4) + (11 perms)

]+NA1A2A3

NBNCND[CA1B(k2)CA2C(k3)CA3D(k4) + (3 perms)

]. (4.101)

Gaussian Scalar Fields

If the scalar field perturbations are Gaussian random fields, then the bispectrum for the fields, BABC , and connected part of thetrispectrum, TABCD, both vanish. In this case the bispectrum of ζ at leading (fourth) order, can be written as

Bζ(k1,k2,k3) =6

5fNL [Pζ(k1)Pζ(k2) + Pζ(k2)Pζ(k3) + Pζ(k3)Pζ(k1)] , fNL =

5

6

NANBNAB

(NCNC)2 . (4.102)

The trispectrum in this case reduces to

Tζ(k1,k2,k3,k4) = NABNACNBNC [P (k13)P (k3)P (k4) + (11 perms)]+NABCN

ANBNC [P (k2)P (k3)P (k4) + (3 perms)] ,(4.103)

Hence we can write the trispectrum as

Tζ(k1,k2,k3,k4) = τNL [Pζ(k13)Pζ(k3)Pζ(k4) + (11 perms)] +54

25gNL [Pζ(k2)Pζ(k3)Pζ(k4) + (3 perms)]

τNL =NABN

ACNBNC(NDND)3

, gNL =25

54

NABCNANBNC

(NDND)3. (4.104)

Single Field Case

The curvature perturbation for a single field is given by

ζ = N ′ϕ+1

2N ′′ϕ2 +

1

6N ′′′ϕ3 + · · · , N ′ = dN/dϕ . (4.105)

58


If the field perturbation is purely Gaussian, ϕ = ϕ1 (no higher order terms), then the non-Gaussianity of the primordial perturbationhas a simple “local form” where the full non-linear perturbation at any point in real space, ζ(x), is a local function of a single Gaussianrandom field, ϕ1. Thus we can write

ζ = ζ1 +3

5fNLζ

21 +

9

25gNLζ

31 + · · · , fNL =

5

6

N ′′

(N ′)2, gNL =

25

54

N ′′′

(N ′)3→ τNL =

(N ′′)2

(N ′)4=

36

25f2

NL . (4.106)

The numerical factors arise because the original definition is given in terms of the Bardeen potential on large scales in mde. Noticethat the bispectrum depends linearly on ζ2 while the trispectrum has both a quadratic dependence upon ζ2 and a linear dependence onζ3. Thus τNL is proportional to f2

NL. However the trispectrum could be large even when the bispectrum is small because of the gNL

term. We have (all the slow-roll parameters in Eq. [4.107] are based on potential, i.e., ε→ εV and so on)

N ′ =H˙ϕ' 1√

2

1

Mpl

1√ε∼ O

(ε−

12

), N ′′ ' −1

2

1

M2pl

1

ε(η − 2ε) ∼ O (1) , N ′′′ ' 1√

2

1

M3pl

1

ε√ε

(εη − η2 +

1

2ξ2

)∼ O(ε

12 ) ,

fNL =5

6(η − 2ε) , τNL = (η − 2ε)2 , gNL =

25

54

(2εη − 2η2 + ξ2

). (4.107)

In single field inflation, ζ is conserved at all orders on superhorizon scales. Therefore no evolution of the bispectrum and trispectrumis possible after Hubble exit to the leading order.

Curvaton scenario

In the curvaton scenario a weakly-coupled field (the curvaton field, χ) which is light, but subdominant during inflation comes tocontribute a significant fraction of the energy density of the universe sometime after inflation. After it eventually decays, it is thefluctuations in this field that produce the primordial curvature perturbation ζ.

In general, the energy density of the curvaton is some function of the field value at Hubble-exit, ρχ ∝ g2(χ∗), and hence t heprimordial curvature perturbation when the curvaton decays is of local form. In the sudden-decay approximation the non-linearityparameters are

fNL =5

4r

(1 +

gg′′

g′2

)−

5

3−

5r

6, gNL =

25

54

[9

4r2

(g2g′′′

g′3+ 3

gg′′

g′2

)−

9

r

(1 +

gg′′

g′2

)+

1

2

(1− 9

gg′′

g′2

)+ 10r + 3r2

], r =

[3ρχ

3ρχ + 4ρr

]decay

,

(4.108)and τNL satisfies the single field relation, ρχ is the density of the curvaton field, and ρr is the density of radiation and hence r satisfies0 < r ≤ 1, .

Applications of δN -Formalism

Let’s use the notation ϕ ≡ δφ for the field perturbation, and indicate the perturbation order as ϕA = ϕA1 + 12ϕ

A2 + 1

6ϕA3 + · · · .

Following Byrnes, Sasaki, and Wands (2006), the power spectrum is defined CAB and to the leading order in slow-roll limit,

〈ϕAkϕBk′〉 = CAB(k)(2π)3δD(k + k′) , s.r. : CAB(k) = δABP (k) , P(k) =4πk3

(2π)3P (k) =

(H∗2π

)2

, (4.109)

where at leading order and in the slow-roll limit the fluctuations are independent, i.e., ∝ δAB . In the case of two field inflation, wehave for the power spectra and cross-correlation respectively,

C11 = Pϕ1 =(2π)3

4πk3

(H∗2π

)2[

1− 2H

H2+ C

(4H

H2+

1

M2pl

φ21

H2− 1

M2pl

φ22

H2− 2M2

pl

Vφ1φ1

V

)], (4.110)

C22 = Pϕ2=

(2π)3

4πk3

(H∗2π

)2[

1− 2H

H2+ C

(4H

H2+

1

M2pl

φ22

H2− 1

M2pl

φ21

H2− 2M2

pl

Vφ2φ2

V

)], (4.111)

C12 =(2π)3

4πk3

(H∗2π

)2

C

(2

M2pl

φ1φ2

H2− 2M2

pl

Vφ1φ2

V

), (4.112)

where C = 2− ln 2− γ ' 0.7296 and γ is the Euler-Mascheroni constant.

59


Higher-Order Spectra of Field Fluctuations

The first signal of non-Gaussianity comes from the bispectrum, which at lowest order is (therefore is fourth order in perturbations)

〈ϕAϕBϕC〉 = 〈ϕA1 ϕB1 ϕC2 〉+ perms , 〈ϕAk1ϕBk2

ϕCk3〉 ≡ BABC(k1, k2, k3)(2π)3δD(k1 + k2 + k3) . (4.113)

The leading order contribution to the connected four-point function (sixth order) comes from two terms,

〈ϕ1ϕ1ϕ1ϕ3〉c 〈ϕ1ϕ1ϕ2ϕ2〉c → 〈ϕAk1ϕBk2

ϕCk3ϕDk4〉c ≡ TABCD(k1,k2,k3,k4)(2π)3δD(k1 + k2 + k3 + k4) . (4.114)

60

5 Applications of the Effective Field Theory

5.1 Basics of EFT

5.1.1 Simple Toy ModelFormally, the heavy fields are integrated out by performing a path integral over the heavy degrees of freedom only. This processresults in an effective action for the light degrees of freedom,

eiSeff (φL) =

∫DφH eiSfull(φL,φH) , Leff(φL) = L∆<4 +

∑i

ciOi(φL)

Λ∆i−4, (5.1)

where ∆i are the dimensions of the operators Oi.1 The scale Λ is the cutoff of the EFT, at which the EFT breaks down. The firstrenormalizable operators also receive contributions from the heavy fields.

A simple toy model is called the sigma model, in which the UV complete Lagrangian hasO(n) rotation symmetry among n-scalarfields:

L = −1

2∂µΣ · ∂µΣ† +

µ2

2ΣΣ† − λ

4![ΣΣ†]2 , Σ =

φ1

...φn

, ∂µΣ · ∂µΣ† ≡ Tr[∂µΣ · ∂µΣ†] . (5.2)

The potential has a classical Mexican hat with vev 〈Σ〉2 = v2 = µ2/λ. So we introduce a radial field ρ and azimuthal fields U :

Σ = [v + ρ(x)]U(x) , U = eiπ(x)/v , L = −1

2(∂µρ)2 + µ2ρ2 +

(v + ρ)2

2∂µU · ∂µU† − λvρ3 − λ

4ρ4 , (5.3)

where [U ] = 1, [π] = [v] = M , and π are n− 1 massless Goldstone bosons. By integrating out the massive field ρ gives the effectiveLagrangian

Leff = −f2π

2∂µU · ∂µU† + c1

[∂µU ∂µU†

]2+ c2(∂µU · ∂νU†)(∂µU · ∂νU†) +O(E4) , fπ ≡ v , c1 ≡

v2

8µ2=

1

8λ, (5.4)

where ci are dimensionless.Generalizing this example, consider a gauge group G that is spontaneously broken to a subgroup H and T a be the generators of

the broken symmetry in the coset G/H. Then the Goldstone modes are and the effective Lagrangian at the loweset order is

U = eiπ(x)/fπ , π ≡ πaT a , Leff = −f2π

2∂µU ·∂µU†+O(E2) −→

π− 1

2(∂µπ)2 +

1

6f2π

[(π∂µπ)2−π2(∂µπ)2

]+ · · · , (5.5)

and non-renormalizable terms are generated, but the coupling constants are determined by fπ (called universal interactions). Athigher order O(E4) but with only single-derivatives, we have non-universal interactions that are proportional to c1 and c2 above

c1[∂µU · ∂µU†

]2 −→π

c14f4π

[(∂µπ)4 − 2

3f2π

(∂µπ)2(π∂µπ)2 + · · ·]. (5.6)

5.2 Effective Field Theory of Inflation

5.2.1 Weinberg 2008: Effective Field Theory for InflationNotation convention is

A→ 2ϕ , B → 2γ , R → ϕδφ , Dij → 2Cij . (5.7)

1Irrelevant operators whose mass dimension is larger than 4 are suppressed by powers of E/Λ, while marginal operators (=4) are important at all scales. Examplesof marginal interactions are φ4, QED, QCD, and Yukawa interactions. In QED, Z-boson contributes to QED process e+e− → e+e− scattering, generating non-renormalizable operators. However, QED is highly successful, not because its main operators are relevant (or renormalizable), but because the Z-boson is veryheavy.


With a dimensionless scalar field ϕ ≡ ϕc/M , the simple Lagrangian density is2

L0 =√g

[− M2

P

2R− M2

2gµν∂µϕ∂νϕ−M2

PU(ϕ)

], U(ϕ) ≡ V (Mϕ)/M2

P . (5.8)

Note that the unperturbed value of U is (3 − ε)H2, so we can think of U as well as ∂µϕ∂µϕ as both being of order H2 at horizonexit. The leading correction will consist of a sum of all generally covariant terms with four spacetime derivatives and coefficients oforder unity. By a judicious weeding out of total derivatives, the most general such correction terms can be put in the form

∆L =√g

[f1(ϕ)

(gµνϕ,µϕ,ν

)2+ f2(ϕ)gρσϕ,ρϕ,σϕ+ f3(ϕ)

(ϕ)2

+ f4(ϕ)Rµνϕ,µϕ,ν + f5(ϕ)Rgµνϕ,µϕ,ν (5.9)

+f6(ϕ)Rϕ+ f7(ϕ)R2 + f8(ϕ)RµνRµν + f9(ϕ)CµνρσCµνρσ

]+ f10(ϕ)εµνρσCµν

κλCρσκλ , ε1230 ≡ +1 .

Writing the last two terms as bilinears in Cµνρσ rather than Rµνρσ has no effect in the last term, and in the penultimate term of coursejust amounts to a different definition of f7 and f8 (Similarly, instead of writing the penultimate term as a bilinear in Cµνρσ or Rµνρσ,we could have written it as the linear combination of curvature bilinears that appears in the Gauss–Bonnet identity; even though thislinear combination is a total derivative, it would affect the field equations because its coefficient f9(ϕ) is not constant).

Using the equation of motion,3 and suitable redefinitions of ϕ, U(ϕ), and f1(ϕ), and with various total derivatives dropped, theleading order corrections are

∆L =√gf1(ϕ)

(gµνϕ,µϕ,ν

)2

+√gf9(ϕ)CµνρσCµνρσ + f10(ϕ)εµνρσCµν

κλCρσκλ . (5.10)

The first term is of the type encountered in theories of “k-inflation”, and the second term (or an equivalent Gauss–Bonnet term) hasbeen considered in connection with inflation and the evolution of dark energy. For a general function f10(ϕ) the final term violatesparity conservation, that is, although the action is invariant under coordinate transformations xµ → x′µ that are “small,” in the sensethat Det(∂x′/∂x) > 0, it is not invariant under inversions. As we are concerned about quadratic perturbations, this saves us fromhaving to calculate the Weyl tensor to second order in perturbations; we have simply

[∆L](2) =

[√gf1(ϕ)

(gµνϕ,µϕ,ν

)2](2)

+ a3f9(ϕ)gµκgνλgρη gσζC(1)κληζC

(1)µνρσ + f10(ϕ)εµνρσ gκη gλζC

(1)µνηζC

(1)ρσκλ . (5.11)

In comparison to Cheung et al. their calculations do not include any of the leading corrections following the first term. Their paperdoes not spell out the rules governing which terms are to be included in the corrections of leading order. They do at first includeterms involving the extrinsic curvature of the spacelike surface with ϕ constant, but later drop these terms. The extrinsic curvature isnot included here, because in a general gauge it does not give a local term in the action. they stick to co-moving gauge, in which theextrinsic curvature can be expanded in a series of local functions, and these do yield some though not all of the terms in this derivation.

• Scalar: The K-inflation action is[−√gM2

P

2R+√gP(− ∂µϕ∂µϕ/2, ϕ

)](2)

= −M2P H

H2a3

[1

c2sR2 − 1

a2(~∇R)2

], c−2

s = 1+2

[X∂2P (X, ϕ)

∂X2

/∂P (X, ϕ)

∂X

]X= ˙ϕ2/2

.

(5.12)The first term in the correction shifts the sound speed and the second term corresponds to additional shift (degenerate)

∆c2s =16HM2

P f1(ϕ)

M4,

[√gf1(ϕ)

(gµνϕ,µϕ,ν

)2](2)

=16M4

Pa3H2f1(ϕ)

M4H2R2 , ∆f1(ϕ) =

M4

3M4P

f9(ϕ) , (5.13)

and the third term f10 vanishes for scalar modes. We conclude that the leading corrections to the Gaussian correlations of R aresolely of the k-inflation type. Bottom line: If we ignore interactions with gravity and simply expand the above calculation beyond the

2As is well known, it is the most general Lagrangian density for gravitation and a single scalar field with no more than two spacetime derivatives. An arbitraryfunction of ϕ multiplying the first term could be eliminated by a redefinition of the metric, and an arbitrary function of ϕ multiplying the second term could beeliminated by a redefinition of ϕ.

3In EFT, there exist numerous higher-order derivative terms suppressed by some cutoff mass scales. These additional terms should be taken as perturbations,literally because EFT is valid below cutoff scales. If one takes the EFT Lagrangian seriously L0 + ∆L as full theory, there exist additional poles at the cutoff massscales and often ghosts, invalidating the theory. The former is rather innocuous, because the theory is not valid at the cutoff anyway, while the latter is not. Whathappens in fact is that the full EoM needs to be expanded in terms of cutoff mass scales, which naturally remove ghosts and extra mass poles (in addition to the originalmass scale m). This procedure is equivalent to using the EoM of L0 and rewriting the perturbed Lagrangian.

62


quadratic terms, we reproduce Cheung et al. results:

√g

[− M2

2gµν∂µϕ∂νϕ−M2

PU(ϕ) + f1(ϕ)(gµν∂µϕ∂νϕ)2

]

= L+ a3M2P H

(−π2 + a−2(~∇π)2

)+

16a3M4P H

2f1(ϕ)

M4

(π2 + π3 − π(~∇π)2

a2+π4

4− π2(~∇π)2

2a2+

(~∇π)4

4a2

). (5.14)

But apparently, Weinberg has only one dof f1, while there are many in Cheung et al.

• Tensor: The first term receives no tensor contribution, and the other two terms are

a3f9(ϕ)gµκgνλgρη gσζC(1)κληζC

(1)µνρσ = f9(ϕ)

[a−5C

(1)ijklC

(1)ijkl − 4a−3C

(1)ijk0C

(1)ijk0 + 4a−1C

(1)i0k0C

(1)i0k0

]= a3f9(ϕ)

Dik

[2H2 + 2∇2/a2]Dik − 4HDik(∇2/a2)Dik + 2Dik(∇4/a4)Dik

, (5.15)

f10(ϕ)εµνρσ gκη gλζC(1)µνηζC

(1)ρσκλ = f10(ϕ)εijk0

[4a−4C

(1)lmijC

(1)lmk0 − 8a−2C

(1)l0ijC

(1)l0k0

]= 4f10(ϕ)εijk0 ∂

∂t

[Dil∂j∇2Dkl

],

where the metric tensor is

hij(x, t) = a2(t)[

expD]ij

(x, t)→ 2Cij , Dii = 0 , ∂iDij = 0 . (5.16)

The field equation for the tensor mode (with the term proportional to f9 dropped for simplicity) is then

Dil+3HDil−(∇2/a2)Dil = −64πGf10a−3(εijk0∂j∇2Dkl+ε

ljk0∂j∇2Dki

), D±+3HD±+(k2/a2)D± = ∓128πG(k/a)3f10D± .

(5.17)

5.2.2 Goldstone Action in Inflationary ModelsConsider a general time-reparametrization (t → t + ξ)4 and a Goldstone field that transforms as π → π − ξ, such that U ≡ t + πremains invariant in a non-perturbative way. The most general Lagrangian can be constructed in term of U :

S =

∫d4x√−gL , L = F

[U, (∂µU)2,U, · · ·

]= Λ4(U)−f4(U)gab∂aU∂bU+· · · , Λ4 = −M2

pl(3H2+H) , f4 = M2

plH ,

(5.18)where we write down the leading order terms. These lowest order terms are completely fixed by the background relation as follows.Consider a single field and assume unitary gauge. Unitary gauge is convenient in the sense that a simple scalar field can be written as

−1

2(∂φ)2 − V (φ) −→

δφ=0−

˙φ(t)2

2g00 − V [φ(t)] = M2

plHg00 −M2

pl(3H2 + H) , (5.19)

˙φ2 = ρ+ p = −2M2plH , 3H2 + H =

ρ− p2M2

pl

=V

M2pl

, (5.20)

where we assume the scalar field is the only energy component. By transforming from unitary gauge t→ t+ π, we derive

g00 → ∂(t+ π)

∂xa∂(t+ π)

∂xbgab = (1+π)2g00+2(1+π)∂απ g

0α+gαβ∂απ∂βπ −→ −1−2π−π2+1

a2(∇π)2 = gab∂aU∂bU , (5.21)

where we ignored mixing with metric perturbations.5 The next leading correction is then

Lcs =1

2M4

2

[1 + gab∂aU∂bU

]2 → 2M42

[π2 + π(∂aπ)2

], Ltot = −

M2plH

c2s

[π2 − c2s(∇π)2 + π(∂aπ)2

],

1

c2s= 1− 2M2

2

M2plH

,

(5.22)4We use a proper time, not conformal time.5This is called decoupling limit, in which Mpl → ∞ and H → 0 with HM2

pl fixed. As long as E2 H , the decoupling limit is satisfied and we can ignoremixing with gravity.

63


where unity is added to keep the background and the leading term affects the sound speed. Small sound speed (M2 6= 0) is related tocubic interaction, which can be probed by non-Gaussianity. Higher-order derivative terms are called k-essence:

LP =∑n

1

n!M4n

[1 + gab∂aU∂bU

]n, M4

n = Xn ∂nP

∂X2

∣∣∣∣0

. (5.23)

Similarly, other higher-order derivative terms including U and its combination with (∂µU) can be implemented.Three energy scales are discussed: Hubble scale is universal for all models and two additional (model-dependent) scales: symme-

try breaking scale Λ4b = 2M2

pl|H|cs and strong coupling scale M4? = M2

pl|H|c−2s /(1− c2s). The former is the scale beyond which the

background evolution can be ignored (i.e., de Sitter symmetry), and the latter is the scale at which the cubic interaction term becomesimportant.

5.2.3 Cheung, Creminelli, Fitzpatrick, Kaplan, Senatore 2008: Effective Field Theory of InflationThey assume unitary gauge (uniform field gauge) and argue that it can depend on certain things, such as Riemann tensor, extrinsiccurvature, and in particular g00.6 With that, the most generic action can be written after matching the given background solutionH(t)as

S =

∫d4x√−g F (Rµνρσ , g

00,Kµν ,∇µ, t)→∫d4x√−g[1

2M2

PlR+M2PlHg

00−M2Pl

(3H2 + H

)+F (2)(g00 + 1, δKµν , δRµνρσ ;∇µ; t)

], (5.24)

where F (2) starts quadratic in the arguments g00 + 1, δKµν = Kµν − a2Hhµν (hµν is the induced spatial metric), and δRµνρσ.Everything else but the Ricci scalar is the matter Lagrangian, and the energy-momentum tensor with Einstein equation fixes thebackground parts. For some reason,7 they consider another time transformation that satisfy t = t + ξ0 and let π = −ξ0, such thatπ + t = π + t in a non-perturbative way. With that, F (2) can be further expanded in terms of its arguments and written in terms ofGoldstone boson π ignoring mixing terms with gravity (H Emix = M2

2Mpl) as

Sπ =

∫d4x√−g[M2

PlH (∂µπ)2 + 2M42

(π2 + π3 − π

1

a2(∂iπ)2

)−

4

3M4

3 π3 −

M2

2

1

a4(∂2i π)2 + . . .

]. (5.25)

In the absence of additional Mi terms, it corresponds to the de-Sitter case when the Goldstone boson is canonically normalized as

π = − 1

Hζ → − 1

Hϕδφ , πc = − ˙φπ =

˙φ

Hζ , 〈πcπc〉 = (2π)3δD(k − k′)

(H2∗

2k3

). (5.26)

The coefficient of the time kinetic term π2 is not completely fixed and has to be positive to avoid instability. Defining the soundspeed, we have (this is the formula often used in literature)

−M2PlH + 2M4

2 > 0 , c−2s ≡ 1−

2M42

M2PlH

, Sπ =

∫d4x√−g[−M2

PlH

c2s

(π2 − c2s

(∂iπ)2

a2

)+M2

PlH

(1−

1

c2s

)(π3 − π

(∂iπ)2

a2

)−

4

3M4

3 π3...

].

(5.27)The calculation of power spectrum is similar with rescaled momentum k = csk. The level of non-Gaussianity from an additionalterm is

Lπ(∇π)2

L2∼Hπ

(Hcsπ)2

H2π2∼ H

c2sπ ∼ 1

c2sζ , (5.28)

and similarly for other terms.

Comparison of Two Approaches

Weinberg approach is general, but perturbative, such that these additional terms are always smaller than the background case and theyare controlled by one additional parameter, while Cheung et al. approach simply focus on perturbations (independent of background),such that these perturbations are in most cases independent and described by numerous additional parameters.

6With uniform field gauge, the scalar field degree vanishes (φ = φ), and all the dofs are in metric perturbations around the FLRW background, i.e., eaten as theysay. Indeed, one has freedoms α, β, ϕ, but Einstein equations eliminate two, such that there is only one, usually ϕδφ = ζ, and they parametrize it with g00. Validonly in a single field case.

7In fact, this is again gauge-transformation, or Stoekelberg trick, to re-introduce the scalar field. In general, a scalar field can be written as φ(t, x) = φ(t, π(t, x)),non-perturbatively defined, but they are concerned at the linear order.

64

6 Modification of Gravity and Dark Energy Mod-els

6.1 Dark Energy ModelsRecent observations of distant supernovas show that the Unvierse is accelerating, rather than decelerating. To achieve such cosmicacceleration, we need an overal energy budget with equation of state w = p/µ, satisfying

a

a= −4πG

3(µ+ 3p) +

Λ

3= −4πG

3µ(1 + 3w) → w < −1

3. (6.1)

Good candidates with w < −1/3 are a cosmological constant and scalar fields dominated by potential. Or it can be argued thatgeneral relativity was never accurately tested on cosmological scales and this cosmic acceleration indicates the breakdown of generalrelativity.

Narrowly speaking, the dark energy model is used to refer to additional energy components to explain the cosmic accelerationwithin the framework of general relativity. Modified gravity theories are simply any gravity theories that are not general relativity orNewtonian gravity. Modification of gravity has its own merit, apart from the cosmic acceleration phenomenon, but here we stick tothose models of modified gravity that attempt to explain the cosmic acceleration without additing any exotic energy component but bymodifying gravity on cosmological scales. Broadly speaking, the dark energy model is often used to encompass these two categories.

• brief discussion of modified gravity theories in the early Universe and the late Universe, solar system tests, screening mechanism,ghosts

6.2 Brans-Dicke Scalar-Tensor TheoryEinstein gravity is basically a tensor theory gµν . The Brans-Dicke theory, known as a scalar-tensor theory, introduces an extrascalar that couples to gravity (hence other matter fields) in a non-minimal way. The Mach principle states that the only (physically)meaningful motion of a particle is the motion relative to the other matter in the Universe. While Einstein was said to be guidedby the principle, GR admits an empty solution or the de Sitter solution. An attempt to fix this incompleteness is the motivation byBrans-Dicke.

By using the dimensional argument, gravity by some mass distribution is

GM

rc2' 1 , φ ∼

∑ m

r∼ φ ∼ 1

G, G = G(φ) ∼ 1

φ, (6.2)

such that the “gravitational constant” should know the mass distribution. The Brans-Dicke (BD) action is

S =

∫d4x√−g[φR

16π+ ω

∂µφ∂µφ

φ+ Lm

], [φ] = L−2 , (6.3)

where we have a non-trivial kinetic term due to the dimension of φ. The equations of motion are

Rµν −1

2gµνR =

8π

φ

(Tmµν + Tφµν

), Tmµν;µ = 0 , (6.4)

Tφµν =ω

φ2

(∂µφ∂νφ−

1

2gµν∂ρφ∂

ρφ

)+

1

φ(φ,µν − gµνφ) , φ =

8π

3 + 2ωTm . (6.5)

Weak field solution

In the solar system, we can derive the simple spherically symmetric solution around the Minkowski background as

g00 = −1 +

(2M

φ0r

)(1 +

1

3 + 2ω

), grr = 1 +

(2M

φ0r

)(1− 1

3 + 2ω

), (6.6)

where φ0 is the constant scalar field value in the Solar system, such that the gravitational constant is

G0 =1

φ0

4 + 2ω

3 + 2ω. (6.7)


The light deflection in BD theory can be computed as

δθ =4G0M

r

(3 + 2ω

4 + 2ω

), (6.8)

and this can be compared to the observational constraint using the parametrized post-Newtonian (PPN) metric

ds2 = −(

1− 2G0M

r

)dt2 +

(1 +

2G0M

rγ

)dr2 + r2dΩ2 . (6.9)

The Solar system constraint is|γ − 1| < 10−5 , ω > 40, 000 , (6.10)

implying that it is virtually identical to GR.

6.3 Scalar-Tensor TheoriesThe most generic action for scalar-tensor theories can be written as

SJ =

∫d4x√−g

M2pl

2f(R) , (6.11)

where the action is written in the Jordan frame. By introducing an auxiliary field Q, we re-write the action

SJ =

∫d4x√−g

M2pl

2[f ′(R)(R−Q) + f(Q)] , (6.12)

where the equation of motion for Q guarantees that the action is equivalent to the original action. Since the Einstein-Hilbert actioncorresponds to f ′ = 1, any deviation from GR can be parametrized by f ′ 6= 1. By re-arranging the action in terms of a new field χ,

χ ≡ f ′(Q) , Q = Q(χ) , SJ =

∫d4x√−g

M2pl

2

[χR− χ Q(χ) + f [Q(χ)]

], (6.13)

we demonstrate that the scalar-tensor theories correspond to the Brans-Dicke theory without the kinetic term (ω = 0) and withnon-trivial interaction potential.

We can further manipulate the action to gain more insight. By re-defining the metric by a conformal transformation

gµν ≡ χgµν ≡ exp

(√2

3

φ

Mpl

)gµν , (6.14)

we can write the action in the Einstein-frame

SE =

∫d4x

√−g

[M2

pl

2R− 1

2gµν∂µφ∂νφ− V (φ) + Lm

], V (φ) =

M2pl

2f ′2(Q) [Qf ′(Q)− f(Q)] , (6.15)

where the gravity section becomes the Einstein-Hilbert action. Note that the relation between the scalar field φ and χ is an implicitfunction to be inverted. Furthermore, in the Einstein frame, the matter Lagrangian Lm depends on the original metric gµν , such thatin terms of the conformally transformed metric gµν , the matter fields couple to the scalar field φ. Therefore, the original scalar-tensortheory becomes a massive scalar field in the Einstein frame, and the deviation (f ′ 6= 1) from GR characterizes the Compton wavelength, beyond which the scalar field is massless and propagates, and below which the scalar field is massive and gravity is enhanced.

The field equations in the Einstein frame are

Rµν −1

2gµνR = 8πG

(Tmµν + Tφµν

), φ =

dV

dφ+

Tm√6Mpl

≡ dVeff

dφ,

(Tmµν + Tφµν

);µ

= 0 , (6.16)

where the conservation law is valid only together with the scalar field. Because of the nonlinear interaction of the scalar field withmatter, we can have non-trivial effective potential Veff , which can suppress the gravity in a dense environment, called the Chameleonmechanism.

Ideally, the Compton wave length of the theory can be set equal to the horizon scale, such that the Universe can accelerate withoutdark energy. However, due the Solar system constraints, one has to resort to some mechanism that suppresses the enhanced gravityin the Solar system, such as the Chameleon or Vainshtein mechanisms.

66


A toy model

Consider a simple toy model:

f(R) = R−µ4

R, V (φ) = µ2M2

pl exp

[−2

√2

3

φ

Mpl

]√√√√exp

[√2

3

φ

Mpl− 1

],

µ4

R2−R

3+µ4

R=

8πGT

3in JF , (6.17)

where the degrees of freedom are evident in the Jordan frame.

6.4 Effective Field Theory of QuintessenceIgnoring the metric coupling, the most general k-essence action can be expanded up to second order in π:

S =

∫d4x√−g P (φ,X) =

∫d4x a3

[P0 + P0π +

1

2P0π

2 + 2PXX0π + 2(PXX0)·ππ + PXX0

(π2 − (∇π)2

a2

)+ 2PXXX

20 π

2

],

(6.18)where 0 denotes the background quantity and we used

φ(t, x) = φ0(t+ π) → φ(t, x) = φ0 + φ0π +1

2φ0π

2 + · · · , ds2 = −dt2 + a2dx2 , X0 = φ20(t) ,

X = −gabφ,aφ,b = φ2(t)− φ20(t+ π)

(∇π)2

a2=[φ0 + φ0π + · · ·

]2 [1− (∇π)2

a2

]= X0 + X0π +

1

2X0π

2 + 2X0π + 2X0ππ +X0

(π2 − (∇π)2

a2

)+ · · · . (6.19)

The linear order part will vanish due to the background EoM, and after some manipulation,

S =

∫d4x a3

[1

2(ρQ + pQ + 4M4)π2 +

1

2(ρQ + pQ)

(∇π)2

a2+

3

2H(ρQ + pQ)π2 − 1

2(ρQ + pQ)hπ

], c2s =

ρQ + pQρQ + pQ + 4M4

,

(6.20)where M4 ≡ PXXX

20 , the background relation for ρQ = 2X0 − PX and pQ = PX is used and metric perturbations in synchronous

gauge are introduced (only one term at the quadratic order). Standard analysis of stability condition may apply, depending on ρQ+pQand M4 (can be negative). Ignoring gravity (i.e., on small scales), higher-order operators can be considered. To fix the background,the leading correction is

LM = −M2

2(φ+ 3H)

2 → − M2

2

(π + 3Hπ − 3Hπ − ∇

2π

a2

)2

. (6.21)

These operators affect the stability condition.

6.5 Weinberg Approach to Cosmic AccelerationWe start with the tree level scalar-tensor theory written in the Jordan frame:

S0 =

∫d4x√−g

[m2p

2Ω2

0(ϕ)R− 1

2Z0(ϕ)gµν∂µϕ∂νϕ− U0(ϕ)

]+ Sm . (6.22)

Variation with respect to metric and scalar field yields

Rµν −1

2gµνR =

1

m2pΩ2

0

[Tµν + Z0∂µϕ∂νϕ− gµν

(1

2Z0g

αβ∂αϕ∂βϕ+ U0

)]+

1

Ω20

(∇µ∇ν − gµν)Ω20 ,

R =1

m2pΩ2

0

[−T + Z0gµν∂µϕ∂νϕ+ 4U0] +

3

Ω20

Ω20 , Z0ϕ+

1

2Z′0g

µν∂µϕ∂νϕ = U ′0 −m2pΩ0Ω′0R ,

∴ Rµν =1

m2pΩ2

0

[Tµν −

1

2gµνT + Z0∂µϕ∂νϕ+ gµνU0

]+

1

Ω20

(∇µ∇ν +

1

2gµν

)Ω2

0 .

(6.23)

67


Then the leading corrections including four derivatives (excluding boundary terms) are

∆S =

∫d4x√−g[α1

Λ4(gµν∂µϕ∂νϕ)2 +

α2

Λ3ϕgµν∂µϕ∂νϕ+

α3

Λ2(ϕ)2 +

b1

Λ2Λ2m

Tµν∂µϕ∂νϕ+b2

Λ2Λ2m

Tgµν∂µϕ∂νϕ+b3

ΛΛ2m

Tϕ

+c1

Λ2Rµν∂µϕ∂νϕ+

c2

Λ2Rgµν∂µϕ∂νϕ+

c3

ΛRϕ+ d1W

µνλρWµνλρ + d2εµνλρWµν

αβWλραβ + d3RµνRµν + d4R

2

+e1

Λ4m

TµνTµν +e2

Λ4m

T 2 +e3

Λ2m

RµνTµν +

e4

Λ2m

RT

],

where αi, bi, ci, di and ei are algebraic functions of ϕ, W is the Weyl tensor. Higher-order time-derivative terms are eliminated byusing the tree-level EoM, e.g.,

α2

Λ3φ(∂φ)2 → α2

Λ3(∂φ)4

[−1

2

Z ′

Z+ · · ·

]+α2

Λ3(∂φ)2

[· · ·]

+α2

Λ3T ab∂aφ∂bφ

[· · ·]

=α1

Λ4(∂φ)4− 1

2Z(∂)2 +

b2Λ2Λ2

m

T ab∂aφ∂bφ ,

(6.24)where tilde represents the renormalized coefficients.1After this (lengthy) procedure, one is left with the most general EFT action withfour derivative terms (still more complicated than the usual). Then noticing that the cut-off scales related to matter Lagrangian islarger that the scale of interest Λm Λ, we are left with the final result:

S =

∫d4x√−g

[m2p

2Ω2(ϕ)R− 1

2Z(ϕ)gµν∂µϕ∂νϕ− U(ϕ) +

α(ϕ)

Λ4(gµν∂µϕ∂νϕ)2

]+ Sm, (6.25)

where all parameters are renormalized due to higher-order corrections. In the end, there is only one additional correction term, asnoted in Weinberg.

6.6 Effective Field Theory of Dark EnergyIn unitary gauge, calculations are greatly simplified, in particular, the single clock (or preferred coordinate) is provided by the scalarfield, defining the preferred (normal) observer and hypersurface:

nµ ≡ −∂µφ√−(∂φ2)

→ −δ0µ√−g00

, X ≡ (∂µφ)2 = φ20g

00 . (6.26)

Using the Stuckelberg trick t→ t+ π, quantities in unitary gauge transform as

c(t) → c(t+ π) = c(t) + c(t)π +1

2c(t)π2 + . . . , g00 → g00 + 2g0µ∂µπ + gµν∂µπ∂νπ , g0i → g0i + gµi∂µπ ,

δKij → δKij − Hπhij − ∂i∂jπ +O(π2) , δK → δK − 3Hπ − a−2∇2π +O(π2) , (6.27)

where the background is removed.Assuming the weak equivalence principle (minimal coupling to matter in Jordan frame), we will consider the most general action:2

S = S0 + S(2)DE , S0 ≡

∫d4x√−g[M2∗

2f(t)R− Λ(t)− c(t)g00

], (6.28)

where S(2)DE contains at least quadratic terms and the full action up to linear order is captured only by three functions. Background

equations are

(fGµν −∇µ∇νf + gµνf)M2∗ + (cg00 + Λ)gµν − 2cδ0

µδ0ν = Tµν ,

G00 = 3H2 + 3k

a2, R = −Gµµ = 12H2 + 6H + 6

k

a2, f(t) = −f − 3Hf , ρm + 3H(ρm + pm) = 0 , (6.29)

where the usual conservation of matter holds in Jordan frame, while the modified Friedman equation defines the dark energy quanti-ties:

H2+k

a2=

1

3fM2∗

(ρm+ρD) , H− k

a2= − 1

2fM2∗

(ρm+ρD+pm+pD) → ρD+3H(ρD+pD) = 3M2∗ f

(H2 +

k

a2

), (6.30)

1In the paper, the bare coefficients are denoted with the subscript 0, while coefficients without subscript are the renormalized.2In the absence of matter fields, f(t) can be re-absorbed by a conformal transformation of gµν . Here, however, the matter sector uniquely singles out the Jordan

metric gµν , in which test particles follow geodesics.

68


which sets the two functions

Λ(t) = M2∗ f

(H + 3H2 + 2

k

a2+

1

2

f

f+

5H

2

f

f

)− 1

2(ρm − pm) =

1

2

[ρD − pD +M2

∗ (5Hf + f)],

c(t) = M2∗ f

(−H +

k

a2− 1

2

f

f+H

2

f

f

)− 1

2(ρm + pm)

1

2

[ρD + pD +M2

∗ (Hf − f)]. (6.31)

Alternative method is often adopted in literature by defining the effective fluid quantities ρeffD and peff

D as

ρD = fρeffD +(f−1)ρm, pD = fpeff

D +(f−1)pm , H2+k

a2=

1

3M2∗

(ρm+ρeffD ) , H− k

a2= − 1

2M2∗

(ρm+ρeffD +pm+peff

D ) .

(6.32)The only freedom left s thus the function f(t), but in practice it is highly constrained by post-Newtonian tests: By expanding in theproper time around the present epoch,

f(t) = 1 + f1H0(t− t0) + f2H20 (t− t0)2 + . . . . (6.33)

They further discuss mixing and stability and present expressions in Newtonian gauge.

6.7 A Unifying Description of Dark EnergyGeneral action principleThe fundamental idea is then to start from a generic action that depends on the basic geometric quantities that appear in an ADMdecomposition of spacetime, with uniform scalar field hypersurfaces as constant time hypersurfaces. Since many dark energy modelsare given explicitly in terms of a scalar field φ, it is useful to write down the correspondance between the various geometrical tensorsand expressions of φ: (Kµν → −Kmine

µν ; K → −Kmine)

nµ = −1

√−X∇µφ , Kµν = nµ;ν = −

1√−X∇µ∇νφ+ 2n(µaν) +

1

2Xnµ nν n

λ∇λX , X = g00φ2(t) = −φ2(t)

N2. (6.34)

Using the Gauss-Codazzi relation

R = KµνKµν −K2 +R(h) + 2∇µ(Knµ − nρ∇ρnµ) , (6.35)

we will consider general gravitational actions which can be written in terms of the geometrical quantities that we have introduced,expressed in ADM coordinates,

Sg =

∫d4x√−g L(N,Kij , R

(h)ij , hij , Di; t) , (6.36)

where Di is the covariant derivative with respect to hij . Note that the above action is automatically invariant under spatial diffeomor-phisms, corresponding to a change of spatial coordinates.

Examples

• General relativity: The Einstein-Hilbert action is

SGR =

∫d4x√−g M

2Pl

2R , LGR =

M2Pl

2

[KijK

ij −K2 +R(h)],

√−g = N

√h , (6.37)

where we used the Gauss-Codazzi expression and get rid of the total derivative term. In contrast with modified gravity theories thatintrinsically contain a scalar degree of freedom, the time slicing in GR is arbitrary since there is no preferred family of spacelikehypersurfaces. This means that the Lagrangian contains an additional symmetry, leading to full four-dimensional invariance, whichis not directly manifest in the ADM form.

• Quintessence and k-essence: The action is

LQ = −1

2∂µφ∂

µφ− V (φ) =φ2(t)

2N2− V (φ(t)) , Lk-essence = P (X) = P

[− φ2(t)

2N2, φ(t)

]. (6.38)

69


The scalar kinetic energy is

X = gab∂aφ∂bφ → g00(φ0)2 =φ2

0(t)

N2, (6.39)

and by reparametrization, the lapse N represents the scalar kinetic energy in unitary gauge.

• F (R) theories: It is easy to verify that the Lagrangian

LF (R) = F (φ) + Fφ(φ)(R− φ) = Fφ(R(h) +KµνKµν −K2) + 2FφφK

√−X + F (φ)− φFφ , (6.40)

where we used the Gauss-Codazzi relation and performed integration by parts. Fφφ term represents the mixing of kinetic term Xwith gravity (kinetic braiding).

• Horndeski theories: The Horndeski Lagrangians are

LH2 [G2] ≡ G2(φ,X) = F2(φ,X) , LH3 [G3] ≡ G3(φ,X)φ = F3(φ,X)K ,

LH4 [G4] ≡ G4(φ,X)R− 2G4X(φ,X)[(φ)2 − (∇µ∇νφ)(∇µ∇νφ)

]= F4(φ,X)R(h) + (2XF4X − F4)(K2 −KµνKµν) ,

LH5 [G5] ≡ G5(φ,X) (4)Gµν∇µ∇νφ+1

3G5X(φ,X)

[(φ)3 − 3φ (∇µ∇νφ)(∇µ∇νφ) + 2 (∇µ∇νφ)(∇σ∇νφ)(∇σ∇µφ)

]= F5(φ,X)GµνK

µν − 1

3XF5X(K3 − 3KKµνK

µν + 2KµνKµσKν

σ) , (6.41)

where the ADM functions are defined as

F2 = G2 −√−X

∫G3φ

2√−X

dX , F3 = −∫G3X

√−X dX − 2

√−XG4φ , (6.42)

F4 = G4 +√−X

∫G5φ

4√−X

dX , F5 = −∫G5X

√−X dX . (6.43)

Ostrogradski theorem says that no more than two time derivatives ensures no ghostlike degrees of freedom, such that terms like∇a∇b∇cφ or ∇cRab are forbidden. It is noted that it is a sufficient condition for healthy theories, such that some healthy theorieshave more than two time derivatives.

• Beyond Horndeski: It has been shown that the following Lagrangians do not lead to Ostrogradski instabilities, in contrast withprevious expectations:

L2 ≡ A2(t,N) , L3 ≡ A3(t,N)K , L4 ≡ A4(t,N)(K2 −KijK

ij)

+B4(t,N)R(h) , (6.44)

L5 ≡ A5(t,N)(K3 − 3KKijK

ij + 2KijKikKj

k

)+B5(t,N)Kij

(R

(h)ij −

1

2hijR

(h)

), (6.45)

where the Lagrangians reduce to Horndeski theories if

A4 = −B4 + 2XB4X , A5 = −1

3XB5X . (6.46)

• Horava-Lifshitz theories: An interesting class of Lorentz-violating gravitational theories has been introduced by Horava with thegoal of obtaining (power counting) renormalizability Horava (2009). These theories, dubbed Horava-Lifshitz gravity, assume theexistence of a preferred foliation of spacelike hypersurfaces. An ADM formulation of these theories is thus very natural, even if acovariant description is also possible, via the introduction of a scalar field, often called “khronon”, that describes the foliation. Thereexist several variants of Horava-Lifshitz gravity in the literature, and the so-called healthy non-projectable extension has been shownto be free of instabitilities. All these theories are describable by a Lagrangian of the form

LHL =M2

Pl

2

[KijK

ij − λK2 + V(R(h)ij ,

1

N∂iN)

]. (6.47)

70

7 CMB Temperature AnisotropiesIn the early Universe, the radiation dominates the overall energy density, and due to high pressure the fluctuations cannot grow withinthe horizon. In particular, the tight-coupling between the baryons and the photons leads to a single fluid, or the baryon-photon fluid,oscillating with a unique sound speed. As a fluid, the density (monopole) and the velocity (dipole) characterize the fluid, and thehigher multipoles are negligible. Once the baryons recombine at later time, the photons are released and free-stream to the observertoday. This free-streaming of the monopole and the dipole generates the temperature anisotropies we measure today, and they showthe acoustic oscillations of the baryon-photon fluid at the recombination epoch.

In this chapter, we will ignore the vector and tensor perturbation, and choose the conformal Newtonian gauge:

α→ αχ , ϕ→ ϕχ , β = γ = χ ≡ 0 , U → vχ , Vα = −vχ,α + v(v)α → −vχ,α . (7.1)

7.1 Collisionless Boltzmann EquationThe Liouville theorem in GR states that the phase-space volume is conserved along the path parametrized by λ with momentum pµ:

0 = ∆(dN) =

(∂f

∂xµ∆xµ +

∂f

∂pµ∆pµ

)dVp =

(pµ

∂f

∂xµ− Γµρσp

ρpσ∂f

∂pµ

)∆λ dVp . (7.2)

This translates into the relativistic collisionless Boltzmann equation:

0 = pµ∂f

∂xµ− Γµρσp

ρpσ∂f

∂pµ, or 0 = pµ

∂f

∂xµ+ Γρµσpρp

σ ∂f

∂pµ, (7.3)

where we used the geodesic equation

0 =d

dλpµ + Γµρσp

ρpσ = pνpµ,ν + Γµρσpρpσ , 0 =

d

dλpµ − Γρµσpρp

σ . (7.4)

Despite the presence of the Christoffel symbol, the equation is indeed invariant under diffeomorphisms. We further need to imposethe on-shell condition in the collisionless Boltzmann equation.

7.1.1 Geodesic EquationTo solve the Boltzmann equation (7.3), we need to know how the physical momentum pµ in FRW coordinates is related to the physicalquantities measured by an observer with four velocity uµ.

[et]µ = uµ =

1

a(1− αχ, V α) , [ei]

µ =1

a

[δβi Vβ , δ

αi (1− ϕχ)

], (7.5)

where we ignored the rotation of tetrad vectors against FRW coordinates. The physical momentum is written in capital letters as

P a = (E,P i) , E = −pµuµ , P i = pµeiµ , E2 = m2 + P 2 . (7.6)

Using the tetrad expression, the physical momentum pµ = P aeµa in FRW coordinates is then obtained as

pη =(1− αχ)E + VjP

j

a, pα =

1

a[Pα + EV α − ϕχ Pα] , (7.7)

and the covariant momentum is

pη = −a(1 + αχ)E − a VjP j , pα = aVαE + aPα(1 + ϕχ) . (7.8)

In the background, the physical momentum is redshifted as Pα ∝ 1/a for both massless and massive particles.1 So, it isconvenient to define the “comoving momentum q” and “comoving energy ε” as

q := aP , ε := aE =√q2 + a2m2 , qi := qni . (7.10)

1The geodesic equation yields

0 = pηpµ′ + Γµρσpρpσ 7→ 0 = pηpη′ +Hpηpη +Hpαpα , 0 = pηpα′ + 2Hpηpα . (7.9)

The last equation says the spatial momentum pα ∝ 1/a2 in the background, i.e., the physical momentum Pα ∝ 1/a for both massless and massive particles in thebackground. In the presence of perturbations, these relations change.


In the background, the comoving momentum and energy are constant, while the momentum in FRW coordinates redshifts as

pη =E

a=

ε

a2, pα =

1

aPα =

1

a2qnα , nα = niδαi . (7.11)

In terms of the comoving quantities, the momentum in FRW coordinates is now

pη =(1− αχ)ε+ Vjq

j

a2, pα =

1

a2(qα + εV α − ϕχqα) . (7.12)

To compute the change in the comoving momentum as the particle propagates, we need to solve the geodesic equation and obtain

dqαdη

= −ε αχ,α − V′αε− qβVα,β −

a2Hm2

εVα − ϕ′χqα −

qβqΓ

ε

(ϕχ,Γδαβ − ϕχ,αδβΓ

), (7.13)

where we use the background geodesic equation (valid for massive & massless)

d

dη=

∂

∂η+qβ

ε

∂

∂xβ→ d

dλ, ε′ =

a2Hm2

ε. (7.14)

At the linear order, the propagation direction is simply the straight path, and only the comoving momentum changes as

d ln q

dη= −ε

q

(αχ,‖ + V ′‖

)− ϕ′χ − Vα,βnαnβ −

a2Hm2

qεV‖ . (7.15)

Indeed, the comoving momentum is constant in the background. For massless particles (m = 0), the comoving momentum is thecomoving energy, changing as

d ln q

dη= −αχ,‖ − ϕ

′χ − V ′‖ − Vα,βn

αnβ . (7.16)

This can be further arranged asd

dη

(ln q + αχ + V‖

)= (αχ − ϕχ)

′, (7.17)

and the whole quantity in the bracket is affected by the structure growth along the path. Note that the gravitational potential αχbecomes more negative as the structure grows in time.

7.1.2 Collisionless Boltzmann Equation for Massless ParticlesThe Boltzmann equation is further simplified, when we switch the variables (η, xα, pµ) to (η, xα, qi), where the on-shell conditionremoves one component of the physical momentum:

0 =df

dΛ∆Λ =

(∂f

∂xµ∆xµ +

∂f

∂qi∆qi

)∆Λ =

(pµ

∂f

∂xµ+dqi

dΛ

∂f

∂qi

)∆Λ , (7.18)

where the partial derivatives fix (xµ, qi), instead of (xµ, pµ) in Eq. (7.3). Splitting the distribution function F , we derive the Boltz-mann equation in the background:

F := f + f , 0 = f ′ , f = f(q) , (7.19)

i.e., the phase-space distribution is constant in time and space, but a function of the comoving momentum only. The perturbationequation can be derived as

0 =

(pηf ′ + pαf,α +

dqi

dΛnidf

dq

)∆Λ +O(2) = pη

(f ′ + nα

∂f

∂xα+

df

d ln q

a2niq2

dqi

dΛ

)∆Λ , (7.20)

where the last term is

d

dη=

1

pηd

dΛ,

dxα

dη=

1

pηdxα

dΛ=pα

pη,

df

d ln q

a2niq2

dqi

dΛ=

df

d ln q

niq

dqi

dη. (7.21)

Therefore, the collisionless Boltzmann equation is at the linear order in perturbations

0 =df

dη− df

d ln q

(αχ,‖ + ϕ′χ + V ′‖ + Vα,βn

αnβ)→ d

dη

[f − df

d ln q

(αχ + V‖

)]= − df

d ln q(αχ − ϕχ)

′. (7.22)

72


7.1.3 Convention for Multipole DecompositionNow we will decompose the Boltzmann equation, but care must be taken in terms of what variables are decomposed. Schematically,we will perform the decomposition of the perturbations in the Boltzmann equation as

P (xµ, qi) =

∫d3k

(2π)3eik·x P (k, qi, η) =

∫d3k

(2π)3

∑lm

(−i)l√

4π

2l + 1Plm(q,k, η)Ylm(n)eik·x , x = rni , (7.23)

where the angular decomposition is (we suppress irrelevant arguments for clarity such as η and k)

P (qi) :=∑lm

(−i)l√

4π

2l + 1Plm(q) Ylm(n) , Plm(q) ≡ il

√2l + 1

4π

∫d2n Y ∗lm(n)P (qi) . (7.24)

Naturally, Plm are helicity eigenstates, such that under a rotation φ→ φ− Φ in a coordinate (k//z) they transform as

Plm = PlmeimΦ . (7.25)

Since we integrate over n and k, we can simply set k//z.In literature, there exists a different convention (up to 2l + 1 factor) for decomposition in terms of the Legendre polynomial, but

it is in fact valid only for a scalar mode (m = 0):

P (k · n) :=∑l

(−i)lPl Ll(n · k) =∑l

(−i)lPl∑m

4π

2l + 1Ylm(n)Y ∗lm(k) , (7.26)

and for a Fourier mode k//z we derive the correspondence to our decomposition convention:

Ylm(z) =

√2l + 1

4πδm0 , P (k · n) =

∑L

(−i)lPl δm0

√4π

2l + 1Ylm(n) , ∴ Pl → Pl0 . (7.27)

So the decomposition with the Legendre polynomial in Eq. (7.26) is valid only for the scalar modes.

• Notation convention in literature.— A common notation convention: ???

P (k · n) :=∑l

(−i)l(2l + 1)Pl Ll(n · k) , ∴ (2l + 1)Pl ≡ Pl ≡ Pl0 . (7.28)

7.1.4 Multipole Expansion of the Boltzmann EquationIn general, the monopole f0 changes only under diffeomorphisms and the dipole f1 changes only under Lorentz transformations ofthe observer, while the higher multipoles are fully gauge-invariant. Therefore, we construct the fully gauge-invariant variables as

fgi := fχ −df

d ln qV‖ , fgi

0 := fχ0 , fgi1 := fχ1 − kvχ

df

d ln q, fgi

l := fl for l ≥ 2 . (7.29)

The collisionless Boltzmann equation is then

0 = f ′0 +k

3f1 −

df

d ln q

(ϕ′χ +

1

3k2vχ

)= fgi′

0 +k

3fgi

1 −df

d ln qϕ′χ , (7.30)

0 = f ′1 + k

(−f0 +

2

5f2

)+

df

d ln q

(kαχ − kv′χ

)= fgi′

1 + k

(−fgi

0 +2

5fgi

2

)+

df

d ln qkαχ . (7.31)

The monopole and the dipole are affected by the scalar (and the vector) perturbations. The quadrupole moment is

0 = f ′2 + k

(−2

3f1 +

3

7f3

)− df

d ln q

(−2

3k2vχ

)= fgi′

2 + k

(−2

3fgi

1 +3

7fgi

3

)(7.32)

independent of scalar perturbations, but affected by the vector and the tensor perturbations. The higher-order multipoles for l > 2 areautomatically gauge-invariant

0 = f ′l + k

(− l

2l − 1fl−1 +

l + 1

2l + 3fl+1

)= fgi′

l + k

(− l

2l − 1fgil−1 +

l + 1

2l + 3fgil+1

), (7.33)

and they are not source by any metric perturbations in the absence of collision (e.g., neutrino distribution), but they are not dampedeither.

73


7.1.5 Massless ParticlesHere we consider photons and neutrinos. Though neutrinos are massive, massless neutrinos are in most cases a good approximation,with which equations are greatly simplified. We define the temperature anisotropies

ρ = aT 4 = aT 4

(1 + 4

δT

T

)+O(2) , Θ(n) :=

δT

T=

1

4

δρ

ρ, (7.34)

and its multipole decomposition

Θl ≡∫dq q3fl

4∫dq q3f

=πg

a4ρ

∫ ∞0

dq q3fl . (7.35)

Note that the temperature anisotropies Θ are related to the distribution function as

F =

[exp

(q

aT (η)[1 + Θ]

)− 1

]−1

' f(

1 + f eq/aTq

aTΘ)

= f

(1− d ln f

d ln qΘ

), ∴ f = − df

d ln qΘ , (7.36)

where we used the relation and the background quantities are

d ln f

d ln q= −f eq/aT q

aT, ρ =

1

3p =

4πg

a4

∫dq q3f . (7.37)

Therefore, the perturbation quantities for massless particles are

δρ =1

3δp =

4πg

a4

∫ ∞0

dq q3f0 = 4ρ Θ0 , kvΓ = kvobs + Θ1 , (7.38)

πij =g

a4

∫dq q3

∫dΩ

(ninj − 1

3δij

)(− df

d ln qΘ

)= 4ρ

∫dΩ

4π

(ninj − 1

3δij

)Θ ≡ Π(0)Q

(0)αβ , (7.39)

where the useful relations are from Eq. (??)

Π(0) =k2

a2Π =

4

5ρΘ2 , Q

(0)αβ :=

1

k2Q

(0)|αβ +

1

3gαβQ

(0) . (7.40)

Finally, noting thatd

dηf = − df

d ln q

dΘ

dη, (7.41)

we show that the collisionless Boltzmann equation for massless particles becomes

Θgi = Θχ + V obs‖ ,

d

dη(Θgi + αχ) = (αχ − ϕχ)

′. (7.42)

Note that the monopole Θ0 changes only under diffeomorphisms and the dipole Θ1m changes only under Lorentz transformations,while the higher-order multipoles (l ≥ 2) are fully gauge-invariant. We construct the fully gauge-invariant variables as

Θgil ≡

πg

a4ρ

∫dq q3 fgi

l , Θgi0 = Θχ

0 , Θgi1 = Θχ

1 + kvobsχ = kvγχ , Θgi

l = Θl for l ≥ 2 . (7.43)

Now, in terms of the multipole coefficients of the temperature anisotropies, the Boltzmann equation becomes

0 = Θ′0 +k

3Θ1 + ϕ′χ +

1

3k2vχ = Θgi′

0 +k

3Θgi

1 + ϕ′χ , (7.44)

0 = Θ′1 + k

(−Θ0 +

2

5Θ2

)− kαχ + kv′χ = Θgi′

1 − kΘgi0 +

2

5kΘgi

2 − kαχ , (7.45)

0 = Θ′2 + k

(−2

3Θ1 +

3

7Θ3

)+−2

3k2vχ = Θgi′

2 + k

(−2

3Θgi

1 +3

7Θgi

3

), (7.46)

where the derivative term df/d ln q is integrated by part to give minus sign. The higher-order multipoles for l > 2 are again gauge-invariant

0 = Θ′l + k

(− l

2l − 1Θl−1 +

l + 1

2l + 3Θl+1

)= Θgi′

l + k

(− l

2l − 1Θgil−1 +

l + 1

2l + 3Θgil+1

). (7.47)

74


7.2 Collisions of the Baryon-Photon Fluid: Thompson ScatteringIn the early Universe (t ' 6 sec), the temperature is already below the electron mass, so that the dominant interaction for thebaryon-photon fluid is the Thompson scattering, low-energy limit of the Compton scattering:

dσTdΩ

=3σT16π

[1 + (pin · pout)

2], σT :=

8π

3r2e = 6.651× 10−25cm2 , (7.48)

where re is the effective radius of electrons (mec2 = e2/re) and pin,out represents the incoming and outgoing directions of the

photons in the rest frame of the electron. The Thompson scattering cross-section for protons is smaller by the mass ratio. Dueto the directional dependence (quadrupole), the Thompson scattering generates the polarization of the scattered photons. Finally,the collisional term should be Lorentz transformed from the electron rest-frame to the FRW coordinate to be put in the collisionalBoltzmann equation.

7.2.1 Collisional Boltzmann Equation for PhotonsUsing the multipole decomposition, the collisional term for the Boltzmann equation is given as

C ≡ df

d ln qΓ

[Θ(n)− n · vb −Θ0 +

1

2L2(µ)

(1

5Θ2 +

1

5Θp

2 + Θp0

)], (7.49)

and the probability of the Thompson scattering until today is expressed in terms of the optical depth τ :

τ(η) :=

∫ η0

η

dη′ aneσT , Γ := aneσT ≡ |τ ′| , (7.50)

where ne is the electron number density. Therefore, the collisional Boltzmann equation for photons is then

d

dη

(fgi −

df

d ln qαχ

)= − df

d ln q(αχ − ϕχ)

′+ C , (7.51)

d


′ − Γ

[Θ(n)− n · vb −Θ0 +

1

2L2(µ)

(1

5Θ2 +

1

5Θp

2 + Θp0

)]. (7.52)

Applying the multipole decomposition on both sides, we derive

0 = Θgi′0 +

k

3Θgi

1 + ϕ′χ , Θgi0 =

1

4δΓχ , Θgi

1 = kvγχ , (7.53)

0 = Θgi′1 − kΘgi

0 +2

5kΘgi

2 − kαχ + Γ(

Θgi1 − kvbχ

), (7.54)

0 = Θgi′2 + k

(−2

3Θgi

1 +3

7Θgi

3

)+ Γ

(− 9

10Θ2 +

1

8ΘP

0 +1

8ΘP

2

).

The higher-order multipoles for l > 2 are again gauge-invariant

0 = Θ′l + k

(− l

2l − 1Θl−1 +

l + 1

2l + 3Θl+1

)+ Γ Θl = Θgi′

l + k

(− l

2l − 1Θgil−1 +

l + 1

2l + 3Θgil+1

)+ Γ Θgi

l . (7.55)

In the early Universe, when the collision is efficient Γ 1, the higher-multipoles are highly suppressed:

Θ′l ∼Θl

η, ΓΘl ∼

τ

ηΘl , ∴ Θ′l ΓΘl , (7.56)

and from the multipole equations we derive

0 ∼ 0− kΘl−1 + 0 + ΓΘl , ∴ Θl ∼kη

τΘl−1 ≈ 0 for l ≥ 2 . (7.57)

75


7.2.2 Collisional Boltzmann Equation for BaryonsThe collisional Boltzmann equation for baryons can be solved exactly the same way the Boltzmann equation for photons is derived.However, it simply reduces to the fluid equation. The number density of baryons is conserved, regardless of the Thompson scatteringwith photons:

δbχ′+ k2vbχ = −3ϕ′χ , (7.58)

but the momentum of the baryon fluid is affected by scattering off photons as

vbχ′+Hvbχ = αχ +

Γ

R

(vΓχ − vbχ

), (7.59)

where the baryon-to-photon momentum density ratio is

R :=˙ρb˙ρΓ≡ ρb + pbρΓ + pΓ

=3ρb4ρΓ

= 0.6( ωb

0.02

)( a

10−3

). (7.60)

Note that the change in baryon velocity is proportional to the relative velocity between photons and baryons and inversely proportionalto the momentum density ratio.

While the interaction between photons and protons is negligible, protons and electrons are tightly coupled by the Coulombinteraction, so that all three components (b, p, Γ) are tightly coupled.

7.3 Initial Conditions for the EvolutionWe would like to set up the initial conditions, with which the Boltzmann equations can be evolved in time. At early time in theradiation-dominated era, the set of the Boltzmann-Einstein equations can be greatly simplified on large scales as

0 = Θ′0 + ϕ′χ , 0 = N ′0 + ϕ′χ , 0 = δ′χ + 3ϕ′χ , ∴ Θ′0 = N ′0 =1

3δ′χ = −ϕ′χ , (7.61)

where we used kη 1. Using the adiabatic condition in Eq. (??), the initial conditions are set

Θ0 = N0 =1

3δ . (7.62)

The Einstein equation (??) in the limit k → 0 becomes

0 = 4πGa2ρδχ + aHκχ , κχ = 3Hαχ , ∴ Θ0 = N0 = −1

2αχ , (7.63)

where the matter density is ignored in the radiation-dominated era. Similarly, the Einstein equation (??) gives the initial conditionsfor the velocity scalars

vγχ = vνχ =αχ2H

, vbχ = vmχ = vΓχ . (7.64)

While the photons are tightly coupled with baryons, the neutrinos decouple at t ' 1 sec (T ' 1 MeV), such that the neutrinodistribution develops non-vanishing quadrupole moment and hence the anisotropic pressure. Using the relation of the anisotropicpressure to the quadrupole moment in Eq. (7.40) and the Einstein equation (??), we derive

8πGΠ =32πG

5

a2

k2ρνN2 =

12

5

H2

k2fνN2 , ∴ N2 = − 5

12

k2

H2

αχ + ϕχfν

, (7.65)

where we used8πGρν = 3H2fν , fν :=

ρνρT

. (7.66)

Finally, the Boltzmann equation for N2 at the leading order of kη becomes

N ′2 '2

3kN1 =

k2

3Hαχ , ∴ N2 =

k2

6H2αχ , (7.67)

where we integrated the equation usingH = 1/η in Eq. (??). Therefore, the relation between two gravitational potentials becomes

− (αχ + ϕχ) = 8πGΠ =2

5fναχ , ϕχ = −

(1 +

2

5fν

)αχ . (7.68)

76


The comoving-gauge curvature perturbation is generated during the inflationary period, and it is conserved on super horizonscales. After the inflationary period ends, the Universe enters into the standard radiation dominated era (under the assumption thatthe reheating period is very short). The conformal Newtonian gauge potential is then related to the curvature perturbation on largescales as

ϕv = ϕχ −1

2

(αχ −

ϕχH

)' −3

2αχ −

2

5fναχ , αχ = − 10

15 + 4fνϕv , (7.69)

where we used w = 1/3 and ignored the time-derivative terms.

7.4 Observed CMB Power Spectrum

7.4.1 Acoustic Oscillation: Tight-Coupling ApproximationBefore the recombination, the photon-baryon fluid is tightly coupled, so that the higher-order multipoles (l ≥ 2) is greatly suppressed.Under this tight-coupling approximation (Θl ≈ 0 for l ≥ 2), we can derive the master evolution equation for the baryon-photon fluid.From the baryon velocity equation in (7.59),

vΓχ − vbχ =

R

Γ

(vbχ′+Hvbχ − αχ

)≈ R

Γ

[vΓχ

′+HvΓ

χ − αχ +O(R

Γ

)], (7.70)

whereR/Γ is a small number, so that we used vbχ = vΓχ +O(R/Γ). Using this tight-coupling approximation, the Boltzmann equation

for the dipole becomes

0 = Θgi′1 − kΘgi

0 +2

5kΘgi

2 − kαχ + Γ(

Θgi1 − kvχ

)≈ Θgi′

1 − kΘgi0 + 0− kαχ +R

(Θgi′

1 +HΘgi1 − kαχ

), (7.71)

such that the derivative of the dipole is

Θgi′1 =

k

1 +RΘgi

0 −HR

1 +RΘgi

1 + kαχ . (7.72)

Taking the derivative of the monopole equation and removing the dipole term, we arrive at the master governing equation for themonopole in the tight-coupling limit:[

d2

dη2+HR

1 +R+ k2c2s

](Θgi

0 + ϕχ

)= k2c2sϕχ −

k2

3αχ , (7.73)

where the (background) sound speed of the baryon-photon fluid is

c2s :=δPbΓδρbΓ

=1

3

˙ρΓ

˙ρΓ + ˙ρb=

1

3

1

1 +R, R′ = HR . (7.74)

It is the simple harmonic oscillator for Θgi0 + ϕχ with the Hubble damping and the driving force.

On small scales, where we can ignore the Hubble expansion, the homogeneous solutions to the equation are

y1 = sin krs , y2 = cos krs , rs :=

∫ η

0

dη′ cs , (7.75)

where rs is the comoving sound horizon and we ignored the time derivative of the sound speed. Given the initial condition from theinflation, only the cosine mode y2 is excited. The particular solution can be found by first computing the Wronskian

W =

∣∣∣∣ y1 y2

y′1 y′2

∣∣∣∣ = −kcs , (7.76)

and by using the variation method as

Θgi0 + ϕχ =

(Θgi

0 + ϕχ

)0

cos krs +

∫ η

0

dηy2(η)y1(η)− y1(η)y2(η)

WF(η)

=(

Θgi0 + ϕχ

)0

cos krs +k√3

∫ η

0

dη (ϕχ − αχ) sin k(rs − rs) , (7.77)

77


where we approximated cs ≈ 1/√

3 and the driving force as

F := k2c2sϕχ −k2

3αχ ≈

k2

3(ϕχ − αχ) . (7.78)

The temperature fluctuation (plus the potential) is a simple oscillation (modulo correction due to the integration) with the frequencyset by the sound horizon at each epoch. The solution for the dipole can be readily obtained as

Θgi1 = −3

k

(Θgi

0 + ϕχ

)′=√

3(

Θgi0 + ϕχ

)0

sin krs − k∫ η

0

dη (ϕχ − αχ) cos k(rs − rs) , (7.79)

and it is out of phase with the monopole.

7.4.2 Diffusion DampingTo a good approximation, the baryon-photon fluid is indeed a fluid. However, this tight-coupling breaks down on small scales, wherephotons simply diffuse out without scattering off the electrons. So the fluctuations in the baryon-photon fluid are damped, and it iscalled the diffusion damping. This scale is closely related to the photon mean-free path, and on this scale our fluid equations are notvalid.

In the tight coupling limit, the baryon-photon fluid oscillates with ω = kcs, so we look for a small deviation that describes thediffusion process as

vbχ ∼ exp

[i

∫dη ω

], ω := ω + δω , ω2 = k2c2s , (7.80)

where the background frequency is obtained in the tight coupling limit (Γ → ∞). On small scales, we can simplify the photonBoltzmann equations by ignoring the gravitational potential fluctuations and again neglecting the higher multipoles (l ≥ 3) andpolarization:

0 = Θgi′0 +

k

3Θgi

1 , 0 = Θgi′1 − kΘgi

0 +2

5kΘgi

2 + Γ(

Θgi1 − kvbχ

), 0 = Θgi′

2 −2

3kΘgi

1 −9

10ΓΘgi

2 , (7.81)

so that we derive the relation among the multipole moments

Θgi0 =

i

3

k

ωΘgi

1 , Θgi2 ' −

20

27

k

ΓΘgi

1 , (7.82)

where we ignored Θgi′2 . Using the baryon velocity equation, we express the baryon velocity in terms of the photon velocity

vΓχ − vbχ =

R

Γ

(vbχ′+Hvbχ − αχ

)≈ R

ΓvΓχ

′= iω

R

Γvbχ → ∴ vbχ = vΓ

χ

(1 + iω

R

Γ

)−1

' vΓχ

(1− iωR

Γ− R2ω2

Γ2

), (7.83)

and the collisional term in the photon dipole equation becomes

Γ(

Θgi1 − kvbχ

)' Θgi

1

(iωR+

R2ω2

Γ

). (7.84)

By plugging these terms in the photon dipole equation, we derive the dispersion relation

ω2(1 +R)− 1

3k2 =

iω

Γ

(R2ω2 − 8

27k2

). (7.85)

Since the background ω2 = k2/3(1 +R) in the limit Γ→∞, we obtain

2

3k2δω =

iω2

Γ

(R2ω2 − 8

27k2

)→ ∴ δω =

ik2

6Γ(1 +R)

(R2

1 +R+

8

9

). (7.86)

Therefore, the photon propagation is now described by

exp

[i

∫dη ω

]= exp

[ik

∫dη cs

]exp

[−k2/k2

d

], (7.87)

and the diffusion scale is

1/k2d ≡

∫dη

[k2

6Γ(1 +R)

(R2

1 +R+

8

9

)]. (7.88)

78


7.4.3 Free Streaming: Line-of-Sight IntegrationHere we derive a formal integral solution by performing the line-of-sight integration. The Boltzmann equation (7.52) for photons is

d


′ − Γ

[Θ(n)− n · vb −Θ0 +

1

2L2(µ)

(1

5Θ2 +

1

5Θp

2 + Θp0

)]. (7.89)

and noting that the derivative along the path in Fourier space becomes

d

dηΘgi + ΓΘgi = Θ′gi + (ikµk − τ ′) Θgi = e−ikµkη+τ(η) d

dη

(Θgi e

ikµkη−τ(η)), (7.90)

the Boltzmann equation can be re-arranged and integrated to yield the line-of-sight integral solution

Θgi =

∫ η0

0

dη e−ikµk(η0−η)−τ(η)

[−ikµkαχ − ϕ′χ + τ ′

(ikµkv

bχ −Θgi

0 +1

2L2(µk)Π

)], (7.91)

where Θgi at the initial time is neglected due to large optical depth and we defined

τ(0) =∞ , τ(η0) = 0 , Π :=1

5Θ2 +

1

5Θp

2 + Θp0 . (7.92)

By replacing the angular dependence with the derivative

ikµk →d

dη, µ2

k → −1

k2

d2

dη2, (7.93)

the solution can be further simplified as

Θgi =

∫ η0

0

dη

[−e−τ

(ϕ′χ + τ ′Θgi

0 +1

4τ ′Π

)− e−τ

(αχ − τ ′vbχ

) ddη− τ ′e−τ 3Π

4k2

d2

dη2

]e−ikµk(η0−η) . (7.94)

Expanding the exponential and performing the multipole decomposition in Eq. (7.23) on both sides, we derive

Θgil

2l + 1=

∫ η0

0

dη

[− e−τ


0 +1

4τ ′Π

)− e−τ



4k2

d2

dη2

]jl[k(η0 − η)]

≈∫ η0

0

dη

[− e−τ


0

)+

d

dη

(e−ταχ

)+ e−ττ ′vbχ

d

dη

]jl[k(η0 − η)] , (7.95)

where we ignored the polarization for the moment and integrated by part for the second term in the square bracket. By defining thevisibility

g(η) := −τ ′e−τ ≡ Γe−τ , (7.96)

the integral solution can be rearranged as

Θgi

2l + 1=

∫ η0

0

dη

[g(

Θgi0 + αχ

)jl(x)− g kvbχj′l(x) + e−τ (αχ − ϕχ)

′jl(x)

], x := k(η0 − η) . (7.97)

Sine the visibility is close to a sharp Dirac delta function at the recombination time

g(η) ' δD(η − η?) , (7.98)

the temperature anisotropies are

Θgi

2l + 1≈(

Θgi0 + αχ

)?jl[k(η0 − η?)]− kvbχ? j

′l [k(η0 − η?)] , (7.99)

where we ignored the time evolution of the potential term with the exponential damping. The observed temperature anisotropies areessentially the “monopole” and the “dipole” of the baryon-photon fluid at the recombination epoch η?, free-streaming to the observerafter the recombination.

On large scales, we can derive the analytic solution for the observed temperature anisotropies. At kη 1, the Boltzmann equationyields

0 = Θgi′0 + ϕ′χ , ∴ Θgi

0 (k, η) = −ϕχ(k, η) + C(k) ≈ −ϕχ(k, η) +3

2ϕχ(k, 0) , (7.100)

79


where the integral constant is fixed by the initial condition. The comoving-gauge curvature perturbation is conserved on large scalesall the time, while the conformal Newtonian gauge curvature transitions its value from rde to mde. Using Eq. (??), we derive

ϕv =5 + 3w

3(1 + w)ϕχ =

3

2ϕχ(0) =

5

3ϕχ(η?) , ∴ ϕχ(0) =

10

9ϕχ(η?) , Θgi

0 (η?) =2

3ϕχ(η?) . (7.101)

The same calculation can be done for the matter density on large scales:

0 = δ′ + 3ϕ′χ , ∴ δ(η?) = −3ϕχ(η?) +9

2ϕχ(0) = 2ϕχ(η?) ,

(Θgi

0 + αχ

)?

= −1

3ϕχ(η?) = −1

6δ(η?) . (7.102)

At the recombination, the overdense region with δ > 0 corresponds to the hotter spot Θ0 > 0, but the observed temperature today iscolder due to the energy loss by the gravitational redshift from the overdense region. Furthermore, given the level that the temperatureanisotropies are ∼ 10−5, the density growth from the recombination epoch z ∼ 1100 will lead only to δ ∼ 10−2, unless it is furtherboosted by the nonlinear growth of dark matter prior to the recombination epoch.

The full line-of-sight integral solution is

Θgil

2l + 1=

∫ η0

0

dη

[− e−τ


0 +1

4τ ′Π

)− e−τ



4k2

d2

dη2

]jl[k(η0 − η)]

=

∫ η0

0

dη

[g

(Θgi

0 + αχ + vb′χ +Π

4+

3Π′′

4k2

)+ e−τ (αχ − ϕχ)

′+ g′

(vbχ +

3Π′

2k2

)+

3g′′Π

4k2

]. (7.103)

7.4.4 CMB Power SpectrumFinally, we need to connect our theoretical predictions to the observation. The observed CMB temperature can be harmonicallydecomposed as

Θ(n) :=∑lm

almYlm(n) , alm ≡∫d2n Y ∗lm(n)Θ(n) , (7.104)

and the observed CMB power spectrum can be obtained as

Cl =1

2l + 1

∑m

|alm|2 . (7.105)

So far, our calculation has been performed by assuming k//z. In a given coordinate, we have k = (θk, φk) and n = (θ, φ), so therelation to the above expressions in a system with k//z is obtained by rotating n into n′ with Rn ≡ Rk, where the above expressionsare indeed with n′, not with n. The observed temperature fluctuation is therefore,

Θ(n) =

∫d3k

(2π)3

∑l

Θl(−i)l√

4π

2l + 1D(Rn)Yl0(n′) =

∫d3k

(2π)3

∑l

Θl(−i)l√

4π

2l + 1

∑m′

Dl∗0m′Ylm′(n) , (7.106)

and the angular multipole and the power spectrum are then

aLM =

∫d3k

(2π)3ΘL(−i)L

√4π

2L+ 1DL∗0M , |m| ≤ 2 , (7.107)

CL =⟨|aLM |2

⟩=

4π

2L+ 1

∫d ln k ∆2

ϕ(k) |ΘL|2 ×(∫

dΩk4π

∣∣DL0M (Rk)∣∣2) = 4π

∫d ln k ∆2

ϕ(k)|ΘL|2

(2L+ 1)2,

where Dlmm′ is the Wigner matrix. Compare to the expressions in literature, where the rotation is neglected:

alm ≡∫

d3k

(2π)3alm(k) , alm(k) = (−i)l

√4π

2l + 1Θl , (7.108)

so that the resulting power spectrum Cl is identical.We can derive a simple approximation to the observed power spectrum, or the Sachs-Wolfe plateau on large scales by assuming

an instantaneous recombination and considering only the monopole contribution:

Θgi

2l + 1≈(

Θgi0 + αχ

)?jl[k(η0 − η?)] ≈ −

1

3ϕχ(η?)jl(kη0) ,

∫ ∞0

dk

kj2l (k) =

1

2l(l + 1), (7.109)

and the power spectrum is then

CSWl ≈ 4π

∫d ln k ∆2

ϕ(k)j2l k(η0)

[Tϕχ(k, η∗)

3

]2

∝ 1

2l(l + 1), l(l + 1)CSW

l = constant , (7.110)

where the transfer function and the initial power spectrum ∆2ϕ are both constant at low k.

80

BibliographyP. J. E. Peebles, The large-scale structure of the universe (Princeton University Press, Princeton, 1980).

R. K. Sheth and G. Tormen, Mon. Not. R. Astron. Soc. 308, 119 (1999), astro-ph/9901122.

A. Jenkins, C. S. Frenk, S. D. M. White, J. M. Colberg, S. Cole, A. E. Evrard, H. M. P. Couchman, and N. Yoshida, Mon. Not. R.Astron. Soc. 321, 372 (2001), astro-ph/0005260.

S. Matarrese, L. Verde, and R. Jimenez, Astrophys. J. 541, 10 (2000), arXiv:astro-ph/0001366.

M. LoVerde, A. Miller, S. Shandera, and L. Verde, J. Cosmol. Astropart. Phys. 4, 14 (2008), 0711.4126.

M. LoVerde and K. M. Smith, J. Cosmol. Astropart. Phys. 8, 3 (2011), 1102.1439.

J. M. Bardeen, J. R. Bond, N. Kaiser, and A. S. Szalay, Astrophys. J. 304, 15 (1986).

N. Kaiser, Astrophys. J. Lett. 284, L9 (1984).

H. D. Politzer and M. B. Wise, Astrophys. J. Lett. 285, L1 (1984).

T. Giannantonio and C. Porciani, Phys. Rev. D 81, 063530 (2010), 0911.0017.

T. Matsubara, Phys. Rev. D 77, 063530 (2008), 0711.2521.

K. B. Fisher, Astrophys. J. 448, 494 (1995), astro-ph/9412081.

R. Scoccimarro, Phys. Rev. D 70, 083007 (2004), arXiv:astro-ph/0407214.

S. Cole, K. B. Fisher, and D. H. Weinberg, Mon. Not. R. Astron. Soc. 275, 515 (1995), arXiv:9412062.

R. Arnowitt, S. Deser, and C. W. Misner, in Gravitation: An Introduction to Current Research, vol. 40 (Wiley, New York, 1962),arXiv:0405109.

J. M. Bardeen, Phys. Rev. D 22, 1882 (1980).

D. Seery and J. E. Lidsey, J. Cosmol. Astropart. Phys. 6, 003 (2005), astro-ph/0503692.

D. H. Lyth, Physical Review Letters 78, 1861 (1997), hep-ph/9606387.

T. Suyama and M. Yamaguchi, Phys. Rev. D 77, 023505 (2008), 0709.2545.

C. T. Byrnes, M. Sasaki, and D. Wands, Phys. Rev. D 74, 123519 (2006), arXiv:astro-ph/0611075.

P. Horava, Phys. Rev. D 79, 084008 (2009), 0901.3775.

J. Kristian and R. K. Sachs, Astrophys. J. 143, 379 (1966).

R. Sachs, Royal Society of London Proceedings Series A 264, 309 (1961).

astro-ph/9901122

astro-ph/0005260

arXiv:astro-ph/0001366

0711.4126

1102.1439

0911.0017

0711.2521

astro-ph/9412081


arXiv:9412062

arXiv:0405109

astro-ph/0503692

hep-ph/9606387

0709.2545


0901.3775

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