parameterizing dark sector perturbations via equations of

Parameterizing dark sector perturbations via equations of state

Jonathan A. Pearson

Centre for Particle Theory, Durham University

With Richard Battye (Manchester)

Adam Moss (Nottingham)

1306.11751301.50421205.3611

1203.0398

(short 4pg letter)

(application to massive gravity)

(conference proceedings)

(initial large paper)

COSMO, Cambridge, Sept 2013

Introduction

Theories

Phenomenological construction

Fundamental physics

Gµ⌫ = 8⇡GTµ⌫ + Uµ⌫

Introduction

Theories


Fundamental physics

Predictions for what such a universe will

look like

... distances, abundance of structure, CMB, lensing...

Observations

case

-by-

case

Gµ⌫ = 8⇡GTµ⌫ + Uµ⌫

Introduction

Theories


Fundamental physics


look like


PhenomenologicalFramework

Characteristic observational signatures of broad classes

of theories

Observations

case

-by-

case

Gµ⌫ = 8⇡GTµ⌫ + Uµ⌫

Introduction

Theories


Fundamental physics


look like




of theories

Observations

case

-by-

case

Imagine r-ns plane for inflation...... but for dark

energy

Gµ⌫ = 8⇡GTµ⌫ + Uµ⌫

Introduction

Theories


Fundamental physics


look like




of theories

Observations

case

-by-

case

Imagine r-ns plane for inflation...... but for dark

energy

Gµ⌫ = 8⇡GTµ⌫ + Uµ⌫

Use observational probes to uncover properties of dark sector theory

Field content

Symmetries & principles

Field content


Complete modification to linearized field equations

Field content


Complete modification to linearized field equations

c.f. particle physicsbispectrum

Theoretical generality

Observational feasibility

Parameterization must realise tension

What can current & future data sets actually do?

Cosmological perturbations

How do we parameterize for evolution of perturbations of the dark sector?

Dark sector has perturbations & will affect growth of structure, CMB lensing, galaxy weak lensing ...

These are important probes used to observationally distinguish gravitational theories

Construct equations of state for perturbations

Well defined (minimal) set of physical assumptions used to obtain distinct way

field equations are modified

Physically meaningful parameterization of perturbations

�Gµ⌫ = 8⇡G�Tµ⌫ + �Uµ⌫

�(rµUµ⌫) =

0

Would like expressions of the form

... these would close the set of perturbation equations.... equations of state

What we are doing is precisely equivalent to P = wρ

�P = A� +B✓ + Ch+ · · · ⇧ = X� + Y ✓ + Z⌘ + · · ·h & η synchronous gauge

metric perturbations


Would like expressions of the form

... these would close the set of perturbation equations.

QUESTIONS: ‣Are there generic forms of these expressions for wide

classes of theories? ‣ Is there a specific form for given field content ...?

... equations of stateWhat we are doing is precisely equivalent to P = wρ

�P = A� +B✓ + Ch+ · · · ⇧ = X� + Y ✓ + Z⌘ + · · ·h & η synchronous gauge

metric perturbations


Perturbed fluid variables

“fundamental” field content of dark sector

�Gµ⌫ = 8⇡G�Tµ⌫ + �Uµ⌫

w� ⌘✓�P

�⇢� w

◆�



Phenomenologically “active” variables

�Gµ⌫ = 8⇡G�Tµ⌫ + �Uµ⌫

w� ⌘✓�P

�⇢� w

◆�



Phenomenologically “active” variables

Free variables

�Gµ⌫ = 8⇡G�Tµ⌫ + �Uµ⌫

w� ⌘✓�P

�⇢� w

◆�

Example

L = L(�, @µ�, @µ@⌫�, gµ⌫ , @↵gµ⌫),

Field content:

w� = B1� +B2✓ +B3h+B4h

B2 = B(0)2 (t) + k2B(1)

2 (t)B1 = B1(t) B3 = B3(t) B4 = B4(t)

Perturbed field equations close under equation of state

where

(this still needs to be made gauge invariant)

linear in second order field equationsSO(1,3) reparameterization invariance

@↵gµ⌫ (for simplicity)

Example

L = L(�, @µ�, @µ@⌫�, gµ⌫ , @↵gµ⌫),

Field content:

w� = B1� +B2✓ +B3h+B4h

B2 = B(0)2 (t) + k2B(1)

2 (t)B1 = B1(t) B3 = B3(t) B4 = B4(t)

Perturbed field equations close under equation of state

where

Gauge invariant entropy perturbation:

w� = (↵� w)

� � 3H(1 + w)�1✓

S � 3H(1 + w)�2

2k2 � 6(H�H2)h

+3H(1 + w)(1� �1 � �2)

6H+ 6H3 � 18HH+ 2k2H h

�.

(this still needs to be made gauge invariant)

linear in second order field equationsSO(1,3) reparameterization invariance

@↵gµ⌫ (for simplicity)

L(gµ⌫ ,�, @µ�)

L(gµ⌫ ,�, @µ�)

Metric only

L(gµ⌫)No derivatives

(Massive gravity)

L(gµ⌫ ,�, @µ�)

Metric only


(Massive gravity)

w⇧ = 32 (c

2s � w)

� � 3(1 + w)⌘

�w� = 0

SO(3, 1)

SO(0, 1)

w⇧ = 0

w� =

✓1

1 + ✏� w

◆� � 3H(1 + w)

✓w(1 + ✏) + 1

w(1 + ✏)� 1

◆✓

�

SO(3, 0)

Elastic dark energy [Battye & Moss 2007]Lorentz violating massive gravity [Rubakov & Tinyakov 2008]

Ghost condensate [Arkhani-Hamed et al 2004]

w = -1 & no perturbations

L(gµ⌫ ,�, @µ�)

Metric only


(Massive gravity)

w⇧ = 32 (c

2s � w)

� � 3(1 + w)⌘

�w� = 0

SO(3, 1)

SO(0, 1)

w⇧ = 0

w� =

✓1

1 + ✏� w

◆� � 3H(1 + w)

✓w(1 + ✏) + 1

w(1 + ✏)� 1

◆✓

�

SO(3, 0)



w = -1 & no perturbationsfull diff.inv

time diff.inv

space diff.inv

L(gµ⌫ ,�, @µ�)

L(�,X )

w⇧ = 0

Lorentzinvariance

Gauge invariance

w� = (c2s � w)

� � 3H(1 + w)✓

�

[Weller & Lewis 2003; Bean & Dore 2004]

[Christopherson & Malik 2009; ...]

c2s = 1

Canonical scalar field theory

L = X � V (�)

Metric only


(Massive gravity)

w⇧ = 32 (c

2s � w)

� � 3(1 + w)⌘

�w� = 0

SO(3, 1)

SO(0, 1)

w⇧ = 0

w� =

✓1

1 + ✏� w

◆� � 3H(1 + w)

✓w(1 + ✏) + 1

w(1 + ✏)� 1

◆✓

�

SO(3, 0)



w = -1 & no perturbationsfull diff.inv

time diff.inv

space diff.inv

L(gµ⌫ ,�, @µ�)

L(�,X )

w⇧ = 0

Lorentzinvariance

Gauge invariance

w� = (c2s � w)

� � 3H(1 + w)✓

�

[Weller & Lewis 2003; Bean & Dore 2004]

[Christopherson & Malik 2009; ...]

c2s = 1

Canonical scalar field theory

L = X � V (�)

Metric only


(Massive gravity)

w⇧ = 32 (c

2s � w)

� � 3(1 + w)⌘

�w� = 0

SO(3, 1)

SO(0, 1)

w⇧ = 0

w� =

✓1

1 + ✏� w

◆� � 3H(1 + w)

✓w(1 + ✏) + 1

w(1 + ✏)� 1

◆✓

�

SO(3, 0)



w = -1 & no perturbations

Coefficients in effective action combine to construct these!

full diff.inv

time diff.inv

space diff.inv

Galaxy weak lensing ALONE can rule out some theory space

-0.4-0.3-0.2-0.1

0 0.1 0.2 0.3

50 100 150 200 250 300 350 400

6j+ /j

+

e

-0.2-0.1

0 0.1 0.2 0.3 0.4 0.5 0.6

50 100

6 C

lTT/C

lTT

l

(1,1)(1,0)(0,0)(0,4)

-0.25-0.2

-0.15-0.1

-0.05 0

0.05 0.1

0.15

0 400 800 1200

6C

ldd/C

ldd

l

Fractional differences in spectrarelative to quintessence

fix α, dial (β1, β2)

Weak galaxy lensing

CMB lensing

AKA elastic dark energy

�4.0 �3.2 �2.4 �1.6 �0.8 0.0

log10 c2s

�1.2

�1.1

�1.0

�0.9

�0.8w

Planck+WP+CMB Lensing +BAOPlanck+WP+CFHTLS+BAO

TDI L(g)

Time-diff invariant theoriesL(gµ⌫)

Constraints on scalar field models

Planck+WP+CMB Lensing+BAO

Planck+WP+CFHTLS+BAO

0.0 0.4 0.8 1.2 1.6

�1

�1.35

�1.20

�1.05

�0.90

w

0 3 6 9 12 15

�2

�1.35

�1.20

�1.05

�0.90

w

�4.0 �3.2 �2.4 �1.6 �0.8 0.0

log10 ↵

�1.35

�1.20

�1.05

�0.90

w

Planck+WP+CMB Lensing +BAOPlanck+WP+CFHTLS+BAO

GSF

Recap

Isolated distinctive way in which field content & symmetry

modify perturbed gravitational field equations

via equations of state

Physically meaningful

On good side of tension between theoretical generality & observational feasibility

Know all scale dependance

Everything can be traced back to interaction terms in Lagrangian for perturbations

Summary

‣ Perturbations in the dark sector need parameterizing

‣ Current methods are not particularly “physical”

Summary



‣ We construct Lagrangian for perturbations after picking a field content

‣ Derive general perturbed dark energy momentum tensor

‣ Identify maximal amount of freedom

Summary






‣ Close fluid equations via equations of state for dark sector perturbations

‣ Derive for different field contents

‣ Compute characteristic signatures... constrain parameters with data

Summary








‣ Compute characteristic signatures... constrain parameters with data

Imagine r-ns plane for inflation...... but for dark energy

Summary








‣ Compute characteristic signatures... constrain parameters with data Lots more still to do...

Really easy to implement in CAMB ... <10lines!

Imagine r-ns plane for inflation...... but for dark energy

Coupled case... Horndeski... vector fields...

Parameterizing dark sector perturbations via equations of state

Jonathan A. Pearson

Centre for Particle Theory, Durham University

With Richard Battye (Manchester)

Adam Moss (Nottingham)

1306.11751301.50421205.3611

1203.0398

(short 4pg letter)

(application to massive gravity)

(conference proceedings)

(initial large paper)

COSMO, Cambridge, Sept 2013

parameterizing dark sector perturbations via equations of

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