Conservation of the non-linear curvature perturbation

in generic single-field inflation

Yukawa Institute for Theoretical PhysicsAtsushi Naruko

In Collaboration with Misao Sasaki Based on : Class. Quantum Grav. 28 072001

The contents of talk1, curvature perturbation and non-Gaussianity

2, gradient expansion approach

3, conservation of non-linear curvature perturbation 3.1, G = 0 (for canonical, k-essential scalar) 3.2, G ≠ 0 (for Galileon scalar) in Einstein gravity

4, summary

curvature perturbation and

non-Gaussianity

Curvature perturbation• Curvature perturbation on a slicing is regarded as Newton potential, therefore it gives the initial condition for Cosmic Microwave Background (CMB).

• Through the observations of CMB, there is a possibility that we can subtract the information of primordial universe.

• Recently, a lot of attention is paid to “non-Gaussianity” in CMB as a new window for primordial universe.

Non-Gaussianity• From the observation by WMAP, we know that temperature fluctuations in CMB are scale invariant and are Gaussianly distributed.

• We parameterise the deviation from Gaussian using “fNL”

• The three point function is sensitive to this type NG.

Gradient expansion approach

non-linear cosmological perturbations

• In order to estimate the non-Gaussianity correctly, we have to make a analysis beyond linear order.

• There are mainly two approaches to nonlinear cosmological perturbations. 1. the standard perturbative approach. 2. the gradient expansion approach • In the gradient expansion, the field equations are expanded in powers of spatial gradients, therefore, it is applicable only to perturbations on superhorizon scales. → However, the full nonlinear effects are taken into account.

Outside Horizon• Thanks to the talk by Yamaguchi-san, Mizuno-san and De Felice-san, we know well about the perturbation in Galileon field at horizon exit.

• To give a theoretical prediction precisely , we need to solve the evolution of curvature perturbation outside Horizon.

• It have been known that curvature perturbation is conserved on superhorizon scales in the case of under .◯◯◯ ◎◎◎

inflation

Lyth, Malik and sasaki• Let’s consider a perfect fluid :

and write down the energy conservation law,

• We take uniform energy density slicing

if P is the function of ρ the rhs become a function of t

curvature per. is conserved

JCAP 0505:004,2005.

P = P (ρ) ?• From the study by LMS, we know that if P is a function of ρ, curvature perturbation is conserved on uniform energy density slice regardless of gravity theory.

• In the case of scalar field, whether / when this condition is satisfied is not so trivial because the relation between P and ρ is complicated.

canonical :

k-essence :

Galileon : very much complicated…

Our goal• We investigate the evolution of curvature perturbation on superhorizon scales in rather generic single-feild inflation using gradient expansion approach.

• We show the conservation of non-linear curvature perturbation

by using scalar field equation.

• We clarify under which circumstances curvature perturbation is conserved in the case of scalar field.

Conservation of

curvature perturbation

metric• We express the metric in the (3 + 1) form,

• We choose the spatial coordinate such that βi vanish and further decompose the spatial metric as

• Ψ is called as curvature perturbation because it corresponds to 3D-ricci scalar at leading order in gradient expansion.

• We define the expansion by the divergence of nμ , which is a vector orthogonal to t = const. surface.

t = const.

nμ

scalar field equation• We consider a rather generic scalar field.

canonical

K-essence, DBI

•　 After taking the variation of the above action, we obtain the scalar field equation

Galileon

• Neglecting spatial derivatives, the equation is rewritten as

G = 0

• If the system has evolved into attractor regime,

• We choose the uniform scalar slicing ,

• The conservation of curvature perturbation is shown without specifying gravity theory.

G ≠ 0• The lowest order equation in gradient expansion is

• There is a second derivative of Ψ (= derivative of K), → we cannot show the conservation W/O Einstein equation.

• Once we invoke Einstein gravity, we can replace K,τ with K.

Again, in the attractor stage, we can show the conservation of curvature perturbation on the uniform scalar field slice.

summary• Using the gradient expansion approach, we have studied the evolution of curvature perturbation on superhorizon scales in the case of single scalar field inflation.

• In the cases of canonical and k-essential scalar field (G = 0), we have shown the conservation of curvature perturbation without specifying gravity theory using scalar field equation.

• The condition for the conservation is whether the system has evolved into a attractor regime.

summary 2• In the case of galileon field, the conservation has not been shown without invoking gravity theory because there appear a second derivative of Ψ in the scalar field equation.

• Once we have used the Einstein equation, we can rewrite the second derivative by first derivative and we can show the conservation of curvature perturbation in the attractor stage.