Download Conservation of the nonlinear curvature perturbation in generic

Conservation of the non-linear
curvature perturbation
in generic single-field inflation
Yukawa Institute for Theoretical Physics
Atsushi Naruko
In Collaboration with Misao Sasaki
Based on : Class. Quantum Grav. 28 072001
The contents of talk
1, 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.
inflation
• 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 ◎◎◎.
Lyth, Malik and sasaki
JCAP 0505:004,2005.
• 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
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.
nμ
t = const.
scalar field equation
• We consider a rather generic scalar field.
canonical
K-essence, DBI
Galileon
• After taking the variation of the above action, we obtain
the scalar field equation
G=0
• Neglecting spatial derivatives, the equation is rewritten as
• 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.