1. N formalism for curvature perturbations from inflation
7 January 2008
Taitung Winter School
N formalism for curvature perturbations from inflation
Misao Sasaki
Yukawa Institute (YITP)
Kyoto University

2. 1. Introduction
2. Linear perturbation theory
metric perturbation & time slicing
dN formalism
3. Nonlinear extension on superhorizon scales
gradient expansion, conservation law
local Friedmann equation
4. Nonlinear DN formula
DN for slowroll inflation
diagrammatic method for DN
5. Summary

3. 1. Introduction
Standard (single-field, slowroll) inflation predicts scale-invariant Gaussian curvature perturbations.
CMB (WMAP) is consistent with the prediction.
Linear perturbation theory seems to be valid.

4. Need to know the dynamics on super-horizon scales
So, why bother doing more on theoretical models?
Because observational data does not exclude other models.
Tensor perturbations have not been detected yet.
T/S ~ ? or smaller?
Inflation may not be so simple.
multi-field, non-slowroll, extra-dim’s, string theory…
future CMB experiments may detect non-Gaussianity
Y=Ygauss+ fNLY2gauss+ ∙∙∙ ; |fNL| ≳ 5?
Pre-bigbang, ekpyrotic, bouncing,....?
Need to know the dynamics on super-horizon scales

5. 2. Linear perturbation theory
Bardeen ‘80, Mukhanov ‘81, Kodama & MS ‘84, ….
metric on a spatially flat background
(g0j=0 for simplicity)
S(t)
S(t+dt) dt
xi = const.
propertime along xi = const.:
curvature perturbation on S(t): R
expansion (Hubble parameter):

6. Choice of time-slicing
comoving slicing
matter-based slices
uniform density slicing
uniform Hubble slicing
geometrical slices
flat slicing
Newton (shear-free) slicing
comoving = uniform r = uniform H on superhorizon scales

7. dN formalism in linear theory
MS & Stewart ’96
e-folding number perturbation between S(t) and S(tfin):
dep only on
ini and fin t
S (tfin), R(tfin)
S0 (tfin)
dN(t,tfin)
N0 (t,tfin)
S0 (t)
S (t), R(t)
xi =const.
dN=O(k2) if both S(t) and S(tfin) are chosen to be ‘flat’ (R=0).

8. By definition, dN(t; tfin) is t-independent
Choose S(t) = flat (R=0) and S(tfin) = comoving:
S(tfin), RC(tfin)
S(t), R(t)=0
xi =const.
on superhorizon scales
curvature perturbation on comoving slice
(suffix ‘C’ for comoving)
The gauge-invariant variable ‘z’ used in the literature
is related to RC as z = -RC or z = RC on superhorizon scales
By definition, dN(t; tfin) is t-independent

9. Example: slow-roll inflation
single-field inflation, no extra degree of freedom
RC becomes constant soon after horizon-crossing (t=th):
log L
RC = const.
L=H-1
inflation
log a
t=tfin
t=th

10. Also dN = H(th) dtF→C , where dtF→C is the time difference
between the comoving and flat slices at t=th.
SC(th) : comoving
df=0, R=RC
dtF→C
R=0, df=dfF
SF(th) : flat
··· dN formula
Starobinsky ‘85
Only the knowledge of the background evolution
is necessary to calculate RC(tfin) .

11. d N for a multi-component scalar: (for slowroll-type inflation)
MS & Stewart ’96
N.B. RC (=z) is no longer constant in time:
··· time varying even on
superhorizon scales
Further extension to non-slowroll case is possible, if
general slow-roll condition is satisfied at horizon-crossing.
Lee, MS, Stewart, Tanaka & Yokoyama ‘05

12. 3. Nonlinear extension Field equations reduce to ODE’s
On superhorizon scales, gradient expansion is valid:
Belinski et al. ’70, Tomita ’72, Salopek & Bond ’90, …
This is a consequence of causality:
light cone
L »H-1
H-1
At lowest order, no signal propagates in spatial directions.
Field equations reduce to ODE’s

13. metric on superhorizon scales
gradient expansion:
metric:
the only non-trivial assumption
contains GW (~ tensor) modes
e.g., choose y (t* ,0) = 0
fiducial `background’

14. um nm Energy momentum tensor: assumption: um – nm = O (e)
(absence of vorticity mode) nm um
t=const.
Local Hubble parameter:
At leading order, local Hubble parameter on any slicing is equivalent to expansion rate of matter flow.
So, hereafter, we redefine H to be ~

15. Local Friedmann equation
xi : comoving (Lagrangean) coordinates.
dt = N dt : proper time along matter flow
exactly the same as the background equations.
“separate universe”
uniform r slice = uniform Hubble slice = comoving slice
as in the case of linear theory
no modifications/backreaction due to super-Hubble
perturbations.
cf. Hirata & Seljak ‘05
Noh & Hwang ‘05

16. 4. Nonlinear DN formula energy conservation:
(applicable to each independent matter component)
e-folding number:
where xi=const. is a comoving worldline.
This definition applies to any choice of time-slicing.
where

17. DN - formula
Lyth & Wands ‘03, Malik, Lyth & MS ‘04,
Lyth & Rodriguez ‘05, Langlois & Vernizzi ‘05
Let us take slicing such that S(t) is flat at t = t1 [ SF (t1) ]
and uniform density/uniform H/comoving at t = t2 [ SC (t1) ] :
( ‘flat’ slice: S (t) on which y = 0 ↔ ea = a(t) )
SC(t2) : uniform density
r (t2)=const.
N (t2,t1;xi) 
SC(t1) : uniform density
r (t1)=const.
DNF
y (t1)=0
SF (t1) : flat

18. Then
suffix C for comoving/uniform r/uniform H
where DNF is equal to e-folding number from SF(t1) to SC(t1):
For slow-roll inflation in linear theory, this reduces to

19. Conserved nonlinear curvature perturbation
Lyth & Wands ’03, Rigopoulos & Shellard ’03, ...
For adiabatic case (P=P(r) ,or single-field slow-roll inflation),
···slice-independent
Lyth, Malik & MS ‘04
non-linear generalization of ‘gauge’-invariant quantity z or Rc
y and r can be evaluated on any time slice
applicable to each decoupled matter component

20. Example: curvaton model
Lyth & Wands ’02
Moroi & Takahashi ‘02
2-field model: inflaton (f) + curvaton (c)
During inflation f dominates.
After inflation, c begins to dominate (if it does not decay).
rf=  a-4 and   a-3, hence  /  a    t
final curvature pert amplitude depends on when c decays.

21. On uniform total density slices, y = z
Before curvaton decay
On uniform total density slices, y = z
nonlinear version of
With sudden decay approx, final curvature pert amp z
is determined by
MS, Valiviita & Wands ‘06
Wc : density fraction of c at the moment of its decay

22. DN for ‘slowroll’ inflation
MS & Tanaka ’98, Lyth & Rodriguez ‘05
In slowroll inflation, all decaying mode solutions of the
(multi-component) inflaton field f die out.
If the value of f determines H uniquely (such as in the
slowroll case) when the scale of our interest leaves the
horizon, N is only a function of f , no matter how
complicated the subsequent evolution would be.
Nonlinear DN for multi-component inflation :
where df =dfF (on flat slice) at horizon-crossing.
(dfF may contain non-gaussianity from subhorizon interactions)
cf. Weinberg ’05, ...

23. Diagrammatic method for nonlinear D N
Byrnes, Koyama, MS & Wands ‘07
‘basic’ 2-pt function:
field space metric
df is assumed to be Gaussian
for non-Gaussian df, there will be basic n-pt functions
connected n-pt function of z:
2-pt function x y A 2! 1 x y A B 3! 1 x y A BC + +
+ ···

24. + + + ··· 3-pt function x y z A B x y z A B C 2! 1 + perm. 2! x x A 2!

25. 8. Summary
Superhorizon scale perturbations can never affect local (horizon-size) dynamics, hence never cause backreaction.
nonlinearity on superhorizon scales are always local.
However, nonlocal nonlinearity (non-Gaussianity) may
appear due to quantum interactions on subhorizon scales.
cf. Weinberg ‘06
There exists a nonlinear generalization of d N formula which is useful in evaluating non-Gaussianity from inflation.
diagrammatic method can by systematically applied.