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

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

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.

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 ~ 0.2 - 0.3? 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

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):

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

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).

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

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

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) .

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

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

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’

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 ~

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

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

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

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

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

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.

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

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, ...

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 + + + ···

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

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.