 N formalism for Superhorizon Perturbations and its Nonlinear Extension Misao Sasaki YITP, Kyoto University Based on MS & E.D. Stewart, PTP95(1996)71 [astro-ph/ ] MS & T. Tanaka, PTP99(1998)763 [gr-qc/ ] D. Lyth, K. Malik & MS, JCAP05(2005)004 [astro-ph/ ] H.C. Lee, MS, E.D. Stewart, T. Tanaka & S. Yokoyama [astro-ph/ ] A. Linde, V. Mukhanov & MS [astro-ph/ ]

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

However,…. PLANCK may detect non-Gaussianity PLANCK may detect non-Gaussianity Inflation may be non-standard Inflation may be non-standard multi-field, non-slowroll, extra-dim’s, … multi-field, non-slowroll, extra-dim’s, … Revisit the dynamics on super-horizon scales Backreaction from/to infrared modes ( H » k / a )? Backreaction from/to infrared modes ( H » k / a )? Bartolo, Kolb & Riotto ‘05, Sloth ‘05 Kolb, Matarrese & Riotto ‘05 Enhancement of tensor-to-scalar ratio? Enhancement of tensor-to-scalar ratio?

2. Cosmological perturbation theory propertime along x i = const.: propertime along x i = const.: curvature perturbation on  ( t ): curvature perturbation on  ( t ):  (t)(t)(t)(t)  ( t+dt ) x i = const. dddd traceless expansion (Hubble parameter): expansion (Hubble parameter): metric (on a spatially flat background)

comoving slicing comoving slicing flat slicing flat slicing Newton (shear-free) slicing Newton (shear-free) slicing uniform density slicing uniform density slicing uniform Hubble slicing uniform Hubble slicing Choice of gauge (time-slicing) matter-based gauge geometrical gauge comoving = uniform  = uniform H on superhorizon scales

Friedmann eq. holds indep’t of slicing, at leading order in  If  is time-indep’t, Friedmann eq. holds up through O(  2 ), local ‘Hubble parameter’ given by with local ‘curvature constant’ given by ‘local’ means ‘measured on scales of Hubble horizon size’ Separate universe approach local Friedmann eq. holds for adiabatic perterbations on comoving / uniform  / uniform H or flat slices.

 N formalism Starobinsky ’85, MS & Stewart ’96, …. e-folding number perturbation between  ( t ) and  ( t fin ): N 0 ( t, t fin )  N ( t, t fin )  ( t fin ),  ( t fin )   ( t fin )  (t) (t) (t) (t) x i =const.  ( t ),  ( t )  N =0 if both  ( t ) and  ( t fin ) are chosen to be ‘flat’ (  =0).

By definition,  N  t  t fin ) is t -independent. The gauge-invariant variable ‘  ’ popularly used in the liturature is related to  C as  = -  C  ( t ),  ( t )=0  (t fin ),  C (t fin ) x i =const. Choose  ( t ) = flat (  =0) and  ( t fin ) = comoving: (suffix ‘C’ for comoving) curvature perturbation on comoving slice

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

Also  N = H ( t h )  t F →C, where  t F→C is the time difference between the comoving and flat slices at t = t h.  F ( t h ) : flat  =0,  =  F  C (t h ) : comoving  t F→C  =0,  =  C ···  N formula No need to solve perturbation equations to calculate  C ( t fin ): No need to solve perturbation equations to calculate  C ( t fin ): Only the knowledge of the background evolution is necessary. Only the knowledge of the background evolution is necessary.

Extension to a multi-component scalar: Extension to a multi-component scalar: (for slow-roll, no isocurvature perturbation) (for slow-roll, no isocurvature perturbation) MS & Stewart ’96, MS & Tanaka ‘98 N.B.  C is no longer constant in time: ··· time varying even on superhorizon scales superhorizon scales Extension to non-slow-roll case is also possible, if general slow-roll condition is satisfied at horizon-crossing. Lee, MS, Stewart, Tanaka & Yokoyama ‘05

Tensor-to-scalar ratio scalar spectrum: scalar spectrum: tensor spectrum: tensor spectrum: tensor spectral index: tensor spectral index: MS & Stewart ‘96 ··· valid for any slow-roll models (‘=’ for a single inflaton model)

(a) Maybe possible in a non-slow-roll /non-Einstein model… (a) Maybe possible in a non-slow-roll /non-Einstein model… But the scalar spectrum must be almost scale-invariant. need both scale-invariance AND large P g /P s ! (b) Suppress P s by some post-inflationary perturbation? This does not work because Enhancement of the ratio P g /P s ? unless  N inf and  N post-inf are strongly correlated. Need an entirely new idea, if seriously want to increase P g /P s But this brings us back to the case (a)! Lyth ’05, Linde, Mukhanov & MS ‘05 Bartolo, Kolb & Riotto ’05, Sloth ‘05

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

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

At leading order, local Hubble parameter is independent of the time slicing,as in linear theory At leading order, local Hubble parameter is independent of the time slicing, as in linear theory nnnn uuuu t =const. assumption u  – n  = O (  ) Energy momentum tensor: Energy momentum tensor: local Hubble parameter: local Hubble parameter: (absence of vorticity mode)

Local Friedmann equation x i : comoving (Lagrangean) coordinates. exactly the same as the background equations. uniform  slice = uniform Hubble slice = comoving slice uniform  slice = uniform Hubble slice = comoving slice d  =  dt : proper time along fluid flow cf. Hirata & Seljak ‘05 Noh & Hwang ‘05 as in the case of linear theory no modifications/backreaction due to super-Hubble no modifications/backreaction due to super-Hubble perturbations. perturbations.

5. Nonlinear  N formula energy conservation: e -folding number: where x i =const. is a comoving worldline. This definition applies to any choice of time-slicing. where Lyth, Malik & MS ’04

 N - formula Let us take slicing such that  ( t ) is flat at t = t 1 [  F ( t 1 ) ] and uniform density/uniform H / comoving at t = t 2 : ( ‘flat’ slice:   ( t ) on which  = 0 ↔ e  = a ( t ) )  F ( t 1 ) : flat  ( t 2 ) : uniform density  ( t 2 )=const.   ( t 1 )=0  ( t 1 )=const.  ( t 1 ) : uniform density N (t2,t1;xi)N (t2,t1;xi) NFNFNFNF

 F  F ( t 1 )  ( t 1 ): where  F is the e -folding number from  F ( t 1 ) to  ( t 1 ): Then For slow-roll inflation In linear theory, this reduces to suffix C for comoving/uniform  /uniform H

For adiabatic ( p = p (  ) ) case, non-linear generalization of conserved ‘gauge’-invariant quantity  or  c ‘gauge’-invariant quantity  or  c  (  and  can be evaluated on any time slice) || Conservation of nonlinear  ···slice-independent

6. Summary Superhorizon scale perturbations can never affect local (horizon-size) dynamics, hence never cause backreaction. There exists a non-linear generalization of  which is conserved for an adiabatic perturbation on superhorizon scales. There exists a non-linear generalization of  N formula, which may be useful in evaluating non-Gaussianity from inflation.

It is impossible to suppress adiabatic perturbations once they are generated. Tensor/scalar ratio could be enhanced in a non-slow-roll/non-Einstein model, but no such model is known. About tensor-to-scalar ratio…