1.   N formalism for curvature perturbations from inflation Yukawa Institute (YITP) Kyoto University Kyoto University Misao Sasaki DESY 27/09/2006

2. 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.

3. Tensor perturbations have not been detected yet.  N-formalism on super-horizon scales  gauss + f NL  2 gauss + ∙∙∙ ; |f NL | ≳ 5? So, why bother doing more on inflation? So, why bother doing more on inflation? PLANCK, CMBpol, … may detect non-Gaussianity PLANCK, CMBpol, … may detect non-Gaussianity Because observational data do not exclude other possibilities. After all, inflation may not be so simple. After all, inflation may not be so simple. multi-field, non-slowroll, extra-dim’s, string theory… multi-field, non-slowroll, extra-dim’s, string theory… T / S (=r) ~ 0.2 - 0.3? or else?

4. 2. Linear perturbation theory propertime along x i = const.: propertime along x i = const.: curvature perturbation on  ( t ): curvature perturbation on  ( t ):  x i = const. (t)(t)(t)(t)  ( t+dt ) dddd expansion (Hubble parameter): expansion (Hubble parameter): metric on a spatially flat background metric on a spatially flat background Bardeen ‘80, Mukhanov ‘81, Kodama & MS ‘84, …. ( g 0 j =0 for simplicity)

5.  N formalism in linear theory 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).

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

7. Example: single-field slowroll inflation Example: single-field slowroll inflation single-field, no extra degree of freedom single-field, 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

8. 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 Only the knowledge of the background evolution is necessary to calculate  C ( t fin ). is necessary to calculate  C ( t fin ). Starobinsky ‘85

9.  N for a multi-field inflation  N for a multi-field inflation MS & Stewart ’96 N.B.  C (=  ) is no longer constant in time: ··· time varying even on superhorizon scales superhorizon scales power spectrum:

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

11. 3. Nonlinear 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.

12. metric on superhorizon scales 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:

13. Local Friedmann equation x i : comoving (Lagrangean) coordinates. comoving slice = uniform  slice = uniform Hubble slice d  =  dt : proper time along matter flow same as linear theory exactly the same as the background equations. exactly the same as the background equations. “separate universe” where

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

15.  N - formula Choose slicing s.t. it is flat at t = t 1 [  F ( t 1 ) ] ( ‘flat’ slice:   ( t ) on which  = 0 ↔ e  = a ( t ) )  F ( t 1 ) : flat  C ( t 2 ) : uniform density  ( t 2 )=const.   ( t 1 )=0  ( t 1 )=const.  C ( t 1 ) : uniform density N (t2,t1;xi)N (t2,t1;xi) NFNFNFNF Lyth & Wands ‘03, Malik, Lyth & MS ‘04, Lyth & Rodriguez ‘05, Langlois & Vernizzi ’05,... and uniform density ( comoving) at t = t 2 [  C ( t 1 ) ] :

16.  F  F ( t 1 )  C ( t 1 ):  F is equal to e -folding number from  F ( t 1 ) to  C ( t 1 ): For slow-roll inflation in linear theory, this reduces to suffix C for comoving (=uniform density) (=uniform density)

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

18.  N for ‘slowroll’ inflation  N for ‘slowroll’ inflation Nonlinear  N for multi-component inflation : MS & Tanaka ’98, Lyth & Rodriguez ‘05 If  is slow rolling when the scale of our interest leaves If  is slow rolling when the scale of our interest leaves the horizon, N is only a function of , no matter how the horizon, N is only a function of , no matter how complicated the subsequent evolution would be. complicated the subsequent evolution would be. where  =  F (on flat slice) at horizon-crossing. (  F may contain non-gaussianity from subhorizon interactions) cf. Maldacena ‘03, Weinberg ’05,...

19. Example: non-gaussianity from curvaton Example: non-gaussianity from curvaton   =    a -4 and    a -3, hence   /    a    t 2-field model: inflaton (  ) + curvaton (  ) After inflation,  begins to dominate (if it does not decay). After inflation,  begins to dominate (if it does not decay). During inflation  dominates. During inflation  dominates. final curvature pert amplitude depends on when  decays. final curvature pert amplitude depends on when  decays. Lyth & Wands ’02, Moroi & Takahashi ‘02 MS, Valiviita & Wands ‘06

20. Before curvaton decay Before curvaton decay With sudden decay approx, final curvature pert amp  With sudden decay approx, final curvature pert amp  is determined by is determined by On uniform total density slices,  C  On uniform total density slices,  C    : density fraction of  at the moment of its decay

21. Diagrammatic method for nonlinear   N Diagrammatic method for nonlinear   N connected n -pt function of  : connected n -pt function of  : ‘basic’ 2-pt function: field space metric Byrnes, MS & Wands in prep. 2-pt function  is assumed to be Gaussian for non-Gaussian , there will be basic (nontrivial) n -pt functions xyA A + ··· x y A B B A + 2! 1 3! 1 + x y A BC BC A

22. 3-pt function (~ bi-spectrum) + perm. xy z A B B C A C + + x y z A B B A x y z A B C A + ··· + perm. 2! x 2! 1 2! 1 The above can be generalized to n -pt function Byrnes, MS & Wands in prep. f NL = O( N A N AB N B /N A N A )

23. IR divergence problem IR divergence problemxy z A B B C A C Loop diagrams like in the r -pt function give rise to IR divergence in the ( r- 1)-spectrum if P(k)~k n  4 with n ≤ 1. eg, the above diagram gives Is this IR cutoff physically observable? (real space 3-pt fcn is perfectly regular if G 0 ( x ) is regular.) Boubekeur & Lyth ‘05 cutoff-dependent!

24. consider a loop diagram of 2-pt function: x y A B BASachs-Wolfe: The divergence disappears by excluding l 1 =0 & l 2 =0. conformal distance to LSS Wigner 3j-symbol Possible prescription for CMB Possible prescription for CMB (need justification)

25. 8. Summary Superhorizon scale perturbations can never affect local (horizon-size) dynamics, hence never cause backreaction. nonlinearity on superhorizon scales are always local. 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  N formula which is useful for evaluation of non-Gaussianity from inflation. diagrammatic method is developed. prescription to remove IR divergence from loop diagrams is given. ∙∙∙ need to be justified. bi-spectrum, tri-spectrum,∙∙∙