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d N formalism for curvature perturbations from inflation

DESY 27/09/2006. d N formalism for curvature perturbations from inflation. Yukawa Institute (YITP) Kyoto University. Misao Sasaki. 1. Introduction. Standard (single-field, slowroll) inflation predicts scale-invariant Gaussian curvature perturbations.

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d N formalism for curvature perturbations from inflation

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

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

  3. So, why bother doing more on inflation? Because observational data do not exclude other possibilities. Tensor perturbations have not been detected yet. T/S (=r) ~ 0.2 - 0.3? or else? After all, inflation may not be so simple. multi-field, non-slowroll, extra-dim’s, string theory… PLANCK, CMBpol, …may detect non-Gaussianity Y=Ygauss+ fNLY2gauss+ ∙∙∙ ; |fNL| ≳ 5? dN-formalism on super-horizon scales

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

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

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

  7. Example: single-field slowroll inflation • single-field, 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

  8. 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) .

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

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

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

  12. 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’

  13. Local Friedmann equation xi: comoving (Lagrangean) coordinates. dt = N dt: proper time along matter flow where • exactly the same as the background equations. “separate universe” • comoving slice=uniform r slice= uniform Hubbleslice same as linear theory

  14. 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

  15. Lyth & Wands ‘03, Malik, Lyth & MS ‘04, Lyth & Rodriguez ‘05, Langlois & Vernizzi ’05, ... Choose slicing s.t. it is flat at t = t1[ SF (t1) ] • DN - formula and uniform density (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) r (t1)=const. SC(t1) : uniform density DNF y (t1)=0 SF (t1) : flat

  16. suffix C for comoving (=uniform density) DNFis equal to e-folding number from SF(t1) to SC(t1): For slow-roll inflation in linear theory, this reduces to

  17. 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 quantityzorRc • yand rcan be evaluated on any time slice • applicable to each decoupled matter component

  18. DN for ‘slowroll’ inflation MS & Tanaka ’98, Lyth & Rodriguez ‘05 • If f is slow rolling 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. Maldacena ‘03, Weinberg ’05, ...

  19. Example: non-gaussianity from curvaton MS, Valiviita & Wands ‘06 2-field model: inflaton (f) + curvaton (c) Lyth & Wands ’02, Moroi & Takahashi ‘02 • 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.

  20. Before curvaton decay • On uniform total density slices, y =yC z • With sudden decay approx, final curvature pert amp z • is determined by Wc : density fraction of c at the moment of its decay

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

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

  23. x A C C A z y B B • IR divergence problem Loop diagrams like in the r-pt function give rise to IR divergence in the (r-1)-spectrum if P(k)~kn-4 with n≤ 1. Boubekeur & Lyth ‘05 eg, the above diagram gives cutoff-dependent! Is this IR cutoff physically observable? (real space 3-pt fcn is perfectly regular if G0(x) is regular.)

  24. A A x y B B (need justification) • Possible prescription for CMB Sachs-Wolfe: conformal distance to LSS consider a loop diagram of 2-pt function: Wigner 3j-symbol The divergence disappears by excluding l1=0 & l2=0.

  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 for evaluation of non-Gaussianity from inflation. diagrammatic method is developed. bi-spectrum, tri-spectrum,∙∙∙ prescription to remove IR divergence from loop diagrams is given. ∙∙∙ need to be justified.

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