Slide 1

N formalism for curvature perturbations from inflation Yukawa Institute (YITP) Kyoto University Kyoto University Misao Sasaki DESY 27/09/2006 Slide 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. Slide 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-dims, string theory multi-field, non-slowroll, extra-dims, string theory T / S (=r) ~ 0.2 - 0.3? or else? Slide 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 ) dddd 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) Slide 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). Slide 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 Slide 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 Slide 8 Also N = H ( t h ) t F C, where t FC 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 FC =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 Slide 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: Slide 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) Slide 11 3. Nonlinear extension This is a consequence of causality: Field equations reduce to ODEs 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. Slide 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: Slide 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 Slide 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 Slide 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) NFNFNFNF Lyth & Wands 03, Malik, Lyth & MS 04, Lyth & Rodriguez 05, Langlois & Vernizzi 05,... and uniform density ( comoving) at t = t 2 [ C ( t 1 ) ] : Slide 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) Slide 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 Slide 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,... Slide 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 Slide 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 Slide 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 Slide 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 ) Slide 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! Slide 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) Slide 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,