δ N δ N formalism for curvature perturbations formalism for curvature perturbations from inflation from inflation Misao Sasaki Misao Sasaki Yukawa Institute (YITP) Yukawa Institute (YITP) Kyoto University Kyoto University
1. Introduction 1. Introduction 2. Linear perturbation theory 2. Linear perturbation theory • metric perturbation & time slicing metric perturbation & time slicing • • δ δ N formalism N formalism • 3. Nonlinear extension on superhorizon scales 3. Nonlinear extension on superhorizon scales • gradient expansion, conservation law gradient expansion, conservation law • • local Friedmann equation local Friedmann equation • Δ N formula 4. Nonlinear Δ N formula 4. Nonlinear • Δ Δ N for slowroll inflation N for slowroll inflation • Δ N diagrammatic method for Δ • diagrammatic method for N • • IR divergence issue IR divergence issue • 5. Summary 5. Summary
1. Introduction 1. Introduction Standard (single- -field, slowroll) inflation predicts scale field, slowroll) inflation predicts scale- - Standard (single invariant Gaussian Gaussian curvature perturbations. curvature perturbations. invariant CMB (WMAP) is consistent with the prediction. Linear perturbation theory seems to be valid.
So, why bother doing more research on inflation? So, why bother doing more research on inflation? Because observational data does not exclude other models. Because observational data does not exclude other models. Tensor perturbations have not been detected yet. Tensor perturbations have not been detected yet. / S S ~ 0.2 ~ 0.2 - - 0.3? or smaller? 0.3? or smaller? T / T In fact, inflation may not be so simple. In fact, inflation may not be so simple. multi- -field, non field, non- -slowroll, extra slowroll, extra- -dim dim’ ’s, string theory s, string theory… … multi PLANCK, CMBpol, … … may detect may detect non non- -Gaussianity Gaussianity PLANCK, CMBpol, Ψ=Ψ Ψ gauss Ψ 2 Ψ= NL Ψ ∙∙∙ ; | ≳ 5? 5? + ∙∙∙ | ≳ 2 + f NL ; |f f NL gauss + f gauss + NL | gauss Nonlinear backreaction on superhorizon scales? Nonlinear backreaction on superhorizon scales? Re- -consider the dynamics on super consider the dynamics on super- -horizon scales horizon scales Re
2. Linear perturbation theory 2. Linear perturbation theory Bardeen ‘ Bardeen ‘80, Mukhanov 80, Mukhanov ‘ ‘81, Kodama & MS 81, Kodama & MS ‘ ‘84, 84, … …. . ( g =0 for simplicity) metric on a spatially flat background ( j =0 for simplicity) metric on a spatially flat background g 0 0 j ( ) ( ) R ⎡ ⎤ = − + + + δ + 2 2 2 i j 1 2 (1 2 ) ds A dt a H dx dx ⎣ ⎦ t ij ij ( ) Σ ( Σ = ∂ ∂ ( t+dt ) t+dt ) H E ij i j scalar τ d τ ( ) d Σ ( = Σ transverse-trace ss le ( t ) t ) H ij tensor x i i = const. = const. x τ = + • propertime along propertime along x i = const.: = const.: (1 ) • x i d A dt (3) 4 (3) Δ R = − Σ ( Ρ curvature perturbation on Σ R • curvature perturbation on ( t ): • t ): 2 a ⎡ ⎤ 1 (3) ( ) % R = − + ∂ + Δ • expansion (Hubble parameter): expansion (Hubble parameter): • 1 H H A E ⎢ ⎥ t ⎣ ⎦ 3
Choice of time- -slicing slicing Choice of time ( ) ( ) μ = φ φ • comoving slicing comoving slicing • 0 = for a scalar field T t i matter- -based slices based slices matter ( ) − ≡ ρ = ρ 0 • uniform density slicing uniform density slicing T t • 0 • uniform Hubble slicing uniform Hubble slicing • ⎡ ⎤ 1 (3) ( ) % R = ⇔ − + ∂ + Δ = 0 H H t H A E ⎢ ⎥ t ⎣ ⎦ 3 geometrical slices geometrical slices (3) (3) 4 R R = − Δ ⇔ = • flat slicing flat slicing • =0 0 R 2 a • Newton (shear Newton (shear- -free) slicing free) slicing • ⎡ ⎤ 1 ( ) 3 scalar ⎡ ⎤ ∂ ≡ ∂ ∂ − δ Δ ∂ = ⇔ ∂ = ⇔ = 0 0 0 H E E E ⎣ ⎦ ⎢ ⎥ t ij i j ij t t ⎣ ⎦ 3 traceless ρ = uniform ρ comoving = = uniform = uniform uniform H H on superhorizon scales on superhorizon scales comoving
δ N formalism in linear theory δ N formalism in linear theory MS & Stewart ’ ’96 96 MS & Stewart Σ ( Σ ( folding number perturbation between Σ ) and Σ e- -folding number perturbation between ( t ( t ): e t ) and t fin fin ): ) ( ( ) ∫ t % ∫ t δ ≡ τ − τ fin fin ; N t t H d H d fin t t background ⎡ ⎤ ( ) 1 (3) ( ) ( ) ∫ t R R R = ∂ + Δ = − + ε fin 2 E dt t t O ⎢ ⎥ t fin ⎣ ⎦ 3 t Σ ( Ρ ( Σ ), Ρ ( t ( t ) t fin fin ), t fin fin ) Σ 0 Σ ( t t fin ) 0 ( fin ) δ N δ ( t , t ) N 0 0 ( t , t fin fin ) N ( t , t ) N ( t , t fin fin ) Σ 0 Σ ( t ) 0 ( t ) Σ ( Ρ ( Σ ), Ρ ( t ( t ) t ), t ) i =const. x i =const. x δ N δ =0 if both Σ Σ ( ) and Σ Σ ( ( Ρ Ρ =0). ( t ( t ) are chosen to be ‘ ‘flat flat’ ’ ( =0). N =0 if both t ) and t fin fin ) are chosen to be
Σ ( Ρ =0) Σ ( Choose Σ ) = flat ( Ρ and Σ ( t =0) and ( t ) = comoving: Choose t ) = flat ( t fin fin ) = comoving: Σ ( Ρ C Σ ), Ρ ( t t fin ( t t fin ) fin ), C ( fin ) Σ ( Ρ ( Σ ), Ρ ( t )=0 ( t t ), t )=0 x i i =const. =const. x ( ) ( ) = R δ ; N t t t fin C fin curvature perturbation on comoving slice curvature perturbation on comoving slice (suffix ‘ ‘C C’ ’ for comoving) for comoving) (suffix ζ ’ ‘ ζ The gauge- -invariant variable invariant variable ‘ ’ used in the literature used in the literature The gauge Ρ C ζ = Ρ C ζ = Ρ C is related to Ρ as ζ - Ρ or ζ = Ρ is related to C as = - C or C δ N ( t ; t By definition, δ N ( t ; ) is t -independent independent By definition, t fin fin ) is t -
Example: slow- -roll inflation roll inflation Example: slow • single • single- -field inflation, no extra degree of freedom field inflation, no extra degree of freedom Ρ C Ρ becomes constant soon after horizon- -crossing ( crossing ( t = t ): C becomes constant soon after horizon t = h ): t h ( ) ( ) ( ) R R δ = = ; N t t t t C C h fin fin h log L log L . C = const. t s n o c = Ρ C Ρ -1 1 = H H - L = L inflation inflation log a log a = t t = t fin t = t t = t h t fin h
δ N ) δ δ t δ t Also δ , where δ N = = H ( t is the time difference Also H ( h ) C , where C is the time difference t h t F t F → C → C F → F → between the comoving and flat slices at t = t . between the comoving and flat slices at t = t h h . Σ C Σ δφ = 0, Ρ = = Ρ Ρ C δφ = 0, Ρ ( t t h C ( ) : comoving h ) : comoving δ t C δ t F → C F → C Ρ = 0, δφ = = δφ δφ F Ρ = 0, δφ F Σ F Σ F ( ( t t h h ) : flat ) : flat ( ) ( ) ( ) & φ + δ = φ δφ + φ δ = i , t t x t 0 t t → → F h F C C h F h F C H ( ) ( ) ( ) R = δ = − δφ ⇐ = − ; dN t N t t t Hdt φ C fin h fin F h / d dt dN ( ) = δφ ··· δ δ N t N formula formula ··· φ F h d Starobinsky ‘ ‘85 85 Starobinsky Only the knowledge of the background evolution knowledge of the background evolution Only the Ρ C is necessary to calculate Ρ ( t ) . is necessary to calculate C ( fin ) . t fin
• δ δ N for a multi multi- -component scalar: component scalar: N for a • (for slowroll inflation) (for slowroll inflation) ∂ N ( ) ∑ ( ) R = δ = δφ a t N t MS & Stewart ’ ’96 96 MS & Stewart ∂ φ C fin F h a a Ρ C ζ ) is no longer constant in time N.B. Ρ (= ζ ) is no longer constant in time: : N.B. C (= & H φ δφ ⋅ ( ) R = − F t ··· time varying even on time varying even on ··· C 2 & φ superhorizon scales superhorizon scales ( ) 2 H t ∂ ( ) R N 2 2 2 2 = ∇ δφ = ∇ h ∇ ≡ ∂ t N N N ( ) C fin F 2 π φ a a 2 Further extension to non- -slowroll case is possible, if slowroll case is possible, if Further extension to non general slow- -roll condition roll condition is satisfied at horizon is satisfied at horizon- -crossing. crossing. general slow Lee, MS, Stewart, Tanaka & Yokoyama ‘ ‘05 05 Lee, MS, Stewart, Tanaka & Yokoyama & & & & & & φ φ φ 2 ( ) ( ) ( ) = ξ = ξ = ξ ξ = , , , ..., 1 O O O & & φ φ 2 2 2 H H H
3. Nonlinear extension 3. Nonlinear extension • On superhorizon scales, gradient expansion is valid: On superhorizon scales, gradient expansion is valid: • ∂ ∂ G ρ = : : ; Q Q HQ H ∂ ∂ i x t Belinski et al. ’70, Tomita ’72, Salopek & Bond ’90, … This is a consequence of causality: -1 1 » H H - L » L -1 1 H - H light cone • At lowest order, no signal propagates in spatial directions. At lowest order, no signal propagates in spatial directions. • Field equations reduce to ODE’s
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