Yukawa Institute for Theoretical Physics Atsushi Naruko Based on : - PowerPoint PPT Presentation

Yukawa Institute for Theoretical Physics Atsushi Naruko Based on : arXiv : 1202.1516

 Inflation is one of the most promising candidates as the generation mechanism of primordial fluctuations.  Inflation can be derived by a scalar field.  We have hundreds or thousands of inflation models. → we have to discriminate those models.  CMB : scale invariant spectrum, Gaussian statistics  Non-Gaussianity may have the key of this puzzle.

 The deviations of CMB from the Gaussian statistics is parameterised by the non-linear parameter “f NL ” . ! � 2 � � � ∆ T ∆ T T ( x ) = ∆ T � � + f NL + · · · � � T T � � Gaussian � Gaussian � 2 � 2 � 3 � �� ∆ T � � ∆ T ( amplitude of 2 nd order perturbation ! ∼ f NL T T � � | � ∆ f local − 10 < f local PLANCK : ! WMAP 7 : ! � < 5 < 74 � � NL NL There exists a possibility to constrain inflation models by f NL !! ! ! We need to go beyond the linear perturbation theory. ! !

 The evolution of the curvature perturbation R (3) k L Newton inflation ! Potential Horizon exit ! H − 1 C M B ! log a ( t ) CMB !  To give a precise theoretical prediction, we need to solve the evolution of R (3) after horizon exit. We focus on superhorizon dynamics of non-linear perturbations. ! !

 There are several approaches for non-linear pert’s. ! 1, higher order perturbation : most general, lengthy ! 2, gradient expansion : superhorizon only, 〜 BG Eqs ! 3, covariant formalism : coordinate-free, geometrical !  What is the relation between No.2 and No.3 ? ! ✓ Equivalence at linear, 2 nd and 3 rd order ! Langlois et al., Enqvist et al,. Lehners et al,. ! ✓ non-linear equivalence in the Einstein gravity ! Suyama et al. !

 On large scales, spatial gradient expansion will be valid. � � � � L � H − 1 � ( � L − 1 Q ) � � ∂ i Q � ∂ t Q � ( � HQ ) � � � � We expand equations in powers of spatial gradients. Full non-linear effects are taken into account.  We express the metric in the ADM form ds 2 = − α 2 dt 2 + ˆ γ ij ( dx i + β i dt )( dx j + β j dt )  The spatial metric is further decomposed curvature γ ij = a 2 ( t ) e 2 ψ γ ij ψ ˆ det | γ ij | = 1 : perturbation !

B.G. e-folding number cf . N ≡  We define the non-linear e-folding number. Hdt N ≡ 1 � � ( H + ∂ t ψ ) dt Θ α dt ∼ Θ � � µ n µ 3 ψ  is given by the difference of “N” δ N formalism ! ! δ N ≡ N − N = ψ ( t fin ) − ψ ( t ini ) ψ ( t fin ) ψ = 0 N ! δ N ψ = 0 x i = const. ! x i = const. ! ψ ( t ini )

final ! ρ = const. !  By choosing the slicings ; initial : flat & final : uniform ρ ψ = 0 ! δ N = ψ ρ initial ! gives the final on the uniform ψ δ N ρ  Let us consider a perfect fluid : T µ ν = ( ρ + P ) u µ u ν + Pg µ ν ψ  The energy cons. law gives the evolution eq. for ∂ t ρ H + ∂ t ψ = − 1 u µ � ν T µ ν = 0 ρ + P 3

 We define the curvature covector. ! ˙ N ζ µ ≡ ∂ µ N − ρ ∂ µ ρ ˙ N ≡ L u N = u µ ∂ µ N ˙ ζ µ  The energy cons. law gives the evolution eq. for , � � ) ˙ Θ P ˙ ζ µ ≡ L u ζ µ = − ∂ µ P − ρ ∂ µ ρ 3( ρ + P ) ˙  Notice !! N ∼ d τ Θ 1, the equation for is valid at all scales. ζ µ 2, there is an ambiguity in the choice of the initial slice, since N is defined in terms of the integration. !

 The relation between “ ζ μ ” in the covariant formalism and “ ψ = δ N ” in the δ N formalism is unclear. !  On the uniform energy density slicing : ρ = ρ (t), ! � � ˙ N � � ] � ψ E ( t, x j ) − ψ ( t ini , x j ) E = ∂ i N − = ∂ i ζ i ρ ∂ i ρ � ˙ E  We choose the initial flat slice as in the δ N formalism, ! � � � � � ] ζ i E = ∂ i ψ E = ∂ i δ N � � T his shows that δ N formalism = covariant formalism. !

 We can show the equivalence between two evolution eqs. ! ζ µ  The evolution eq. for on large scales ! ) � � ˙ Θ P ˙ ζ µ = − ∂ µ P − ρ ∂ µ ρ 3( ρ + P ) ˙ 3( ρ + P ) − ∂ i ρ ( ρ � + P � ) 1 � ∂ i ρ � 3( ρ + P ) 2 ] 1 ρ � � ∂ i P − P � � ) ∂ i ψ � + ρ � ∂ i ρ α α 3( ρ + P ) 2 3( ρ + P ) ] � ρ � ∂ i ψ � = − ∂ i T his also shows that δ N formalism = covariant formalism. !

 We have shown that the non-linear equivalence between the δ N and covariant formalisms on superhorizon scales.  In the proof, we have not assumed the gravity theory, which means the equivalence holds in any gravity theory. !

R (3) � � R  Let us consider perturbations around the FLRW universe. ! dx i dx j ] ds 2 = a 2 ( η ) � (1 + 2 A ) d η 2 � 2 � − 1 / 2 B ,i dx i d η + [ � � ( 1 + 2 R ) δ ij � 2 � − 1 C ,ij R ∼ Φ N ∼ ∆ T/T (gravitational redshift) ! photon ! Φ N  The Einstein equations the master equation for R ! c + 2 z � R c : R in δφ = 0 R �� z R � c � � R c = 0 � � ) φ � z ≡ a × 0 / H λ � H − 1  On superhorizon scales , ! Slow-roll ! c ∝ z � 2 ∼ a � 2 R c = const . and ! R �

 We define the e-folding number, which is the integration of the expansion along an integral curve of u μ , ! N � 1 � d τ Θ = 1 � d τ � µ u µ 3 3  We define the curvature covector, which is one of the most important quantities in the covariant formalism. ! ˙ N ζ µ ≡ ∂ µ N − ρ ∂ µ ρ ˙ where the dot denotes the Lie derivative with respect to u μ , ! ˙ ˙ ζ µ ≡ L u ζ µ = u ν ∂ ν ζ µ + ζ ν ∂ µ u ν N ≡ L u N = u µ ∂ µ N

 The energy cons. law gives the evolution eq. for CC, � � ) ˙ Θ P ˙ ζ µ = − ∂ µ P − ρ ∂ µ ρ 3( ρ + P ) ˙  When the adiabatic condition “P = P ( ρ )” is satisfied, the RHS vanishes and CC is conserved.  Notice !! 1, the above equation is valid at all scales. 2, there is an ambiguity in the choice of the initial slice, since N is defined in terms of the integration. ! N ∼ d τ Θ