Multi-disformal invariance of nonlinear primordial perturbations

Transcription

1 Multi-disformal invariance of nonlinear primordial perturbations Yuki Watanabe Natl. Inst. Tech., Gunma Coll.) with Atsushi Naruko and Misao Sasaki accepted in EPL [arxiv: ] 2nd RESCEU-APCosPA Summer School on Cosmology and Particle Astrophysics Nikko, Tochigi, Japan 2nd August 2015

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3 Introduction & goals anti-correlation of TE spectrum -> super-h curvature perturbations (future) BB spectrum -> super-h gravitational-wave perturbations The primordial linear tensor power spectrum from inflation can be always cast into the standard form at leading order in derivatives with suitable conformal and disformal transformations in EFT of inflation. [Creminelli et al ] Invariance of the curvature perturbation under disformal transformations has been shown at linear order. [Minamitsuji , Tsujikawa ] We extend the invariance of the curvature and GW perturbations to fully nonlinear order. We further show the invariance under a new type of disformal transformation, dubbed multi-disformal transformation, generated by a multi-component scalar field.

4 Decomposing spacetime ( )( ds 2 = g µν dx µ dx ν = α 2 dt 2 +ˆγ ij ( dx i + β i dt )( dx j + β j dt ) α is the lapse function, β i is the shift vector ˆγ ij = a 2 (t)e 2ψ γ ij, det γ ij =1 { [ ( χ 3 4 1{ i[ e 3ψ j( e 3ψ )]} (γ ij δ ij ) is the flat 3-dimensional Laplacian and i = δ ij j R R

5 ( )( Decomposing spacetime and curvature pert n ds 2 = g µν dx µ dx ν = α 2 dt 2 +ˆγ ij ( dx i + β i dt )( dx j + β j dt ) The uniform φ slicing (comoving slicing if φ dominates): α is the lapse function, β i is the shift vector ˆγ ij = a 2 (t)e 2ψ γ ij, det γ ij =1 χ 3 4 1{ i[ e 3ψ j( e 3ψ )]} (γ ij δ ij ) φ = φ(t) { [ ( is the flat 3-dimensional Laplacian and i = δ ij j R R ( )]} where is the fl (R c ψ c + χ c /3) order and R its n R c is the (comoving) curvature perturbation at linear 2 order and R c its non-linear generalization

6 ( )( Decomposing spacetime and GW pert n ds 2 = g µν dx µ dx ν = α 2 dt 2 +ˆγ ij ( dx i + β i dt )( dx j + β j dt ) [ ( χ 3 )]} 4 1{ i[ e 3ψ j( e 3ψ )]} (γ ij δ ij ) The uniform φ slicing: φ = φ(t) part of γ ij with respect to the flat background metric δ ij. We is independent of the time-slicing condition at linear order γ TT ij α is the lapse function, β i is the shift vector ˆγ ij = a 2 (t)e 2ψ γ ij, det γ ij =1 alization as j γ TT = 0. ij but is slice-dependent at higher orders. { [ ( is the flat 3-dimensional Laplacian and i = δ ij j R R

7 General disformal transformation g µν = A(φ,X) g µν + B(φ,X) µ φ ν φ, X g µν µ φ ν φ/2 A = 1, B 0: Disformal transformation A 1, B = 0: Conformal transformation

8 Invariance of nonlinear perturbations g µν = A(φ,X) g µν + B(φ,X) µ φ ν φ, X g µν µ φ ν φ/2 A = 1, B 0: Disformal transformation The uniform φ slicing: φ = φ(t) g µν = g µν + B φ 2 δ µ 0 δ ν 0 Only the lapse function is affected by α 2 = α 2 B φ 2 the disformal transformation! Thus, the spatial metric & shift vector are invariant to fully nonlinear order -> Invariance of curvature and GW perturbations

9 Invariance of nonlinear perturbations g µν = A(φ,X) g µν + B(φ,X) µ φ ν φ, X g µν µ φ ν φ/2 A 1, B = 0: Conformal transformation The uniform φ slicing: φ = φ(t) g µν = Ag µν = Āg µν + δa ḡ µν + δa δg e easily notices that the se µν R R + δa/(2ā)+ δa is sourced by δx, but is vanishing on large scales b/c X c = 1 2 φ 2 (t) δα c = O(ε 2 ) urbation is sol Unimodular part (GW) is invariant to fully nonlinear order α 2 c(t,x) = 1 2 φ 2 (t) δα c (t,x) φ 2 (t)+ s, because th Ṙ c Hδα c, ε represents the terms of 1st order in spatial deriv s. Curvature pert n is nonlinearly invariant on super-h scales

10 Transformation of linear MS equations GR: R 1 1 z 2 d dη ( z 2 ( ) ) ( ) d dη R c + c 2sk 2 R c =0, z a φ ( ) H is the background value of the lapse function and c s is the sound velocity, Under disformal transformation, interpret: ( 1 1 d z 2 ) ( z 2 dη ) It takes the same form if On the other hand ) α 2 = α 2 B φ 2 ) d dη R c + c 2 sk 2 R c =0 H, there are two ways to dτ = dt and d τ = dt (dt = adη) 1 1 z 2 d dη ( z 2 d dη R c ) + c 2 sk 2 R c =0, α0 z c s c s, z It can be interpreted as the one in modified gravity with this redefinition of c_s and z.

11 Transformation of nonlinear equations in spatial gradient expansion GR [Takamizu et al]: 1 1 z 2 η ( z 2 ( ) η R c ) ( ) + c2 s 4 [ ] (3) R [ e 2ψ γ ij ] = O(ε 4 ) ψ = R c + O(ϵ 2 ) and (3) R is the spatial scalar curvature. By the same reasoning as in the linear case, it takes the same form if the proper time is rescaled, or is interpreted as the one in modified gravity with rescaled c_s and z. c s c s, z α0 z

12 Transformation of nonlinear equations in spatial gradient expansion GR: ( ) 1 1 z 2 t η η γtt ij + 1 ( ( e 2ψ (3) [ ) R ij e 2ψ ] ) ( TT γ ij = O(ε 4 [ ) 4 ( ) ) ( [ ] z ) t a and ( ) TT ( denotes the transverse-traceless [ ] ) projection z 2 t By the same reasoning as in the linear case, it takes the same form if the proper time is rescaled, or is interpreted as the one in modified gravity with rescaled c_s and z. 1 z 2 t 1 η c t, ( ) z 2 t η γtt ij z t + c2 t 4 α0 z t = α0 a ( e 2ψ (3) R ij [ e 2ψ γ ij ] ) TT = O(ε 4 )

13 Multi-disformal transformation Suppose there are N component scalar field, φ I (I =1,, N ). consider g µν = A(φ I,X IJ ) g µν + B KL (φ I,X IJ ) µ φ K ν φ L, X IJ = 1 2 gµν µ φ I ν φ J φ I = φ I (ϕ) Adiabatic [ limit: ] ] g µν = g µν + B KL [φ I (ϕ),x IJ (ϕ, ϕ) (φ K ) (φ L ) µ ϕ ν ϕ, (φ I ) dφi dϕ The uniform ϕ slicing: ] g µν = g µν + B KL [φ I (ϕ),x IJ (ϕ, ϕ/α) (φ K ) (φ L ) ϕ 2 δ µ 0 δ ν 0 Since the multi-disformal transformation only affects the lapse function, we can apply the same argument as before!

14 Summary The curvature and tensor perturbations on the uniform φ slicing are fully nonlinearly invariant under the disformal transformation. The same conclusion can be drawn for a multi-component extension of the disformal transformation, dubbed multi-disformal transformation, on the uniform ϕ slicing in the adiabatic limit. Once a 2nd order differential eq. is obtained in modified gravity or EFT, one can map it into the same form as the one in GR by a suitable disformal transformation.