Problem of definition: Jordan vs Einstein frame, beyond slow-roll - - PowerPoint PPT Presentation

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion

Problem of definition: Jordan vs Einstein frame, beyond slow-roll

Godfrey Leung

godfrey.leung@apctp.org

Asia-Pacific Centre for Theoretical Physics collaboration work with Jonathan White [arXiv:1509.xxxxx] 3rd - 5th August 2015 APCTP-TUS joint workshop on Dark Energy

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Einstein’s General Relativity Non-minimal Coupled Models Frame dependence or Independence? Caveat: Curvature Perturbation ζ Isocurvature Perturbation ζ = ˜ ζ in general

Einstein’s General Relativity

In Einstein gravity, it is assumed matter is minimally coupled to gravity S =

• d4x√−g(M2

pR/2) + Sm[gµν, φ]

R= Ricci Scalar, Sm = matter action For example, single scalar field φ Sm =

• d4x√−g
• − 1

2 gµν∂µφ∂νφ + V (φ)

• Godfrey Leung

Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Einstein’s General Relativity Non-minimal Coupled Models Frame dependence or Independence? Caveat: Curvature Perturbation ζ Isocurvature Perturbation ζ = ˜ ζ in general

Einstein’s General Relativity (con’t)

Strong/weak equivalence principle Strong: The gravitational motion of a small test body depends only on its initial position in spacetime and velocity, and not on its constitution.

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Einstein’s General Relativity Non-minimal Coupled Models Frame dependence or Independence? Caveat: Curvature Perturbation ζ Isocurvature Perturbation ζ = ˜ ζ in general

Non-minimal Coupled Models

Can we violate the equivalence principle? Yes

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Einstein’s General Relativity Non-minimal Coupled Models Frame dependence or Independence? Caveat: Curvature Perturbation ζ Isocurvature Perturbation ζ = ˜ ζ in general

Non-minimal Coupled Models

Can we violate the equivalence principle? Yes Example, introducing non-minimal coupling to gravity SJ =

• d4x√−g(f (ψ)M2

pR/2) + Sm[gµν, ψ]

generic in modified gravity and unified theories, such as string theory, f(R), Chameleons, TeVeS... conformally related to SE =

• d4x√−gE (M2

p ˜

R/2) + ˜ Sm[(gµν)E , ψ] by the conformal transformation gµν → (gµν)E = f (ψ)gµν They are mathematically equivalent Question: But are they physically equivalent?

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Einstein’s General Relativity Non-minimal Coupled Models Frame dependence or Independence? Caveat: Curvature Perturbation ζ Isocurvature Perturbation ζ = ˜ ζ in general

Physics should be frame independent!

Conformal transformation = field redefinition More precisely, conformal transformation = change of scale 1 meter is only meaningful with respect to a reference scale

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Einstein’s General Relativity Non-minimal Coupled Models Frame dependence or Independence? Caveat: Curvature Perturbation ζ Isocurvature Perturbation ζ = ˜ ζ in general

But...

In cosmology, density fluctuations are usually quantified in terms of ζ q ζ ≡ −ϕ + Hδρ ˙ ρ can be defined in both conformal frames, where ρ is the effective energy density from Gµν = Tµν/M2

p

For instance

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Einstein’s General Relativity Non-minimal Coupled Models Frame dependence or Independence? Caveat: Curvature Perturbation ζ Isocurvature Perturbation ζ = ˜ ζ in general

But...

In cosmology, density fluctuations are usually quantified in terms of ζ q ζ ≡ −ϕ + Hδρ ˙ ρ can be defined in both conformal frames, where ρ is the effective energy density from Gµν = Tµν/M2

p

For instance dimensionless and gauge invariant, but not frame independent as we will see...

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Einstein’s General Relativity Non-minimal Coupled Models Frame dependence or Independence? Caveat: Curvature Perturbation ζ Isocurvature Perturbation ζ = ˜ ζ in general

Aside: Isocurvature perturbation

Perturbation is purely adiabatic if δP =

˙ P ˙ ρ δρ. Not always true though...

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Einstein’s General Relativity Non-minimal Coupled Models Frame dependence or Independence? Caveat: Curvature Perturbation ζ Isocurvature Perturbation ζ = ˜ ζ in general

Aside: Isocurvature perturbation

Perturbation is purely adiabatic if δP =

˙ P ˙ ρ δρ. Not always true though...

Entropic/isocurvature perturbations = perturbations ⊥ background trajectory, natural in multifield inflation models some hints in 2013 Planck results

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Einstein’s General Relativity Non-minimal Coupled Models Frame dependence or Independence? Caveat: Curvature Perturbation ζ Isocurvature Perturbation ζ = ˜ ζ in general

Inequivalence of ζ in Einstein and Jordan frames

It was found that ζ is frame-dependent in the presence of isocurvature perturbation [White et al. 12, arXiv:1205.0656], [Chiba and Yamaguchi 13] reason: isocurvature perturbation is frame-dependent (artificial)

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Einstein’s General Relativity Non-minimal Coupled Models Frame dependence or Independence? Caveat: Curvature Perturbation ζ Isocurvature Perturbation ζ = ˜ ζ in general

Inequivalence of ζ in Einstein and Jordan frames

Examples, in multifield models ζ − ˜ ζ ≈ AJK KJK + BJK ˙ KJK , KJK ≡ δφJ ˙ φK − δφK ˙ φJ where KJK is a measure of the isocurvature perturbation ζ − ˜ ζ → 0 only if isocurvature vanishes in general

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion ζ ↔ ˜ ζ Relation between observables

Relation between ζ and ˜ ζ

linear and non-linear order using the separate universe assumption, we can write ζ and ˜ ζ in terms of the δN formalism [White, Minamitsuji and Sasaki et al.] ζ =NI δφI + NIJδφI

˜ RδφJ ˜ R+

˜ ζ = ˜ NI δφI + ˜ NIJδφI

˜ RδφJ ˜ R + ...

δφI

˜ R = flat-gauge field perturbations in Einstein frame

• bservables can be expressed in terms of δN coefficients, e.g. in Jordan frame

ns − 1 = −2(˜ ǫH)∗ − 2 NI N I + 2 3 ˜ H2

∗

N K N L NI NJSIJ

∗

• ˜

∇K ˜ ∇L ˜ V − ˜ RKLPQ dφP d˜ t dφQ d˜ t

• ∗

Pζ = N I NI ˜ H 2π 2

∗

and r = 8 N I NI

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion ζ ↔ ˜ ζ Relation between observables

Relation between ζ and ˜ ζ (con’t)

general relation between N and ˜ N N = ω

R

Hdη = ω

R

• ˜

H − f ′ 2f

• dη = ˜

N(ω, R) − 1 2 ln fω fR

• consider a simplified case where f ′ ≈ 0 at the time of interest (late time)

using the δN formalism, the first and second order δN coefficients are related by NI ≈ ˜ NI − 1 2 + c fJ f

• ⋄

∂φJ

⋄

∂φI

∗

• ω

NIJ ≈ ˜ NIJ − 1 2 + c fKL f − fK fL f 2

• ⋄

∂φK

⋄

∂φI

∗

• ω

∂φL

⋄

∂φJ

∗

• ω

+ fK f

• ⋄

∂2φK

⋄

∂φI

∗∂φJ ∗

• ω
• we have assumed ǫf ≡ |f ′/Hf | ≪ 1

c ≡

˜ H ˜ ρ′ ˜ ρ, equals to −1/3 (matter era) and −1/4 (radiation era)

ζ − ˜ ζ can be arbitrarily large depending on f , but how about observables?

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion ζ ↔ ˜ ζ Relation between observables

Difference between primordial observables beyond slowroll

Defining the fractional difference between the power spectra amplitudes and the spectral indices ∆Pζ ≡ Pζ − ˜ Pζ ˜ Pζ and slightly different definition for ns and fNL ns − 1 + 2(˜ ǫH)∗ ˜ ns − 1 + 2(˜ ǫH)∗ = (1 + ∆Pζ)−1(1 + ∆ns) fNL ˜ fNL = (1 + ∆Pζ)−2 (1 + ∆fNL) using the asymptotic relation between the δN coefficients, we therefore have ∆Pζ = − 1 ˜ NP ˜ NP

• (1 + 2c)

fK f

• ⋄

∂φK

⋄

∂φI

∗

• ω

˜ NI − 1 2 + c 2 fK fL f 2

• ⋄

∂φK

⋄

∂φI

∗

• ω

∂φL

⋄

∂φJ

∗

• ω

SIJ

∗

• and similarly for ∆ns and ∆fNL

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion

Model considered

we consider the multifield model with the following action in the Jordan frame SJ =

• d4x√−g
• f (Φ)R − 1

2 GIJ(Φ)gµν∂φI ∂φJ − V (Φ)

• r in the Einstein frame

SE =

• d4x
• −˜

g

• M2

pR

2 − 1 2 SIJ(Φ)˜ gµν∂φI ∂φJ − ˜ V (Φ)

• note that the field space metric and scalar potential are related by

SIJ = M2

p

2f

• GIJ + 3 fI fJ

f

• and

˜ V = VM4

p

4f 2 after inflation ends, reheating is modeled by adding a friction to the field EOM in Einstein frame (φI )′′ + ˜ ΓI

JK (φJ)′(φK )′ + 2( ˜

H + ˜ a(I)Γ)(φI )′ + ˜ a2SIJ ˜ VJ = 0 ρ′

γ + 4Hργ = (I)Γ

˜ a SIJ(φI )′(φJ)′

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion

Simple example: Non-minimal coupling f ′ = 0 always

to illustrate, we consider the class of two-field models simple explicit example: f = f (χ) and χ is a frozen spectator field Sχχ ≡ M2

p

2f

• Gχχ + 3

f 2

χ

f

• and

˜ V = V (φ)M2

p

we further assume Gφχ = 0 and Gφφ = f such that there is no mixing in the kinetic term Einstein frame results = simple chaotic inflation ˜ ns − 1 = −6˜ ǫ∗ + 2˜ η∗ , ˜ r = 16˜ ǫ∗

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion

Simple example: Non-minimal coupling f ′ = 0 always (con’t)

results during slow-roll inflationary regime, consistent with previous analytic work

0.964 0.966 0.968 0.97 0.972 0.974 0.976 0.978 0.98 0.982 10 20 30 40 50 60 Spectral index ˜ N

Jordan, β = 1, χ∗ = 10−3Mpl Jordan, β = 10, χ∗ = 10−3Mpl Jordan, β = 20, χ∗ = 10−3Mpl Jordan, β = 10, χ∗ = 10Mpl Einstein

0.05 0.1 0.15 0.2 10 20 30 40 50 60 Tensor-to-scalar ratio ˜ N

Jordan, β = 1, χ∗ = 10−3Mpl Jordan, β = 10, χ∗ = 10−3Mpl Jordan, β = 20, χ∗ = 10−3Mpl Jordan, β = 10, χ∗ = 10Mpl Einstein

model choice: ˜ V (φ) = 1

2 m2φ2, 2f /M2 p = e−βχ/Mp with φ∗ = 15.0Mp

• bservables seem coincide at the end of inflation

however, ζ still evolve in Jordan frame...

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion

Simple example: Non-minimal coupling f ′ = 0 always (con’t)

how about beyond slow-roll, particularly after reheating?

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion

Simple example: Non-minimal coupling f ′ = 0 always (con’t)

how about beyond slow-roll, particularly after reheating? for this particular model, since the non-minimal coupled field χ is frozen, things simplify the fractional difference ∆Pζ = 1 16

• 1

˜ N2

φ

fχ f 2

∗

Sχχ

∗

˜ ns − ns ns = ∆Pζ ns − 1 + 2(˜ ǫH)∗ ns

• difference can only be large if ∆Pζ ≫ O(1)

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion

Simple example: Non-minimal coupling f ′ = 0 always (con’t)

Beyond slow-roll regime

0.964 0.966 0.968 0.97 0.972 0.974 0.976 0.978 0.98 55.5 56 56.5 57 57.5 58 58.5 59 59.5 60 Spectral index ˜ N

Jordan, β = 1, χ∗ = 10−3Mpl Jordan, β = 10, χ∗ = 10−3Mpl Jordan, β = 20, χ∗ = 10−3Mpl Jordan, β = 10, χ∗ = 10Mpl Einstein

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 55.5 56 56.5 57 57.5 58 58.5 59 59.5 60 Tensor-to-scalar ratio ˜ N

Jordan, β = 1, χ∗ = 10−3Mpl Jordan, β = 10, χ∗ = 10−3Mpl Jordan, β = 20, χ∗ = 10−3Mpl Jordan, β = 10, χ∗ = 10Mpl Einstein

with reheating parameter Γφ = 0.1(m/ √ 2) evolution terminates when Ωγ > 0.9999

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion

Difference is negligible after reheating

we see the fractional difference between observables are negligible even ζ − ˜ ζ is large why?

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion

Difference is negligible after reheating

we see the fractional difference between observables are negligible even ζ − ˜ ζ is large why? recall ∆Pζ = − 1 ˜ NP ˜ NP

• (1 + 2c)

fK f

• ⋄

∂φK

⋄

∂φI

∗

• ω

˜ NI − 1 2 + c 2 fK fL f 2

• ⋄

∂φK

⋄

∂φI

∗

• ω

∂φL

⋄

∂φJ

∗

• ω

SIJ

∗

• reason: Einstein frame field space metric SIJ

∗ also depends on f

Sχχ ≡ M2

p

2f

• Gχχ + 3

f 2

χ

f

• Godfrey Leung

Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Caveat I Caveat II

Caveat I: negative Jordan frame field space metric

We may tune (Gχχ)∗ → (f 2

χ/f )∗, with Sχχ remains positive

Example: Gχχ = −b1(f 2

χ/f ), ˜

V (φ) = 1

2 m2φ2 and 2f /M2 p = e−βχ/Mp , with b1 ≤ 3

0.964 0.965 0.966 0.967 0.968 0.969 0.97 0.971 55.5 56 56.5 57 57.5 58 58.5 59 59.5 60 Spectral index ˜ N

Jordan, b1 = 1 Jordan, b1 = 2.5 Jordan, b1 = 2.9 Einstein

0.04 0.06 0.08 0.1 0.12 0.14 0.16 55.5 56 56.5 57 57.5 58 58.5 59 59.5 60 Tensor-to-scalar ratio ˜ N

Jordan, b1 = 1 Jordan, b1 = 2.5 Jordan, b1 = 2.9 Einstein

• nly in the very fine-tuned limit the difference becomes significant

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion Caveat I Caveat II

Caveat II: non-frozen f

more generic case: f evolves the model choice: ˜ V (φ) = 1

2 m2φ2 exp(−λχ2/M2 p), Sχχ = Sφφ = 1 and

2f /M2

p = exp(−0.5λχ2/M2 p).

λ = {0.05, 0.06}, initial conditions χ∗ = 10−3Mp and φ∗ = 15.0Mp.

0.7 0.75 0.8 0.85 0.9 0.95 1 55 56 57 58 59 60 61 62 Spectral index ˜ N

Einstein, λ = 0.05 Jordan, λ = 0.05 Einstein, λ = 0.06 Jordan, λ = 0.06

0.02 0.04 0.06 0.08 0.1 0.12 0.14 55 56 57 58 59 60 61 62 Tensor-to-scalar ratio ˜ N

Einstein, λ = 0.05 Jordan, λ = 0.05 Einstein, λ = 0.06 Jordan, λ = 0.06

special case: potential ∼ ridge like, initial conditions close to top of the ridge

Godfrey Leung Frame (In)dependence, Non-minimal coupled models

Non-minimal Coupled Models Asymptotic relation beyond slow roll Simple explicit example Counter-example Discussion and Conclusion

Discussion and Conclusion

Take home message conventional definition of curvature perturbation is a not frame-dependent quantity in theory, using the wrong definition can lead to very different results e.g. ζ − ˜ ζ can be arbitrarily large however asymptotically the difference between observables are negligible in general after reheating possible to realise counter examples, but need fine-tuned initial conditions Ongoing and Future Directions study the correlation between large (local) non-Gaussianity and the fractional difference decay rates are generically modulated in non-minimal coupled models even in simple perturbative reheating Γ → Γ(χ) At quantum level? see Steinwachs

Godfrey Leung Frame (In)dependence, Non-minimal coupled models