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
Einstein’s General Relativity Non-minimal Coupled Models Non-minimal Coupled Models Asymptotic relation beyond slow roll Frame dependence or Independence? Simple explicit example Caveat: Curvature Perturbation ζ Counter-example Isocurvature Perturbation Discussion and Conclusion ζ � = ˜ ζ in general Einstein’s General Relativity In Einstein gravity, it is assumed matter is minimally coupled to gravity d 4 x √− g ( M 2 � S = p R / 2) + S m [ g µν , φ ] R = Ricci Scalar, S m = matter action For example, single scalar field φ d 4 x √− g � � − 1 � 2 g µν ∂ µ φ∂ ν φ + V ( φ ) S m = Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Einstein’s General Relativity Non-minimal Coupled Models Non-minimal Coupled Models Asymptotic relation beyond slow roll Frame dependence or Independence? Simple explicit example Caveat: Curvature Perturbation ζ Counter-example Isocurvature Perturbation Discussion and Conclusion 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
Einstein’s General Relativity Non-minimal Coupled Models Non-minimal Coupled Models Asymptotic relation beyond slow roll Frame dependence or Independence? Simple explicit example Caveat: Curvature Perturbation ζ Counter-example Isocurvature Perturbation Discussion and Conclusion ζ � = ˜ ζ in general Non-minimal Coupled Models Can we violate the equivalence principle? Yes Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Einstein’s General Relativity Non-minimal Coupled Models Non-minimal Coupled Models Asymptotic relation beyond slow roll Frame dependence or Independence? Simple explicit example Caveat: Curvature Perturbation ζ Counter-example Isocurvature Perturbation Discussion and Conclusion ζ � = ˜ ζ in general Non-minimal Coupled Models Can we violate the equivalence principle? Yes Example, introducing non-minimal coupling to gravity d 4 x √− g ( f ( ψ ) M 2 � S J = p R / 2) + S m [ g µν , ψ ] generic in modified gravity and unified theories, such as string theory, f(R), Chameleons, TeVeS... conformally related to d 4 x √− g E ( M 2 � p ˜ R / 2) + ˜ S E = S m [( g µν ) E , ψ ] by the conformal transformation g µν → ( g µν ) E = f ( ψ ) g µν They are mathematically equivalent Question: But are they physically equivalent? Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Einstein’s General Relativity Non-minimal Coupled Models Non-minimal Coupled Models Asymptotic relation beyond slow roll Frame dependence or Independence? Simple explicit example Caveat: Curvature Perturbation ζ Counter-example Isocurvature Perturbation Discussion and Conclusion 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
Einstein’s General Relativity Non-minimal Coupled Models Non-minimal Coupled Models Asymptotic relation beyond slow roll Frame dependence or Independence? Simple explicit example Caveat: Curvature Perturbation ζ Counter-example Isocurvature Perturbation Discussion and Conclusion ζ � = ˜ ζ in general But... In cosmology, density fluctuations are usually quantified in terms of ζ q ζ ≡ − ϕ + H δρ ρ ˙ can be defined in both conformal frames, where ρ is the effective energy density from G µν = T µν / M 2 p For instance Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Einstein’s General Relativity Non-minimal Coupled Models Non-minimal Coupled Models Asymptotic relation beyond slow roll Frame dependence or Independence? Simple explicit example Caveat: Curvature Perturbation ζ Counter-example Isocurvature Perturbation Discussion and Conclusion ζ � = ˜ ζ in general But... In cosmology, density fluctuations are usually quantified in terms of ζ q ζ ≡ − ϕ + H δρ ρ ˙ can be defined in both conformal frames, where ρ is the effective energy density from G µν = T µν / M 2 p For instance dimensionless and gauge invariant, but not frame independent as we will see... Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Einstein’s General Relativity Non-minimal Coupled Models Non-minimal Coupled Models Asymptotic relation beyond slow roll Frame dependence or Independence? Simple explicit example Caveat: Curvature Perturbation ζ Counter-example Isocurvature Perturbation Discussion and Conclusion ζ � = ˜ ζ in general Aside: Isocurvature perturbation ˙ P Perturbation is purely adiabatic if δ P = ρ δρ . Not always true though... ˙ Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Einstein’s General Relativity Non-minimal Coupled Models Non-minimal Coupled Models Asymptotic relation beyond slow roll Frame dependence or Independence? Simple explicit example Caveat: Curvature Perturbation ζ Counter-example Isocurvature Perturbation Discussion and Conclusion ζ � = ˜ ζ in general Aside: Isocurvature perturbation ˙ P Perturbation is purely adiabatic if δ 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
Einstein’s General Relativity Non-minimal Coupled Models Non-minimal Coupled Models Asymptotic relation beyond slow roll Frame dependence or Independence? Simple explicit example Caveat: Curvature Perturbation ζ Counter-example Isocurvature Perturbation Discussion and Conclusion 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
Einstein’s General Relativity Non-minimal Coupled Models Non-minimal Coupled Models Asymptotic relation beyond slow roll Frame dependence or Independence? Simple explicit example Caveat: Curvature Perturbation ζ Counter-example Isocurvature Perturbation Discussion and Conclusion ζ � = ˜ ζ in general Inequivalence of ζ in Einstein and Jordan frames Examples, in multifield models ζ ≈ A JK K JK + B JK ˙ K JK , K JK ≡ δφ J ˙ φ K − δφ K ˙ ζ − ˜ φ J where K JK 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 ζ ↔ ˜ ζ Simple explicit example Relation between observables Counter-example Discussion and Conclusion 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 δ N formalism [White, Minamitsuji and Sasaki et al.] ζ = N I δφ I + N IJ δφ I R δφ J R + ˜ ˜ N I δφ I + ˜ ζ = ˜ ˜ N IJ δφ I R δφ J R + ... ˜ ˜ δφ I R = flat-gauge field perturbations in Einstein frame ˜ observables can be expressed in terms of δ N coefficients, e.g. in Jordan frame N K N L d φ P d φ Q 2 2 � � ∇ K ˜ ˜ ∇ L ˜ V − ˜ n s − 1 = − 2(˜ ǫ H ) ∗ − N I N I + R KLPQ 3 ˜ d ˜ d ˜ H 2 N I N J S IJ t t ∗ ∗ ∗ � ˜ � 2 H 8 P ζ = N I N I and r = N I N I 2 π ∗ Godfrey Leung Frame (In)dependence, Non-minimal coupled models
Non-minimal Coupled Models Asymptotic relation beyond slow roll ζ ↔ ˜ ζ Simple explicit example Relation between observables Counter-example Discussion and Conclusion Relation between ζ and ˜ ζ (con’t) general relation between N and ˜ N � f ω � ω � ω H − f ′ � � N ( ω, R ) − 1 � ˜ d η = ˜ N = H d η = 2 ln 2 f f R R R consider a simplified case where f ′ ≈ 0 at the time of interest (late time) using the δ N formalism, the first and second order δ N coefficients are related by � 1 � � f J � ∂φ J � � N I ≈ ˜ ⋄ N I − 2 + c ∂φ I f ⋄ ∗ ω � 1 � �� f KL � ∂φ K � ∂φ L � f K � ∂ 2 φ K − f K f L � � � � � � N IJ ≈ ˜ ⋄ ⋄ ⋄ N IJ − 2 + c + f 2 ∂φ I ∂φ J ∂φ I ∗ ∂φ J f f ⋄ ∗ ω ∗ ω ⋄ ∗ ω we have assumed ǫ f ≡ | f ′ / H f | ≪ 1 ˜ H c ≡ ρ , equals to − 1 / 3 (matter era) and − 1 / 4 (radiation era) ρ ′ ˜ ˜ ζ − ˜ ζ can be arbitrarily large depending on f , but how about observables? Godfrey Leung Frame (In)dependence, Non-minimal coupled models
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