1. 1 Loop corrections to the primordial perturbations Yuko Urakawa (Waseda university) Kei-ichi Maeda (Waseda university)

2. 2 Motivation Non-linear perturbations Quantum fluc. of inflaton Transition from Quantum fluctuation to Classical perturbation Observable quantity [Inflation model] Minimally coupled single scalar field + Einstein – Hilbert action Loop corrections from “ Stochastic gravity ”. More information about the inflation modelNon-linear perturbations

3. 3 Closed Time Path formalism B.L.Hu and E.Verdaguer (1999) Stochastic gravity hh φ φ h hh h Interacting system : Scalar field φ & Gravitational fieldFluc. h Evolution of the in-in expectation value. Effective action in the CTP formalism Stochastic gravity [ Effective action in CTP] Sub-Planck region Quantum fluc. of scalar φ ＞＞ Quantum fluc. h Integrate out only φ time “ Coarse – graining ” h ∈ External line, h ∈ Internal line h ～ Classical external field @ Path integral of Γ CTP

4. 4 A.A.Starobinsky (1987) Evolution of Gravitational field ← Quantum φ @ Sub-Planck region Evolution of Long-wave mode, φ sp ← Quantum fluc. of Short-wave mode, φ sb Imaginary part in Γ CTP [g] Quantum Fluc. of φ Stochastic inflation Stochastic gravity Quantum Fluc. of φ sb integrated out φ sp φ sb Self-interaction of φ g ab φ integrated out Interaction between φ and g Imaginary part in Γ CTP [φ sp ] → Stochastic variable ξab Transition from Quantum fluc. to Classical perturbations Γ CTP with “ Coarse – graining ” “ Loop corrections “ Langevin type equation → Stochastic variable ξ

5. 5 Application to the inflationary universe ξ ab → Fluc. of T ab for φ on g Memory term H abcd (x, y) ← Re[ Γ CTP ] Quantum effect of φ Background g : Slow-roll inflation Fluctuations (h, φ ) → Γ CTP hh φ φ hh φ hh φφ φ etc δΓ CTP / δh ab = 0 N abcd (x, y) ← Im[ Γ CTP ]

6. 6 Perturbation Non-linear effect of φ → Couples these tree modes Coupling 1. Stochastic variable ξ ab has also Vector and Tensor part. 2. Memory term scalar + vector + tensor δg ab One loop corrections to Scalar & Tensor perturbations Flat slicing hh φ φ Metric ansatz scalartensorscalar Coupling among the three modes: scalar,vector, and tensor

7. 7 UV divergence Renormalization IR divergence [ Initial condition ] for － k τ i > 1 IR divergence hh φ φq k - q kk Neglection Mode eq. for φ I in Interacting picture UV divergent part ・・・ Decaying mode in superhorizon D.Podolsky and A.A.Starobinsky (1996) Unphysical initial condition Beginning of Inflation τ i subhorizonsuperhorizon Quantum effect ： Like in Minkowski sp. Cut off Need not care about UV divergence in “ Observable quantity ” → Quantum fluc. ～ Classical stochastic fluc.(Observable) ∝ k -3 q HiHi ０

8. 8 Scalar perturbations superhorizon limit Gauge invariant ζ ∝ δT / T ( Leading part of Loop corrections ) / (Linear perturbation) ～ (H/m pl ) 2. [ Results ] N k < exp[1/2(ε － η V )]If 2 (ε － η V ) log(k/H i ) < 1 Amplified by the N k Similar ampfilication @ S.Weinberg (2005) & M.S.Sloth(2006). τ τiτi -1/k e-foldings N k for η V log k|τ| ＜＜ 1

9. 9 Tensor perturbations c.f. Linear perturbation ( LHS ) Evolution eq. for H T (t) in Linear perturbation ( RHS ) Amplification from Quantum φ (Due to Non-linear interactions) [ Results ] No amplification in terms of the e-foldings. No IR divergence. ( Leading part of the loop corrections ) / (Linear perturbation) ～ (H/m pl ) 2.

10. 10 Summary Stochastic gravity Amplitude ∝ (H/m pl ) 4 Amplified by N k ・ Non-linear quantum effect ・ Transition from Quantum fluc. to Classical perturbations One Loop corrections Both the scalar perturbations and the tensor perturbations Scalar perturbations Tensor perturbations No Amplification by N k No IR divergence. hh φ φq k - q kk

11. 11 Effective action 〔 Quantum correction of φ 〕 Global feature of the inflation model quantum fluctuation １． Vertex operator α m = O ( (ε SR ) m/2 ) ← We can prove by the mathematical deduction. In case, the slow-roll condition are satisfied. ε SR ≡ε ， η V, η H, η ２． Propagator ~ H 2 η V ≡ V’’/ κ 2 V As its coefficient, the vertex operators include the information of the potential. In principle, the higher loop correction include the more global information of the potential. The propagator depends on the evolution of the background spacetime.

12. 12 Effective action Loop corrections α m = O ((ε SR ) m/2 ) ２． Propagator ~ H 2 １． Vertex hh ψ ψ ◆ (κH) 2 [tree graph] coupling between g and φ α1α1 hh ψ α1 α1 (κH) 2 V’(φ), H vertex h ・ ψ ← This interaction is included in linear analysis. ◆ (κH) 4 [loop graph] ◆ (κH) 6 [loop graph] α2α2 α2α2 (κH) 2 + V’’(φ) + V (3) (φ)+ V (4) (φ) (κH) 2 hh ψ ψ ψ α3α3 α3α3 α1 α1 h α 4 α1α1 h (κH) 2

13. 13 Global time dependence among Loop corrections Up to second order perturbations S. Weinberg (2005), (2006) Comparison with preceding researches 【 Interaction 】 massless scalar field （ → σ ）, fermions, etc In most of standard inflation models, although regularization problem has been left, there are no global time dependence among loop corrections. ( Loop corrections for fixed internal momentum) 【 Results 】 scalar perturbations & inflaton → curvature perturbation in comoving slicing ζ tensor perturbations → γ ζζ σ σ hh φ φ ζ ζ ζ Stochastic gravity Comparison Number of σ field Constant number c.f.

14. 14 Secondary Motivation The influence of taking care of decoherence Squeezing due to inflationary expansion dominate the decoherence process It is sufficient to consider the classical evolution of perturbations. Ref Lyth and Lidsey (2006) ＠ super horizon scale ( – k η << 1 ) hh φ φ vertex hφφ vertex φ 4 etc 1. Non – linear effect (Loop corrections) y k (η) = f (η) e k 2. Influence from horizon crossing time Gaussian As long as the squeezing parameter r is not so large and the decaying mode cannot be neglected, the classical evolution due to the decomposition is not guaranteed. Until enough super horizon scale ( especially – k η ～ O (1) ) Are there any possibilities that assuming the classical behavior of perturbations also for this region induce some differences, corresponding to the decoherence process. ・ Comparison the distribution function in Ref [1] to the dynamical result described by the evolution of RDM or Langevin-type equation. (Especially the decomposition can be realized as a result of its evolution.) h : gravitational field φ: Inflaton [2] 「 Pointer states for primordial fluctuations in inflationary cosmology 」 [1] 「 Semiclassicality and decoherence of cosmological perturbations 」

15. 15 Linear Perturbations [ super horizon region ] Ordinary analysis in linear perturbations horizon crosssuper horizonsub horizon during inflation Wands et.al. (2000) adiabatic perturbation [ sub horizon region ] Mukhanov-Sasaki variable mass less scalar field in Minkowski space time red tilted spectrum M.Sasaki Brandenburgar et.al.

16. 16 AB xy Reduced density matrix The evolution of the reduced density matrix Assumption coarse – graining B ( trace out ) Influence function (→ represents the effect from A to B ) S IF is the effective action, which represents the effect of B Influence functional

17. 17 BA g φ The gravitational field, affected by the quantum scalar field Effective action Perturbation g → g + h In S eff [ ｈ ２ ], there exists the imaginary part → “stochastic interpretation” （ We can interpret as an stochastic source ） P [ξ] ： Gaussian Stochastic gravity B.L.Hu and E.Verdaguer (1999) coarse – graining φ