Loop corrections to the primordial perturbations

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

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

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

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

5. Application to the inflationary universe φ h h φ φ h h φ φ h φ h Background g :Slow-roll inflation Fluctuations (h , φ )　→　ΓCTP δΓCTP /δhab = 0 etc Quantum effect of φ Memory term Nabcd (x , y) ←Im[ΓCTP] Habcd (x , y) ←Re[ΓCTP ] ξab → Fluc. of Tab for φ ong

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

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

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

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

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