Loop corrections to the primordial perturbations YukoUrakawa Waseda

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

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

Stochastic gravity B. L. Hu and E. Verdaguer (1999) Closed Time Path formalism Evolution of the in-in expectation value. < in　| ** | in >　 time Effective action in the CTP formalism Stochastic gravity 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 ” h ∈ External line, h φ h ～ Classical external field h φ h h ∈　Internal line h h h 3

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

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

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

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

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

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. 9

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