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Loop corrections to the primordial perturbations

Loop corrections to the primordial perturbations. Yuko Urakawa (Waseda university). Kei-ichi Maeda (Waseda university). Non-linear perturbations. More information about the inflation model. Non-linear perturbations. Transition from Quantum fluctuation to Classical perturbation.

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Loop corrections to the primordial perturbations

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  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.

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