Loop corrections to the primordial perturbations

1. Loop corrections to the primordial perturbations YukoUrakawa (Waseda university) Keiichi Maeda

2. Global dependence on the potential of inflaton among loop corrections φ h h φ Motivation Linear perturbations Two point correlation function of curvature perturbation can be determined from the behavior on the horizon crossing time We can pick up only local information about the inflation model. (ex) V(φ)，　V’(φ)， V’’(φ)．．． How about non-linear perturbations ? Loop effect to two point correlation function of the curvature perturbations and tensor perturbations from Stochastic gravity vertexhφφ vertexφ4 etc To Search Global time dependence among loop effects 1. Dependence on Vertex operators 2. Dependence on the Background field evolution

3. There is some region where we can approximate as, 10-33cm 10-13cm matter field 　→　fully quantized gravitational field　→　partially quantized Planck scale Compton wavelength φ h h h h h great difference φ h Basic idea of Stochastic gravity Effective action in stochastic gravity Stochastic gravity B.L.Hu and E.Verdaguer (1999) Gravitational field is treated as a external field. It does not contribute as the propagator of the internal line. Feynman-Vernon’s influence function (or IN-IN formalism) Effective action SIF , which describes the quantum effect of the matter field [Total effective action] Causal equation Gauge invariant equation

4. Einstein-Langevin equation Coupling among the three modes: scalar ,vector, and tensor Due to the direct change of the gravitational field Due to the back reaction throughφ Stochastic gravity These quantum effects are described by the propagator of the scalar field. These three modes are not independent each other due to the non-linear effect of the scalar field. Coupling １．Stochastic variable ξab　（←Quantum fluctuations of the scalar field ） Tensor type and vector type equation also have an-isotropic pressure of ξab. ２．Memory term scalar + vector + tensor δgab

5. Effective action ~ H2 ηV ≡ V’’/ κ2 V Global feature of the inflation model 〔Quantum correction ofφ〕 quantum fluctuation As its coefficient, the vertex operators include the information of the potential. １．Vertex operator In principle, the higher loop correction include the more global information of the potential. In case, the slow-roll condition are satisfied.εSR ≡ε，ηV, ηH, η αm = O ((εSR)m/2) ← We can prove by the mathematical deduction. ２．Propagator The propagator depends on the evolution of the background spacetime.

6. １．Vertex αm = O ((εSR)m/2) ２．Propagator ~ H2 coupling between g and φ ψ h h ψ (κH)2 Effective action α1 α1 h ψ h α1 (κH)2 α1 ψ (κH)2 h h h h ψ (κH)2 (κH)2 α3 α3 ψ (κH)2 α4 Loop corrections ◆(κH)2 [tree graph] vertexh ・ ψ　←　This interaction is included in linear analysis. V’(φ), H ◆(κH)4 [loop graph] (κH)2 + V’’(φ) α2 α2 (κH)2 ◆(κH)6 [loop graph] (κH)2 + V(3)(φ) + V(4)(φ)

7. Scalar perturbations Metric ansatz ・ Gauge condition for scalar perturbations two-component Einstein equation Perturbations of Einstein-Langevin equation ・ We have neglected vector perturbations. superhorizon limit Memory term includes the metric perturbations. But they are suppressed by slow-roll parameters.

8. Tensor perturbations Metric ansatz ・ Gauge condition for scalar perturbations Perturbations of Einstein-Langevin equation + Slow-roll parameter constant ・ We have neglected vector perturbations. c.f. Linear perturbation → source free

9. Comparison with preceding researches Global time dependence among Loop corrections Comparison ζ φ σ h ζ ζ h ζ σ φ ζ S. Weinberg (2005), (2006) Up to second order perturbations scalar perturbations & inflaton → curvature perturbation in comoving slicingζ tensor perturbations → γ massless scalar field　（→ σ） , fermions, etc 【 Interaction 】 Stochastic gravity 【Results 】 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) Number of σ field Constant number c.f.

10. ηV ≡ V’’/ κ2 V ψ ψ q q k k k k h h h h Infrared divergence Ultraviolet divergence ψ ψ k - q k - q Renormalization [ Initial condition ] in superhorizon region – k τ<< 1 This infrared divergence is due to the unphysical initial condition. We cannot impose this initial condition to the mode which was outside horizon on the beginning of the inflation. To avoid this unphysical divergence, we have introduced the cutoff Hi = H (τi) . (i.e. q ≧ Hi) Based on the discussion with A.A.Starobinsky If m > 0, this ultraviolet divergence part shall decay in superhorizon region. We have neglected this decaying part in superhorizon region. c.f. Physical meaning of the neglection of the decaying mode A.A.Starobinsky C.Q.G. 13 (1996) 377

11. Global time dependence on the potential of inflaton among loop corrections Summary ・ Vertex operator ◆(κH)2 [tree graph] V’(φ), H 　　　←　線形摂動 ◆(κH)4 [loop graph] + V’’(φ) ◆(κH)6 [loop graph] + V(3)(φ)＆　V(4)(φ) ・ Evolution of the background field ◆(κH)4 [loop graph] Scalar perturbation もTensor perturbationも、super horizon でほぼ一定。 Tensor perturbationの方がslow-roll parameterに対する依存性が弱い。