1. IRGAC 2006 1 Cosmological perturbations in stochastic gravity Yuko Urakawa with Kei-ichi Maeda

2. IRGAC 20062 Quantum effects of the scalar field during inflation [main topic] How to evaluate the quantum effects on the primordial perturbations? ・ Quantization of Mukhanov – Sasaki variable action canonical quantization There is the quantum effect of the scalar field up to the linear order. 1. The higher order quantum effect of the scalar field “Can we neglect the higher order quantum effect ?? ” Questions 2. The decoherence problem “How to relate the quantum correlation to the classical correlation which represents the space-time anisotropy ?? ” The treatment in open quantum system

3. IRGAC 20063 Open quantum system AB in case interested only in A AB coarse – graining B ( trace out ) ・ A is described by Langevin-type equation in semi-classical region, which includes higher order quantum effects. A.O.Caldeira, A.J.Leggett (1983) etc… during Inflation interacting system ( scalar field φ, gravitational field g) The treatment in Open quantum system interacting system ( A, B ) Gell-Mann and Hartle (1993) ・ In many cases, the decoherence is induced on A. “stochastic interpretation” → classical correlation Question 1Question 2

4. IRGAC 20064 BA g φ The gravitational field, which is affected by the quantum scalar field Stochastic gravity B.L.Hu and E.Verdaguer (1999) coarse – graining φ Einstein–Langevin equation Semi-classical Einstein equation back ground equation linear order equation (for the gravitational field) This includes the fluctuation of the energy-momentum tensor. ( → represented by the stochastic variable ξ) effective action → Langevin type equation

5. IRGAC 20065 Einstein-Langevin equation g → g + h the change of the gravitational field back reaction from g to φ x memory term The characteristic quantum effect of φ, described by Einstein-Langevin equation is…. 1. stochastic source 2. memory term ( This represents the fluctuation of ) bi-tensor H abcd & N abcd are given by the background quantity ( g, φ, |φ ＞ ) B.L.Hu and E.Verdaguer (1999)

6. IRGAC 20066 We can interpret as the ordinary two-component Einstein equation. Analysis of the Einstein – Langevin equation Set up massive scalar fieldminimal coupling back ground solution gravitational field → de Sitter State of the scalar field → Bunch – Davies vacuum linear perturbation Ordinary cosmological perturbation for two-component system can be applied !!

7. IRGAC 20067 → index (ex) ① Introduction of perturbed variables metric @ Longitudinal gauge Φ and Ψ are gauge invariant variables. matter (two-component system) → index ( ξ ) ② back ground p = - ρ back ground none

8. IRGAC 20068 horizon crosssub-horizon super-horizon variable transformation Linear perturbations 1 from perturbed Einstein equation A, B ・・・ constant variables Then, let us substitute the explicit form of pπ ex and δp ex.

9. IRGAC 20069 Linear perturbations 2 so complicated …. noise memory term

10. IRGAC 200610 Quantum fluctuation noise N abcd is given by the background quantity ( g, φ, |φ ＞ ). from coincidence limit N abcd (x,x) ・ x・ x ・ y・ y If x ~ y, distribution function is independent on x or y. t r [assumption]

11. IRGAC 200611 Linear perturbations 3 Quantum effect of the scalar field 1 : noise term anisotropic pressure isotropic pressure Anisotropic pressure is more effective as the quantum effect of the scalar field. x memory Quantum effect of the scalar field 2 : memory term C, D ・・・ constant variables now analyzing memory term The memory term plays an important role, which cannot be neglected.

12. IRGAC 200612 Future work Analysis of the memory term ・ To proceed the estimatation of this effect on the evolution of the curvature perturbation ・ To check the possibility of the amplification of the tensor mode. In case we solve the tensor type Einstein – Langevin equation, there might exist the tensor part of T (m)ab,which is amplified by h (s)ab. Then, the quantum fluctuation of the scalar field might be amplify the tensor perturbation. But, it is just possibility,then we should check it.

13. IRGAC 200613 AB xy Reduced density matrix The evolution of the reduced density matrix Assumption The description of open quantum system 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

14. IRGAC 200614 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

15. IRGAC 200615 Einstein–Langevin equation stochastic variable ξ （ Einstein-Langevin equation ） Effective action stochastic interpretation variation δ/ δ g Langevin type equation The basic equation in stochastic gravity back ground equation Semi-classical Einstein equation linear order equation This includes the fluctuation of the energy-momentum tensor. ( → represented by the stochastic variable ξ)

16. IRGAC 200616 The short summary Question to the ordinal method 1. The higher order quantum effect of the scalar field 2. The decoherence problem Stochastic gravity Einstein –Langevin equation can describe these quantum effect. Stochastic interpretation make it possible to get the classical stochastic correlation of the metric perturbation naturally. Einstein–Langevin equation ・ This includes the fluctuation of the energy-momentum tensor. ・ There are two characteristic quantum effect, noise term and memory term, described by bi-tensor H abcd and N abcd.