Quantum Field Theory in de Sitter space

1. Quantum Field Theory in de Sitter space Hiroyuki Kitamoto (Sokendai) with Yoshihisa Kitazawa (KEK,Sokendai) based on arXiv:1004.2451 [hep-th]

2. Introduction • Quantum field theory in de Sitter space concerns deep mysteries: inflation in the early universe and dark energy of the present universe • In a time dependent background like dS space, there is no stable vacuum • In such a background, Feynman-Dyson perturbation theory breaks down and we need to use Schwinger-Keldysh formalismN.C. Tsamis, R.P. Woodard S. Weinberg A.M. Polyakov

3. Our problem is related to non-equilibrium physics, for example, Boltzmann equation A.M. Polyakov • We derive a Boltzmann equation in dS space from a Schwinger-Dyson equation • We investigate the energy-momentum tensor of an interacting field theory to estimate the effective cosmological constant

4. Scalar field theory in dS space Poincare coordinate We rescale the field

5. Infra-red divergence If we apply Feynman rules to deal with the interaction, the integrations over time give rise to IR divergences at the infinite future Feynman-Dyson perturbation theory breaks down in dS space

6. Premise for Feynman rule In the Feynman-Dyson formalism, the vacuum expectation value is given by the transition amplitude from to in-out formalism This is because due to the time translational invariance

7. Schwinger-Keldysh formalism There is no time translational symmetry in dS space, and so we can’t prefix In this case, we can evaluate the vev only with respect to in-in formalism

8. Because there are two time indices （＋,－）, the propagator has 4 components At time vertexes, we sum (＋,－) indices like products of matrices The integration over time is manifestly finite due to causality

9. Boltzmann equation In a time dependent background, we need to consider excited states in general It is possible that the distribution function has time dependence due to the interaction

10. Intuitively, we can derive a Boltzmann equation from transition amplitude

11. If , in Minkowski space in dS space because energy doesn’t conserve This indicates the instability of dS space? A.M. Polyakov

12. Transition amplitude is based on in-out formalism • We should derive the Boltzmann equation on dS background in in-in formalism

13. Schwinger-Dyson equation = + ∑

14. We focus on the （－＋） component of the propagator There is explicit time dependence at the integration

15. We consider the following identity from Schwinger-Dyson equation Boltzmann equation can be derived from this identity L.P. Kadanoff, G. Baym L.V. Keldysh T. Kita A. Hohenegger, A. Kartavtsev, M. Lindner

16. Assumption We investigate propagators well inside the cosmological horizon: Second term is treated perturbatively Full propagator is

17. The left hand side: Time derivative

18. The right hand side: Collision term At the vertexes,

19. Because there is no time translational symmetry in dS space, the collision term has the on-shell part and the off-shell part In Minkowski space,

20. The on-shell term

21. The off-shell term

22. Infra-red effect The on-shell and off-shell parts have IR divergences at So we redefine the on-shell and off-shell parts by transferring the contribution of within the energy resolution to When , IR divergences cancel out in this procedure

23. Thermal distribution case When , IR divergence remains We focus on the case that the initial distribution function is thermal

26. Spectral weight The on-shell state weight is reduced to compensate the weight of off-shell states

27. Change of distribution function In the case, logarithmic divergent term remains but this term has a cut-off It leads to the change of distribution function

28. Mass renormalization Here we adopt a fixed physical UV cut-off counter term: virtuality:

29. We represent these results by physical quantities Explicit time dependence disappears when it is expressed by physical quantities

30. Effective cosmological constant Einstein equation: Is it possible that has time dependence?

31. Contribution from free field Conformal anomaly: From :

32. Contribution from the interaction gives growing time dependence to ? • Such effects screen the cosmological constant?

33. Contribution from inside the cosmological horizon We substitute the results of the Boltzmann equation to Although this effect reduces the cosmological constant, it vanishes as the universe cools down with time

34. Contribution from outside the cosmological horizon Inside the cosmological horizon, the degrees of freedom are constant because we adopt a fixed physical UV cut-off Outside the cosmological horizon, the degrees of freedom increase as time goes on : IR cut-off constant increase

35. We estimate this effect from outside the cosmological horizon in the case The leading contribution is log order in massless case

36. Increase in the degrees of freedom screens the cosmological constant is screened in theory The cosmological constant is also screened in theory N.C. Tsamis, R.P. Woodard

37. Conclusion • We have investigated an interacting scalar field theory in dS space in Schwinger-Keldysh formalism • We have investigated the time dependence of the propagator well inside the cosmological horizon by Boltzmann equation • We have found the nontrivial change of matter distribution function and spectral weight

38. However, explicit time dependence disappears when it is expressed by physical quantities • Contribution from inside the cosmological horizon doesn’t give time dependence to the cosmological constant except for cooling down • Increase in the degree of freedom outside the cosmological horizon screens the cosmological constant, and this effect grows as time goes on

39. Future work • Non-thermal distribution case • Physics around and beyond the cosmological horizon • Quantum effects of gravity • Non-perturbative effects