Yoshiharu Tanaka (YITP)

1. Finnish-Japanese Workshop on Particle Cosmology @ Helsinki, 9 March, 2007 Gradient expansion approach to nonlinear superhorizon perturbations Yoshiharu Tanaka (YITP) Y. Tanaka & M. Sasaki, gr-qc/0612191(to be published in PTP) Y. Tanaka & M. Sasaki, in preparation

2. 1.Introduction Deviations from Gaussian statics of CMB anisotropy can be a powerful probe for the early universe Gravitational potential (which relates directly to CMB anisotropy ) (WMAP+SDSS) Standard single fieldslow-roll inflation predicts very small Other scenarios Curvaton scenario, inhomogeneous reheating scenario, ghost inflation DBI inflation …… can make large non-Gaussianity! Non-Gaussianity will be a smoking gun for these inflation models ! Non-Gaussianity is produced by interaction of fields Thus, we need to go beyond linear theory !

3. We take gradient expansion approach toward non-linear theory We consider fluctuations whose typical scale L is larger than Hubble horizon scale, 1/H Expand equations as a power series in εand solve iteratively ε= expansion parameter The solutions are effective only on superhorizon scales, but full non-linear !

4. Previous works ( Lifshitz & Khalatnikov ’60, Tomita ’72、 ’75、Muller et al. ’89, Salopek & Bond ’90・・・・・) ･　Most authors worked in the synchronous gauge. The gauge doesn’t fix time coordinate uniquely. Gauge modes appear. ∙∙∙ inconvenient ･　On the other hand, there exists a convenient gauge (as uniform Hubble slicing) in which gauge invariant nonlinear scalar perturbation is conserved on superhorizon for adiabatic case, neglecting all the spatial gradients. cf. Lyth, Malik, Sasaki ’04 But, gradient expansion on the covenient gauge, keeping second order gradients is still not formulated. is important to study non slow-roll models. ･　Scalar, vector, and tensor modes have not been identified clearly. Correspondence to gauge-inv. linear pert. theory was unclear. Further investigations on nonlinear perturbations in gradient expansion are needed. We formulate gradient expansion on appropriate slicing to and study the correspondence to gauge-inv. linear pert. theory.

5. terms are important to study non slow-roll models Linear perturbation equation for curvature perturbation, R superhorizon scales: Slow-roll Non slow-roll In non slow-roll regime, R is not conserved, but enhanced, or damped on superhorizon.

6. 2. Gradient expansion for a single-scalar system for local Friedmann eq. to hold Assumptions Fixing , limit As ε→ 0, locally observable universe becomes homogeneous and isotropic universe For simplicity, Cf. Lyth, Malik & Sasaki ’04 Stress-energy tensor

7. Onuniform Hubble slicing = uniform which fixes the time coordinate uniquely, so time dependent gauge modes do not appear Einstein equations yield Cf. Shibata & Sasaki ’99

8. Basic equations Klein Gordon equation on uniform Hubble slicing with

9. Basic equations (continued) Einstein equations on uniform Hubble slicing with Hamiltonian constraint Momentum constraint Evolution equations

10. Solution represented by four arbitrary spatial dependent scalars and tensors satisfy Friedmann equation

11. Mode identification (scalar and tensor modes; no vector for a scalar) Momentum constraint Gravitational waves should not contribute to R.H.S. of this constraint. can be decomposed to longitudinal part and Transverse-Traceless part uniquely (Gravitational waves) (Cf. York 1972) GWs are conformally invariant, determined non-locally and can be generated by nonlinear interactions of only scalar modes

12. Counting the physical degrees of freedom Counting d.o.f. contained in four arbitrary scalars and tensors 1 Momentum constraints relate to : 5 5 1 Total: 9 d.o.f. Counting the physical d.o.f. Scalar field : growing mode 1 + decaying mode 1 = 2 d.o.f. GW from metric : 2 d.o.f. GW from extrinsic curvature : 2 d.o.f. Total: 6 d.o.f. Remaining 3 d.o.f. are spatial gauge freedom: they are contained in Thus, : 1 (scalar growing mode) 2 (GW)=5 – 3 (spatial gauge) 2 (GW)=5 – 3 (constraints) : 1 (scalar decaying mode)

13. is the nonlinear generalization of gauge inv. linear scalar curvature perturbation Cf. Lyth, Malik & Sasaki ’04 In Starobinsky model (’93), analytic solution in slow-roll Friction-dominated …. Non slow-roll period ⇒ later, slow-roll again

14. Non slow-roll period terms decay during slow-roll, butmay become constant even on superhorizon scales if non slow-roll If terms were constant at horizon crossing, the curvature perturbation would change from its value at horizon crossing on superhorizon scales, because of terms’ decay at late times.

15. 3. Summary • We obtained the general solution to for the metric, scalar field, and especially the nonlinear scalar curvature perturbation on uniform Hubble slice with for a single-scalar system. • We identified the scalar and tensor modes in the general solution to in gradient expansion . • GWs are conformally invariant, and can be generated by nonlinear interactions of only scalar modes. • Issues: Calculation of non-Gaussianity generated in non slow-roll model. Extension to multi-scalar fields.