Nonlinear curvature perturbations in two-field hybrid inflation

1. cifar08/Lindefest 5-9 March, 2008 Nonlinear curvature perturbations in two-field hybrid inflation -- dN in exactly soluble models -- Yukawa Institute (YITP) Kyoto University Misao Sasaki

2. Happy Kanreki, Andrei! kanreki = one cycle of chinese zodiacal calendar 祝還暦 60 yrs = 12 animals x 5 elements chan-chanko clothing this (Andrei’s) year Red signifies a baby mouse x earth also, Red is a color of happiness reliability productivity A person is regarded as reborn on the 60th birthday

3. 1. Success of Inflation (slow-roll) inflation Linde ’82, ... explains the origin of cosmological perturbations. .... finally observed by COBE & WMAP! Inflation from string theory KKLMMT ’03, ... brane/DBI inflation, moduli inflation, ... may lead to large non-Gaussianity. −9 < fNLlocal < 111 (WMAP 5yr) may be detected in the very near future

4. Why care about soluble models? In this talk, I analyze full nonlinear curvature perturbations in exactly soluble models of slow-roll inflation Using dN formalism, (dN··· e-folding number perturbation) • We can explicitly see how and when curvature • perturbations are generated. • Non-Gaussianity can be explicitly evaluated. We may deepen our understanding of cosmological perturbations. tatemae (建前) hon-ne (本音) Because it’s fun!

6. 2. dN for curvature perturbations Starobinsky ‘85, MS & Stewart ‘96, MS & Tanaka '98, Lyth, Malik & MS ‘04,.... Separate universe (gradient expansion) approach Belinski et al. ’70, Tomita ’72, Salopek & Bond ’90, … • each Hubble region evolves independently • perturbations on superhorizon scales • ≈ difference between different FLRW universes • calculate dN for different FLRW universes ٠٠٠ curvature perturbation on comoving slice defined by an integral =non-local quantity → gauge invariance

7. dN for ‘slowroll-type’ inflation MS & Tanaka ’98, Lyth & Rodriguez ’05, ... • In slowroll inflation, all decaying mode solutions of the • (multi-component) scalar field f die out. → N=indep. of df/dt where df =dfF (on flat slice) at horizon-crossing. • The above formula is valid for any model in which • N is only a function of f but not of df/dt. (e.g., power-law inflation)

8. 3. Exactly soluble slow-roll models (a,b= 1,2,...,n) Slow-roll equations of motion e-folding number from the end of inflation

9. An exactly soluble class If for each a (a= 1,2,...,n) this is sufficient (slowroll unnecessary) Then ... new field space coordinates ··· na is conserved solvability  (n-1) constants of integration

10. simple examples product: sum:

11. trajectories are radial in space qa na q N=0 qb N=const.

12. e-folding number • Nonlinear dN is • Linear curvature perturbation is given by where (q, na) and (dq, dna) are the values at horizon crossing during inflation

13. curvature perturbation generated from adiabatic & “entropy” perturbations perturbations orthogonal to dqa/dN adiabatic perturbation entropy perturbation during inflation Polarski & Starobinsky ’94, Mukhanov & Steinhardt ’97, MS & Tanaka ’98,... entropy perturbation at the end of inflation Bernardeau, Kofman & Uzan ‘04, Lyth ‘05, ...

14. V f inflation 4. Two (plus one)-field hybrid inflation ∙ can be easily generalized to n fields “n-brid inflation” • slow-roll eom: • transformation of variables:

15. Assume that inflation ends at and the universe is thermalized instantaneously. realized by Parametrize orbits by an angle at the end of inflation

16. (∙∙∙ const of motion) This determines g in terms of f1 & f2 . whereg=g(f1,f2) • dN valid to full nonlinear order is simply given by

17. To be precise, one has to add a correction term to adjust • the energy density difference at the end of inflation where (assuming instantaneous thermalization) However, this correction is negligible if

18. dN to 2nd order in df: • comoving curvature perturbation spectrum spectral index: tensor/scalar: • single-field case No non-Gaussianity if dfis Gaussian

19. Let “true” entropy perturbation linear entropy perturbation contributes at 2nd order practically any non-Gaussianity is possible

20. numbers just in case... (respecting WMAP5yr) model parameters: outputs:

21. So, here is a birthday present for you, Andrei. hope you like it...

22. 5. Summary • Exactly soluble models are useful in understanding • generation of (nonlinear) curvature perturbations • Curvature perturbation may be generated from both • adiabatic and entropy perturbations during inflation • In multi-field models, final amplitude of curvature • perturbation depends crucially on how inflation ends. • n-brid inflation looks like an almighty model! (classically. negligible quantum corrections assumed)