Eiichiro Komatsu & Yuki Watanabe University of Texas at Austin

1. Reheating of the Universe after Inflation with f(f)R Gravity:Spontaneous Decay of Inflatons to Bosons, Fermions, and Gauge Bosons Eiichiro Komatsu & Yuki Watanabe University of Texas at Austin Caltech High Energy Physics Seminar, March 28, 2007 Reference: Phys. Rev. D 75, 061301(R) (2007)

2. Why Study Reheating? • The universe was left cold and empty after inflation. • But, we need a hot Big Bang cosmology. • The universe must reheat after inflation. • Successful inflation must transfer energy in inflaton to radiation, and heat the universe to at least ~1 MeV for successful nucleosynthesis. • …however, little is known about this important epoch…. • Outstanding Questions • Can one reheat universe successfully/naturally? • How much do we know about reheating? • What can we learn from observations (if possible at all)? • Can we use reheating to constrain inflationary models? • Can we use inflation to constrain reheating mechanism?

3. Standard Picture Slow-roll Inflation: potential shape is arbitrary here, as long as it is flat. Oscillation Phase: around the potential minimum at the end of inflation Energetics: What determines“energy-conversion efficiency factor”, g?

4. Perturbative ReheatingDolgov & Linde (1982); Abbott, Farhi & Wise (1982); Albrecht et al. (1982) Inflaton decays and thermalizes through the tree-level interactions like: c y f f c y Inflaton can decay if allowed kinematically with the widths given by Pauli blocking Bose condensate Thermal medium effect

5. Reheating Temperature from Energetics Coupling constants determine the decay width,  But, what determines coupling constants?

6. Fine-tuning Problem? • To relax fine-tuning, one needs: • High reheat temperature -> unwanted relics (e.g., gravitinos), • Very low-scale inflation (H~10-18 Mpl~10 GeV) -> worse fine-tuning, or • Natural explanation for the smallness of g.

7. What are coupling constants? Problem: arbitrariness of the nature of inflaton fields • Inflation works very well as a concept, but we do not understand the nature (including interaction properties) of inflaton. • Arbitrariness of inflaton = Arbitrariness of couplings • Can we say anything generic about reheating? Universal reheating? Universal coupling? e.g. Higgs-like scalar fields, Axion-like fields, Flat directions, RH sneutrino, Moduli fields, Distances between branes, and many more… Gravitational coupling is universal -> however, too weak to cause reheating with GR. In the early universe, however, GR would be modified. What happens to “gravitational decay channel”, when GR is modified?

8. Conventional Einstein gravity during inflation Einstein-Hilbert term generates GR. Inflaton minimally couples to gravity. Conventionally one had to introduce explicit couplings between inflaton and matter fields by hand.

9. Modifying Einstein gravity during inflation Instead of introducing explicit couplings by hand, Non-minimal gravitational coupling: common in effective Lagrangian from e.g., extra dimensional theories. In order to ensure GR after inflation, Matter (everything but gravity and inflaton) completely decouples from inflaton and minimally coupled to gravity as usual.

10. Field equations: GR Linearized field equation: Wave modes are gravitational waves. To identify the wave modes, we usually define Harmonic (Lorenz) gauge:

11. Field equations: f()R gravity Linearized field equation during coherent oscillation Wave modes are mixed up. To extract “true” gravitational degrees of freedom, we define Harmonic (Lorenz) gauge:

12. New decay channel through “scalar gravity waves” y s Fermionic (spinor) matter field: y Yukawa interaction Bosonic (scalar) matter field: c s Three-legged interaction c

13. Spontaneous emergence of Yukawa interaction: analog of spontaneous symmetry breaking -v v Expanding  around the vev, one gets New term appeared.

14. Spontaneous emergence of Yukawa interaction: analog of spontaneous symmetry breaking e.g. Expanding  around the vev, one gets Wave mode mixing in the linear perturbations (appearance of scalar gravity waves) Yukawa interactions are induced.

15. Magnitude of Yukawa coupling • For f()=g=(v/Mpl)(m/Mpl) • Natural to obtain g~10-7 for e.g., m~10-7 Mpl • The induced Yukawa coupling vanishes for massless fermions: conformal invariance of massless fermions. • Massless, minimally-coupled scalar fields are not conformally invariant. Therefore, the three-legged interaction does not vanish even for massless scalar fields:

16. The Results So Far… Phys. Rev. D 75, 061301(R) (2007) • After inflation with f()R gravity, inflatons decay spontaneously into: • Massive fermions, • Massive scalar bosons, or • Massless scalar bosons with non-conformal coupling. • The smallness of coupling can be explained naturally. • Inflaton decay is “built-in” and the coupling constrants can be calculated explicitly from a single function, f(). • Rates of decay to fermions and bosons are related. • This mechanism allows inflatons to decay into any fields that are not conformally coupled. • Other possibilities?

17. Breaking of conformal invariance by anomaly Conformally coupled fields at the tree-level may not be conformally invariant when loops are included. Example: decay to massless gauge bosons, F F f F (c.f.) two-photon decay of the Higgs

18. Conformal anomaly: Lowest order decay channel to massless gauge fields F f F Inflaton -> 2 gauge fields

19. Decay Width Summary Fermions Scalar Bosons Probably the most dominant decay channels Gauge Bosons

20. Constraint on f(f)R gravity models from reheating Constraints from chaotic inflation e.g.

21. Connection to Supergravity? • Similar effects have been pointed out by Endo, Takahashi and Yanagida (2006; 2007) in the context of supergravity • Inflatons decay into any fields even if inflatons are not coupled directly with these fields in the superpotential • A correspondence may be made as • f()R gravity <-> Kahler potential • Conformal anomaly <-> Super-Weyl anomaly • Our model is simpler and does not require explicit use of supergravity -- hence more general. • It may also give physical (rather than mathematical) insight into their effects.

22. Conclusions • A natural mechanism for reheating after inflation with f(f)R gravity: Why natural? • Inflaton quanta decay spontaneously into any matter fields (spin-0, ½, 1) without explicit interactions in the original Lagrangian • Conformal invariance must be broken at the tree-level or by loops • Reheating spontaneously occurs in any theories with f(f)R gravity • Predictability • All the decay widths are related through a single function, f(f). • A constraint on f(f) from the reheat temperature can be found • A possible limit on the reheat temperature can constrain the form of f(f), or vice versa. • These constraints on f(f) are totally independent of the other constraints from inflation and density fluctuations Further Study… • Preheating? F(,R) gravity?