Preheating in Derivatively-Coupled Inﬂation Models Cristian Armendariz-Picon * , Mark Trodden † , Eric J. West ‡ Cosmology Group, Department of Physics, Syracuse University, Syracuse, NY 13244-1130, USA We study preheating in theories where the inﬂaton couples derivatively to scalar and gauge ﬁelds. Such couplings may dominate in natural models of inﬂation, in which the ﬂatness of the inﬂaton potential is related to an approximate shift symmetry of the inﬂaton. We compare our results with previously studied models with non-derivative couplings. For suﬃciently heavy scalar matter, parametric resonance is ineﬀective in reheating the universe, because the couplings of the inﬂaton to matter are very weak. If scalar matter ﬁelds are light, derivative couplings lead to a mild long- wavelength instability that drives matter ﬁelds to non-zero expectation values. In this case however, long-wavelength ﬂuctuations of the light scalar are produced during inﬂation, leading to a host of cosmological problems. In contrast, axion-like couplings of the inﬂaton to a gauge ﬁeld do not lead to production of long-wavelength ﬂuctuations during inﬂation. However, again because of the weakness of the couplings to the inﬂaton, parametric resonance is not eﬀective in producing gauge ﬁeld quanta. I. INTRODUCTION During a suﬃciently long epoch of inﬂation, matter is diluted away by the quasi-exponential expansion of the universe. The re-population of the universe with radiation after the end of inﬂation is then typically achieved by the decay of the inﬂaton into ordinary mat- ter particles. The perturbative theory of this process— reheating [1, 2]—was reﬁned by the discovery [3, 4, 5] that in many models the dynamics would proceed through a stage of parametric resonance, leading to the extremely eﬃcient decay of the inﬂaton into a far from equilibrium distribution of matter. The understanding of this prelim- inary stage of reheating—preheating—has since been de- veloped by many authors [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Most work on preheating has focused on models with direct, non-derivative, couplings to matter. Little is known about the strength of these couplings, but the- oretical arguments suggest that they have to be rather weak. Indeed, in order for the inﬂaton to drive a phe- nomenologically acceptable stage of inﬂation, its poten- tial has to be extremely “ﬂat”. A variety of diﬀerent ways to stabilize these ﬂat potentials have been stud- ied [16], but perhaps the most compelling idea that has emerged to date is that inﬂation is driven by a pseudo- Nambu-Goldstone boson [17]. In the simplest realization of this idea, the inﬂaton sector is invariant under an ap- proximate global U (1) symmetry that shifts the inﬂaton ﬁeld by a constant. Because only a constant potential is invariant under shifts of the inﬂaton, this approximate symmetry guarantees that deviations from ﬂatness are small. But at the same time, because direct couplings between the inﬂaton and matter violate the shift sym- metry, they are are also expected to be extremely weak, and perhaps even negligible. * armen@physics.syr.edu † trodden@physics.syr.edu ‡ ejwest@physics.syr.edu On the other hand, derivative couplings of the inﬂaton to matter do satisfy the shift symmetry, and so there is no reason for them to be particularly weak. It is there- fore entirely possible that derivative couplings could be more important than non-derivative ones during reheat- ing. Studies of preheating in models with derivative cou- plings [18, 19] have been less extensive than those of di- rect couplings, and therefore our aim here is to carry out one such study: examining preheating in these mod- els and determining whether they lead to a qualitatively diﬀerent picture of the end of inﬂation and the onset of reheating. In the next section we will review the standard ideas of preheating in models with direct couplings between the inﬂaton and matter. In section III we then motivate the study of derivatively coupled models and carry out the related preheating calculations, including couplings to both scalar and gauge ﬁelds, before concluding. II. PARAMETRIC RESONANCE IN CANONICAL MODELS We begin by reviewing the basic results about preheat- ing in a simple model with a direct coupling between the inﬂaton and matter ﬁelds. For detailed methods and re- sults regarding the physics of reheating and preheating, we refer the reader to [15, 20]. For simplicity we assume inﬂation is driven by a single real scalar ﬁeld φ, slow-rolling down a quadratic eﬀec- tive potential, and coupled to a massive real scalar ﬁeld χ (representing matter ﬁelds) through a quartic interac- tion, L = − 1 2 (∂ μ φ)∂ μ φ− 1 2 (∂ μ χ)∂ μ χ− m 2 φ 2 φ 2 − m 2 χ 2 χ 2 − g 2 2 φ 2 χ 2 . (1) Throughout we treat the inﬂaton ﬁeld as spa- tially homogeneous and work in a spatially ﬂat Friedmann-Robertson-Walker universe with metric ds 2 = −dt 2 + a 2 (t) dx 2 . arXiv:0707.2177v2 [hep-ph] 27 Nov 2007