Cosmological perturbations in coherent oscillating scalar field models

HJE Cosmological perturbations in coherent oscillating J.A.R. Cembranos 0 1 A.L. Maroto 0 1 S.J. Nun~ez Jaren~o 0 1 0 Avenida Complutense s/n , Madrid , Spain 1 Departamento de F sica Teorica I, Universidad Complutense de Madrid The fact that fast oscillating homogeneous scalar elds behave as perfect uids in average and their intrinsic isotropy have made these models very fruitful in cosmology. In this work we will analyse the perturbations dynamics in these theories assuming general power law potentials V ( ) = j jn=n. At leading order in the wavenumber expansion, a simple expression for the e ective sound speed of perturbations is obtained ce2 = ! = Cosmology of Theories beyond the SM; Classical Theories of Gravity - 2)=(n + 2) with ! the e ective equation of state. We also obtain the rst order correction in k2=!e2 , when the wavenumber k of the perturbations is much smaller than the background oscillation frequency, !e . For the standard massive case we have also analysed general anharmonic contributions to the e ective sound speed. These results are reached through a perturbed version of the generalized virial theorem and also studying the exact system both in the super-Hubble limit, deriving the natural ansatz for ; and for sub-Hubble modes, exploiting Floquet's theorem. 1 Introduction 2 3 4 5 6 4.1 4.2 6.1 6.2 6.3 7 Conclusions 1 Introduction Perturbation of power law potential theories Comparing with the non-averaged solutions Super-Hubble analytic approach Sub-Hubble analytic approach High-k modes Rapidly evolving coherent scalar elds have been widely studied in cosmology. Their dynamics is not only important during the reheating epoch after in ation, but they can also support periods of accelerated expansion in both the early universe [1{5] or at late times [6{ 12]. Concerning the dark matter problem, non-thermal candidates like the axion [13{21] or other massive scalar [22{25] or pseudoscalar elds [26{32] also fall in this class. These models can be interpreted as Bose-Einstein condensates, where the scalar particles occupy the lowest quantum state of the potential [33{48]. Finally, the possibility of ultra-light elds as dark matter candidates has been explored in di erent works [49{58] by tuning appropriately the potential and initial conditions [54{58]. The general analysis of a homogeneous oscillating scalar eld in an expanding universe was performed by Turner in [59]. For a power-law potential V ( ) = scalar oscillations around the minimum of such a potential behave as a perfect uid with an e ective equation of state ! = (n 2)=(n + 2). His results can be recovered by means of a generalization of the virial theorem [60]. Recently, it has been shown that a fast oscillating abelian vector [61], non-abelian vector [62, 63] or arbitrary spin eld [64] will behave in a j jn=n, the rapid very similar way. The purpose of this work is to analyse the growth of perturbations in these coherent oscillating scalar theories for arbitrary power law potential. This subject has been mainly { 1 { studied for harmonic potential models that mimic the standard dark matter case [65{71], as it happens for the axion eld [72, 73]. It has been proved by using the linear perturbation theory that the axion was equivalent to CDM for high enough masses [74{77]. However, gravitational instabilities of oscillations in a harmonic potential are suppressed on small scales [72, 73, 78{80]. This analysis determines the cut-o in the matter power spectrum and its deviations with respect to the CDM phenomenology. On the other hand, the dynamical stability (ignoring metric perturbations) of general coherent oscillating scalar dark energy models has been analysed in di erent works [1, 11, 12, 60, 81{84], even by considering nonlinear evolutions [ 85 ]. They conclude that potentials supporting accelerated expansion are generically unstable with respect to the growth of inhomogeneities. This work is organized as follows: we will brie y review the standard average approach for the background evolution of a scalar eld under a power law potential (section 2), as well as set the equations that rule its perturbations (section 3). After the preliminary discussion, we will analyse the well-known case of a massive scalar by means of an adiabatic expansion approach (section 4). The perturbations evolution of power-law potential models will be studied following the average approach (section 5). Firstly, we will compute the e ective sound speed, which is in general the quantity that rules the evolution, using the perturbed version of the generalized virial theorem. This method allows to extend previous results to an arbitrary power-law potential. Also, exploiting this equation, we will be able to derive a general expression for a possible anharmonic correction in a massive scalar theory. After that (section 6), we will check the validity of the result for the e ective sound speed by studying the exact system of equations (non-a (...truncated)