Gauge-invariant perturbations in a scalar field dominated universe

Abstract

The authors propose a new covariant and gauge-invariant (GI) treatment of perturbations in a Robertson-Walker universe dominated by a classical scalar field phi . They first set up the formalism, based on the natural slicing of the problem by the surfaces phi =constant, and introduce a set of covariantly defined GI variables. In their approach the whole inhomogeneity of the matter field is incorporated in the GI spatial fluctuations of the momentum psi of phi ; then the GI density perturbations are simply proportional to the momentum perturbations. The inhomogeneity of the geometry is characterized by GI fluctuations of the 3-curvature scalar of the surfaces phi =constant. The time evolution of the matter and curvature perturbations are coupled by a pair of first-order linear differential equations. Correspondingly, each GI variable satisfies a second-order linear homogeneous differential equation. When the background curvature vanishes, k=0, the curvature variable is conserved for perturbation scales larger than the horizon, but this is no longer true in general if k not=0. They discuss simple examples, including the case when more than one scalar field is present, recovering standard results for inflationary universe models. They also demonstrate that in coasting solutions with k=-1, inhomogeneities are damped out on all scales.

Gauge invariant perturbations of scalar-tensor cosmologies: The vacuum case

“…which is equivalent to the result found in [11], and we conclude that, in this case, ∆ characterises scalar field clumping. However, this is not true in general for a non-minimally coupled scalar field because (47) is related to the variable C, the details of the coupling function F and its derivative, as well as specific features of the background like Θ and the sign of φ.…”

“…and its comoving divergence Φ defined in [11]. This quantity is equal, up to a constant, to the combination…”

Braneworld dynamics of inflationary cosmologies with exponential potentials

“…the energy -momentum tensor T µν for a scalar field takes the form of a perfect fluid (See page 17 in [18] for details):…”

Adiabatic and entropy perturbations from inflation

“…[23]. We will work in the metric based approach of Bardeen [24] in order to define gaugeinvariant cosmological perturbations, but our formalism can also be applied to the study of multiple scalar fields in other approaches [25][26][27].…”