Superhorizon perturbations and preheating

We discuss the evolution of linear perturbations about a Friedmann– Robertson–Walker background metric, using only the local conservation of energy– momentum. We show that on sufficiently large scales the curvature perturbation on spatial hypersurfaces of uniform-density is constant when the non-adiabatic pressure perturbation is negligible. We clarify the conditions under which super-horizon curvature perturbations may vary, using preheating as an example. Structure in the Universe is generally supposed to originate from the quantum fluctuation of the inflaton field. As each scale leaves the horizon during inflation, the fluctuation freezes in, to become a perturbation of the classical field. The resulting cosmological inhomogeneity is commonly characterized by the intrinsic curvature of spatial hypersurfaces defined with respect to the matter. This metric perturbation is a crucial quantity, because at approach of horizon re-entry after inflation it determines the adiabatic perturbations of the various components of the cosmic fluid, which seem to give a good account of large-scale structure [1]. To compare the inflationary prediction for the curvature perturbation with observation, we need to know its evolution outside the horizon, through the end of inflation, until re-entry on each cosmologically relevant scale. The standard assumption is that the curvature perturbation is practically constant. This has recently been called into question in the context of preheating models [2] at the end of inflation where non-inflaton perturbations can be resonantly amplified [3,4]. We investigate the circumstances under which the curvature perturbation may vary. Using only the local conservation of energy–momentum, we show that the rate of change of the curvature perturbation on uniform-density hypersurfaces, ζ , on large scales is due to the non-adiabatic part of the pressure perturbation. This result is independent of the form of the gravitational field equations, demonstrating for the first time that the curvature perturbation remains constant on large scales for purely adiabatic perturbations in any relativistic theory of gravity where the energy–momentum tensor is covariantly conserved. In contrast with the usual approach to cosmological perturbation theory, we shall not invoke any gravitational field equations. General coordinate invariance implies the energy-momentum conservation law T ν;μ = 0, without invoking the Einstein field equations. The pressure perturbation must be adiabatic if there is a definite equation of state for the pressure as a function of density, which is the case during both radiation domination and matter domination. On the other hand, a change in ζ on superhorizon scales will occur during the transition from matter to radiation domination if there is an isocurvature matter density perturbation [6,7]. A simple intuitive understanding of how the curvature perturbation on large scales changes, due to the different integrated expansion in locally homogeneous but causally-disconnected regions of the universe, can be obtained within the ‘separate universes’ picture which we described in [8]. I LINEAR SCALAR PERTURBATIONS The line element allowing arbitrary linear scalar perturbations of a Friedmann– Robertson–Walker (FRW) background can be written [9–12] ds = −(1 + 2A)dt + 2a(t)∇iB dx dt+ a(t) [(1− 2ψ)γij + 2∇i∇jE] dx dx . (1) The unperturbed spatial metric for a space of constant curvature κ is given by γij and covariant derivatives with respect to this metric are denoted by ∇i. The curvature perturbation on fixed-t hypersurfaces, ψ, is a gauge-dependent quantity and under an arbitrary linear coordinate transformation, t → t + δt, it transforms as ψ → ψ + Hδt. On uniform-density hypersurfaces the curvature perturbation can be written as −ζ = ψ +H δρ ρ̇ . (2) The curvature perturbation on uniform-density hypersurfaces, ζ , is often chosen as a convenient gauge-invariant definition of the scalar metric perturbation on large scales. These hypersurfaces become ill-defined if the density is not strictly decreasing, as can occur in a scalar field dominated universe when the kinetic energy of the scalar field vanishes. In this case one can instead work in terms of the density 1) The “conserved quantity” ζ was originally defined in Bardeen, Steinhardt and Turner [5], but constructed from perturbations defined in the uniform Hubble-constant gauge. 2) The sign of ζ is chosen here to coincide with Refs. [5]. perturbation on uniform-curvature hypersurfaces, δρψ = δρ+ ρ̇ψ/H, which remains finite. The pressure perturbation (in any gauge) can be split into adiabatic and entropic (non-adiabatic) parts, by writing δp = c2sδρ+ ṗΓ , (3) where c2s ≡ ṗ/ρ̇. The non-adiabatic part is δpnad ≡ ṗΓ, and Γ ≡ δp/ṗ− δρ/ρ̇. The entropy perturbation Γ, defined in this way, is gauge-invariant, and represents the displacement between hypersurfaces of uniform pressure and uniform density. II EVOLUTION OF THE CURVATURE PERTURBATION The energy conservation equation nT ν;μ = 0 for first-order density perturbations gives δ̇ρ = −3H(δρ+ δp) + (ρ+ p) [