A Concise Introduction to Perturbation Theory in Cosmology

We give a concise, self-contained introduction to perturbation theory in cosmology at linear and second order, striking a balance between mathematical rigour and usability. In particular we discuss gauge issues and the active and passive approach to calculating gauge transformations. We also construct gauge-invariant variables, including the second order tensor perturbation on uniform curvature hypersurfaces.Comment: revtex4, 16 pages, 3 figures; v2: minor changes, typos corrected, reference added, version accepted by CQ

Nonlinear curvature perturbations in an exactly soluble model of multi-component slow-roll inflation

Using the nonlinear $\delta N$ formalism, we consider a simple exactly soluble model of multi-component slow-roll inflation in which the nonlinear curvature perturbation can be evaluated analytically.Comment: 4 pages, no figure, typos corrected, references added, final version to be published in CQ

Adiabatic Modes in Cosmology

We show that the field equations for cosmological perturbations in Newtonian gauge always have an adiabatic solution, for which a quantity ${\cal R}$ is non-zero and constant in all eras in the limit of large wavelength, so that it can be used to connect observed cosmological fluctuations in this mode with those at very early times. There is also a second adiabatic mode, for which ${\cal R}$ vanishes for large wavelength, and in general there may be non-adiabatic modes as well. These conclusions apply in all eras and whatever the constituents of the universe, under only a mild technical assumption about the wavelength dependence of the field equations for large wave length. In the absence of anisotropic inertia, the perturbations in the adiabatic modes are given for large wavelength by universal formulas in terms of the Robertson--Walker scale factor. We discuss an apparent discrepancy between these results and what appears to be a conservation law in all modes found for large wavelength in synchronous gauge: it turns out that, although equivalent, synchronous and Newtonian gauges suggest inequivalent assumptions about the behavior of the perturbations for large wavelength.Comment: 24 pages, Latex, no special macro

Non-linear isocurvature perturbations and non-Gaussianities

We study non-linear primordial adiabatic and isocurvature perturbations and their non-Gaussianity. After giving a general formulation in the context of an extended delta N-formalism, we analyse in detail two illustrative examples. The first is a mixed curvaton-inflaton scenario in which fluctuations of both the inflaton and a curvaton (a light isocurvature field during inflation) contribute to the primordial density perturbation. The second example is that of double inflation involving two decoupled massive scalar fields during inflation. In the mixed curvaton-inflaton scenario we find that the bispectrum of primordial isocurvature perturbations may be large and comparable to the bispectrum of adiabatic curvature perturbations.Comment: 24 pages, typos corrected, references adde

Non-gaussianity from the inflationary trispectrum

We present an estimate for the non-linear parameter \tau_NL, which measures the non-gaussianity imprinted in the trispectrum of the comoving curvature perturbation, \zeta. Our estimate is valid throughout the inflationary era, until the slow-roll approximation breaks down, and takes into account the evolution of perturbations on superhorizon scales. We find that the non-gaussianity is always small if the field values at the end of inflation are negligible when compared to their values at horizon crossing. Under the same assumption, we show that in Nflation-type scenarios, where the potential is a sum of monomials, the non-gaussianity measured by \tau_NL is independent of the couplings and initial conditions.Comment: 15 pages, uses iopart.sty. Replaced with version accepted by JCAP; journal reference adde

Inhomogeneous vacuum energy

Vacuum energy remains the simplest model of dark energy which could drive the accelerated expansion of the Universe without necessarily introducing any new degrees of freedom. Inhomogeneous vacuum energy is necessarily interacting in general relativity. Although the four-velocity of vacuum energy is undefined, an interacting vacuum has an energy transfer and the vacuum energy defines a particular foliation of spacetime with spatially homogeneous vacuum energy in cosmological solutions. It is possible to give a consistent description of vacuum dynamics and in particular the relativistic equations of motion for inhomogeneous perturbations given a covariant prescription for the vacuum energy, or equivalently the energy transfer four-vector, and we construct gauge-invariant vacuum perturbations. We show that any dark energy cosmology can be decomposed into an interacting vacuum+matter cosmology whose inhomogeneous perturbations obey simple first-order equations.Comment: 8 pages; v2 clarified discussion of Chaplygin gas model, references adde

Non-Gaussian perturbations from multi-field inflation

We show how the primordial bispectrum of density perturbations from inflation may be characterised in terms of manifestly gauge-invariant cosmological perturbations at second order. The primordial metric perturbation, zeta, describing the perturbed expansion of uniform-density hypersurfaces on large scales is related to scalar field perturbations on unperturbed (spatially-flat) hypersurfaces at first- and second-order. The bispectrum of the metric perturbation is thus composed of (i) a local contribution due to the second-order gauge-transformation, and (ii) the instrinsic bispectrum of the field perturbations on spatially flat hypersurfaces. We generalise previous results to allow for scale-dependence of the scalar field power spectra and correlations that can develop between fields on super-Hubble scales.Comment: 11 pages, RevTex; minor changes to text; conclusions unchanged; version to appear in JCA

Non-Gaussianities in two-field inflation

We study the bispectrum of the curvature perturbation on uniform energy density hypersurfaces in models of inflation with two scalar fields evolving simultaneously. In the case of a separable potential, it is possible to compute the curvature perturbation up to second order in the perturbations, generated on large scales due to the presence of non-adiabatic perturbations, by employing the $\delta N$-formalism, in the slow-roll approximation. In this case, we provide an analytic formula for the nonlinear parameter $f_{NL}$. We apply this formula to double inflation with two massive fields, showing that it does not generate significant non-Gaussianity; the nonlinear parameter at the end of inflation is slow-roll suppressed. Finally, we develop a numerical method for generic two-field models of inflation, which allows us to go beyond the slow-roll approximation and confirms our analytic results for double inflation.Comment: 29 pages, 6 figures. v2, comparison with previous estimates. v3, JCAP version; Revisions based on Referee's comment, corrected typos, added few eqs and refs, conclusions unchange

The inflationary trispectrum

We calculate the trispectrum of the primordial curvature perturbation generated by an epoch of slow-roll inflation in the early universe, and demonstrate that the non-gaussian signature imprinted at horizon crossing is unobservably small, of order tau_NL < r/50, where r < 1 is the tensor-to-scalar ratio. Therefore any primordial non-gaussianity observed in future microwave background experiments is likely to have been synthesized by gravitational effects on superhorizon scales. We discuss the application of Maldacena's consistency condition to the trispectrum.Comment: 23 pages, 2 diagrams drawn with feynmp.sty, uses iopart.cls. v2, replaced with version accepted by JCAP. Estimate of maximal tau_NL refined in Section 5, resulting in smaller numerical value. Sign errors in Eq. (44) and Eq. (48) corrected. Some minor notational change

Evolution of Second-Order Cosmological Perturbations and Non-Gaussianity

We present a second-order gauge-invariant formalism to study the evolution of curvature perturbations in a Friedmann-Robertson-Walker universe filled by multiple interacting fluids. We apply such a general formalism to describe the evolution of the second-order curvature perturbations in the standard one-single field inflation, in the curvaton and in the inhomogeneous reheating scenarios for the generation of the cosmological perturbations. Moreover, we provide the exact expression for the second-order temperature anisotropies on large scales, including second-order gravitational effects and extend the well-known formula for the Sachs-Wolfe effect at linear order. Our findings clarify what is the exact non-linearity parameter f_NL entering in the determination of higher-order statistics such as the bispectrum of Cosmic Microwave Background temperature anisotropies. Finally, we compute the level of non-Gaussianity in each scenario for the creation of cosmological perturbations.Comment: 14 pages, LaTeX file. Further comments adde