Second order density perturbations for dust cosmologies

Abstract

We present simple expressions for the relativistic first and second order fractional density perturbations for FL cosmologies with dust, in four different gauges: the Poisson, uniform curvature, total matter and synchronous gauges. We include a cosmological constant and arbitrary spatial curvature in the background. A distinctive feature of our approach is our description of the spatial dependence of the perturbations using a canonical set of quadratic differential expressions involving an arbitrary spatial function that arises as a conserved quantity. This enables us to unify, simplify and extend previous seemingly disparate results. We use the primordial matter and metric perturbations that emerge at the end of the inflationary epoch to determine the additional arbitrary spatial function that arises when integrating the second order perturbation equations. This introduces a non-Gaussianity parameter into the expressions for the second order density perturbation. In the special case of zero spatial curvature we show that the time evolution simplifies significantly, and requires the use of only two non-elementary functions, the so-called growth supression factor at the linear level, and one new function at the second order level. We expect that the results will be useful in applications, for example, studying the effects of primordial non-Gaussianity on the large scale structure of the universe.

28 pages; changes in the introduction and some simplified expressions in the main text together with some added references

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Cited by in corpus (10)

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• Dynamics of cosmological perturbations at first and second order
• Second order cosmological perturbations: simplified gauge change formulas
• fNL - gNL mixing in the matter density field at higher orders
• Scalar field evolution at background and perturbation levels for a broad class of potentials
• Second-order Gauge-invariant Cosmological Perturbation Theory: Current Status updated in 2019
• Relativistic perturbations in $\Lambda$CDM: Eulerian & Lagrangian approaches