Cosmology on all scales: A two-parameter perturbation expansion

Abstract

We propose and construct a two-parameter perturbative expansion around a Friedmann-Lemaître-Robertson-Walker geometry that can be used to model high-order gravitational effects in the presence of non-linear structure. This framework reduces to the weak-field and slow-motion postNewtonian treatment of gravity in the appropriate limits, but also includes the low-amplitude large-scale fluctuations that are important for cosmological modelling. We derive a set of field equations that can be applied to the late Universe, where non-linear structure exists on supercluster scales, and perform a detailed investigation of the associated gauge problem. This allows us to identify a consistent set of perturbed quantities in both the gravitational and matter sectors, and to construct a set of gauge-invariant quantities that correspond to each of them. The field equations, written in terms of these quantities, take on a relatively simple form, and allow the effects of small-scale structure on the large-scale properties of the Universe to be clearly identified. We find that inhomogeneous structures source the global expansion, that there exist new field equations at new orders, and that there is vector gravitational potential that is a hundred times larger than one might naively expect from cosmological perturbation theory. Finally, we expect our formalism to be of use for calculating relativistic effects in upcoming ultra-large-scale surveys, as the form of the gravitational coupling between small and large scales depends on the non-linearity of Einstein's equations, and occurs at what is normally thought of as first order in cosmological perturbations.PACS numbers: 98.80. Jk, 98.65.Dx, 04.25.Nx

Multi-scale perturbation theory. Part I. Methodology and leading-order bispectrum corrections in the matter-dominated era

“…The most pressing calculation, however, is that of δ (3) 2PPT . Using the results presented here, such a calculation is eminently feasible, and would provide a reasonable prediction for the relativistic P (k) 1−loop = P (k) 22 + P (k) 13 , at least down to k ∼ 10 −3 Mpc −1 . This quantity is of great interest to upcoming ultra-large surveys, since it is one of the easiest quantities to actually measure, and the 2PPT approach shows great promise in providing a framework in which a calculation of this quantity could realistically be performed.…”

“…In particular, calculating P (k) 1−loop = P (k) 22 +P (k) 13 in full relativistic CPT is an extremely challenging proposal, requiring a third-order calculation of δ 3 in Poisson gauge. On the other hand, 2PPT naturally provides a framework in which only the most relevant terms from third-order are included.…”

“…The 2PPT approach in an Einstein-de Sitter universe was first presented in Ref. [13], and we refer the reader to that paper for further details on what we present here. For the purposes of this paper, we will consider the following metric:…”

“…This appendix contains the full 2PPT field equations in terms of the gauge-invariant variables, including vectors and tensors, as derived in Ref. [13]. The choice of relations between , η, L C and L N is as given in Eq.…”

“…This question was studied in the context of an approach named two-parameter perturbation theory ("2PPT", for short) in Refs. [13][14][15]. These papers have shown that this set-up has a well-defined gauge problem, in which the field equations and conservation can be written in terms of gauge-invariant variables.…”