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Cutting out the cosmological middle man: general relativity in the light-cone coordinates

G. Fanizza, Grimm,. N., D. R. Yocum

Abstract
Analytical computations in relativistic cosmology can be split into two sets: time evolution relating the initial conditions to the observer's light-cone and light propagation to obtain observables. Cosmological perturbation theory in the Friedmann–Lemaître–Robertson–Walker (FLRW) coordinates constitutes an efficient tool for the former task, but the latter is dramatically simpler in light-cone-adapted coordinates that trivialize the light rays toward the observer world-line. Here we point out that time evolution and observable reconstruction can be combined into a single computation that relates directly initial conditions to observables. This is possible if one works uniquely in such light-cone coordinates, thus completely bypassing the FLRW 'middle-man' coordinates. We first present in detail these light-cone coordinates, extending and generalizing the presently available material in the literature, and construct a particularly convenient subset for cosmological perturbation theory. We then express the Einstein and energy–momentum conservation equations in these coordinates at the fully non-linear level. This is achieved through a careful 2 + 1 + 1 decomposition which leads to relatively compact expressions and provides good control over the geometrical interpretation of the involved quantities. Finally, we consider cosmological perturbation theory to linear order, paying attention to the remaining gauge symmetries and consistently obtaining gauge-invariant equations. Moreover, we show that it is possible to implement statistical homogeneity on stochastic fluctuations, despite the fact that the coordinate system privileges the observer world-line.

Keywords
general relativity; cosmological perturbation theory; cosmological observables; General Relativity and Quantum Cosmology; Astrophysics - Cosmology and Nongalactic Astrophysics

Classical and Quantum Gravity
Volume 38, Number 0550, Page 46
2021 January

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Instituto de Astrofísica e Ciências do Espaço Universidade do Porto Faculdade de Ciências da Universidade de Lisboa
Fundação para a Ciência e a Tecnologia COMPETE 2020 PORTUGAL 2020 União Europeia