Focus issue: Relativistic effects in cosmology

The general relativistic description of galaxy clustering provides a complete and unified treatment of all the effects in galaxy clustering such as the redshift-space distortion, gravitational lensing, Sachs–Wolfe effects, and their relativistic effects. In particular, the relativistic description resolves the gauge issues in the standard Newtonian description of galaxy clustering by providing the gauge-invariant expression for the observed galaxy number density. The relativistic effect in galaxy clustering is significant on large scales, in which dark energy models or alternative theories of modified gravity deviate from general relativity. In this paper, we review the relativistic effect in galaxy clustering by providing a pedagogical derivation of the relativistic formula and by computing the observed galaxy two-point statistics. The relativistic description of galaxy clustering is an essential tool for testing general relativity and probing the early Universe on large scales in the era of precision cosmology.

We present a fully relativistic calculation of the observed galaxy number counts in the linear regime. We show that besides the density fluctuations and redshift-space distortions, various relativistic effects contribute to observations at large scales. These effects all have the same physical origin: they result from the fact that our coordinate system, namely the galaxy redshift and the incoming photons' direction, is distorted by inhomogeneities in our Universe. We then discuss the impact of the relativistic effects on the angular power spectrum and on the two-point correlation function in configuration space. We show that the latter is very well adapted to isolate the relativistic effects since it naturally makes use of the symmetries of the different contributions. In particular, we discuss how the Doppler effect and the gravitational redshift distortions can be isolated by looking for a dipole in the cross-correlation function between a bright and a faint population of galaxies.

We review recent studies that rigorously define several key observables of the large-scale structure of the Universe in a general relativistic context. Specifically, we consider (i) redshift perturbation of cosmic clock events; (ii) distortion of cosmic rulers, including weak lensing shear and magnification; and (iii) observed number density of tracers of the large-scale structure. We provide covariant and gauge-invariant expressions of these observables. Our expressions are given for a linearly perturbed flat Friedmann–Robertson–Walker metric including scalar, vector, and tensor metric perturbations. While we restrict ourselves to linear order in perturbation theory, the approach can be straightforwardly generalized to higher order.

Extremely well! In the ΛCDM model, the spacetime metric, g_ab, of our Universe is approximated by an FLRW metric, $g_{ab}^{(0)}$, to about one part in 10⁴ or better on both large and small scales, except in the immediate vicinity of very strong field objects, such as black holes. However, derivatives of g_ab are not close to derivatives of $g_{ab}^{(0)}$, so there can be significant differences in the behavior of geodesics and huge differences in curvature. Consequently, observable quantities in the actual Universe may differ significantly from the corresponding observables in the FLRW model. Nevertheless, as we shall review here, we have proven general results showing that—within the framework of our approach to treating backreaction—the large matter inhomogeneities that occur on small scales cannot produce significant effects on large scales, so $g_{ab}^{(0)}$ satisfies Einsteinʼs equation with the averaged stress–energy tensor of matter as its source. We discuss the flaws in some other approaches that have suggested that large backreaction effects may occur. As we also will review here, with a suitable 'dictionary,' Newtonian cosmologies provide excellent approximations to cosmological solutions to Einsteinʼs equation (with dust and a cosmological constant) on all scales. Our results thereby provide strong justification for the mathematical consistency and validity of the ΛCDM model within the context of general relativistic cosmology.

We discuss the relation between the output of Newtonian N-body simulations on scales that approach or exceed the particle horizon to the description of general relativity. At leading order, the Zeldovich approximation is correct on large scales, coinciding with the general relativistic result. At second order in the initial metric potential, the trajectories of particles deviate from the second order Newtonian result and hence the validity of second order Lagrangian perturbation theory initial conditions should be reassessed when used in very large simulations. We also advocate using the expression for the synchronous gauge density as a well behaved measure of density fluctuations on such scales.

General relativistic cosmology cannot be reduced to linear relativistic perturbations superposed on an isotropic and homogeneous (Friedmann–Robertson–Walker) background, even though such a simple scheme has been successfully applied to analyse a large variety of phenomena (such as cosmic microwave background primary anisotropies, matter clustering on large scales, weak gravitational lensing, etc). The general idea of going beyond this simple paradigm is what characterizes most of the efforts made in recent years: the study of second and higher-order cosmological perturbations including all general relativistic contributions—also in connection with primordial non-Gaussianities—the idea of defining large-scale structure observables directly from a general relativistic perspective, the various attempts to go beyond the Newtonian approximation in the study of nonlinear gravitational dynamics, by using e.g., post-Newtonian treatments, are all examples of this general trend. Here we summarize some of these directions of investigation, with the aim of emphasizing future prospects in this area of cosmology, both from a theoretical and observational point of view.

We present a framework for general relativistic N-body simulations in the regime of weak gravitational fields. In this approach, Einsteinʼs equations are expanded in terms of metric perturbations about a Friedmann–Lemaître background, which are assumed to remain small. The metric perturbations themselves are only kept to linear order, but we keep their first spatial derivatives to second order and treat their second spatial derivatives as well as sources of stress–energy fully non-perturbatively. The evolution of matter is modelled by an N-body ensemble which can consist of free-streaming nonrelativistic (e.g. cold dark matter) or relativistic particle species (e.g. cosmic neutrinos), but the framework is fully general and also allows for other sources of stress–energy, in particular additional relativistic sources like modified-gravity models or topological defects. We compare our method with the traditional Newtonian approach and argue that relativistic methods are conceptually more robust and flexible, at the cost of a moderate increase of numerical difficulty. However, for a $\Lambda {\rm CDM}$ cosmology, where nonrelativistic matter is the only source of perturbations, the relativistic corrections are expected to be small. We quantify this statement by extracting post-Newtonian estimates from Newtonian N-body simulations.