The averaging problem on the past null cone in inhomogeneous dust cosmologies

Abstract

Cosmological models typically neglect the complicated nature of the spacetime mani-fold at small scales in order to hypothesize idealized general relativistic solutions for describing the average dynamics of the Universe. Although these solutions are remarkably successful in accounting for data, they introduce a number of puzzles in cosmology, and their foundational assumptions are therefore important to test. In this paper, we go beyond the usual assumptions in cosmology and propose a formalism for averaging the local general relativistic spacetime on an observer’s past null cone: we formulate average properties of light fronts as they propagate from a cosmological emitter to an observer. The energy-momentum tensor is composed of an irrotational dust source and a cosmological constant—the same components as in the $\varLambda$CDM model for late cosmic times—but the metric solution is not a priori constrained to be locally homogeneous or isotropic. This generally makes the large-scale dynamics depart from that of a simple Friedmann–Lemaître–Robertson–Walker solution through ‘backreaction’ effects. Our formalism quantifies such departures through a fully covariant system of area-averaged equations on the light fronts propagating towards an observer, which can be directly applied to analytical and numerical investigations of cosmic observables. For this purpose, we formulate light front averages of observable quantities, including the effective angular diameter distance and the cosmological redshift drift and we also discuss the backreaction effects for these observables.

Light front-adapted spacetime metric

We shall consider the embedding of the light fronts into the cosmological spacetime, and write the metric tensor in local coordinates that are adapted to the screen space, $x^\mu = (\tau , V, x^A)$. We require that the local coordinates $x^{A}$, with $A = 2,3$, satisfy the propagation law $k^{\rho }\partial _{\rho }x^A = 0$. The propagation laws for $\tau$ and V are also fixed through $k^{\rho }\partial _{\rho }\tau = E$ and $k^{\rho }\partial _{\rho }V = 0$ (see Eqs. (24) and (7)), and the full adapted system of local coordinates $x^\mu = (\tau ,V, x^A)$ is thus specified throughout the past null cone domain, once initial conditions for the local coordinates are fixed at a screen space. In these local coordinates, we have that

$$\begin{aligned}&k_\mu = (0, {1}, 0,0) ; \quad u_\mu = ( {-1},0,0,0) ; \end{aligned}$$

(A.1)

$$\begin{aligned}&k^\mu = ( {E},0, 0,0); \quad u^\mu = ( {1, - E, EU^A}) , \end{aligned}$$

(A.2)

where $U^A := u^{\rho }\partial _{\rho }(x^A) /E$ defines the drift of the screen space coordinates in the matter frame. The one-form components in (A.1) follow directly from the definitions $k_\mu := \partial _\mu V$ and $u_\mu := - \partial _\mu \tau$. The vector components in (A.2) follow from the definition of the energy function $E := - k^\mu u_\mu$ and the transport rules (shift vectors) $k^{\rho }\partial _{\rho }(x^A) = 0$ and $u^{\rho }\partial _{\rho }(x^A) = E U^A$. We may now write the projection tensor onto the light fronts (11) in the adapted local coordinate system $x^\mu = (\tau , V, x^A)$:

$$\begin{aligned} p_{\mu \nu } = \left( \begin{array}{ccc} 0 &{} 0 &{} \textbf{0} \\ 0 &{} \, \, U^2 \, \, &{} U_B \\ \textbf{0} &{} U_A &{} p_{AB} \\ \end{array} \right) ;\quad p^{\mu \nu } = \left( \begin{array}{ccc} 0 &{} 0 &{} \textbf{0} \\ 0 &{} \, \, 0 \, \, &{} \textbf{0} \\ \textbf{0} &{} \textbf{0} &{} p^{AB} \\ \end{array} \right) , \end{aligned}$$

(A.3)

where $U_A := p_{AB} U^B$ and $U^2 := p_{AB} U^A U^B$, and where the area-adapted screen space metric $p_{AB}$ has inverse $p^{AB}$. The tensor components in (A.3) follow from the orthogonality conditions $p_{\mu \nu } k^\nu = p_{\mu \nu } u^\nu = 0$ and $p^{\mu \nu } k_\nu = p^{\mu \nu } u_\nu = 0$, respectively. Note that in general the values of the components $p_{11}$, $p_{1A}$ and $p_{A1}$ are non-zero, which comes from generally non-zero values for $u^1$ and $u^A$ in (A.2).

Using the definitions (10a) and (11), we might formulate the metric tensor for the cosmological spacetime as $g_{\mu \nu } = k_\mu k_\nu /E^2 - (k_\mu u_\nu + u_\mu k_\nu )/E + p_{\mu \nu }$, and insert Eqs. (A.1), (A.2), and (A.3) in this formulation to obtain

$$\begin{aligned} { g_{\mu \nu } = \left( \begin{array}{ccc} 0 &{} 1/E &{} \textbf{0} \\ 1/E &{} \, (1/E^2) \! + \! U^2 \, &{} U_B \\ \textbf{0} &{} {U_A} &{} p_{AB} \\ \end{array} \right) ;\quad g^{\mu \nu } = \left( \begin{array}{ccc} -1 &{} \, E \, &{} {-} E U^B \\ E &{} 0 &{} \textbf{0} \\ {-} E U^{A} &{} \textbf{0} &{} p^{AB} \\ \end{array} \right) .} \end{aligned}$$

(A.4)