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Abstract
We present a general gauge invariant formalism for defining cosmological averages that are relevant for observations based on light-like signals. Such averages involve either null hypersurfaces corresponding to a family of past light-cones or compact surfaces given by their intersection with timelike hypersurfaces. Generalized Buchert-Ehlers commutation rules for derivatives of these light-cone averages are given. After introducing some adapted ``geodesic light-cone'' coordinates, we give explicit expressions for averaging the redshift to luminosity-distance relation and the so-called ``redshift drift'' in a generic inhomogeneous Universe.
Highlights
It is well-known that averaging solutions of the full inhomogeneous Einstein equations leads, in general, to different results from those obtained by solving the averaged Einstein equations
We present a general gauge invariant formalism for defining cosmological averages that are relevant for observations based on light-like signals
The averaging procedure does not commute with the non-linear differential operators appearing in the Einstein equations and, as a result, the dynamics of the averaged geometry is affected by socalled “backreaction” terms, originating from the inhomogeneities present in the geometry and in the various components of the cosmic fluid
Summary
It is well-known (see, for example, [1]) that averaging solutions of the full inhomogeneous Einstein equations leads, in general, to different results from those obtained by solving the averaged (i.e. homogeneous) Einstein equations. Following the discovery of cosmic acceleration on large scales, interest in the possible effects of inhomogeneities for interpreting the data themselves has considerably risen (see [2, 3] for some recent review papers). The aim of this paper is to introduce a general (covariant and gauge invariant) prescription for averaging scalar objects on null hypersurfaces, to apply it to the past light-cone of a generic observer in the context of an inhomogeneous cosmological metric, and to provide the analog of the Buchert-Ehlers commutation rules [9] for the derivatives of light-cone averaged quantities.
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