Light propagation in a homogeneous and anisotropic universe

Abstract

This article proposes a comprehensive analysis of light propagation in an anisotropic and spatially homogeneous Bianchi I universe. After recalling that null geodesics are easily determined in such a spacetime, we derive the expressions of the redshift and direction drifts of light sources; by solving analytically the Sachs equation, we then obtain an explicit expression of the Jacobi matrix describing the propagation of narrow light beams. As a by-product, we recover the old formula by Saunders for the angular diameter distance in a Bianchi I spacetime, but our derivation goes further since it also provides the optical shear and rotation. These results pave the way to the analysis of both supernovae data and weak lensing by the large-scale structure in Bianchi universes.

Geodesic-light-cone coordinates and the Bianchi I spacetime

“…The equality between those two expressions for D A is not to be proven, as our GLC expression of the Jacobi matrix agrees with the one of ref. [45], which itself agrees with Saunders' formula (4.50).…”

“…2.4, we finally get the following semi-explicit expressions for the GLC quantities of the Bianchi I spacetime: 2), we conclude that the redshift in a Bianchi I universe reads 30) in agreement with ref. [45]. Introducing the inverse of [I i a ], denoted [I a i ] and defined by the relations 31) it is easy to show that the photoshift vector reads…”

“…In ref. [45], the Jacobi matrix in a conformal Bianchi I spacetime is found to be 33) where the integral matrix I ij is defined by…”

“…Let us now start from the GLC expression (3.7) of the Jacobi matrix 37) where the inverse of the constant matrix reads C A a = (ds A a /dλ) o . Reference [45] has proposed various techniques and explicit examples to determine the components s i A of the Sachs basis with respect to the comoving basis ∂ i . There is no need to repeat them here.…”

“…[44] and recently rederived in refs. [32,45], which involves only three different integrals (J i ). The equality between those two expressions for D A is not to be proven, as our GLC expression of the Jacobi matrix agrees with the one of ref.…”