Geodesic-light-cone coordinates and the Bianchi I spacetime

Abstract

Abstract. The geodesic-light-cone (GLC) coordinates are a useful tool to analyse light propagation and observations in cosmological models. In this article, we propose a detailed, pedagogical, and rigorous introduction to this coordinate system, explore its gauge degrees of freedom, and emphasize its interest when geometric optics is at stake. We then apply the GLC formalism to the homogeneous and anisotropic Bianchi I cosmology. More than a simple illustration, this application (i) allows us to show that the Weinberg conjecture according to which gravitational lensing does not affect the proper area of constant-redshift surfaces is significantly violated in a globally anisotropic universe; and (ii) offers a glimpse into new ways to constrain cosmic isotropy from the Hubble diagram.

CMB lensing beyond the leading order: Temperature and polarization anisotropies

“…On the one hand, w represent the fully nonlinear potential for the photon four-momentum k µ . On the other hand, the fact thatθ a remain constant along the photon path implies that they can be identified, up to some internal degrees of freedom 2 [49,55], with the incoming photon directions, i.e. the observed direction of the source.…”

Non-linear general relativistic effects in the observed redshift

“…The only constraints regard the gauge fixing allowed by diffeomorphism invariance in general relativity. This is evident from the metric in which we have six arbitrary functions depending on all the four coordinates(see [15,52] for the construction and the discussion about the geometrical properties of this line element). The physical advantage of the line element in Eq.…”

“…Just as found for δw, here we have the free function δθ a o , which must satisfy the condition (∂ η − ∂ r )δθ a o = 0. This means that δθ a o can be only function the angleθ a and the combination η + r, which is exactly the same symmetry allowed by a residual gauge freedom within the GLC gauge [52,53]. This means that the initial condition for the evolution of the angles perturbation δθ a o can be chosen in order to fix the so-called observational gauge 6 .…”

“…It means that it can contribute to the dipole in the physical observables, hence its angular dependence is not null. Then, our normalization implies that δw o can depend on the angles and this choice regards a class of residual gauge freedom within the GLC which is different from the one exploited in [52,53]. In particular, we have that our coordinate system is invariant under the redefinition The knowledge of δw allows us to obtain also the expression for the radial shift at source presented in Eq.…”

Light-cone observables and gauge-invariance in the geodesic light-cone formalism

“…The last factor (which is unity if the GLC angles are matched to the observed angles) can be conveniently written as 17) and the two Jacobian determinants of the transformations θ a →θ a and θ a obs → θ a can be calculated according to the relations between the different angles (θ a = θ a obs + δθ a ,θ a = θ a + δθ a ):…”