Observed angles and geodesic light-cone coordinates

Abstract

We discuss the interpretation of the angles in the Geodesic Light-Cone (GLC) coordinates. In particular, we clarify the way in which these angles can be identified with the observed ones. We show that, although this identification is always possible in principle, one cannot implement it in the usual gauge-fixing way, i.e. through a set of conditions on the GLC metric. Rather, one needs to invoke a tetrad at the observer and a Cartesian-like coordinate system in order to obtain the desired map globally on the observed sky. *

Non-linear general relativistic effects in the observed redshift

“…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.…”

CMB lensing beyond the leading order: Temperature and polarization anisotropies

“…These internal degrees of freedom can lead to some misalignement with the observed angles if not properly addressed[50]. However, this misalignement can just appear as some corrections at the observer position and these are completely sub-leading with respect to the lensing terms here considered.…”

Observation angles, Fermi coordinates, and the Geodesic-Light-Cone gauge

“…This is the temporal gauge discussed in [9]. Here we have shown that, thanks to the other residual gauge freedom [5,10], θ a → θ a (w, θ), it is also possible to define the GLC angles along the observer's geodesic to be those of the Fermi coordinates along that same geodesic. The global definition of the GLC angles then simply consists in saying that they are constant (modulo the occurrence of caustics?…”

“…We can also check, for consistency, the behavior of the GLC metric around the geodesic in terms of the new coordinates y µ defined by Eqs. (3.12) and (3.13)(see [10] for the derivation of the whole set of gauge transformations). We then find that the component g GLC τ w = −Υ keeps unchanged under the given gauge transformation, so that Υ = Υ = 1 + Υ 1 (w, θ)(w − τ ) + • • • .…”