Tetrad formalism for exact cosmological observables

Abstract

The standard description of cosmological observables is incomplete, because it does not take into account the correct angular parametrization of the sky, i.e. the one determined by the observer frame. The corresponding corrections must be taken into account for reliable results at non-linear orders. This can be accomplished by introducing an orthonormal basis, or "tetrad", at the observer point, representing the frame with respect to which observations are performed. In this paper we consider the tetrad formalism of General Relativity, thus associating tetrads to sources as well, and develop a new formalism for describing cosmological observables. It is based on a manifold which we call the "observer space-time", whose coordinates are the proper time, redshift and angles an observer uses to parametrize measurements, and on which the rest of the observables are defined. This manifold does not have to be diffeomorphic to the true space-time and allows us to resolve caustics in the latter, in contrast with similar coordinate-based formalisms. As a concrete example, we work out the definitions and equations for the angular diameter distance, weak lensing and number count observables. As for the observables associated to the CMB, they lie inside the phase space matrix distribution of the photon fluid evaluated at the observer point and pulled back on the observer sky. The second part of this paper is therefore devoted to general-relativistic matrix kinetic theory. Here too the tetrad formalism appears as the natural approach for relating the macroscopic dynamics to the microscopic QFT and therefore for constructing the matrix Boltzmann equations. Part of the presented material is known and is included for completeness, but we provide more detailed discussions over some subtle issues and we also consider an alternative construction of the collision term which deviates from the standard one at higher order in the loop corrections. In summary, the present paper contains all the required structures for computations in cosmology with exact and model-independent cosmological observables. The associated linear perturbation theory will be given in a companion paper.

Second-order gauge-invariant formalism for the cosmological observables: Complete verification of their gauge-invariance

“…With the theoretical descriptions in a general metric representation, we have next performed the second-order gauge-transformation of the metric perturbation variables in those theoretical descriptions. By comparing the direct gaugetransformation properties to the expectations from the nonlinear relations, we have explicitly verified the consistency of our theoretical expressions, among which only the expressions for the physical volume and area occupied by the source are gauge-invariant [104,108].…”

“…With δΓ η ηη we denote the nonlinear perturbation δΓ η ηη := Γ η ηη − H, where H = a /a is the conformal Hubble parameter. We are now in a position to resume the discussion about tetrads (see [104] for the tetrad formalism in cosmology). In the previous subsection, we stated that we want to equip the null light path γ with tetrad basis vectors at each point; however, there are many possible ways to do this, actually infinite.…”

“…i.e. invariant under diffeomorphism [104,108]. These are valid at any perturbative order, while the second-order gauge transformations can be read from Eq.…”

“…Drawing on the previous work, here we develop a nonlinear theoretical framework to describe the cosmological observables, present their second-order perturbative expressions, and verify the gauge-transformation properties of the second-order expressions. Adopting a geometric approach to describing the light propagation and observations and using the tetrad formalism [104], we describe the observations in our rest frame and physical quantities in the source rest frame, and their connection through light propagation was computed in a nonlinear way. Thereby, we can clarify the separation between the background quantities and the nonlinear perturbative quantities in the theoretical descriptions of the cosmological observables.…”

Non-Gaussianity in the Squeezed Three-Point Correlation from the Relativistic Effects

“…Indeed, observables can be defined in a completely coordinate-independent way and at the fully non-linear level, i.e. without any reference to the background solution (see, e.g., [89]), so they do not see the difference between standard and large gauge transformations. The fact that some coordinate system can be given a physical interpretation does not mean that one has to commit to that system to obtain the correct answer to a physical question -the essence of general covariance.…”