\(1+3\) covariant cosmic microwave background anisotropies. II: The almost Friedmann-Lemaître model (Q1581166)

The first part (see Zbl 0979.83043 above) of this series of papers on cosmic microwave background anisotropies from a \(1+3\) covariant and gauge-invariant point of view was devoted to algebraic fundamentals. Now, in the second part, the dynamical equations are taken into account, i.e., the Boltzmann equation for the photon gas and Einstein's field equation, with an energy-momentum tensor containing a matter source in addition to the photon gas, for the metric tensor. The collision term in the Boltzmann equation is given in terms of an optical depth appropriate for modeling Thompson scattering. It is the goal to develop a scenario for the cosmic microwave background radiation before and after decoupling, thereby clarifying in which way the present-day temperature anisotropies arise from inhomogeneities in the early universe. To that end the authors make full use of the multipole and mode expansions which were the subject of the first part, and they analyze the dynamical equations by linearizing around a Friedman-Lemaître universe. Several types of additional approximative assumptions are considered in different parts of the paper, among them the matter-dominated case (applying if the ratio of radiation energy density to matter energy density is small), the tight-coupling case (if the Thompson scattering time is small) and the slow-decoupling case (if the differential optical depth is small). The scenario of our universe considered in this paper consists of four different eras. Going backwards in time, in the first era the universe is assumed to be matter-dominated and the photons are assumed to be collisionless. In the second era the universe is again matter-dominated but Thompson scattering has to be taken into account, with the slow-decoupling approximation being valid. The remaining two eras are tight-coupling eras, divided from each other by the moment when the energy densities of matter and radiation are equal. The temporal development starts at the beginning of the tight-coupling eras where initial matter perturbations have to be seeded by earlier conditions, e.g. inflation. Within their \(1+3\) covariant and gauge-invariant framework, the authors are able to reproduce several results known from other approaches. In addition, the paper provides a basis from which to tackle nonlinear and gravitational-wave effects. From a mathematical point of view, the use of the linearized Boltzmann equation in a time-integrated form and the derivation of its mode form deserves particular mention. Quite generally, this voluminous (73 pages!) second part of the series is to be viewed as a very valuable contribution to the subject, although the presentation is less transparent and the paper is much more difficult to read than the first part.