224 The cosmic microwave background
the currently popularCDM models), the redshift of matter–radiation equality
occurs late enough that the gravitational potentials are still evolving significantly
when the microwave background radiation decouples, leading to a non-negligible
Integrated Sachs–Wolfe effect. The same situation also occurs at late times in
these models; gravitational potentials begin to evolve again as the universe makes
a transition from matter domination to either vacuum energy domination or a
significantly curved background spatial metric, giving an additional Integrated
Sachs–Wolfe contribution.
7.2.2 A formal description
The early universe at the epoch when the microwave background radiation begins
propagating freely, around a redshift ofz=1100, is a conceptually simple place.
Its constituents are ‘baryons’ (including protons, helium nuclei and electrons,
even though electrons are not baryons), neutrinos, photons and DM particles.
The neutrinos and DM can be treated as interacting only gravitationally since
their weak interaction cross sections are too small at this energy scale to be
dynamically or thermodynamically relevant. The photons and baryons interact
electromagnetically, primarily via Compton scattering of the radiation from the
electrons. The typical interaction energies are low enough for the scattering to
be well approximated by the simple Thomson cross section. All other scattering
processes (e.g. Thomson scattering from protons, Rayleigh scattering of radiation
from neutral hydrogen) have small enough cross-sections to be insignificant, so
we have four species of matter with only one relevant (and simple) interaction
process among them. The universe is also very close to being homogeneous and
isotropic, with small perturbations in density and velocity on the order of a part in
105. The tiny size of the perturbations guarantees that linear perturbation theory
around a homogeneous and isotropic background universe will be an excellent
approximation.
Conceptually, the formal description of the universe at this epoch is quite
simple. The unperturbed background cosmology is described by the Friedmann–
Robertson–Walker (FRW) metric, and the evolution of the cosmological scale
factora(t)in this metric is given by the Friedmann equation (see the lectures
by Peacock in this volume). The evolution of the free electron densityneis
determined by the detailed atomic physics describing the recombination of neutral
hydrogen and helium; see Seageret al(2000) for a detailed discussion. At a
temperature of around 0.5 eV, the electrons combine with the protons and helium
nuclei to make neutral atoms. As a result, the photons cease Thomson scattering
and propagate freely to us. The microwave background is essentially an image of
the ‘surface of last scattering’. Recombination must be calculated quite precisely
because the temperature and thickness of this surface depend sensitively on the
ionization history through the recombination process.
The evolution of first-order perturbations in the various energy density
components and the metric are described with the following sets of equations: