226 The cosmic microwave background
coupled, first-order linear ordinary differential equations which form a well-posed
initial value problem. Initial conditions must be specified. Generally they are
taken to be so-called adiabatic perturbations: initial curvature perturbations with
equal fractional perturbations in each matter species. Such perturbations arise
naturally from the simplest inflationary scenarios. Alternatively, isocurvature
perturbations can also be considered: these initial conditions have fractional
density perturbations in two or more matter species whose total spatial curvature
perturbation cancels. The issue of numerically determining initial conditions is
discussed later in section 7.4.2.
The set of equations are numerically stiff before last scattering, since
they contain the two widely discrepant time scales: the Thomson scattering
time for electrons and photons and the (much longer) Hubble time.
Initial conditions must be set with high accuracy and an appropriate stiff
integrator must be employed. A variety of numerical techniques have
been developed for evolving the equations. Particularly important is the
line-of-sight algorithm first developed by Seljak and Zaldarriaga (1996) and
then implemented by them in the publicly available CMBFAST code (see
http://www.sns.ias.edu/∼matiasz/CMBFAST/cmbfast.html)..)
This discussion is intentionally heuristic and somewhat vague because many
of the issues involved are technical and not particularly illuminating. My main
point is an appreciation for the detailed and precise physics which goes into
computing microwave background fluctuations. However, all of this formalism
should not obscure several basic physical processes which determine the ultimate
form of the fluctuations. A widespread understanding of most of the physical
processes detailed have followed from a seminal paper by Hu and Sugiyama
(1996), a classic of the microwave background literature.
7.2.3 Tight coupling
Two basic time scales enter into the evolution of the microwave background.
The first is the photon scattering time scalets, the mean time between Thomson
scatterings. The other is the expansion time scale of the universe,H−^1 ,where
H = ̇a/ais the Hubble parameter. At temperatures significantly greater than
0.5 eV, hydrogen and helium are completely ionized andts H−^1 .The
Thomson scatterings which couple the electrons and photons occur much more
rapidly than the expansion of the universe; as a result, the baryons and photons
behave as a single ‘tightly coupled’ fluid. During this period, the fluctuations
in the photons mirror the fluctuations in the baryons. (Note that recombination
occurs at around 0.5 eV rather than 13.6 eV because of the huge photon–baryon
ratio; the universe contains somewhere around 10^9 photons for each baryon, as
we know from primordial nucleosynthesis. It is a useful exercise to work out the
approximate recombination temperature.)
The photon distribution function for scalar perturbations can be written
as'(k,μ,t)whereμ = kˆ·ˆnand the scalar character of the fluctuations