Physics of temperature fluctuations 225- The photons and neutrinos are described by distribution functionsf(x,p,t).
A fundamental simplifying assumption is that the energy dependence of
both is given by the blackbody distribution. The space dependence is
generally Fourier transformed, so the distribution functions can be written as
'(k,nˆ,t), where the function has been normalized to the temperature of the
blackbody distribution andnˆrepresents the direction in which the radiation
propagates. The time evolution of each is given by the Boltzmann equation.
For neutrinos, collisions are unimportant so the Boltzmann collision term on
the right hand side is zero; for photons, Thomson scattering off electrons
must be included. - The DM and baryons are, in principle, described by Boltzmann equations
as well, but a fluid description incorporating only the lowest two velocity
moments of the distribution functions is adequate. Thus each is described
by the Euler and continuity equations for their densities and velocities. The
baryon Euler equation must include the coupling to photons via Thomson
scattering. - Metric perturbation evolution and the connection of the metric perturbations
to the matter perturbations are both contained in the Einstein equations.
This is where the subtleties arise. A general metric perturbation has 10
degrees of freedom, but four of these are unphysical gauge modes. The
physical perturbations include two degrees of freedom constructed from
scalar functions, two from a vector, and two remaining tensor perturbations
(Mukhanovet al1992). Physically, the scalar perturbations correspond
to gravitational potential and anisotropic stress perturbations; the vector
perturbations correspond to vorticity and shear perturbations; and the tensor
perturbations are two polarizations of gravitational radiation. Tensor and
vector perturbations do not couple to matter evolving only under gravitation;
in the absence of a ‘stiff source’ of stress energy, like cosmic defects or
magnetic fields, the tensor and vector perturbations decouple from the linear
perturbations in the matter.
A variety of different variable choices and methods for eliminating the gauge
freedom have been developed. The subject can be fairly complicated. A detailed
discussion and comparison between the Newtonian and synchronous gauges,
along with a complete set of equations, can be found in Ma and Bertschinger
(1995); also see Huet al(1998). An elegant and physically appealing formalism
based on an entirely covariant and gauge-invariant description of all physical
quantities has been developed for the microwave background by Challinor and
Lasenby (1999) and Gebbieet al(2000), based on earlier work by Ehlers (1993)
and Ellis and Bruni (1989). A more conventional gauge-invariant approach was
originated by Bardeen (1980) and developed by Kodama and Sasaki (1984).
The Boltzmann equations are partial differential equations, which can be
converted to hierarchies of ordinary differential equations by expanding their
directional dependence in Legendre polynomials. The result is a large set of