242 The cosmic microwave background
describing scalar perturbations determine the size of the Sachs–Wolfe effect and
also magnitude of the forces driving the acoustic oscillations. The potentials
are determined by 0 h^2 , the matter density as a fraction of critical density.
The baryon density,bh^2 , determines the degree to which the acoustic peak
amplitudes are modulated as previousy described in section 7.4.4.
The time of matter–radiation equality is obviously determined solely by the
total matter density 0 h^2. This quantity affects the size of the DM fluctuations,
since DM starts to collapse gravitationally only after matter–radiation equality.
Also, the gravitational potentials evolve in time during radiation domination and
not during matter domination: the later matter–radiation equality occurs, the
greater the time evolution of the potentials at decoupling, increasing the Integrated
Sachs–Wolfe effect. The power spectrum also has a weak dependence on 0 in
models with 0 significantly less than unity, because at late times the evolution
of the background cosmology will be dominated not by matter, but rather by
vacuum energy (for a flat universe with) or by curvature (for an open universe).
In either case, the gravitational potentials once again begin to evolve with time,
giving an additional late-time integrated Sachs–Wolfe contribution, but this tends
to affect only the largest scales for which the constraints from measurements are
least restrictive due to cosmic variance (see the discussion in section 7.5.4).
The sound speed, which sets the sound horizon and thus affects the
wavelength of the acoustic modes (cf equation (7.28)), is completely determined
by the baryon densitybh^2. The horizon size at recombination, which sets the
overall scale of the acoustic oscillations, depends only on the total mass density
0 h^2. The damping scale for diffusion damping depends almost solely on the
baryon densitybh^2 , although numerical fits give a slight dependence onb
alone (Hu and Sugiyama 1996). Finally, the angular diameter distance to the last-
scattering surface is determined by 0 handh; the angular diameter sets the
angular scale on the sky of the acoustic oscillations.
In summary, the physical dependence of the temperature perturbations at
last scattering depends on 0 h^2 ,bh^2 , 0 h,andhinstead of the individual
cosmological parameters 0 , b,hand. When analysing constraints on
cosmological models from microwave background power spectra, it may be more
meaningful and powerful to constrain these physical parameters rather than the
cosmological ones.
7.5.3 Power spectrum degeneracies
As might be expected from the previous discussion, not all of the parameters
considered here are independent. In fact, one nearly exact degeneracy exists if
0 ,b,handare taken as independent parameters. To see this, consider a
shift in 0. In isolation, such a shift will produce a corresponding stretching
of the power spectrum inl-space. But this effect can be compensated by first
shiftinghto keep 0 h^2 constant, then shiftingbto keepbh^2 constant, and
finally shiftingto keep the angular diameter distance constant. This set of