8.2. Horizon-angle degeneracy
As we have seen, the geometrical degeneracy can be broken either by additional information (such as a limit on h), or by invoking a theoretical prejudice in favour of flatness. Even for flat models, however, there still exists a version of the same degeneracy. What determines the CMB peak locations for flat models? The horizon size at last scattering is DHLS = 184 (m h2)-1/2 Mpc. The angular scale of these peaks depends on the ratio between the horizon size at last scattering and the present-day horizon size for flat models:
(using the approximation of Vittorio & Silk 1985). This yields an angle scaling as m-0.1, so that the scale of the acoustic peaks is apparently almost independent of the main parameters.
However, this argument is incomplete because the earlier expression for DH(zLS) assumes that the universe is completely matter dominated at last scattering, and this is not perfectly true. The comoving sound horizon size at last scattering is defined by (e.g. Hu & Sugiyama 1995)
where vacuum energy is neglected at these high redshifts; the expansion factor a (1 + z)-1 and aLS, aeq are the values at last scattering and matter-radiation equality respectively. In practice, zLS 1100 independent of the matter and baryon densities, and cS is fixed by b. Thus the main effect is that aeq depends on m. Dividing by DH(z = 0) therefore gives the angle subtended today by the light horizon as
where zLS = 1100 and aeq = (23900 m)-1. This remarkably simple result captures most of the parameter dependence of CMB peak locations within flat CDM models. Differentiating this equation near a fiducial m = 0.147 gives
in good agreement with the numerical derivatives in Eq. (A15) of Hu et al. (2001).
Thus for moderate variations from a `fiducial' model, the CMB peak multipole number scales approximately as peak m-0.14 h-0.48, i.e. the condition for constant CMB peak location is well approximated as
However, information about the peak heights does alter this degeneracy slightly; the relative peak heights are preserved at constant m, hence the actual likelihood ridge is a `compromise' between constant peak location (constant m h3.4) and constant relative heights (constant m h2); the peak locations have more weight in this compromise, leading to a likelihood ridge along approximately m h3.0 const (Percival et al. 2002). It is now clear how LSS data combines with the CMB: m h3.4 is measured to very high accuracy already, and Percival et al. deduced m h3.4 = 0.078 with an error of about 6% using pre-WMAP CMB data. The first-year WMAP results in fact prefer m h3.4 = 0.084 (Spergel et al. 2003); the slight increase arises because WMAP indicates that previous datasets around the peak were on average calibrated low.