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 D_H^LS = 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:

(152)

(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 D_H(z_LS) 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)

(153)

where vacuum energy is neglected at these high redshifts; the expansion factor a (1 + z)^-1 and a_LS, a_eq are the values at last scattering and matter-radiation equality respectively. In practice, z_LS 1100 independent of the matter and baryon densities, and c_S is fixed by _b. Thus the main effect is that a_eq depends on _m. Dividing by D_H(z = 0) therefore gives the angle subtended today by the light horizon as

(154)

where z_LS = 1100 and a_eq = (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

(155)

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

(156)

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 h^3.4) and constant relative heights (constant _m h²); the peak locations have more weight in this compromise, leading to a likelihood ridge along approximately _m h^3.0 const (Percival et al. 2002). It is now clear how LSS data combines with the CMB: _m h^3.4 is measured to very high accuracy already, and Percival et al. deduced _m h^3.4 = 0.078 with an error of about 6% using pre-WMAP CMB data. The first-year WMAP results in fact prefer _m h^3.4 = 0.084 (Spergel et al. 2003); the slight increase arises because WMAP indicates that previous datasets around the peak were on average calibrated low.