**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}
*h*^{2})^{-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}
*h*^{2}); 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.