6.2. The modern angular size test: CMB-ology
Although it is not my purpose here to discuss the CMB anisotropies, it is necessary to say a few words on the preferred angular scale of the longest wavelength acoustic oscillations, the "first peak", because this is now the primary evidence for a flat Universe (k = 0). In Fig. 4 we see again the now very familiar plot of the angular power spectrum of anisotropies as observed by WMAP  (in my opinion, of all the WMAP papers, this reference provides the clearest discussion of the physics behind the peak amplitudes and positions). The solid line is the concordance model- not a fit, but just the predicted angular power spectrum (via CMBFAST ) from the m = 0.3, = 0.7 model Universe with an optical depth of 0.17 to the surface of last scattering. I must admit that the agreement is impressive.
I remind you that the harmonic index on the horizontal axis is related to angular scale as
so the first peak, at l 220, would correspond to an angular scale of about one degree. I also remind you that the first peak corresponds to those density inhomogeneities which entered the horizon sometime before decoupling (at z = 1000); enough before so that they have had time to collapse to maximum compression (or expand to maximum rarefaction) just at the moment of hydrogen recombination. Therefore, the linear scale of these inhomogeneities is very nearly given by the sound horizon at decoupling, that is
where tdec is the age of the Universe at decoupling.
So one might say, the test is simple: we have a known linear scale lh which corresponds to an observed angular scale ( 0.014 rad) so we can determine the geometry of the Universe. It is not quite so simple because the linear scale, lh depends, via tdec on the matter content of the Universe (m); basically, the larger m, the sooner matter dominates the expansion, and the earlier decoupling with a correspondingly smaller lh. This comoving linear scale is shown in Fig. 5 as a function of m ( hardly matters here, because the vacuum energy density which dominates today has no effect at the epoch of decoupling). Another complication is that the angular size distance to the surface of last scattering not only depends upon the geometry, but also upon the expansion history. This is evident in Fig. 6 which shows the comoving angular size distance (in Gpc) to the surface of last scattering as a function of m for three values of tot = m + (i.e., k = 1 - tot). Note that the comoving angular size distance, DA(1 + z), is the same as the radial comoving coordinate r.
Figure 5. The comoving linear scale of the perturbation corresponding to the first peak as a function of m.
Figure 6. The angular size distance (Gpc) to the last scattering surface (z = 1000) as a function of m for various values of tot.
We can combine Figs. 5 and 6 to plot the expected angular size (or harmonic index) of the first peak as a function of m and tot, and this is shown in Fig. 7 with the dashed line giving the observed l of the first peak. We see that a model with tot 1.1 (a closed universe) is clearly ruled out, but it would be possible to have an open model with tot = 0.9 and m = 0.8 from the position of the first peak alone; the predicted peak amplitude, however, would be about 40% too low. The bottom line of all of this is that the position of the first peak does not uniquely define the geometry of the Universe because of a degeneracy with m (I haven't mentioned the degeneracy with h taken here to be 0.72). To determine whether or not we live in a flat Universe we need an independent handle on m and that is provided, in WMAP data, by the amplitudes of the first two peaks (the more non-baryonic matter, the deeper the forming potential wells, and the lower the amplitudes). From this it is found that m 0.3, and from Fig. 7 we see that the model Universe should be near flat (tot 1.0). Of course if the Universe is near flat with m = 0.3 then the rest must be in dark energy; this is the indirect evidence from the CMB anisotropies for dark energy.
Figure 7. The harmonic index expected for the first peak as a function of m for various values of tot.
I just add here that the observed peak amplitudes (given the optical depth to z = 1000 determined from WMAP polarization results ), is taken now as definitive evidence for CDM. However, alternative physics which affects the amplitude and positions of peaks (e.g.  could weaken this conclusion, as well as affect the derived cosmological parameters. Even taking the peak amplitudes as prima facie evidence for the existence of cold dark matter, it is only evidence for CDM at the epoch of recombination (z = 1000) and not in the present Universe. To address the cosmic coincidence problem, models have been suggested in which dark matter transmutes into dark energy (e.g. ).
Now I turn to the direct evidence for dark energy.