The 2dFGRS power spectrum contains important information about the key parameters of the cosmological model, but we have seen that additional assumptions are needed, in particular the values of n and h. Observations of CMB anisotropies can in principle measure most of the cosmological parameters, and combination with the 2dFGRS can lift most of the degeneracies inherent in the CMB-only analysis. It is therefore of interest to see what emerges from a joint analysis.
The clearest immediate result is that the geometrical degeneracy becomes broken (Efstathiou et al. 2002). A 95% confidence upper limit on any curvature can be set at | - 1| < 0.05. We can therefore be confident that the universe is very nearly flat so it is defensible to assume hereafter that this is exactly true. The importance of tensors will of course be one of the key questions for cosmology over the next several years, but it is interesting to consider the limit in which these are negligible. In this case, the standard model for structure formation contains a vector of only 6 parameters:
Of these, the optical depth to last scattering, , is almost entirely degenerate with the normalization, Q - and indeed with the bias parameter; we discuss this below. The remaining four parameters are pinned down very precisely: using a compilation of pre-WMAP CMB data plus the 2dFRGS power spectrum, Percival et al. (2002) obtained
or an overall density parameter of m = 0.313 ± 0.055.
It is remarkable how well these figures agree with completely independent determinations: h = 0.72 ± 0.08 from the HST key project (Mould et al. 2000; Freedman et al. 2001); b h2 = 0.020± 0.001 (Burles et al. 2001). This gives confidence that the tensor component must indeed be sub-dominant.
This analysis was published in Percival et al. (2002), and is based on the preliminary version of the 2dFGRS power spectrum, from Percival et al. (2001). We can make a first estimate of how this is likely to change using the m h = 0.18± 0.02 from the preliminary analysis of P(k) from the final dataset. In combination with the WMAP m h3.4 = 0.084 from the CMB peak degeneracy, this yields
as the preferred current figures from an analysis of this type. The matter density remains frustratingly imprecise, and it is clear that it will be very hard to measure h accurately enough to cure this problem. However, complementary constraints on m exist at similar precision (e.g. m = 0.28 ± 18% for a flat model from the SNe Ia Hubble diagram; Tonry et al. 2003). With new results from gravitational lensing, m should be measured to better than 10% precision within a year.
Perhaps the most striking conclusion from these results concerns the nature of the primordial fluctuations, which remain consistent with the n = 1 scale-invariant form. The WMAP analysis of Spergel et al. (2003) yields 0.97 ± 0.03 from CMB plus 2dFGRS (cf. 0.96 ± 0.04 from Percival et al. 2002). The WMAP team also consider adding data from the Lyman- forest, which pushes the solution away from a pure power-law:
This evidence for running of n is at best marginal, and disappears completely when systematic uncertainties in the Lyman- data are considered (Seljak, McDonald & Makarov 2003). It would in any case be surprising if true, since simple inflation models suggest that dn / d ln k should be second order in (n - 1). Although the tensor degeneracy prevents any very strong statements, the data are best described by pure scalar fluctuations, and Percival et al. (2002) set an upper limit of 0.7 to the tensor-to-scalar ratio.
The agreement with pure scalar n = 1 is not yet a strong embarrassment for inflation, but it is starting to bite on some inflationary models. Leach & Liddle (2003) show that CMB plus 2dFGRS are inconsistent with the V = 4 model at just about 95% confidence. It is possible to set up inflation models in which tilt and tensors are both negligible, but there has been a long-standing hope for more substantial signs of inflationary dynamics; if these are not seen soon, it will be a major disappointment.