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9.1. Matter fluctuation amplitude and bias

The above conclusions were obtained by considering the shapes of the CMB and galaxy power spectra. However, it is also of great interest to consider the amplitude of mass fluctuations, since a comparison with the galaxy power spectrum allows us to infer the degree of bias directly. This analysis was performed by Lahav et al. (2002). Given assumed values for the cosmological parameters, the present-day linear normalization of the mass spectrum (e.g. sigma8) can be inferred. It is convenient to define a corresponding measure for the galaxies, sigma8g, such that we can express the bias parameter as

Equation 161 (161)

In practice, we define sigma8g to be the value required to fit a CDM model to the power-spectrum data on linear scales (0.02 < k < 0.15 h Mpc-1). The amplitude of 2dFGRS galaxies in real space estimated by Lahav et al. (2002) is sigma8gR(L*) = 0.76, with a negligibly small random error. This assumes no evolution in sigma8g, plus the luminosity dependence of clustering measured by Norberg et al. (2001).

The value of sigma8 for the dark matter can be deduced from the CMB fits. Percival et al. (2002) obtain

Equation 162 (162)

where the error bar includes both data errors and theory uncertainty. The WMAP number here is almost identical: sigma8 exp(- tau) = 0.71, but no error is quoted (Spergel et al. 2003). The unsatisfactory feature is the degeneracy with the optical depth to last scattering. For reionization at redshift 8, we would have tau appeq 0.05; it is not expected theoretically that tau can be hugely larger, and popular models would place reionization between z = 10 and z = 15, or tau appeq 0.1 (e.g. Loeb & Barkana 2001). One of the many impressive aspects of the WMAP results is that they are able to infer tau = 0.17± 0.04 from large-scale polarization. Taken at face value, tau = 0.17 would argue for reionization at z = 20, but the error means that more conventional figures are far from being ruled out. Taking all this together, it seems reasonable to assume that the true value of sigma8 is within a few % of 0.80. Given the 2dFGRS figure of sigma8gR = 0.76, this implies that L* galaxies are very nearly exactly unbiased. Since there are substantial variations in the clustering amplitude with galaxy type, this outcome must be something of a coincidence. This conclusion of near-unity bias was reinforced in a completely independent way, by using the measurements of the bispectrum of galaxies in the 2dFGRS (Verde et al. 2002). As it is based on three-point correlations, this statistic is sensitive to the filamentary nature of the galaxy distribution - which is a signature of nonlinear evolution. One can therefore split the degeneracy between the amplitude of dark-matter fluctuations and the amount of bias.

These conclusions point the way towards a possible limit on the tensor contribution: a large contribution of tensors to the COBE signal would lower the required scalar amplitude. As an extreme example, a scalar-to-tensor ratio of 1 would reduce the `apparent' sigma8 by roughly a factor of 21/2, to 0.5. Even for an implausibly large value of tau, this would be hard to reconcile with the level of galaxy clustering plus the requirement of a low degree of bias. Also, more direct limits on sigma8 are now being derived from large-scale gravitational lensing surveys, with sigma8 appeq 0.7 to 0.8 being favoured (e.g. Brown et al. 2003; Jarvis et al. 2003).

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