Reconstructing the linear power spectrum of cosmological mass fluctuations

Abstract

We describe an attempt to reconstruct the initial conditions for the formation of cosmological large-scale structure, under the assumption of gravitational instability in a Gaussian density field. Information on the power spectrum of the primordial fluctuations is provided by a variety of autocorrelation and cross-correlation analyses on samples of different classes of galaxy and galaxy clusters. These results differ from the desired linear power spectrum because of three modifying effects: bias, non-linear evolution and redshift-space distortions. We show how the latter two effects can be corrected for analytically, allowing the linear mass spectrum to be recovered provided that the bias is independent of scale for a given class of galaxy. We argue that this is a good assumption for large scales, which is well verified in practice. We apply this method to eight independent data sets, and obtain excellent agreement in the estimated linear power spectra for wavelengths |$\lambda \gtrsim 10h^{-1}$| Mpc, given the following conditions. First, the relative bias factors for Abell clusters, radio galaxies, optical galaxies and IRAS galaxies must be in the ratios |$b_\text{A} : b_\text{R} : b_\text{O} : b_\text{I} = 4.5 : 1.9 : 1.3 :1$|⁠, to within 6 per cent rms. Secondly, the data require a significant degree of redshift-space distortion: |$\Omega^{0.6}/b_\text{I} = 1.0 \pm 0.2$|⁠. Thirdly, low values of Ω and bias are disfavoured because non-linear evolution would spoil the agreement in shape between galaxy and cluster power spectra. The amplitude of the preferred linear power spectrum is only weakly dependent on Ω and agrees well at large wavelengths with the normalization demanded by the COBE data for a scale-invariant primordial spectrum, provided that Ω = 1 and gravity-wave anisotropies are negligible. In this case, the shape of the spectrum is extremely well described by a CDM transfer function with an apparent value of the fitting parameter |$\Omega h = 0.25$|⁠. Tilted models, for which inflation requires a large gravity-wave contribution to the COBE data, predict too little power at 100-Mpc wavelengths.