The Power Spectrum of Matter

We calculate the mean power spectrum of galaxies using published power spectra of galaxies and clusters of galaxies. The mean power spectrum of cluster samples as well as APM 3-D, IRAS QDOT, and SSRS+CfA2 galaxy samples has a relatively sharp maximum at wavenumber k = 0.05±0.01 h Mpc, followed by an almost exact power-law spectrum of index n ≈ −1.9 toward smaller scales. The power spectrum found from APM 2-D galaxy distribution and from LCRS and IRAS 1.2 Jy surveys is flatter around the maximum. We show that power spectra of galaxies and matter are similar in shape for a wide class of assumptions, the bias parameter of galaxies relative to matter is fixed by the fraction of clustered matter associated with galaxies; and find bgal = 1.3± 0.1. We compare the empiric power spectrum of matter with analytical power spectra and show that the primordial power spectrum has a break in amplitude and a spike if presently available galaxy and cluster samples represent the true mass distribution of the Universe. 1 The power spectrum of galaxies and clusters There exist a large body of observational determinations of the power spectrum of galaxies and clusters of galaxies. In this talk I shall review the determination of the mean galaxy power spectrum (Einasto et al1998a, hereafter E98a). The mean galaxy power spectrum shall be reduced to that of matter (Einasto et al1998b, E98b). This semi-empirical matter power spectrum shall be used to determine the primordial power spectrum (Einasto et al1998c, E98c). Recent determinations of power spectra for large galaxy and cluster samples are plotted in Figure 1. The compilation in based on summary by E98a. Spectra are shifted in amplitude to match the amplitude of the power spectrum of APM galaxies on wavenumber k = 0.1 h Mpc. The APM galaxy spectrum is a reconstruction of the 3-dimensional spectrum based on deep 2-dimensional distribution of over 2 millions of galaxies, thus the cosmic error is smaller here than in available 3-D surveys. The APM galaxy spectrum is also free of redshift distortions. We see that after vertical scaling there is little scatter between individual determinations of power spectra on medium and small scales. On large scales around the maximum the scatter is much larger. We have formed two mean power spectra. One spectrum is based on samples having power spectra with a high amplitude near the maximum. Such samples are Abell-ACO and APM cluster surveys, the redshift survey of APM galaxies, and the SSRS-CfA2-130 galaxy survey. These samples cover large regions in space where both highand mediumdensity regions are well represented, thus we use the notation PHD for this mean power spectrum (HD for high-density). This mean power spectrum has a relatively sharp maximum at k = 0.05 ± 0.01 h Mpc, followed by an almost exact powerlaw spectrum of index n ≈ −1.9 toward smaller scales. The other mean spectrum is based on samples which have a power spectrum with a shallower turnover; such samples are the LCRS survey, IRAS QDOT galaxies, and the APM 2-D sample of galaxies. In LCRS and IRAS QDOT surveys mediumdensity regions are well present but not regions of highest density (very rich superclusters). We use the notation PMD for this mean power spectrum (MD for medium-density). On medium and small scales it coincides with the previous spectrum, but it has a maximum of lower amplitude. Presently it is not clear, whether the difference between these two spectra is real or partly due to some