Modelling the Power Spectrum of Density Fluctuations: A Phenomenological Approach

We show how, based on considerations on the observed form of the galaxy 2-point spatial correlation function xi®, a very simplified – yet surprisingly effective – model for the linear density fluctuations power spectrum can be constructed. We first relate the observed large-scale shape of xi® to a power-law form for the power spectrum, P(k)\propto k^{-2.2}. For a plausible value of the bias parameter b = 1/sigma_8 ~ 1.8, one has (delta_rho / rho){rms} ~ 1 r ~ 3.5/h Mpc, suggesting that the change of slope observed in xi® around this scale marks the transition between the linear and nonlinear gravitational regimes. Under this working hypothesis, we use a simple analytical form to fit the large-scale correlations constraints together with the COBE CMB anisotropy measurement, thus constructing a simple phenomenological model for the linear power spectrum. Despite its simplicity, the model fits remarkably well directly estimated power spectra from different optical galaxy samples, and when evolved through an N-body simulation it provides a good match to the observed galaxy correlations. One of the most interesting features of the model is the small-scale one-dimensional velocity dispersion produced: sigma{1d} = 450 Km s^{-1} at 0.5/h Mpc and sigma_{1d} = 350 Km s^{-1} for separations larger than ~ 2/h Mpc.