Observational estimates of the initial power spectrum at small scale from Lyman-$\alpha$ absorbers

We present a new method of measuring the power spectrum of initial perturbations to an unprecedently small scale of $\sim$ 10$h^{-1}$ kpc. We apply this method to a sample of 4500 Ly-$\alpha$ absorbers and recover the cold dark matter (CDM) like power spectrum at scales $\geq 300h^{-1}$kpc with a precision of $\sim$ 10%. However at scales $\sim 10 - 300 h^{-1}$kpc the measured and CDM--like spectra are noticeable different. This result suggests a complex inflation with generation of excess power at small scales. The magnitude and reliability of these deviations depend also upon the possible incompleteness of our sample and poorly understood process of formation of weak absorbers. Confirmation of the CDM--like shape of the initial power spectrum or detection of its distortions at small scales are equally important for widely discussed problems of physics of the early Universe, galaxy formation, and reheating of the Universe. Our method links the observed mass function of absorbers with the correlation function of the initial velocity field and therefore it avoids the Nyquist restrictions limiting the investigations based on the smoothed flux or density fields. The physical model of absorbers adopted here asserts that they are formed in the course of both linear and nonlinear adiabatic or shock compression of dark matter (DM) and gaseous matter. At scales $\geq 1h^{-1}$Mpc all characteristics of the DM component and, in particular, their redshift distribution are found to be consistent with theoretical expectations for Gaussian initial perturbations with a CDM--like power spectrum.