The local power spectrum and correlation hierarchy of the cosmic mass field

We analyze the power spectrum of a QSO's Ly alpha -transmitted flux in the discrete wavelet transform (DWT) representation. Although the mean DWT powe r spectrum is consistent with its counterpart in Fourier representation, th e spatial distribution of the local power varies greatly; i.e., the local D WT power spectra show remarkably spiky structures on small scales. To measu re these spiky features, we introduce the quantities roughness of the local power spectrum and correlation between spikes on different scales. We then test the predictions made by the correlation hierarchy model on the roughn ess and the scale-scale correlations of the local power spectrum. Using the Ly alpha -transmitted flux of the QSO HS 1700, we find that the underlying cosmic mass field of the transmitted flux at redshift around z similar or equal to 2.2 can be described by the hierarchical clustering model on physi cal scales from 2.5 h(-1) Mpc to a few tens h(-1) kpc in an Einstein-de Sit ter universe. However, the nonlinear features of the clustering show differ ences on different scale ranges: (1) On physical scales larger than similar to1.3 h(-1) Mpc, the field is almost Gaussian. (2) On scales 1.3-0.3 h(-1) Mpc, the field is consistent with the correlation hierarchy with a constan t value for the coefficient Q(4). (3) On scales less than 300 h(-1) kpc, th e field is no longer Gaussian but essentially intermittent. In this case, t he field can still be fitted by the correlation hierarchy, but the coeffici ent, Q4, should be scale dependent. These three points are strongly support ed by the following result : the scale dependencies of Q(4) given by two st atistically independent measures, i.e., Q(4)(R) by roughness and Q(4)(C) by scale-scale correlation, are the same in the entire scale range considered .