The Power Spectrum in a Strongly Inhomogenous Universe

A crucial issue in cosmology is the determination of the fluctuation powerspectrum.The standard picture of the matter clustering, the Cold Dark Mattermodel (and its variant), assumes that,on scales smaller than a certain``flattening scale'' $\lambda_f$, the power spectrum increases with the scale,while on much larger scales it decreases so to match the tiny fluctuationsobserved in the microwave background. The standard picture also assumes that,once a statistically homogeneous sample is reached, the power spectrumamplitude is fixed, and any major variation should be attributed to luminositysegregation. However, the determination of $\lambda_f$ and of the absoluteamplitude, if any, is still matter of debate. In particular, there is noconsensus on whether the turnaround has been detected or not, and on the actualimportance of the luminosity segregation effect. We show that, due to thefiniteness of the sample the standard analysis of self-similar (fractal)distributions yields a turnaround for scales close to the survey scale, and asystematic amplitude shift with the survey scale. We point out that bothfeatures, bending and scaling, are in agreement with recent determination ofthe power spectrum, in particular with the CfA2 spectrum. We remark that thestandard power spectrum is not a well defined statistical tool to characterizethe galaxy distribution when a homogeneity scale has not been reached. In orderto perform an analysis that does not imply any a priori assumption one shouldstudy the PS of the density, rather than the PS of the density contrast.Comment: 11 pages Latex file. 3 .ps figures are available by anonymous ftp in ftp://oarhp2.rm.astro.it/pub/amendola/ as psfig1.ps psfig2.ps and psfig3.p