New and Old Tests of Cosmological Models and Evolution of Galaxies
  We describe the classical cosmological tests, such as the Log$N$-Log$S$, redshift-magnitude and angular diameter tests, and propose some new tests of the evolution of galaxies and the universe. Most analyses of these tests treat the problem in terms of a luminosity function and its evolution which can lead to incorrect conclusions when dealing with high redshift sources. We develop a proper treatment in three parts. In the first part we describe these tests based on the isophotal values of the quantities such as flux, size or surface brightness. We show the shortcomings of the simple point source approximation based solely on the luminosity function and consideration of the flux limit. We emphasize the multivariate nature of the problem and quantify the effects of other selection biases due to the surface brightness and angular size limitations. In these considerations the surface brightness profile plays a critical role. In the second part we show that considerable simplification over the complicated isophotal scheme is achieved if these test are carried out in some sort of metric scheme, for example that suggested by Petrosian (1976). This scheme, however, is limited to well resolved sources. Finally, we describe the new tests, which use the data to a fuller extent than the isophotal or metric based tests, and amount to simply counting the pixels or adding their intensities as a function of the pixel surface brightness, instead of dealing with surface brightness, sizes and fluxes of individual galaxies. We show that the data analysis and its comparison with the theoretical models of the distributions and evolution of galaxies has the simplicity of the metric test and utilizes the data more fully than the isophotal test. 
Vahe' Petrosian (Center for Space Science and Astrophysics, Stanford   University)