New and Old Tests of Cosmological Models and the 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 ofthe evolution of galaxies and the universe. Most analyses of these tests treatthe problem in terms of a luminosity function and its evolution which can leadto incorrect conclusions when dealing with high redshift sources. We develop aproper treatment in three parts. In the first part we describe these testsbased on the isophotal values of the quantities such as flux, size or surfacebrightness. We show the shortcomings of the simple point source approximationbased solely on the luminosity function and consideration of the flux limit. Weemphasize the multivariate nature of the problem and quantify the effects ofother selection biases due to the surface brightness and angular sizelimitations. In these considerations the surface brightness profile plays acritical role. In the second part we show that considerable simplification overthe complicated isophotal scheme is achieved if these test are carried out insome sort of metric scheme, for example that suggested by Petrosian (1976).This scheme, however, is limited to well resolved sources. Finally, we describethe new tests, which use the data to a fuller extent than the isophotal ormetric based tests, and amount to simply counting the pixels or adding theirintensities as a function of the pixel surface brightness, instead of dealingwith surface brightness, sizes and fluxes of individual galaxies. We show thatthe data analysis and its comparison with the theoretical models of thedistributions and evolution of galaxies has the simplicity of the metric testand utilizes the data more fully than the isophotal test.