We discuss the lace-time evolution of the gravitational clustering in
an expanding universe, based on the nonlinear scaling relations (NSR)
that connect the nonlinear and linear two-point correlation functions.
The existence of critical indices for the NSR suggests that the evolu
tion may proceed toward a universal profile that does not change its s
hape at late times. We begin by clarifying the relation between the de
nsity profiles of the individual halos and the slope of the correlatio
n function, and we discuss the conditions under which the slopes of th
e correlation function at the extreme nonlinear end can be independent
of the initial power spectrum. If the evolution should lead to a prof
ile that preserves the shape at late times, then the correlation funct
ion should grow as a(2) (in a Omega = 1 universe), even at nonlinear s
cales. We prove that such exact solutions do not exist; however, there
exists a class of solutions (''psuedolinear profiles'') that evolve a
s a(2) to a good approximation. It turns out that pseudolinear profile
s are the correlation functions that arise if the individual halos are
assumed to be isothermal spheres. They are also configurations of mas
s in which the nonlinear effects of gravitational clustering are a min
imum, and hence they can act as building blocks of the nonlinear unive
rse. We discuss the implications of this result.