  We prove that the Einstein equations in Standard Schwarzschild Coordinates
close to form a system of three ordinary differential equations for a family of
spherically symmetric, self-similar expansion waves, and the critical ($k=0$)
Friedmann universe associated with the pure radiation phase of the Standard
Model of Cosmology (FRW), is embedded as a single point in this family.
Removing a scaling law and imposing regularity at the center, we prove that the
family reduces to an implicitly defined one parameter family of distinct
spacetimes determined by the value of a new {\it acceleration parameter} $a$,
such that $a=1$ corresponds to FRW. We prove that all self-similar spacetimes
in the family are distinct from the non-critical $k\neq0$ Friedmann spacetimes,
thereby {\it characterizing} FRW as the unique spacetime lying at the
intersection of these two one-parameter families. Expanding solutions about the
center in fractional distance to the Hubble Length, we obtain the exact leading
order quadratic and cubic corrections to the redshift vs luminosity relation
for an observer at the center. It follows by continuity that corrections to the
redshift vs luminosity relation observed after the radiation phase of the Big
Bang can be accounted for, at the leading order quadratic level, by adjustment
of the free parameter $a$. The third order correction is then a prediction.
Since self-similar expanding waves represent possible time-asymptotic wave
patterns for the conservation laws associated with the highly nonlinear
radiation phase, we propose to further investigate the possibility that these
corrections to the Standard Model might be the source of the anomalous
acceleration of the galaxies, an explanation wholly within Einstein's equations
with classical sources, and not requiring Dark Energy or the cosmological
constant.
