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We prove that the Einstein equations in Standard Schwarzschild Coordinatesclose to form a system of three ordinary differential equations for a family ofspherically symmetric, self-similar expansion waves, and the critical ($k=0$)Friedmann universe associated with the pure radiation phase of the StandardModel 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 thefamily reduces to an implicitly defined one parameter family of distinctspacetimes determined by the value of a new {\it acceleration parameter} $a$,such that $a=1$ corresponds to FRW. We prove that all self-similar spacetimesin the family are distinct from the non-critical $k\neq0$ Friedmann spacetimes,thereby {\it characterizing} FRW as the unique spacetime lying at theintersection of these two one-parameter families. Expanding solutions about thecenter in fractional distance to the Hubble Length, we obtain the exact leadingorder quadratic and cubic corrections to the redshift vs luminosity relationfor an observer at the center. It follows by continuity that corrections to theredshift vs luminosity relation observed after the radiation phase of the BigBang can be accounted for, at the leading order quadratic level, by adjustmentof the free parameter $a$. The third order correction is then a prediction.Since self-similar expanding waves represent possible time-asymptotic wavepatterns for the conservation laws associated with the highly nonlinearradiation phase, we propose to further investigate the possibility that thesecorrections to the Standard Model might be the source of the anomalousacceleration of the galaxies, an explanation wholly within Einstein's equationswith classical sources, and not requiring Dark Energy or the cosmologicalconstant.

General relativistic self-similar waves that induce an anomalous acceleration into the standard model of cosmology