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General Relativistic Self-Similar Waves that induce an Anomalous Acceleration into the Standard Model of Cosmology

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.