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

About this Title

Joel Smoller, Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 and Blake Temple, Department of Mathematics, University of California, Davis, Davis, California 95616

Publication: Memoirs of the American Mathematical Society
Publication Year 2012: Volume 218, Number 1025
ISBNs: 978-0-8218-5358-0 (print); 978-0-8218-9012-7 (online)
Published electronically: November 3, 2011
MSC (2010): Primary 34A05, 76L05, 83F05, 85A40

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Self-similar coordinates for the $k=0$ FRW spacetime
  • Chapter 3. The expanding wave equations
  • Chapter 4. Canonical co-moving coordinates and comparison with the $k \ne 0$ FRW spacetimes
  • Chapter 5. Leading order corrections to the standard model induced by the expanding waves
  • Chapter 6. A foliation of the expanding wave spacetimes into flat spacelike hypersurfaces with modified scale factor $R(t) = t^a$.
  • Chapter 7. Expanding wave corrections to the standard model in approximate co-moving coordinates
  • Chapter 8. Redshift vs luminosity relations and the anomalous acceleration
  • Chapter 9. Appendix: The mirror problem
  • Chapter 10. Concluding remarks


We prove that the Einstein equations for a spherically symmetric spacetime in Standard Schwarzschild Coordinates (SSC) close to form a system of three ordinary differential equations for a family of self-similar expansion waves, and the critical ($k=0$) Friedmann universe associated with the pure radiation phase of the Standard Model of Cosmology 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 acceleration parameter $a$, such that $a=1$ corresponds to the Standard Model. We prove that all of the self-similar spacetimes in the family are distinct from the non-critical $k\neq 0$ Friedmann spacetimes, thereby characterizing the critical $k=0$ Friedmann universe as the unique spacetime lying at the intersection of these two one-parameter families. We then present a mathematically rigorous analysis of solutions near the singular point at the center, deriving the expansion of solutions up to fourth order in the fractional distance to the Hubble Length. Finally, we use these rigorous estimates to calculate 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.

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