# 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
2011: Volume 218, Number 1025

ISBNs: 978-0-8218-5358-0 (print); 978-0-8218-9012-7 (online)

DOI: http://dx.doi.org/10.1090/S0065-9266-2011-00641-6

Published electronically: November 3, 2011

MSC (2010): Primary 34A05, 76L05, 83F05, 85A40

### Table of Contents

**Chapters**

- Chapter 1. Introduction
- Chapter 2. Self-similar coordinates for the FRW spacetime
- Chapter 3. The expanding wave equations
- Chapter 4. Canonical co-moving coordinates and comparison with the 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 .
- 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

### Abstract

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 () 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* ,
such that corresponds to the Standard
Model. We prove that all of the self-similar spacetimes in the family are
distinct from the non-critical
Friedmann
spacetimes, thereby *characterizing* the critical
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 . 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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