Surveys in Differential Geometry

Volume 20 (2015)

On proving future stability of cosmological solutions with accelerated expansion

Pages: 249 – 266



Hans Ringström (Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden)


In the late 90’s, the standard perspective on how to model the universe changed dramatically; observational data concerning supernovae, obtained in 98–99, indicate that our universe is expanding at an accelerated rate. As a consequence, it is of interest to prove that cosmological solutions to Einstein’s equations with accelerated expansion are future stable. That is the topic of the present contribution. The current standard models of the universe include different types of matter, but it turns out that many of the essential difficulties appear already in the vacuum setting. As a consequence, we here focus on giving a rough outline of how to prove future stability in the case of Einstein’s vacuum equations with a positive cosmological constant. However, we also wish to give an overview of the stability results that have been obtained more generally, and to give an idea of how to arrive at the conclusion that the universe is expanding at an accelerated rate on the basis of observations.


cosmological solutions, accelerated expansion of the universe, Einstein equations, stability, Cauchy problem for Einstein equations, supernovae, cosmological constant

2010 Mathematics Subject Classification

35Q76, 83C05, 83F05

Published 7 July 2015