Matrix model approach to cosmology

Abstract

We perform a systematic search for rotationally invariant cosmological solutions to matrix models, or more specifically the bosonic sector of Lorentzian IKKT-type matrix models, in dimensions d less than ten, specifically d = 3 and d = 5. After taking a continuum (or commutative) limit they yield d − 1 dimensional space-time surfaces, with an attached Poisson structure, which can be associated with closed, open or static cosmologies. For d = 3, we obtain recursion relations from which it is possible to generate rotationally invariant matrix solutions which yield open universes in the continuum limit. Specific examples of matrix solutions have also been found which are associated with closed and static two-dimensional space-times in the continuum limit. The solutions provide for a matrix resolution of cosmological singularities. The commutative limit reveals other desirable features, such as a solution describing a smooth transition from an initial inflation to a noninflationary era. Many of the d = 3 solutions have analogues in higher dimensions. The case of d = 5, in particular, has the potential for yielding realistic four-dimensional cosmologies in the continuum limit. We find four-dimensional de Sitter dS 4 or anti-de Sitter AdS 4 solutions when a totally antisymmetric term is included in the matrix action. A nontrivial Poisson structure is attached to these manifolds which represents the lowest order effect of noncommutativity. For the case of AdS 4 , we find one particlular limit where the lowest order noncommutativity vanishes at the boundary, but not in the interior.

Cosmological space-times with resolved Big Bang in Yang-Mills matrix models

“…Also, equation (4.38) is rather strange compared with the Friedmann equations. On the other hand, the analysis of the 13 Note that the mechanism in [44] is different from the present one and essentially relies on the embedding metric, which can be the effective one only assuming a certain complexification. 14 This aspect is somewhat reminiscent of Vilenkin's "tunneling from nothing" proposal [20].…”

“…[12][13][14][15][16][17]. Even compact solutions were found in [18,19], where it was pointed out that the induced metric on the brane can change from Euclidean to Minkowski signature.…”

“…Nevertheless, it suggests that some mild corrections of the metric may easily modify this conclusion; see e.g. [13] for a related discussion in 2 dimensions. Some other aspects of this solutions will be discussed below.…”

“…Here we show that such a term is not necessary in the presence of a mass term. The 4-dimensional cosmologies in [13] are not fully covariant but carry Poisson-structures which break SO(4) invariance. This is avoided here.…”

FuzzyCP2spacetimes

“…[1]- [8] In a previous work, [9] we showed that a fuzzy sphere embedded in a Minkowski background, which we denote by S 2,L F , can serve as a two-dimensional toy model of a closed noncommutative cosmology. Noncommutative or matrix cosmologies have been of interest for some time, and they possess a limit, the 'commutative' limit, where a space-time manifold is recovered from the matrix configurations.…”