Cosmological Solutions of Emergent Noncommutative Gravity

Abstract

Matrix models of Yang-Mills type lead to an emergent gravity theory, which may not require fine-tuning of a cosmological constant. We find cosmological solutions of Friedmann-RobertsonWalker type. They generically have a big bounce, and an early inflation-like phase with graceful exit. The mechanism is purely geometrical, no ad-hoc scalar fields are introduced. The solutions are stabilized through vacuum fluctuations and are thus compatible with quantum mechanics. This leads to a Milne-like universe after inflation, which appears to be in remarkably good agreement with observation and may provide an alternative to standard cosmology.PACS numbers: 98.80.Qc, 98.80.Cq Quantum field theory and general relativity provide the basis of our present understanding of fundamental forces and matter. However, there is up to now no satisfactory way to reconcile them in a consistent quantum theory. General arguments based on quantum mechanics and general relativity suggest a "foam-like" or quantum structure at the Planck scale 10 −33 cm. This problem has its most dramatic manifestation in the cosmological constant problem: the small but non-vanishing cosmological constant in the currently accepted ΛCDM model is in striking contradiction with quantum mechanical expectations, which are off by at least 60 orders of magnitude. No satisfactory solution of this problem within the conventional frameworks has been found.We point out in this letter that emergent gravity on non-commutative (NC) spaces may provide a resolution of these problems. The starting point are matrix models of Yang-Mills typesupplemented by suitable fermionic terms. Such models have been discussed in the context of NC gauge theory and string theory. Here X a , a = 1, ..., D are (infinite-dimensional) hermitian matrices, and η ab = diag(1, 1, ..., ±1) in the Euclidean resp. Minkowski case. The basic hypothesis of our approach is that space-time is realized as 3+1-dimensional NC "brane" solution of such a model. The effective geometry and gravity on such a brane was clarified recently [1][2][3]: the effective metric is not fundamental but depends on the embedding and the Poisson structure, reminiscent of the open string metric [7]. This leads to an emergent gravity closely related to NC gauge theory, as anticipated in [5,6]. Among these models, the IKKT model [8] is singled out by maximal supersymmetry, required by consistency at the quantum level. This implies in particular D = 10, which we assume from now on. This model is a candidate for a theory of all fundamental interactions and matter. * Electronic address: daniela.klammer@univie.ac.at † Electronic address: harold.steinacker@univie.ac.atThe model admits 4-dimensional noncommutative spaces M θ ⊂ R 10 as solution, interpreted as space-time embedded in 10 dimensions. This can be seen by splitting the matrices aswhere the "scalar fields" φ i = φ i (X µ ) are assumed to be functions of X µ . We only consider the semi-classical limit of such a quantum space, indicated by ∼. Then X µ ∼ x µ is interpreted as qua...

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].…”

“…It may also be possible to find a twisted embedding along the lines of [44] to obtain a matrix realization of a covariant space-time with big bounce. 13 …”

Topics in Noncommutative Geometry Inspired Physics

“…This perturbative expansion is however an essential feature of incorporating NC effects in gravity. In the last few years an interesting study has been done in the literature [21,22,[215][216][217][218][219][220][221] where gravity is interpreted in a completely different manner within a noncommutative framework. Here gravity is not incorporated in the theory by hand rather it turns out to be an emergent structure from the electromagnetic phenomena in noncommutative spacetime.…”

Spinning squashed extra dimensions and chiral gauge theory from N=4 SYM

“…The present results are therefore also very encouraging towards considering matrix models as fundamental theories, cf. [26][27][28] for interesting recent results. Furthermore, given the much-studied relation between N = 4 SYM and string theory on AdS 5 × S 5 [29], it is natural to ask about a dual description in terms of supergravity or string theory.…”