Length-preserving biconnection gravity and its cosmological implications

Abstract

We consider a length preserving biconnection gravitational theory, inspired by information geometry, which extends general relativity, by using the mutual curvature as the fundamental object describing gravity. The two connections used to build up the theory are the Schrödinger connection, and its dual. In our geometric approach it can be seen that the dual of a non-metric Schrödinger connection possesses torsion, even if the Schrödinger connection itself does not, and consequently the pair (M,g,∇*) is a quasi-statistical manifold. The field equations are postulated to have the form of the standard Einstein equations, but with the Ricci tensor- and scalar replaced with the mutual curvature tensor, and the mutual curvature scalar, resulting in additional torsion-dependent terms. The covariant divergence of the matter energy-momentum does not vanish in this theory. We derive the equation of motion for massive particles, which shows the presence of an extra force, depending on the torsion vector. The Newtonian limit of the equations of motion is also considered. We explore the cosmological implications by deriving the generalized Friedmann equations for the Friedmann-Lemaitre-Robertson-Walker (FLRW geometry). They contain additional terms that can be interpreted as describing an effective, geometric type dark energy. We examine two cosmological models: one with conserved matter, and one where dark energy and pressure are related by a linear equation of state. The predictions of both models are compared with a set of observational values of the Hubble function, and with the standard ΛCDM model. Length-preserving biconnection gravity models fit well the observational data, and also align with ΛCDM at low redshifts (z < 3). The obtained results suggest that a modified biconnection geometry could explain the late-time acceleration through an effective geometric dark energy, as well as the formation of the supermassive black holes, as they predict a different age of our Universe as compared to standard cosmology.