Abstract
We propose an extension of the symmetric teleparallel gravity, in which the gravitational action L is given by an arbitrary function f of the non-metricity Q and of the trace of the matter-energy-momentum tensor T, so that L=f(Q,T). The field equations of the theory are obtained by varying the gravitational action with respect to both metric and connection. The covariant divergence of the field equations is obtained, with the geometry–matter coupling leading to the nonconservation of the energy-momentum tensor. We investigate the cosmological implications of the theory, and we obtain the cosmological evolution equations for a flat, homogeneous and isotropic geometry, which generalize the Friedmann equations of general relativity. We consider several cosmological models by imposing some simple functional forms of the function f(Q, T), corresponding to additive expressions of f(Q, T) of the form f(Q,T)=alpha Q+beta T, f(Q,T)=alpha Q^{n+1}+beta T, and f(Q,T)=-alpha Q-beta T^2. The Hubble function, the deceleration parameter, and the matter-energy density are obtained as a function of the redshift by using analytical and numerical techniques. For all considered cases the Universe experiences an accelerating expansion, ending with a de Sitter type evolution. The theoretical predictions are also compared with the results of the standard Lambda CDM model.
Highlights
The detection of the gravitational waves [12] did give the opportunity to evaluate the predictions of general relativity in the final stages of binary black hole coalescence, corresponding to the limiting case of strong gravitational fields
The same observations indicate the surprising result that around 95–96% of the content of the Universe is in the form of two mysterious components, called dark energy and dark matter, respectively, with only about 4–5% of the total composition represented by baryonic matter [21,22]
Standard general relativity may not be the ultimate theory of the gravitational force, since it cannot give satisfactory explanations to the two fundamental problems present day cosmology is confronted with: the dark matter problem and the dark energy problem, respectively
Summary
Weyl introduced an important generalization of the Riemannian geometry, representing the mathematical basis of general relativity, by assuming that during the parallel transport around a closed path, an arbitrary vector will be subject to a change of its direction, but it will experience a modification of its length [79]. By performing a local scaling of lengths of the form l = σ (x)l, the field wμ changes as wμ = wμ + (ln σ ),μ, while the metric tensor coefficients are modified according to the conformal transformations gμν = σ 2gμν and gμν = σ −2gμν, respectively [99] Another important property of the Weyl geometry is the existence of the semi-metric connection,. In a Weyl–Cartan spacetime we can introduce a symmetric metric tensor gμν, which defines the length of a vector, and an asymmetric connectionλμν, which determines the law of the parallel transport as dvμ = −vσμσ νd xν [86,99]. The gravitational effects occur not because of the rotation of the angle between two vectors in the parallel transport, but because of the variation of the length of the vector itself
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