Weyl-Eddington-Einstein affine gravity in the context of modern cosmology

Abstract

We propose new models of an 'affine' theory of gravity in D-dimensional space-times with symmetric connections. They are based on ideas of Weyl, Eddington and Einstein and, in particular, on Einstein's proposal to specify the space -time geometry by use of the Hamilton principle. More specifically, the connection coefficients are derived by varying a 'geometric' Lagrangian that is supposed to be an arbitrary function of the generalized (non-symmetric) Ricci curvature tensor (and, possibly, of other fundamental tensors) expressed in terms of the connection coefficients regarded as independent variables. In addition to the standard Einstein gravity, such a theory predicts dark energy (the cosmological constant, in the first approximation), a neutral massive (or, tachyonic) vector field, and massive (or, tachyonic) scalar fields. These fields couple only to gravity and may generate dark matter and/or inflation. The masses (real or imaginary) have geometric origin and one cannot avoid their appearance in any concrete model. Further details of the theory -such as the nature of the vector and scalar fields that can describe massive particles, tachyons, or even 'phantoms' -depend on the concrete choice of the geometric Lagrangian. In 'natural' geometric theories, which are discussed here, dark energy is also unavoidable. Main parameters -mass, cosmological constant, possible dimensionless constants -cannot be predicted, but, in the framework of modern 'multiverse' ideology, this is rather a virtue than a drawback of the theory. To better understand possible applications of the theory we discuss some further extensions of the affine models and analyze in more detail approximate ('physical') Lagrangians that can be applied to cosmology of the early Universe.

Polynomial integrals of motion in dilaton gravity theories

“…Obviously, the Hamiltonian itself is also a quadratic integral; therefore, only integrals with a zero component b 12 are reasonable to consider. Unlike the linear case, where we solve system (10) in steps, we directly pass to solving the final equation (14), where the parameters v i are eliminated.…”

“…It is likely that the simplest mechanisms such as inflation with a single scalar field (inflaton) and dark matter with the cosmological constant are insufficient for describing the observable dynamics [3]- [6]. At the same time, progress in developing new models of supergravity, string theory, and many other theories generalizing the Einstein gravity theory by introducing new fields [7]- [10] and other versions of action [11], [12] or space-time geometry [13], [14] leads to understanding that the complicated nonlinear and nontrivial constructions are not exotic and can be used practically to describe the real Universe.…”

“…3, we study dilaton gravity models with a single scalar field for the presence of linear and quadratic integrals of motion. This class of dilaton gravity models can be used, for instance, to describe an affine generalization of the Einstein theory [25], cosmology with a scalar and vector field [14], black holes and solutions of the gravity-wave type [26]- [28], and so on. Knowing a linear integral of motion, we can select the cyclic coordinate and reduce the configuration space, thus simplifying the analysis of the model.…”

Vector fields in cosmology

“…The motivation for this modification was, at first, to derive electromagnetism from gravity, and later just to remove the restriction of the Riemannian geometry in the GR formalism which has no fundamental reasoning. The theory was recently generalized on case of arbitrary dimensions [3]. Again, the three-dimensional case is the most simple, since the theory becomes linear.…”

“…We just mention that even in the simplest D = 3 theory with vector fields coming from a geometry (cylindrical dimensional reduction, non-Riemannian geometry) there arise a rather non-trivial dynamics, which can be useful in cosmology 3 . A bit more detailed consideration will be given in the next section for the realistic four-dimensional isotropic theory.…”