There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Over the past decade, f(R) theories have been extensively studied as one of the simplest modifications to General Relativity. In this article we review various applications of f(R) theories to cosmology and gravity - such as inflation, dark energy, local gravity constraints, cosmological perturbations, and spherically symmetric solutions in weak and strong gravitational backgrounds. We present a number of ways to distinguish those theories from General Relativity observationally and experimentally. We also discuss the extension to other modified gravity theories such as Brans-Dicke theory and Gauss-Bonnet gravity, and address models that can satisfy both cosmological and local gravity constraints.

f(R) theories

Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution

We review recent progress in massive gravity. We start by showing how different theories of massive gravity emerge from a higher-dimensional theory of general relativity, leading to the Dvali-Gabadadze-Porrati model, cascading gravity and ghost-free massive gravity. We then explore their theoretical and phenomenological consistency, proving the absence of Boulware-Deser ghosts and reviewing the Vainshtein mechanism and the cosmological solutions in these models. Finally we present alternative and related models of massive gravity such as new massive gravity, Lorentz-violating massive gravity and non-local massive gravity.

Massive Gravity

Beyond the cosmological standard model

We consider a gravitational theory in which matter is nonminimally coupled to geometry, with the effective Lagrangian of the gravitational field being given by an arbitrary function of the Ricci scalar, the trace of the matter energy-momentum tensor, and the contraction of the Ricci tensor with the matter energy-momentum tensor. The field equations of the theory are obtained in the metric formalism, and the equation of motion of a massive test particle is derived. In this type of theory the matter energy-momentum tensor is generally not conserved, and this nonconservation determines the appearance of an extra force acting on the particles in motion in the gravitational field. It is interesting to note that in the present gravitational theory, the extra force explicitly depends on the Ricci tensor, which entails a relevant deviation from the geodesic motion, especially for strong gravitational fields, thus rendering the possibility of a space-time curvature enhancement by the ${R}_{\ensuremath{\mu}\ensuremath{\nu}}{T}^{\ensuremath{\mu}\ensuremath{\nu}}$ coupling. The Newtonian limit of the theory is also considered, and an explicit expression for the extra acceleration that depends on the matter density is obtained in the small velocity limit for dust particles. We also analyze in detail the so-called Dolgov-Kawasaki instability and obtain the stability conditions of the theory with respect to local perturbations. A particular class of gravitational field equations can be obtained by imposing the conservation of the energy-momentum tensor. We derive the corresponding field equations for the conservative case by using a Lagrange multiplier method, from a gravitational action that explicitly contains an independent parameter multiplying the divergence of the energy-momentum tensor. The cosmological implications of the theory are investigated in detail for both the conservative and the nonconservative cases, and several classes of exact analytical and approximate solutions are obtained.

Further matters in space-time geometry: f(R,T,RμνTμν) gravity

A generalization to the theory of massive gravity is presented which includes three dynamical metrics. It is shown that at the linear level, the theory predicts a massless spin-2 field which is decoupled from the other two gravitons, which are massive and interacting. In this regime, the matter should naturally couple to massless gravitons which introduce a preferred metric that is the average of the primary metrics. The cosmological solution of the theory shows the de Sitter behavior with a function of mass as its cosmological constant. Surprisingly, it lacks any nontrivial solution when one of the metrics is taken to be Minkowskian and seems to enhance the predictions which suggest that there is no homogeneous, isotropic, and flat solution for the standard massive cosmology.

Multi-Metric Gravity via Massive Gravity

We consider a gravitational model in a Weyl-Cartan space-time, in which the Weitzenbock condition of the vanishing of the sum of the curvature and torsion scalar is also imposed. Moreover, a kinetic term for the torsion is also included in the gravitational action. The field equations of the model are obtained from a Hilbert-Einstein type variational principle, and they lead to a complete description of the gravitational field in terms of two fields, the Weyl vector and the torsion, respectively, defined in a curved background. The cosmological applications of the model are investigated for a particular choice of the free parameters in which the torsion vector is proportional to the Weyl vector. Depending on the numerical values of the parameters of the cosmological model, a large variety of dynamic evolutions can be obtained, ranging from inflationary/accelerated expansions to non-inflationary behaviors. In particular we show that a de Sitter type late time evolution can be naturally obtained from the field equations of the model. Therefore the present model leads to the possibility of a purely geometrical description of the dark energy, in which the late time acceleration of the Universe is determined by the intrinsic geometry of the space-time.

Weyl-Cartan-Weitzenb\"{o}ck gravity as a generalization of teleparallel gravity

We consider a gravitational model in a Weyl-Cartan space-time in which the Weitzenbock condition of the vanishing of the sum of the curvature and torsion scalar is imposed. In contrast to the standard teleparallel theories, our model is formulated in a four-dimensional curved spacetime. The properties of the gravitational field are then described by the torsion tensor and Weyl vector fields. A kinetic term for the torsion is also included in the gravitational action. The field equations of the model are obtained from a Hilbert-Einstein type variational principle, and they lead to a complete description of the gravitational field in terms of two fields, the Weyl vector and the torsion, respectively, defined in a curved background. The cosmological applications of the model are investigated for a particular choice of the free parameters in which the torsion vector is proportional to the Weyl vector. The Newtonian limit of the model is also considered, and it is shown that the Poisson equation can be recovered in the weak field approximation. Depending on the numerical values of the parameters of the cosmological model, a large variety of dynamic evolutions can be obtained, ranging from inflationary/accelerated expansions to non-inflationary behaviors. In particular we show that a de Sitter type late time evolution can be naturally obtained from the field equations of the model. Therefore the present model leads to the possibility of a purely geometrical description of the dark energy, in which the late time acceleration of the Universe is determined by the intrinsic geometry of the space-time.

/pdf/weyl-cartan-weitzenbock-gravity-as-a-generalization-of-8x8abg3x7x.pdf

Weyl-Cartan-Weitzenböck gravity as a generalization of teleparallel gravity

We consider an f(Q, T) type gravity model in which the scalar non-metricity $$Q_{\alpha \mu \nu }$$ of the space-time is expressed in its standard Weyl form, and it is fully determined by a vector field $$w_{\mu }$$. The field equations of the theory are obtained under the assumption of the vanishing of the total scalar curvature, a condition which is added into the gravitational action via a Lagrange multiplier. The gravitational field equations are obtained from a variational principle, and they explicitly depend on the scalar nonmetricity and on the Lagrange multiplier. The covariant divergence of the matter energy-momentum tensor is also determined, and it follows that the nonmetricity-matter coupling leads to the nonconservation of the energy and momentum. The energy and momentum balance equations are explicitly calculated, and the expressions of the energy source term and of the extra force are found. 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 standard general relativity. We consider several cosmological models by imposing some simple functional forms of the function f(Q, T), and we compare the predictions of the theory with the standard $$\Lambda $$CDM model.

/pdf/weyl-type-f-q-t-gravity-and-its-cosmological-implications-1vfbzhw7kt.pdf

Shahab Shahidi

Papers

Further matters in space-time geometry: f(R,T,RμνTμν) gravity

Multi-Metric Gravity via Massive Gravity

Weyl-Cartan-Weitzenb\"{o}ck gravity as a generalization of teleparallel gravity

Weyl-Cartan-Weitzenböck gravity as a generalization of teleparallel gravity

Weyl type f(Q, T) gravity, and its cosmological implications