Cosmological implications of a class of hybrid metric-Palatini gravity with a non-minimal matter-geometry coupling is considered. The theory contains a metric curvature tensor, together with a curvature tensor constructed from an independent affine connection. We will show that the model could be written as a bi-scalar–tensor gravity with a non-minimal coupling between matter sector and a scalar field. The theory will then be confronted with observational data from Cosmic Chronometers, BAO dataset from DESI and the Pantheon $$^+$$ dataset. We will show that the theory could be a good alternative to the $$\Lambda $$ CDM model with the difference that the conservation of the baryonic matter sector holds only at the background level. The statefinder analysis will also be applied to the theory and it is observed that the DE behavior of the theory exhibits a quintessence to phantom transition occurs at redshifts around $$z\approx 0.86$$ .

Iran

Reza Jalali

Shahab Shahidi

Mohammad Haghighi

Cosmology in generalized hybrid metric-Palatini with matter-geometry coupling

The hybrid metric-Palatini gravity with Lagrangian density $$L =R+f(\mathcal {R},\mathcal {R}_{\mu \nu }\mathcal {R}^{\mu \nu })$$ is considered, where $\textit{R}$ is the metric Ricci scalar and $$\mathcal {R}_{\mu \nu }$$ is the Palatini Ricci tensor. Contrary to the standard hybrid metric-Palatini theory, because of the term $$\mathcal {R}_{\mu \nu }\mathcal {R}^{\mu \nu }$$ in the action, the model can not be analytically transformed to a scalar-tensor theory. However, on top of a maximally symmetric space-times like the FRW universe, there is a way to solve for a metric compatible connection which we will follow in this paper. The cosmological implications of the resulting model will then be fully considered. The best fit values of the model and cosmological parameters will be obtained by confronting the model with the recent observational data on the Hubble parameter. We will see that the observational data can be explained very good in this model, but, significant deviations from the standard $$\Lambda $$ CDM model could be seen in derivatives of the Hubble parameter $\textit{q}$, $\textit{j}$ and $\textit{s}$. We will perform a statefinder analysis for the model and show that its behavior differs from that of the $$\Lambda $$ CDM model. Also, we will consider the recently proposed $\textit{Om}$ diagnostics to categorize the dark energy type of the model and obtain the $$\omega $$ -varying alternative of the model that mimics the Hubble flow.

Shiva Kayedi

Cosmological implications of hybrid metric-Palatini $$f(\mathcal {R},\mathcal {R}_{\mu \nu }\mathcal {R}^{\mu \nu })$$ gravity

We consider the cosmological implications of the Weyl geometric gravity theory. The basic action of the model is obtained from the simplest conformally invariant gravitational action, constructed, in Weyl geometry, from the square of the Weyl scalar, the strength of the Weyl vector, and a matter term, respectively. The total action is linearized in the Weyl scalar by introducing an auxiliary scalar field. To maintain the conformal invariance of the action the trace condition is imposed on the matter energy–momentum tensor, thus making the matter sector of the action conformally invariant. The field equations are derived by varying the action with respect to the metric tensor, the Weyl vector field, and the scalar field, respectively. We investigate the cosmological implications of the theory, and we obtain first the cosmological evolution equations for a flat, homogeneous and isotropic geometry, described by Friedmann–Lemaitre–Robertson–Walker metric, which generalize the Friedmann equations of standard general relativity. In this context we consider two cosmological models, corresponding to the vacuum state, and to the presence of matter described by a linear barotropic equation of state. In both cases we perform a detailed comparison of the predictions of the theory with the cosmological observational data, and with the standard  $$\Lambda $$  <math> <mi>Λ</mi> </math>   CDM model. By assuming that the presence of the Weyl geometric effects induce small perturbations in the homogeneous and isotropic cosmological background, and that the anisotropy parameter is small, the equations of the cosmological perturbations due to the presence of the Weyl geometric effects are derived. The time evolution of the metric and matter perturbations are explicitly obtained. Therefore, if Weyl geometric effects are present, the Universe would acquire some anisotropic characteristics, and its geometry will deviate from the standard FLRW one.

Romania

Tiberiu Harko

Cosmological implications of the Weyl geometric gravity theory

We consider numerical black hole solutions in the Weyl conformal geometry and its associated conformally invariant Weyl quadratic gravity. In this model, Einstein gravity (with a positive cosmological constant) is recovered in the spontaneously broken phase of Weyl gravity after the Weyl gauge field ( $$\omega _{\mu }$$  <math> <msub> <mi>ω</mi> <mi>μ</mi> </msub> </math>  ) becomes massive through a Stueckelberg mechanism and it decouples. As a first step in our investigations, we write down the conformally invariant gravitational action, containing a scalar degree of freedom and the Weyl vector. The field equations are derived from the variational principle in the absence of matter. By adopting a static spherically symmetric geometry, the vacuum field equations for the gravitational, scalar, and Weyl fields are obtained. After reformulating the field equations in a dimensionless form, and by introducing a suitable independent radial coordinate, we obtain their solutions numerically. We detect the formation of a black hole from the presence of a Killing horizon for the timelike Killing vector in the metric tensor components, indicating the existence of the singularity in the metric. Several models corresponding to different functional forms of the Weyl vector are considered. An exact black hole model corresponding to a Weyl vector having only a radial spacelike component is also obtained. The thermodynamic properties of the Weyl geometric type black holes (horizon temperature, specific heat, entropy, and evaporation time due to Hawking luminosity) are also analyzed in detail.

China

Jin-Zhao Yang

Black hole solutions in the quadratic Weyl conformal geometric theory of gravity

We investigate the field equations of the conformally invariant models of gravity with curvature-matter coupling, constructed in Weyl geometry, using the Palatini formalism. We consider the case in which the Lagrangian is given by the sum of the square of the Weyl scalar, the strength of the field associated to the Weyl vector, and a conformally invariant geometry-matter coupling term, constructed from the matter Lagrangian and the Weyl scalar. After substituting the Weyl scalar in terms of its Riemannian counterpart, the quadratic action is defined in Riemann geometry and involves a nonminimal coupling between the Ricci scalar and the matter Lagrangian. For the sake of generality, a more general Lagrangian, in which the Weyl vector is nonminimally coupled with an arbitrary function of the Ricci scalar, is also considered. By varying the action independently with respect to the metric and the connection, the independent connection can be expressed as the Levi-Civita connection of an auxiliary Ricci scalar- and Weyl vector-dependent metric, which is related to the physical metric by means of a conformal transformation. The field equations are obtained in both the metric and the Palatini formulations. The cosmological implications of the Palatini field equations are investigated for three distinct models corresponding to different forms of the coupling functions. A comparison with the standard  $$\Lambda $$  <math> <mi>Λ</mi> </math>  CDM model is also performed, and we find that the Palatini type cosmological models can give an acceptable description of the observations.

Palatini formulation of the conformally invariant  $$f\left( R,L_m\right) $$  <math> <mrow> <mi>f</mi> <mfenced> <mi>R</mi> <mo>,</mo> <msub> <mi>L</mi> <mi>m</mi> </msub> </mfenced> </mrow> </math>   gravity theory

Recently a new 4D Einstein–Gauss–Bonnet theory has been introduced (Glavan and Lin in Phys Rev Lett 124: 081301, 2020) with a serious debate that it does not possess a covariant equation of motion in 4D. This feature, makes impossible to consider non-symetric space-times in this model, such as anisotropic cosmology. In this note, we will present a new proposal to make this happen, by introducing a Lagrange multiplier to the action which eliminates the higher dimensional term from the equation of motion. The theory has then a covariant 4D equation of motion which is useful to study the less symmetric metrics. On top of FRW universe, the constraint theory is equivalent to the original 4D Einstein–Gauss–Bonnet gravity. We will then consider the anisotropic cosmology of the model and compare the theory with observational data. We will see that the theory becomes non-conservative and the matter density abundance falls more rapidly at larger redshifts compared to the conservative matter sources.

Nima Khosravi

Anisotropy in constraint 4D Gauss–Bonnet gravity

We investigate the coupling of matter to geometry in conformal quadratic Weyl gravity, by assuming a coupling term of the form  $$L_m{\tilde{R}}^2$$  <math> <mrow> <msub> <mi>L</mi> <mi>m</mi> </msub> <msup> <mrow> <mover> <mi>R</mi> <mo>~</mo> </mover> </mrow> <mn>2</mn> </msup> </mrow> </math>  , where  $$L_m$$  <math> <msub> <mi>L</mi> <mi>m</mi> </msub> </math>   is the ordinary matter Lagrangian, and  $${\tilde{R}}$$  <math> <mover> <mi>R</mi> <mo>~</mo> </mover> </math>   is the Weyl scalar. The coupling explicitly satisfies the conformal invariance of the theory. By expressing  $${\tilde{R}}^2$$  <math> <msup> <mrow> <mover> <mi>R</mi> <mo>~</mo> </mover> </mrow> <mn>2</mn> </msup> </math>   with the help of an auxiliary scalar field and of the Weyl scalar, the gravitational action can be linearized, leading in the Riemann space to a conformally invariant  $$f\left( R,L_m\right) $$  <math> <mrow> <mi>f</mi> <mfenced> <mi>R</mi> <mo>,</mo> <msub> <mi>L</mi> <mi>m</mi> </msub> </mfenced> </mrow> </math>   type theory, with the matter Lagrangian nonminimally coupled to the Ricci scalar. We obtain the gravitational field equations of the theory, as well as the energy–momentum balance equations. The divergence of the matter energy–momentum tensor does not vanish, and an extra force, depending on the Weyl vector, and matter Lagrangian is generated. The thermodynamic interpretation of the theory is also discussed. The generalized Poisson equation is derived, and the Newtonian limit of the equations of motion is considered in detail. The perihelion precession of a planet in the presence of an extra force is also considered, and constraints on the magnitude of the Weyl vector in the Solar System are obtained from the observational data of Mercury. The cosmological implications of the theory are also considered for the case of a flat, homogeneous and isotropic Friedmann–Lemaitre–Robertson–Walker geometry, and it is shown that the model can give a good description of the observational data for the Hubble function up to a redshift of the order of  $$z\approx 3$$  <math> <mrow> <mi>z</mi> <mo>≈</mo> <mn>3</mn> </mrow> </math>  .

Coupling matter and curvature in Weyl geometry: conformally invariant  $$f\left( R,L_m\right) $$  <math> <mrow> <mi>f</mi> <mfenced> <mi>R</mi> <mo>,</mo> <msub> <mi>L</mi> <mi>m</mi> </msub> </mfenced> </mrow> </math>   gravity

We consider the cosmological evolution in an osculating point Barthel–Randers type geometry, in which to each point of the space-time manifold an arbitrary point vector field is associated. This Finsler type geometry is assumed to describe the physical properties of the gravitational field, as well as the cosmological dynamics. For the Barthel–Randers geometry the connection is given by the Levi-Civita connection of the associated Riemann metric. The generalized Friedmann equations in the Barthel–Randers geometry are obtained by considering that the background Riemannian metric in the Randers line element is of Friedmann–Lemaitre–Robertson–Walker type. The matter energy balance equation is derived, and it is interpreted from the point of view of the thermodynamics of irreversible processes in the presence of particle creation. The cosmological properties of the model are investigated in detail, and it is shown that the model admits a de Sitter type solution, and that an effective cosmological constant can also be generated. Several exact cosmological solutions are also obtained. A comparison of three specific models with the observational data and with the standard  $$\Lambda $$  <math> <mi>Λ</mi> </math>  CDM model is also performed by fitting the observed values of the Hubble parameter, with the models giving a satisfactory description of the observations.

Thailand

Rattanasak Hama

Japan

Sorin Sabau

Cosmological evolution and dark energy in osculating Barthel–Randers geometry

We consider a gravitational theory with an additional non-minimal coupling between baryonic matter fields and geometry. The coupling is second order in the energy momentum tensor and can be seen as a generalization of the energy–momentum squared gravity model. We will add a constraint through a Lagrange multiplier to ensure the conservation of the energy–momentum tensor. Background cosmological implications together with its dynamical system analysis will be investigated in details. Also we will consider the growth of matter perturbations at first order, and estimate the model parameter from observations on H and also  $$f\sigma _8$$  <math> <mrow> <mi>f</mi> <msub> <mi>σ</mi> <mn>8</mn> </msub> </mrow> </math>  . We will show that the model parameter should be small and positive in 2 $$\sigma $$  <math> <mi>σ</mi> </math>   confidence interval. The theory is shown to be in a good agreement with observational data.

Non-minimal energy–momentum squared gravity

We consider the geodesic deviation equation, describing the relative accelerations of nearby particles, and the Raychaudhuri equation, giving the evolution of the kinematical quantities associated with deformations (expansion, shear and rotation) in the Weyl-type f(Q, T) gravity, in which the non-metricity Q is represented in the standard Weyl form, fully determined by the Weyl vector, while T represents the trace of the matter energy–momentum tensor. The effects of the Weyl geometry and of the extra force induced by the non-metricity–matter coupling are explicitly taken into account. The Newtonian limit of the theory is investigated, and the generalized Poisson equation, containing correction terms coming from the Weyl geometry, and from the geometry matter coupling, is derived. As a physical application of the geodesic deviation equation the modifications of the tidal forces, due to the non-metricity–matter coupling, are obtained in the weak-field approximation. The tidal motion of test particles is directly influenced by the gradients of the extra force, and of the Weyl vector. As a concrete astrophysical example we obtain the expression of the Roche limit (the orbital distance at which a satellite begins to be tidally torn apart by the body it orbits) in the Weyl-type f(Q, T) gravity.