Generalized Friedmann Equation Research Articles

Semi-symmetric metric gravity: From the Friedmann–Schouten geometry with torsion to dynamical dark energy models

In this paper, we develop a geometric generalization of General Relativity based on the semi-symmetric metric connection introduced by Friedmann and Schouten in 1924. Although the mathematical properties of this connection, which allows for torsion, have been extensively studied, its physical implications remain underexplored. We provide a detailed exposition of the differential geometric aspects of semi-symmetric connections and formulate the corresponding field equations induced by the specific form of torsion we are investigating. We consider the cosmological applications of the theory by deriving the generalized Friedmann equations in a flat, homogeneous and isotropic geometry. The Friedmann equations also include some supplementary terms as compared to their general relativistic counterparts, which can be interpreted as a geometric type dark energy. To evaluate the proposed theory, we consider three cosmological models — the first with constant effective density and pressure, the second with the dark energy satisfying a linear equation of state, and a third one with a polytropic equation of state. We also compare the predictions of the semi-symmetric metric gravitational theory with the observational data for the Hubble function, and with the predictions of the ΛCDM model. Our findings indicate that the semi-symmetric metric cosmological models give a good description of the observational data, and for certain values of the model parameters, they can reproduce almost exactly the predictions of the ΛCDM paradigm. Consequently, Friedmann’s initially proposed geometry emerges as a credible alternative to standard general relativity, in which dark energy has a purely geometric origin.

Warm inflation in a Universe with a Weylian boundary

We investigate the influence of boundary terms in the warm inflationary scenario, by considering that in the Einstein–Hilbert action the boundary can be described in terms of a Weyl-type geometry. The gravitational action, as well as the field equations, are thus extended to include new geometrical terms, coming from the non-metric nature of the boundary, and depending on the Weyl vector, and its covariant derivatives. We investigate the effects of these new boundary terms by considering the warm inflationary scenario of the early evolution of the Universe, in the presence of a scalar field. We obtain the generalized Friedmann equations in the Universe with a Weylian boundary by considering the Friedmann–Lemaitre–Robertson–Walker metric. We consider the simultaneous decay of the scalar field, and of the creation of radiation, by appropriately splitting the general conservation equation through the introduction of the dissipation coefficient, which can depend on both the scalar field, and the Weyl vector. We consider three distinct warm inflationary models, in which the dissipation coefficients are chosen as different functions of the scalar field and of the Weyl vector. The numerical solutions of the cosmological evolution equations show that the radiation is created during the very early phases of expansion, and, after the radiation reaches its maximum value, the transition from an accelerating inflationary phase to a decelerating one takes place. Moreover, it turns out that the Weyl vector, describing the boundary effects on the cosmological evolution, plays a significant role during the process of radiation creation.

Finsler–Randers–Sasaki gravity and cosmology

We present for the first time a Friedmann-like construction in the framework of an osculating Finsler–Randers–Sasaki (F–R–S) geometry. In particular, we consider a vector field in the metric on a Lorentz tangent bundle, and thus the curvatures of horizontal and vertical spaces, as well as the extra contributions of torsion and non-linear connection, provide an intrinsic richer geometrical structure, with additional degrees of freedom, that lead to extra terms in the field equations. Applying these modified field equations at a cosmological setup we extract the generalized Friedmann equations for the horizontal and vertical space, showing that we obtain an effective dark energy sector arising from the richer underlying structure of the tangent bundle. Additionally, as it is common in Finsler-like constructions, we obtain an effective interaction between matter and geometry. Finally, we consider a specific model and we show that it can describe the sequence of matter and dark-energy epochs, and that the dark-energy equation of state can lie in the quintessence or phantom regimes, or cross the phantom divide.

Dissipative quintessence and its cosmological implications

We consider a generalization of the quintessence type scalar field cosmological models, by adding a multiplicative dissipative term in the scalar field Lagrangian, which generally is represented in an exponential form. The generalized dissipative Klein-Gordon equation is obtained in a general covariant form in Riemann geometry, from the variational principle with the help of the Euler-Lagrange equations. The energy-momentum tensor of the dissipative scalar field is also obtained from the dissipative Lagrangian, and its properties are discussed in detail. Several applications of the general formalism are presented for the case of the cosmological Friedmann-Lema\^{\i}tre-Robertson-Walker metric. The generalized Friedmann equations in the presence of the dissipative scalar field are obtained for a specific form of dissipation, with the dissipation exponent represented as the time integral of the product of the Hubble function, and of a function describing the dissipative properties of the scalar field. For this case the Friedmann equations reduce to a system of differential-integral equations, which, by means of some appropriate transformation, can be represented in the redshift space as a first order dynamical system. Several cosmological models, corresponding to different choices of the dissipation function, and of the scalar field potential, are considered in detail. For the different values of the model parameters the evolution of the cosmological parameters (scale factor, Hubble function, deceleration parameter, the effective density and pressure of the scalar field, and the parameter of the dark energy equation of state, respectively) is considered in detail by using both analytical and numerical techniques. A comparison with the observational data for the Hubble function and with the predictions of the standard $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ paradigm is presented for each dissipative scalar field model. In the large time limit the model describes an accelerating universe, with the effective negative pressure induced by the dissipative effects associated with the scalar field. Accelerated expansion in the absence of the scalar field potential is also possible, with the kinetic term dominating the expansionary evolution. The dissipative scalar field models describe well the data, with the model free parameters obtained by a trial and error method. The obtained results show the dissipative scalar field model offers an effective dynamical possibility for replacing the cosmological constant and for explaining the recent cosmological observational data.

Cosmological tests of the osculating Barthel–Kropina dark energy model

We further investigate the dark energy model based on the Finsler geometry inspired osculating Barthel–Kropina cosmology. The Barthel–Kropina cosmological approach is based on the introduction of a Barthel connection in an osculating Finsler geometry, with the connection having the property that it is the Levi-Civita connection of a Riemannian metric. From the generalized Friedmann equations of the Barthel–Kropina model, obtained by assuming that the background Riemannian metric is of the Friedmann–Lemaitre–Robertson–Walker type, an effective geometric dark energy component can be generated, with the effective, geometric type pressure, satisfying a linear barotropic type equation of state. The cosmological tests, and comparisons with observational data of this dark energy model are considered in detail. To constrain the Barthel–Kropina model parameters, and the parameter of the equation of state, we use 57 Hubble data points, and the Pantheon Supernovae Type Ia data sample. The st statistical analysis is performed by using Markov Chain Monte Carlo (MCMC) simulations. A detailed comparison with the standard Lambda CDM model is also performed, with the Akaike information criterion (AIC), and the Bayesian information criterion (BIC) used as the two model selection tools. The statefinder diagnostics consisting of jerk and snap parameters, and the Om(z) diagnostics are also considered for the comparative study of the Barthel–Kropina and Lambda CDM cosmologies. Our results indicate that the Barthel–Kropina dark energy model gives a good description of the observational data, and thus it can be considered a viable alternative of the Lambda CDM model.

Novel relationship between shear and energy density at the bounce in nonsingular Bianchi I spacetimes

In classical Bianchi I spacetimes, underlying conditions for what dictates the singularity structure---whether it is anisotropic shear or energy density, can be easily determined from the generalized Friedmann equation. However, in nonsingular bouncing anisotropic models these insights are difficult to obtain in the quantum gravity regime where the singularity is resolved at a nonvanishing mean volume which can be large compared to the Planck volume, depending on the initial conditions. Such nonsingular models may also lack a generalized Friedmann equation making the task even more difficult. We address this problem in an effective spacetime description of loop quantum cosmology (LQC) where energy density and anisotropic shear are universally bounded due to quantum geometry effects, but a generalized Friedmann equation has been difficult to derive due to the underlying complexity. Performing extensive numerical simulations of effective Hamiltonian dynamics, we bring to light a surprising, seemingly universal relationship between energy density and anisotropic shear at the bounce in LQC. For a variety of initial conditions for a massless scalar field, an inflationary potential, and two types of ekpyrotic potentials we find that the values of energy density and the anisotropic shear at the quantum bounce follow a novel parabolic relationship which reveals some surprising results about the anisotropic nature of the bounce, such as that the maximum value of the anisotropic shear at the bounce is reached when the energy density reaches approximately half of its maximum allowed value. The relationship we find can prove very useful for developing our understanding of the degree of anisotropy of the bounce, isotropization of the postbounce universe, and discovering the modified generalized Friedmann equation in Bianchi I models with quantum gravity corrections.

Effects of the matter Lagrangian degeneracy in f(Q, T) gravity

In this paper, we investigated the theoretical and cosmological effects of the matter Lagrangian degeneracy in an extension of the Symmetric Teleparallel Equivalent of General Relativity, denoted as $f (Q, T )$ gravity. This degeneracy comes from the fact that both $\mathcal{L}_m = p$ and $\mathcal{L}_m = -\rho$ can give rise to the stress-energy tensor of a perfect fluid. The $f(Q,T)$ equations depend on the form of the matter Lagrangian and hence they also have a degeneracy that has influence on the density evolution of dust matter and radiation, on the form of the generalized Friedmann equations and also shows changes in the cosmological parameters confidence regions when performing Monte Carlo Markov Chains analyses. Our results suggest that since changing the matter Lagrangian causes different theoretical and cosmological results, future studies in $f(Q,T)$ gravity should consider both forms of the matter Lagrangians to account for the different mathematical and observational results.

Dark energy and accelerating cosmological evolution from osculating Barthel–Kropina geometry

Finsler geometry is an important extension of Riemann geometry, in which each point of the spacetime manifold is associated with an arbitrary internal variable. Two interesting Finsler geometries with many physical applications are the Randers and Kropina type geometries. A subclass of Finsler geometries is represented by the osculating Finsler spaces, in which the internal variable is a function of the base manifold coordinates only. In an osculating Finsler geometry, we introduce the Barthel connection, with the remarkable property that it is the Levi–Civita connection of a Riemannian metric. In the present work we consider the gravitational and cosmological implications of a Barthel–Kropina type geometry. We assume that in this geometry the Ricci type curvatures are related to the matter energy–momentum tensor by the standard Einstein equations. The generalized Friedmann equations in the Barthel–Kropina geometry are obtained by considering that the background Riemannian metric is of Friedmann–Lemaitre–Robertson–Walker type. The matter energy balance equation is also derived. 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 dark energy component can also be generated. Several cosmological solutions are also obtained by numerically integrating the generalized Friedmann equations. A comparison of two specific classes of models with the observational data and with the standard Lambda CDM model is also performed, and it is found that the Barthel–Kropina type models give a satisfactory description of the observations.

Emergent cosmology from quantum gravity in the Lorentzian Barrett-Crane tensorial group field theory model

We study the cosmological sector of the Lorentzian Barrett-Crane (BC) model coupled to a free massless scalar field in its Group Field Theory (GFT) formulation, corresponding to the mean-field hydrodynamics obtained from coherent condensate states. The relational evolution of the condensate with respect to the scalar field yields effective dynamics of homogeneous and isotropic cosmologies, similar to those previously obtained in SU(2)-based EPRL-like models. Also in this manifestly Lorentzian setting, in which only continuous SL(2,ℂ)-representations are used, we obtain generalized Friedmann equations that generically exhibit a quantum bounce, and can reproduce all of the features of the cosmological dynamics of EPRL-like models. This lends support to the expectation that the EPRL-like and BC models may lie in the same continuum universality class, and that the quantum gravity mechanism producing effective bouncing scenarios may not depend directly on the discretization of geometric observables.

Non-minimal geometry–matter couplings in Weyl–Cartan space–times: f(R,T,Q,Tm) gravity

We consider an extension of standard General Relativity in which the Hilbert-Einstein action is replaced by an arbitrary function of the Ricci scalar, nonmetricity, torsion, and the trace of the matter energy-momentum tensor. By construction, the action involves a non-minimal coupling between matter and geometry. The field equations of the model are obtained, and they lead to the nonconservation of the matter energy-momentum tensor. A thermodynamic interpretation of the nonconservation of the energy-momentum tensor is also developed in the framework of the thermodynamics of the irreversible processes in open systems. The Newtonian limit of the theory is considered, and the generalized Poisson equation is obtained in the low velocity and weak fields limits. The nonmetricity, the Weyl vector, and the matter couplings generate an effective gravitational coupling in the Poisson equation. We investigate the cosmological implications of the theory for two different choices of the gravitational action, corresponding to an additive and a multiplicative algebraic structure of the function $f$, respectively. We obtain the generalized Friedmann equations, and we compare the theoretical predictions with the observational data. \te{We find that the cosmological models can give a good descriptions of the observations up to a redshift of $z=2$, and, for some cases, up to a redshift of $z=3$.

Effects of Quantum Metric Fluctuations on the Cosmological Evolution in Friedmann-Lemaitre-Robertson-Walker Geometries

In this paper, the effects of the quantum metric fluctuations on the background cosmological dynamics of the universe are considered. To describe the quantum effects, the metric is assumed to be given by the sum of a classical component and a fluctuating component of quantum origin . At the classical level, the Einstein gravitational field equations are equivalent to a modified gravity theory, containing a non-minimal coupling between matter and geometry. The gravitational dynamics is determined by the expectation value of the fluctuating quantum correction term, which can be expressed in terms of an arbitrary tensor Kμν. To fix the functional form of the fluctuation tensor, the Newtonian limit of the theory is considered, from which the generalized Poisson equation is derived. The compatibility of the Newtonian limit with the Solar System tests allows us to fix the form of Kμν. Using these observationally consistent forms of Kμν, the generalized Friedmann equations are obtained in the presence of quantum fluctuations of the metric for the case of a flat homogeneous and isotropic geometry. The corresponding cosmological models are analyzed using both analytical and numerical method. One finds that a large variety of cosmological models can be formulated. Depending on the numerical values of the model parameters, both accelerating and decelerating behaviors can be obtained. The obtained results are compared with the standard ΛCDM (Λ Cold Dark Matter) model.

Cosmological evolution and dark energy in osculating Barthel\u2013Randers geometry

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 CDM model is also performed by fitting the observed values of the Hubble parameter, with the models giving a satisfactory description of the observations.

Non-Riemannian cosmology: The role of shear hypermomentum

We consider the usual Einstein–Hilbert action in a Metric-Affine setup and in the presence of a Perfect Hyperfluid. In order to decode the role of shear hypermomentum, we impose vanishing spin and dilation parts on the sources and allow only for non-vanishing shear. We then consider an FLRW background and derive the generalized Friedmann equations in the presence of shear hypermomentum. By providing one equation of state among the shear variables we study the cases for which shear has an accelerating/decelerating effect on Universe’s expansion. In particular, we see that shear offers a possibility to prevent the initial singularity formation. We also provide some exact solutions in the shear-dominated era and discuss the physical significance of the shear current.

К моделированию динамической эволюции Вселенной под действием энтропийной силы, связанной с модифицированной энтропией Шарма-Миттала

Using the Verlind formalism, the paper considers several scenarios of the evolution of the Friedman-Robertson-Walker Universe, which arise in the framework of entropic cosmology based on the formulated new modification of the Sharma-Mittal entropy. The research, carried out in the framework of non-Gaussian statistical theory, uses several entropies associated with the surface of the horizon of the Universe due to the holographic information stored there. A set of new generalized Friedmann equations is obtained, in which, instead of the cosmological constant, control forces appear based on the Bekenstein-Hawking, Tsallis-Chirto and Barrow entropies, as well as modified Sharma-Mittal and Renyi entropies containing additional nonextensity parameters. The proposed approach, associated with the use of probabilistic nonextensive aspects of the Hubble horizon of the surface of the Universe, meets all the basic requirements for thermodynamic modeling of the dynamic behavior of outer space without involving the concept of dark energy.

Thermodynamic aspects of entropic cosmology with viscosity

We describe the evolution of the early and late universe from thermodynamic considerations, using the generalized nonextensive Tsallis entropy with a variable exponent. A new element in our analysis is the inclusion of a bulk viscosity in the description of the cosmic fluid. Using the generalized Friedmann equation, a description of the early and the late universe is obtained.

Higher derivative and mimetic models on non flat FLRW space–times

Abstract An effective Lagrangian approach, partly inspired by Quantum Loop Cosmology (QLC), is presented and formulated in a non flat FLRW space–times, making use of modified gravitational models. The models considered are non generic, and their choice is dictated by the necessity to have at least second order differential equations of motion in a non flat FLRW space–time. This is accomplished by a class of Lagrangian which are not analytic in the curvature invariants, or making use of a mimetic gravitational scalar field. In this paper we want to show that, for some effective models, although the associated generalized Friedmann equation may admit non singular metrics as solutions, the de Sitter space–time is present only for a restricted class, which includes General Relativity and Lovelock gravity. The other models admit pseudo de Sitter solutions, namely FLRW metrics, such that for vanishing spatial curvature looks like flat de Sitter patch, but for non vanishing spatial curvature are regular bounce metrics or, when the spatial curvature is negative, Big-Bang singular metrics.

Viscous Extended Cosmic Chaplygin Gas with Varying Cosmological Constant in FRW Universe

In this paper, we study the viscous extended cosmic Chaplygin gas whose equation of state reduces to extended Chaplygin gas in the limit $$\omega \rightarrow 0$$ with varying cosmological constant in flat FRW universe. In this framework, we assume the bulk viscosity $$\zeta $$ and cosmological constant $$\Lambda $$ as a linear combination of two terms, one is constant and other is a function of dark energy density $$\rho $$. We obtain generalized Friedmann equations due to bulk viscosity, cosmological constant and extended cosmic Chaplygin gas. We calculate the time-dependent dark energy density $$\rho $$ for various values of n and $$\alpha = 1/2$$ both analytically and numerically. We analyse the behaviour of scale factor, Hubble expansion parameter and deceleration parameter graphically and discuss the stability of the model by using square of speed of sound.

Extended ΛCDM model and viscous dark energy: a Bayesian analysis

We propose an approach considering the nonextensive effects in the context of the Verlinde theory in order to address an extended cosmological model in the context of viscous dark energy. Specifically, this model leads to a tiny perturbation in the dynamics of the expansion of the universe through the generalized Friedmann equations so-called the extended ΛCDM model. From the observational test standpoint, we make a Bayesian analysis of the models of bulk viscosity for dark energy which follows the Eckart theory of bulk viscosity. These models are investigated through the context of both models ΛCDM and extended ΛCDM. The Bayesian analysis is performed using the data of CMB Distance priors, Baryon Acoustic Oscillations Measurements, Cosmic Chronometers, and SNe Ia distance measurements.

Extended ΛCDM model

In this work we discuss a general approach for the dissipative dark matter considering a nonextensive bulk viscosity and taking into account the role of generalized Friedmann equations. This generalized ΛCDM model encompasses a flat universe with a dissipative nonextensive viscous dark matter component, following the Eckart theory of bulk viscosity. In order to compare models and constrain cosmological parameters, we perform Bayesian analysis using one of the most recent observations of Type Ia Supernova, baryon acoustic oscillations, and cosmic microwave background data.

Palatini formulation of f(R,\xa0T) gravity theory, and its cosmological implications

We consider the Palatini formulation of f(R, T) gravity theory, in which a non-minimal coupling between the Ricci scalar and the trace of the energy-momentum tensor is introduced, by considering the metric and the affine connection as independent field variables. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy-momentum trace dependent metric, related to the physical metric by a conformal transformation. Similar to the metric case, the field equations impose the non-conservation of the energy-momentum tensor. We obtain the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra force, which is identical to the one obtained in the metric case. The thermodynamic interpretation of the theory is also briefly discussed. We investigate in detail the cosmological implications of the theory, and we obtain the generalized Friedmann equations of the f(R, T) gravity in the Palatini formulation. Cosmological models with Lagrangians of the type f=R-alpha ^2/R+g(T) and f=R+alpha ^2R^2+g(T) are investigated. These models lead to evolution equations whose solutions describe accelerating Universes at late times.