Power-law Solutions Research Articles

Atypicality of power-law solutions to Emden–Fowler type higher order equations

Atypicality of power-law solutions to Emden–Fowler type higher order equations

Mott quantum criticality in the one-band Hubbard model: Dynamical mean-field theory, power-law spectra, and scaling

Recent studies of electrical transport, both theoretical and experimental, near the bandwidth-tuned Mott metal-insulator transition have uncovered apparent quantum critical scaling of the electrical resistivity at elevated temperatures, despite the fact that the actual low-temperature phase transition is of first order. This raises the question whether there is a hidden Mott quantum critical point. Here we argue that the dynamical mean-field theory of the Hubbard model admits, in the low-temperature limit, asymptotically scale-invariant (i.e. power-law) solutions, corresponding to the metastable insulator at the boundary of metal-insulator coexistence region, which can be linked to the physics of the pseudogap Anderson model. While our state-of-the-art numerical renormalization group calculations reveal that this asymptotic regime is restricted to very small energies and temperatures and hence difficult to access numerically, we uncover the existence of a wide crossover regime where the single-particle spectrum displays a \emph{different} power law. We show that it is this power-law regime, corresponding to approximate local quantum criticality, which is continuously connected to and responsible for the apparent quantum critical scaling above the classical critical end point. We connect our findings to experiments on tunable Mott materials.

A comparative study of some cosmological models with time varying G and Λ: A symmetry approach

We study how the constants G and Λ may vary in four different theoretical models: general relativity with time-varying constants (Y.-K. Lau. Aust. J. Phys. 38, 547 (1985). doi: 10.1071/PH850547 ), the model proposed by Lu et al. (Phys Rev D, 89, 063526 (2014). doi: 10.1103/PhysRevD.89.063526 ), the model proposed by Bonanno et al. (Class. Quant. Grav. 24, 1443 (2007). doi: 10.1088/0264-9381/24/6/005 ), and the Brans–Dicke model with Λ([Formula: see text]) [ 25 ]. To carry out this study, we work under the self-similar hypothesis and we assume the same metric, a flat Friedmann–Robertson–Walker metric, and the same matter source, a perfect fluid. We put special emphasis on mathematical and formal aspects, which allows us to calculate exact power-law solutions through symmetry methods, matter collineation, and Noether symmetries. This enables us to compare the solutions of each model and in the same way to contrast the results with some observational data.

Energy conditions in non-local gravity

We investigate the different energy conditions in non-local gravity, which is obtained by adding an arbitrary function of d’Alembertian operator, [Formula: see text], to the Hilbert–Einstein action. We analyze the validity of four different energy conditions and illustrate the different constraints over parameters of the power-law solution as well as de Sitter solution.

Superpotential method for chiral cosmological models connected with modified gravity

We consider the Chiral Cosmological Models (CCMs) and modified gravity theories associated with them. Generalization of the superpotential method for a general CCM with several scalar fields is performed, and the method of construction CCMs admitting exact solutions is developed. New classes of exact solutions in the two-component CCM connected with an $f(R)$ gravity model with an additional scalar field have been constructed. We construct new cosmological solutions for a diagonal metric of the target space, including modified power-law solutions. In particular, we propose the reconstruction procedure based on the superpotential method and present examples of kinetic part reconstruction for periodic and hyperbolic Hubble parameters. We also focus on a cyclic type of Universe dubbed the Quasi-Steady State (QSS) model, with the aim of constructing single- and double-field potentials for one and the same behaviour of the Hubble parameter using the developed superpotential method for the CCM. The realization of this task includes a new set of solutions for a CCM with a scale factor characterized by the QSS theory. We also propose a method for reducing the two-field CCM to the single scalar field model.

Chiral Cosmological Model of f(R) Gravity with a Kinetic Curvature Scalar

We consider modified f(R) gravity with a kinetic curvature scalar, which can be reduced to a chiral cosmological model of special kind. A detailed derivation is presented for the action of a chiral cosmological model as an equivalent to a gravitational model with higher derivatives with respect to the Ricci scalar using Lagrange multipliers and a transition from the Jordan frame to the Einstein one. The equations of the model are written in the spatially flat Friedmann-Robertson-Walker metric on the basis of the constructed chiral cosmological model. Examples of solutions are found, corresponding to a special choice of the field χ = χ* = const, and its fixed value $$\chi_0=-\sqrt{3/2}\;\text{ln}2$$ . For this value of χ0, in the case of the canonical inclusion of the kinetic component of the Ricci scalar, we have obtained a nonlinear second-order differential equation with respect to H, which is not amenable to analytic solution. Therefore we implement a transition to a noncanonical form of the kinetic term. Using a fixed value of χ0, an exact solution is obtained for power-law inflation. We have considered a transition from the de Sitter and power-law solutions specified in the Jordan frame to the Einstein frame for comparison with the results obtained in f(R) gravity with higher derivatives. It is proved that there is a Weyl conformal transformation which transforms the de Sitter and power-law solutions in one frame to similar solutions in the other.

Scalar Tensor Cosmology With Kinetic, Gauss-Bonnet and Nonminimal Derivative Couplings and Supersymmetric Loop Corrected Potential

We discuss a particular four-dimensional cosmology based on non-minimal scalar tensor theories characterized by a supersymmetric loop corrected potential and a Hubble parameter defined as a function of the scalar field. Power-law solutions are obtained in the FRW background giving rise to acceleratedly expanding universe characterized by a scale factor and a scalar field depending both on the non-minimal coupling parameter ξ. Based on SNeIa data and on Hubble data X-ray gas mass fraction measurements, we find 0.116 < ξ < 0.225 which results on a universe dominated by vacuum energy.

No Small Hairs in Anisotropic Power-law Gauss-Bonnet Inflation

We will examine whether anisotropic hairs exist in a string-inspired scalar-Gauss-Bonnet gravity model with the absence of potential of scalar field during the inflationary phase. As a result, we are able to obtain the Bianchi type I power-law solution to this model under the assumption that the scalar field acts as the phantom field, whose kinetic is negative definite. However, the obtained anisotropic hair of this model turns out to be large, which is inconsistent with the observational data. We will therefore introduce a nontrivial coupling between scalar and vector fields such as \(f^2(\phi)F_{\mu\nu}F^{\mu\nu}\) into the scalar-Gauss-Bonnet model with the expectation that the anisotropic hair would be reduced to a small one. Unfortunately, the magnitude of the obtained anisotropic hair is still large. These results indicate that the scalar-Gauss-Bonnet gravity model with the absence of potential of scalar field might not be suitable to generate small anisotropic hairs during the inflationary phase.

Dressed power-law inflation with a cuscuton

We study dressed inflation with a cuscuton and find a novel exact power-law solution. It is well known that the conventional power-law inflation is inconsistent with the Planck data. In contrast to this standard lore, we find that power-law inflation with a cuscuton can be reconciled with the Planck data. Moreover, we argue that the cuscuton generally ameliorates inflation models so that predictions are consistent with observations.

On the evolution and stability of cosmological perturbations in higher order scalar–tensor theories of gravity

We study a new type of extended theory of gravity in the framework of general scalar–tensor theories in which the higher order terms of curvature are coupled with a scalar field and its derivatives. We analyze the stability and evolution of cosmological perturbations in this setup. For this purpose, we perturb the Hubble parameter, matter density, and scalar field to check stability and evolution of perturbations to first order. In this framework, we investigate stability conditions for de Sitter and power law solutions and we examine viability of cosmological evolution of these perturbations. We consider some specific f(R) models and show that the stability analysis gives some constraints on the parameters of these models.

Nonlinear diffusion models for gravitational wave turbulence

A fourth-order and a second-order nonlinear diffusion model in spectral space are proposed to describe gravitational wave turbulence in the approximation of strongly local interactions. We show analytically that the model equations satisfy the conservation of energy and wave action, and reproduce the power law solutions previously derived from the kinetic equations with a direct cascade of energy and an explosive inverse cascade of wave action. In the latter case, we show numerically by computing the second-order diffusion model that the non-stationary regime exhibits an anomalous scaling which is understood as a self-similar solution of the second kind with a front propagation following the law kf∼(t∗−t)3.296, with t<t∗. These results are relevant to better understand the dynamics of the primordial universe where potent sources of gravitational waves may produce space–time turbulence.

Long-profile evolution of transport-limited gravel-bed rivers

Abstract. Alluvial and transport-limited bedrock rivers constitute the majority of fluvial systems on Earth. Their long profiles hold clues to their present state and past evolution. We currently possess first-principles-based governing equations for flow, sediment transport, and channel morphodynamics in these systems, which we lack for detachment-limited bedrock rivers. Here we formally couple these equations for transport-limited gravel-bed river long-profile evolution. The result is a new predictive relationship whose functional form and parameters are grounded in theory and defined through experimental data. From this, we produce a power-law analytical solution and a finite-difference numerical solution to long-profile evolution. Steady-state channel concavity and steepness are diagnostic of external drivers: concavity decreases with increasing uplift rate, and steepness increases with an increasing sediment-to-water supply ratio. Constraining free parameters explains common observations of river form: to match observed channel concavities, gravel-sized sediments must weather and fine – typically rapidly – and valleys typically should widen gradually. To match the empirical square-root width–discharge scaling in equilibrium-width gravel-bed rivers, downstream fining must occur. The ability to assign a cause to such observations is the direct result of a deductive approach to developing equations for landscape evolution.

Holographic dark energy model in unimodular f(T) gravity

This work deals with holographic dark energy in the context of unimodular [Formula: see text] gravity, which is a modification of teleparallel gravity. We develop the general reconstruction procedure of the [Formula: see text] form that can yield the holographic feature of the dark energy. We fit the reconstructed model with the [Formula: see text] data and our results show a perfect agreement with the WMAP9 cosmological observational data, at least for the range [Formula: see text]. We investigate the consistency of the reconstructed model by studying its stability against linear gravitational and matter perturbations, fixing [Formula: see text] to [Formula: see text]. The model presents stability for both de Sitter and power-law solutions and we conclude that it is a good candidate as alternative viable model for characterizing holographic dark energy.

Cosmological constant in chameleon Brans–Dicke theory

We consider a generalized Brans–Dicke model in which the scalar field has a self-interacting potential function. The scalar field is also allowed to couple nonminimally with the matter part. We assume that it has a chameleon behavior in the sense that it acquires a density-dependent effective mass. We consider two different types of matter systems which couple with the chameleon, dust and vacuum. In the first case, we find a set of exact solutions when the potential has an exponential form. In the second case, we find a power-law exact solution for the scale factor. In this case, we will show that the vacuum density decays during expansion due to coupling with the chameleon.

De Sitter and power-law solutions in non-local Gauss–Bonnet gravity

The cosmological dynamics of a non-locally corrected gravity theory, involving a power of the inverse d’Alembertian, is investigated. Casting the dynamical equations into local form, the fixed points of the models are derived, as well as corresponding de Sitter and power-law solutions. Necessary and sufficient conditions on the model parameters for the existence of de Sitter solutions are obtained. The possible existence of power-law solutions is investigated, and it is proven that models with de Sitter solutions have no power-law solutions. A model is found, which allows to describe the matter-dominated phase of the Universe evolution.

Finsler–Randers Cosmological Models in Modified Gravity Theories

Finsler–Randers (FR) cosmological models in modified gravity theories have been investigated in this paper. The de Sitter and power law solutions are assumed for the scale factor in the corresponding cosmological models. All energy conditions have been discussed in detail for each scenario. Behaviour of FR cosmological models in modified theories of gravity like Einstein theory, Hoyle–Narlikar creation field theory, lyra geometry and general class of scalar–tensor theories have also been investigated.

Particle creation in the framework of f ( G ) $f(G)$ gravity

In this paper, we study the problem of massless particle creation in a flat, homogeneous and isotropic universe in the framework of $f(G)$ gravity. The Bogolyubov coefficients are calculated for the accelerating power-law solutions of the model in a matter dominated universe, from which the total number of created particle per unit volume of space can be obtained. It is proved that the total particle density always has a finite value. Therefore, the Bogolyubov transformations are well-defined and the Hilbert spaces spanned by the vacuum states at different times are unitarily equivalent. We find that the particles with small values of the mode $k$ are produced in the past and particles with large values of $k$ are produced only in the future. The negative pressure resulting from the gravitational particle creation is also determined. It is then argued that this pressure even in the presence of energy density and thermal pressure may affect significantly the cosmic expansion.

The speed of gravitational waves and power-law solutions in a scalar-tensor model

One of the most relevant solutions in any cosmological model concerning the evolution of the universe is the power-law solution. For the scalar-tensor model of dark energy with kinetic and Gauss Bonnet couplings, it is shown that we can conserve the power-law solution and at the same time meet the recent observational bound on the speed of gravitational waves. In the FRW background the anomalous contribution to the speed of gravitational waves, coming from the kinetic and Gauss-Bonnet couplings, cancel each other for power-law solutions. It is shown that by simple restriction on the model parameters we can achieve a non-time-dependent cancellation of the defect in the velocity of the gravitational waves. The model can realize the cosmic expansion with contributions from the kinetic and Gauss-Bonnet couplings of the order of O(1) to the dark energy density parameter. The results are valid on the homogeneous FRW background and the limitations of the approach are discussed.

Anisotropic power-law inflation of the five dimensional scalar\u2013vector and scalar-Kalb\u2013Ramond model

We will study the cosmological implications of the five dimensional scalar–vector and scalar-Kalb–Ramond model. In particular, a new set of Bianchi type I power-law analytic solution will be obtained for this model. The cosmic no-hair conjecture can be shown to break down in the presence of the scalar–vector and scalar-Kalb–Ramond couplings. The effect of the Kalb–Ramond field in the presence of the power-law solution will be shown explicitly. We will also show that the presence of a phantom field does, however, destabilize the corresponding Bianchi type I power-law inflationary solutions.

Observational constraints on Gauss–Bonnet cosmology

We analyze a fully geometric approach to dark energy in the framework of [Formula: see text] theories of gravity, where [Formula: see text] is the Ricci curvature scalar and [Formula: see text] is the Gauss–Bonnet topological invariant. The latter invariant naturally exhausts, together with [Formula: see text], the whole curvature content related to curvature invariants coming from the Riemann tensor. In particular, we study a class of [Formula: see text] models with power law solutions and find that, depending on the value of the geometrical parameter, a shift in the anisotropy peaks position of the temperature power spectrum is produced, as well as an increasing in the matter power spectrum amplitude. This fact could be extremely relevant to fix the form of the [Formula: see text] model. We also perform an MCMC analysis using both Cosmic Microwave Background data by the Planck (2015) release and the Joint Light-Curve Analysis of the SNLS–SDSS collaborative effort, combined with the current local measurements of the Hubble value, [Formula: see text], and galaxy data from the Sloan Digital Sky Survey (BOSS CMASS DR11). We show that such a model can describe the CMB data with slightly high [Formula: see text] values, and the prediction on the amplitude matter spectrum value is proved to be in accordance with the observed matter distribution of the universe. At the same time, the value constrained for the geometric parameter implies a density evolution of such a component that is growing with time.