Exact Power-law Solutions Research Articles

Exact Solutions and Cosmological Constraints in Fractional Cosmology

This paper investigates exact solutions of cosmological interest in fractional cosmology. Given μ, the order of Caputo’s fractional derivative, and w, the matter equation of state, we present specific exact power-law solutions. We discuss the exact general solution of the Riccati Equation, where the solution for the scale factor is a combination of power laws. Using cosmological data, we estimate the free parameters. An analysis of type Ia supernovae (SNe Ia) data and the observational Hubble parameter data (OHD), also known as cosmic chronometers, and a joint analysis with data from SNe Ia + OHD leads to best-fit values for the free parameters calculated at 1σ, 2σ and 3σ confidence levels (CLs). On the other hand, these best-fit values are used to calculate the age of the Universe, the current deceleration parameter (both at 3σ CL) and the current matter density parameter at 1σ CL. Finding a Universe roughly twice as old as the one of ΛCDM is a distinction of fractional cosmology. Focusing our analysis on these results, we can conclude that the region in which μ>2 is not ruled out by observations. This parameter region is relevant because fractional cosmology gives a power-law solution without matter, which is accelerated for μ>2. We present a fractional origin model that leads to an accelerated state without appealing to Λ or dark energy.

Energy Spectrum of Two-Dimensional Acoustic Turbulence.

We report an exact unique constant-flux power-law analytical solution of the wave kinetic equation for the turbulent energy spectrum, E(k)=C_{1}sqrt[ϵac_{s}]/k, of acoustic waves in 2D with almost linear dispersion law, ω_{k}=c_{s}k[1+(ak)^{2}], ak≪1. Here, ϵ is the energy flux over scales, and C_{1} is the universal constant which was found analytically. Our theory describes, for example, acoustic turbulence in 2D Bose-Einstein condensates. The corresponding 3D counterpart of turbulent acoustic spectrum was found over half a century ago, however, due to the singularity in 2D, no solution has been obtained until now. We show the spectrum E(k) is realizable in direct numerical simulations of forced-dissipated Gross-Pitaevskii equation in the presence of strong condensate.

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.

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.

Steady states in Leith's model of turbulence

We present a comprehensive study and full classification of the stationary solutions in Leith’s model of turbulence with a generalised viscosity. Three typical types of boundary value problems are considered: Problems 1 and 2 with a finite positive value of the spectrum at the left (right) and zero at the right (left) boundaries of a wave number range, and Problem 3 with finite positive values of the spectrum at both boundaries. Settings of these problems and analysis of existence of their solutions are based on a phase–space analysis of orbits of the underlying dynamical system. One of the two fixed points of the underlying dynamical system is found to correspond to a ‘sharp front’ where the energy flux and the spectrum vanish at the same wave number. The other fixed point corresponds to the only exact power-law solution—the so-called dissipative scaling solution. The roles of the Kolmogorov, dissipative and thermodynamic scaling, as well as of sharp front solutions, are discussed.

Designing anisotropic inflation with form fields

We study inflation with anisotropic hair induced by form fields. In four dimensions, the relevant form fields are gauge (one-form) fields and two-form fields. Assuming the exponential form of potential and gauge kinetic functions, we find new exact power-law solutions endowed with anisotropic hair. We also explore the phase space of anisotropic inflation and find fixed points corresponding to the exact power-law solutions. Moreover, we perform the stability analysis around the fixed points to reveal the structure of the phase space. It turns out that one of the fixed points becomes an attractor and others (if any) are saddle points. In particular, the one corresponding to anisotropic inflation becomes an attractor when it exists. We also argue that various anisotropic inflation models can be designed by choosing coupling constants.

Power-law solution for anisotropic universe in f(G) gravity

We try to study the theory of modified Gauss-Bonnet gravity in non-isotope universe. It is considered the exact power-law solution in modified gravity models. A f(G) function corresponding with power law solutions for given scale factor are calculated. We show that BI-like power-law solutions only exist for a very special class of f(G) theories. It is shown that transition to phantom phase is happened by applied some bound on free parameters. We also explore the stability issue of modified gravitational models.

A holographic reconstruction of the modified f(R) Horava-Lifshitz gravity with scale factor in power-law form

This paper is devoted to the study of holographic reconstruction of the modified f(R) Horava-Lifshitz gravity in the flat universe. For this purpose, we assume a scale factor in the form of exact power-law solution. A reconstructed model is obtained which may provide reasonable qualitative description of some cosmological era satisfying the sufficient condition for a realistic model. We represent the graphical description of this model exhibiting increasing behavior. We also discuss the evolution of the accelerated expansion of the universe through equation of state parameter (EoS) for the reconstructed model. Its graphical representation depicts about a quintessence dominated era in near future. The stability of the model is observed initially, however it becomes unstable after a small interval of time.

Particle creation in flat Friedmann–Robertson–Walker (FRW) universe in the framework of f(T) gravity

We study particle production in a flat Friedmann–Robertson–Walker universe in the framework of f(T) gravity. An exact power-law solution is obtained by solving the Friedmann equations and assuming that matter is minimally coupled with gravitation. The torsion scalar, T, appears to plays the same role as the curvature (Ricci scalar) in general relativity (GR) and its modified theories, f(R). Particularly, in the phantom phase, we observe that the vacuum state corresponds to a vanishing torsion scalar and particle production becomes important as the torsion scalar diverges. This aspect not only provides the equivalence between teleparallel gravity and GR, but also between their respective modified versions, f(T) and f(R), in the view of massless particle production phenomenon when matter is minimally coupled with gravity. However, when the gravitational and scalar fields are not minimally coupled, it appears that this similarity between the teleparallel gravity and GR may break down, because the torsion scalar no longer has the same time-dependent expression as the Ricci scalar.

Energy Conditions Constraints and Stability of Power Law Solutions inf(R,T) Gravity

The energy conditions are derived in the context of $f(R,T)$ gravity, where $R$ is the Ricci scalar and $T$ is the trace of the energy-momentum tensor, which can reduce to the well-known conditions in $f(R)$ gravity and general relativity. We present the general inequalities set by the energy conditions in terms of Hubble, deceleration, jerk and snap parameters. In this study, we concentrate on two particular models of $f(R,T)$ gravity namely, $f(R)+\lambda{T}$ and $R+2f(T)$. The exact power-law solutions are obtained for these two cases in homogeneous and isotropic $f(R,T)$ cosmology. Finally, we find certain constraints which have to be satisfied to ensure that power law solutions may be stable and match the bounds prescribed by the energy conditions.

Particle Creation in Higher Dimensional Space-time with Variable G and Λ

In this paper we consider higher dimensional FRW type cosmological models with time-varying gravitational constant G=G(t) and decaying vacuum energy Λ=Λ(t) in context of the open thermodynamical system of particle creation. The particle creation rate is determined by variable G and Λ. An Exact solution of the gravitational field equations is obtained by assuming the exact power-law solution of the scale factor of the universe. We evaluate the various cosmological parameters and discuss their physical significance in detail for radiation and matter-dominated phases of the universe. The solutions are also relevant for quintessence (accelerating) universe. We also discuss the relevance of the solutions for 5-dimensional Kaluza-Klein FRW type model with and without particle creation in radiation and matter-dominated phases as special case. We obtain the solution where the gravitational constant increases with time during no particle creation. The cosmological constant Λ, however, decreases with time in all the cases.

Bianchi I in scalar and scalar-tensor cosmologies

Abstract We study how the constants G and Λ may vary in different theoretical models (general relativity (GR) with a perfect fluid, scalar cosmological models (SM) (“quintessence”) with and without interacting scalar and matter fields and three scalar-tensor theories (STT) with a dynamical Λ) in order to explain some observational results. We apply the program outlined in section II to study the Bianchi I models, under the self-similarity hypothesis. We put special emphasis on calculating exact power-law solutions which allow us to compare the different models. In all the studied cases we conclude that the solutions are isotropic and noninflationary. We also arrive at the conclusion that in the GR model with time-varying constants, Λ vanishes while G is constant. In the SM all the solutions are massless i.e. the potential vanishes and all the interacting models are inconsistent from the thermodynamical point of view. The solutions obtained in the STT collapse to the perfect fluid one obtained in the GR model where G is a true constant and Λ vanishes as in the GR and SM frameworks.

Power-law solutions in f (T) gravity

We have considered an action of the form $T+f(T)+L_m$ describing Einstein's gravity plus a function of the torsion scalar. By considering an exact power-law solution we have obtained the Friedmann equation as a differential equation for the function $f(T)$ in spatially flat universe and obtained the real valued solutions of this equation for some power-law solutions. We have also studied the power-law solutions when the universe enters a Phantom phase and shown that such solutions may exist for some f(T) solutions.

(An)Isotropic models in scalar and scalar-tensor cosmologies

We study how the constants $G$ and $\Lambda$ may vary in different theoretical models (general relativity with a perfect fluid, scalar cosmological models (\textquotedblleft quintessence\textquotedblright) with and without interacting scalar and matter fields and a scalar-tensor model with a dynamical $\Lambda$) in order to explain some observational results. We apply the program outlined in section II to study three different geometries which generalize the FRW ones, which are Bianchi \textrm{V}, \textrm{VII}$_{0}$ and \textrm{IX}, under the self-similarity hypothesis. We put special emphasis on calculating exact power-law solutions which allow us to compare the different models. In all the studied cases we arrive to the conclusion that the solutions are isotropic and noninflationary while the cosmological constant behaves as a positive decreasing time function (in agreement with the current observations) and the gravitational constant behaves as a growing time function.

Power-law solutions and accelerated expansion in scalar-tensor theories

We find exact power-law solutions for scalar-tensor theories and clarify the conditions under which they can account for an accelerated expansion of the Universe. These solutions have the property that the signs of both the Hubble rate and the deceleration parameter in the Jordan frame may be different from the signs of their Einstein-frame counterparts. For special parameter combinations we identify these solutions with asymptotic attractors that have been obtained in the literature through dynamical-system analysis. We establish an effective general-relativistic description for which the geometrical equivalent of dark energy is associated with a time dependent equation of state. The present value of the latter is consistent with the observed cosmological ``constant". We demonstrate that this type of power-law solutions for accelerated expansion cannot be realized in f(R) theories.

Note on an exact solution for the pipe flow of a third-grade fluid

We obtain a new exact power law solution for the pipe flow of a third-grade fluid. Moreover, we provide general analytical expressions from which all previous known solutions can be constructed.

Wave turbulence in incompressible Hall magnetohydrodynamics

We investigate the steepening of the magnetic fluctuation power law spectra observed in the inner solar wind for frequencies higher than 0.5 Hz. This high frequency part of the spectrum may be attributed to dispersive nonlinear processes. In that context, the long-time behavior of weakly interacting waves is examined in the framework of 3D incompressible Hall MHD turbulence. The Hall term added to the standard MHD equations makes the Alfv\'en waves dispersive and circularly polarized. We introduce the generalized Els\"asser variables and, using a complex helicity decomposition, we derive for three-wave interaction processes the general wave kinetic equations; they describe the nonlinear dynamics of Alfv\'en, whistler and ion cyclotron wave turbulence in the presence of a strong uniform magnetic field $B_0 \ep$. Hall MHD turbulence is characterized by anisotropies of different strength. We show that electron and standard MHD turbulence can be seen as two frequency limits of the present theory but the standard MHD limit is singular; additionally, we analyze in detail the ion MHD turbulence limit. Exact power law solutions of the master wave kinetic equations are given in the small and large scale limits for which we have, respectively, the total energy spectra $E(\kpn,\kpa) \sim \kpn^{-5/2} |\kpa|^{-1/2}$ and $E(\kpn,\kpa) \sim \kpn^{-2}$. An anisotropic phenomenology is developed to describe continuously the different scaling laws of the energy spectrum; one predicts $E(\kpn,\kpa) \sim \kpn^{-2} |\kpa|^{-1/2} (1+\kpn^2d_i^2)^{-1/4}$. Nonlocal interactions between Alfv\'en, whistler and ion cyclotron waves are investigated; a non trivial dynamics exists only when a discrepancy from the equipartition between the large scale kinetic and magnetic energies happens.

Anisotropic wave turbulence in electron MHD

It is well known that the magnetic and the velocity field fluctuations in the solar wind possess many features expected of fully developed magnetohydrodynamic (MHD) turbulence. However, for frequencies higher than 0.1 Hz the situation is different and clear discrepancies have been recently observed with, for example, a steepening of the magnetic fluctuation power law spectrum. In order to give a satisfactory description of the high frequency solar wind it is necessary to adopt a new description like the electron MHD approximation in which the dynamics is entirely governed by electrons. In that context, we review and discuss recent theoretical results obtained in anisotropic whistler wave turbulence for electron MHD in the presence of a strong and uniform external magnetic field. Using helicity decomposition, the wave kinetic equations for energy and magnetic helicity are derived at the level of three-wave interactions between whistler waves. It is shown that nonlinear interactions of whistler waves transfer energy and magnetic helicity mainly in the direction perpendicular to the external magnetic field. The anisotropic turbulence thus generated has exact stationary power law solutions which scale as for the energy spectrum and for the magnetic helicity spectrum. A strong analogy is found with the problem of rotating turbulence for incompressible neutral flows which share almost all the same properties.

Letter: On Tilted Perfect Fluid Bianchi Type VI0Self-Similar Models

We show that the tilted perfect fluid Bianchi VI$_0$ family of self-similar models found by Rosquist and Jantzen [K. Rosquist and R. T. Jantzen, \emph{% Exact power law solutions of the Einstein equations}, 1985 Phys. Lett. \textbf{107}A 29-32] is the most general class of tilted self-similar models but the state parameter $\gamma $ lies in the interval $(\frac 65,\frac 32) $. The model has a four dimensional stable manifold indicating the possibility that it may be future attractor, at least for the subclass of tilted Bianchi VI$_0$ models satisfying $n_\alpha ^\alpha =0$ in which it belongs. In addition the angle of tilt is asymptotically significant at late times suggesting that for the above subclasses of models the tilt is asymptotically extreme.

Anisotropic weak whistler wave turbulence in electron magnetohydrodynamics

A weak wave turbulence theory is given for electron magnetohydrodynamics in the presence of a strong and uniform external magnetic field. Using helicity decomposition, the wave kinetic equations for energy and magnetic helicity are derived at the level of three-wave interactions between whistler waves. It is shown that nonlinear interactions of whistler waves transfer energy and magnetic helicity mainly in the direction perpendicular to the external magnetic field. The anisotropic turbulence thus generated has exact stationary power law solutions which scale as k⊥−5/2k∥−1/2 for the energy spectrum and k⊥−7/2k∥−1/2 for the magnetic helicity spectrum.