Attractor Solution Research Articles

Growth of matter perturbations in nonminimal teleparallel dark energy

We study the growth rate of matter perturbations in the context of teleparallel dark energy in a flat universe. We investigate the dynamics of different theoretical scenarios based on specific forms of the scalar field potential. Allowing for non-minimal coupling between torsion scalar and scalar field, we perform a phase-space analysis of the autonomous systems of equations through the study of critical points. We thus analyze the stability of the critical points, and discuss the cosmological implications searching for possible attractor solutions at late times. Furthermore, combing the growth rate data and the Hubble rate measurements, we place observational constraints on the cosmological parameters of the models through Monte Carlo numerical method. We find that the scenario with a non-minimal coupling is favoured with respect to the standard quintessence case. Adopting the best-fit results, we show that the dark energy equation of state parameter can cross the phantom divide. Finally, we compare our results with the predictions of the concordance $\Lambda$CDM paradigm by performing Bayesian model selection.

Asymptotic regularity of trajectory attractor and trajectory statistical solution for the 3D globally modified Navier–Stokes equations

We first prove the existence and regularity of the trajectory attractor for a three-dimensional system of globally modified Navier–Stokes equations. Then we use the natural translation semigroup and trajectory attractor to construct the trajectory statistical solutions in the trajectory space. In our construction the trajectory statistical solution is an invariant Borel probability measure, which is supported by the trajectory attractor and is invariant under the action of the translation semigroup. As a byproduct of the regularity of the trajectory attractor, we obtain the asymptotic regularity of the trajectory statistical solution in the sense that it is supported by a set in the trajectory space in which all weak solutions are in fact strong solutions.

Power-law inflation with minimal and nonminimal coupling

Power-law inflationary scenarios based on minimal and non-minimal coupling of a scalar field to gravity with exponential- and power-law potential, respectively, are studied using the symmetry-based approach. In particular, we obtained only one parameter Lie point symmetry for both the minimal and non-minimal coupling cases and it is interesting to note that the non-minimal coupled equation is invariant under a scale transformation. We find the exact analytical group invariant solutions from invariant curve condition for both the minimal and non-minimal cases of the power-law inflationary model. The solutions obtained are consistent with the Friedmann equations subject to constraints on the inflationary potential parameter $ \lambda$ for the minimal case and on the coupling parameter $ \zeta$ for the non-minimal case. In this scenario, we find transformation relations for various inflationary parameters e.g. amplitude of the scalar power spectrum, spectral index, slow-roll parameters (SRP), tensor-to-scalar perturbation ratio, equation of state parameters, non-Gaussianity parameter as well as the form of the potential in two different frames, namely the Jordan and Einstein frames by making use of the conformal transformation. The results for various inflationary parameters for the non-minimal case are presented in the background of Planck2015 and Planck2018 and are in good agreement with the cosmological observations if the non-minimal coupling parameter is chosen properly. We treat minimally and non-minimally coupled scalar field equations by the dynamical system theory and present critical point analysis. By checking the stability of the critical points in the phase space for both cases we have shown that the solutions obtained from the Lie symmetry approach are the stable attractor solutions.

Phantom Dirac-Born-Infeld dark energy

Motivated by the apparent discrepancy between Cosmic Microwave Background measurements of the Hubble constant and measurements from Type-Ia supernovae, we construct a model for Dark Energy with equation of state $w = p / \rho < -1$, violating the Null Energy Condition. Naive canonical models of so-called "Phantom" Dark Energy require a negative scalar kinetic term, resulting in a Hamiltonian unbounded from below and associated vacuum instability. We construct a scalar field model for Dark Energy with $w < -1$, which nonetheless has a Hamiltonian bounded from below in the comoving reference frame, {\it i.e.} in the rest frame of the fluid. We demonstrate that the solution is a cosmological attractor, and find that early-time cosmological boundary conditions consist of a "frozen" scalar field, which relaxes to the attractor solution once the Dark Energy component dominates the cosmological energy density. We consider the model in an arbitrary choice of gauge, and find that, unlike the case of comoving gauge, the fluid Hamiltonian is in fact unbounded from below in the reference frame of a highly boosted observer, corresponding to a nonlinear gradient instability. We discuss this in the context of general NEC-violating perfect fluids, for which this instability is a general property.

Reconstruction of cosmological evolution in the presence of extra dimensions

The model of Physics resulting from a Kaluza-Klein dimensional reduction procedure offers very good dark matter candidates in the form of Light Kaluza-Klein Particles, and thus becomes relevant to Cosmology. In this work we utilize an analytically known, Kasner-type, attractor solution. It is shown that for a plethora of pairs of initial conditions of the usual and extra spatial Hubble parameters, one can recover a late-time cosmological picture, similar to that of the ΛCDM, in a multidimensional scenario, in the general framework of Universal Extra Dimensions. The phenomenology of fundamental interactions dictates the stabilization of the extra dimensional evolution from a very early epoch in these scenarios. Without an explicit mechanism, this is achieved through particular behaviors of the usual and extra spatial fluids, which have to be motivated by a more fundamental theory.

Dynamical systems perspective of cosmological finite-time singularities in f(R) gravity and interacting multifluid cosmology

In this work we shall investigate the occurrence of future cosmological finite-time singularities in the dynamical system corresponding to two cosmological theories, namely that of vacuum $f(R)$ gravity and that of three fluids. The vacuum $f(R)$ gravity is an example for which the variables we will choose to quantify the phase space dynamics, do not necessarily blow-up near a cosmological singularity. After appropriately choosing the variables, we shall investigate the behavior of the corresponding dynamical system near some types of cosmological finite-time singularities, for some limiting cases in which we can produce analytic solutions for the dynamical variables. The most interesting case from both a mathematical and physical point of view, is the Big Rip case, and particularly in the limiting case of a very strong singularity. The physically appealing outcome is that the resulting non-autonomous dynamical system is attracted asymptotically to an accelerating attractor solution, with equation of state parameter $w_{eff}=-1$. Our analytic results, show that an extremely strong Big Rip singularity in vacuum $f(R)$ gravity theories is always related to an accelerating solution, or tends to acceleration. The converse statement though may not be true. The second cosmology we shall study is a multifluid cosmology, consisting of three fluids, the interacting dark matter and dark energy fluids, and the baryonic fluid. By appropriately choosing the variables, we will show that the dynamical system can become an autonomous polynomial dynamical system, in which case, by using a dominant balance analysis, we shall investigate the occurrence of finite-time singularities in this system.

Reconstructing the Equation of State in Holographic Dark Energy in 5D Brans-Dicke Theory, and Comparison with ΛCDM Model and CPL Parametrization

We study Holographic dark energy in 5D Brans-Dicke cosmology. To investigate the stability and attractor solutions, we fit the model with the SNIa and Gamma-Ray Burst (GRB) observational data. We also examine the holographic nature of dark energy in the model through the equation of state (EoS). In addition, a reconstructed form of the EoS from holographic dark energy on the model is driven The model strongly exhibits the present cosmic accelerating. It also shows that the universe starts from an unstable radiation-dominated epoch and reaches a stable late-time accelerated phase originating from holographic dark energy in the Brans-Dicke model.

Axions from strings: the attractive solution

We study the system of axion strings that forms in the early Universe if the Peccei-Quinn symmetry is restored after inflation. Using numerical simulations, we establish the existence of an asymptotic solution to which the system is attracted independently of the initial conditions. We study in detail the properties of this solution, including the average number of strings per Hubble patch, the distribution of loops and long strings, the way that different types of radiation are emitted, and the shape of the spectrum of axions produced. We find clear evidence of logarithmic violations of the scaling properties of the attractor solution. We also find that, while most of the axions are emitted with momenta of order Hubble, most of the axion energy density is contained in axions with energy of order the string core scale, at least in the parameter range available in the simulation. While such a spectrum would lead to a negligible number density of relic axions from strings when extrapolated to the physical parameter region, we show that the presence of small logarithmic corrections to the spectrum shape could completely alter such a conclusion. A detailed understanding of the evolution of the axion spectrum is therefore crucial for a reliable estimate of the relic axion abundance from strings.

Large- η constant-roll inflation is never an attractor

Slow roll solutions to inflationary potentials have been widely believed to be the only universal attractor. Over the last few years there has been growing interest in a new class of inflationary models known as Constant-Roll Inflation. Constant roll solutions are a generalization of "ultra-slow roll" dynamics, where the first slow roll parameter is small, but the second slow roll parameter $\eta$ is larger than unity. In Ultra-slow Roll Inflation, the large-$\eta$ solution is a dynamical transient, relaxing exponentially to the attractor de Sitter solution. In the constant roll generalization, recent papers have concluded that Constant-Roll Inflation represents a new class of non-slow roll attractor solutions. In this paper we show that these attractor solutions are actually the usual slow roll attractor, disguised by a parameter duality, and that the large-$\eta$ solutions, as in the case of ultra-slow roll, represent a dynamical transient.

Structure formation beyond shell-crossing: nonperturbative expansions and late-time attractors

Structure formation in 1+1 dimensions is considered, with emphasis on the effects of shell-crossing. The breakdown of the perturbative expansion beyond shell-crossing is discussed, and it is shown, in a simple example, that the perturbative series can be extended to a transseries including nonperturbative terms. The latter converges to the exact result well beyond the range of validity of perturbation theory. The crucial role of the divergences induced by shell-crossing is discussed. They provide constraints on the structure of the transseries and act as a bridge between the perturbative and the nonperturbative sectors. Then, we show that the dynamics in the deep multistreaming regime is governed by attractors. In the case of simple initial conditions, these attractors coincide with the asymptotic configurations of the adhesion model, but in general they may differ. These results are applied to a cosmological setting, and an algorithm to build the attractor solution starting from the Zel'dovich approximation is developed. Finally, this algorithm is applied to the search of `haloes' and the results are compared with those obtained from the exact dynamical equations.

Coupled kinetic equations for fermions and bosons in the relaxation-time approximation

Kinetic equations for quarks and gluons are solved numerically in the relaxation time approximation for the case of one-dimensional boost-invariant geometry. Quarks are massive and described by the Fermi-Dirac statistics, while gluons are massless and obey Bose-Einstein statistics. The conservation laws for the baryon number, energy, and momentum lead to two Landau matching conditions which specify the coupling between the quark and gluon sectors and determine the proper-time dependence of the effective temperature and baryon chemical potential of the system. The numerical results illustrate how a non-equlibrium mixture of quarks and gluons approaches hydrodynamic regime described by the Navier-Stokes equations with appropriate forms of the kinetic coefficients. The shear viscosity of a mixture is the sum of the shear viscosities of quark and gluon components, while the bulk viscosity is given by the formula known for a gas of quarks, however, with the thermodynamic variables characterising the mixture. Thus, we find that massless gluons contribute in a non-trivial way to the bulk viscosity of a mixture, provided quarks are massive. We further observe the hydrodynamization effect which takes place earlier in the shear sector than in the bulk one. The numerical studies of the ratio of the longitudinal and transverse pressures show, to a good approximation, that it depends on the ratio of the relaxation and proper times only. This behaviour is connected with the existence of an attractor solution for conformal systems.

Observational constraints on tachyonic chameleon dark energy model

It has been recently shown that tachyonic chameleon model of dark energy in which tachyon scalar field non-minimally coupled to the matter admits stable scaling attractor solution that could give rise to the late-time accelerated expansion of the universe and hence alleviate the coincidence problem. In the present work, we use data from Type Ia supernova (SN Ia) and Baryon Acoustic Oscillations to place constraints on the model parameters. In our analysis we consider in general exponential and non-exponential forms for the non-minimal coupling function and tachyonic potential and show that the scenario is compatible with observations.

Dynamical analysis for a scalar–tensor model with kinetic and nonminimal couplings

We study the autonomous system for a scalar–tensor model of dark energy with nonminimal coupling to curvature and nonminimal kinetic coupling to the Einstein tensor. The critical points describe important stable asymptotic scenarios including quintessence, phantom and de Sitter attractor solutions. Two functional forms for the coupling functions and the scalar potential were considered: power-law and exponential functions of the scalar field. For power-law couplings, the restrictions on stable quintessence and phantom solutions lead to asymptotic freedom regime for the gravitational interaction. For the exponential functions, the stable quintessence, phantom or de Sitter solutions allow asymptotic behaviors where the effective Newtonian coupling can reach either the asymptotic freedom regime or constant value. The phantom solutions could be realized without appealing to ghost degrees of freedom. Transient inflationary and radiation dominated phases can also be described.

Relativistic Fluid Dynamics Far From Local Equilibrium

Fluid dynamics is traditionally thought to apply only to systems near local equilibrium. In this case, the effective theory of fluid dynamics can be constructed as a gradient series. Recent applications of resurgence suggest that this gradient series diverges, but can be Borel resummed, giving rise to a hydrodynamic attractor solution which is well defined even for large gradients. Arbitrary initial data quickly approaches this attractor via nonhydrodynamic mode decay. This suggests the existence of a new theory of far-from-equilibrium fluid dynamics. In this Letter, the framework of fluid dynamics far from local equilibrium for a conformal system is introduced, and the hydrodynamic attractor solutions for resummed Baier-Romatschke-Son-Starinets-Stephanov theory, kinetic theory in the relaxation time approximation, and strongly coupled N=4 super Yang-Mills theory are identified for a system undergoing Bjorken flow.

Does anisotropic inflation produce a small statistical anisotropy?

Anisotropic inflation is an interesting model with an \\U(1) gauge field and it predicts the statistical anisotropy of the curvature perturbation characterized by a parameter g*. However, we find that the background gauge field does not follow the classical attractor solution due to the stochastic effect. We develop the stochastic formalism of a vector field and solve Langevin and Fokker-Planck equations. It is shown that this model is excluded by the CMB constraint g*≤ 10−2 with a high probability about 99.999%.

Time-varying q-deformed dark energy interacts with dark matter

We propose a new model for studying the dark constituents of the universe by regarding the dark energy as a [Formula: see text]-deformed scalar field interacting with the dark matter, in the framework of standard general relativity. Here we assume that the number of particles in each mode of the [Formula: see text]-deformed scalar field varies in time by the particle creation and annihilation. We first describe the [Formula: see text]-deformed scalar field dark energy quantum-field theoretically, then construct the action and the dynamical structure of these interacting dark sectors, in order to study the dynamics of the model. We perform the phase space analysis of the model to confirm and interpret our proposal by searching the stable attractor solutions implying the late-time accelerating phase of the universe. We then obtain the result that when interaction and equation-of-state parameter of the dark matter evolve from the present day values into a particular value, the dark energy turns out to be a [Formula: see text]-deformed scalar field.

Relativistic hydrodynamic attractors with broken symmetries: non-conformal and non-homogeneous

Standard textbooks will state that hydrodynamics requires near-equilibrium to be applicable. Recently, however, out-of-equilibrium attractor solutions for hydrodynamics have been found in kinetic theory and holography in systems with a high degree of symmetry, suggesting the possibility of a genuine out-of-equilibrium formulation of hydrodynamics. This work demonstrates that attractor solutions also occur in non-conformal kinetic theory and spatially non-homogeneous systems, potentially having important implications for the interpretation of experimental data in heavy-ion and proton-proton collisions and relativistic fluid dynamics as a whole.

The accelerating pace of star formation

We study the temporal and spatial distribution of star formation rates in four well-studied star-forming regions in local molecular clouds(MCs): Taurus, Perseus, $\rho$ Ophiuchi, and Orion A. Using published mass and age estimates for young stellar objects in each system, we show that the rate of star formation over the last 10 Myrs has been accelerating and is (roughly) consistent with a $t^2$ power law. This is in line with previous studies of the star formation history of molecular clouds and with recent theoretical studies. We further study the clustering of star formation in the Orion Nebula Cluster(ONC). We examine the distribution of young stellar objects as a function of their age by computing an effective half-light radius for these young stars subdivided into age bins. We show that the distribution of young stellar objects is broadly consistent with the star formation being entirely localized within the central region. We also find a slow radial expansion of the newly formed stars at a velocity of $v=0.17\,{\rm km\,s}^{-1}$, which is roughly the sound speed of the cold molecular gas. This strongly suggests the dense structures that form stars persist much longer than the local dynamical time. We argue that this structure is quasi-static in nature and is likely the result of the density profile approaching an attractor solution as suggested by recent analytic and numerical analysis.

Dynamical analysis for a scalar\u2013tensor model with Gauss\u2013Bonnet and non-minimal couplings

We study the autonomous system for a scalar–tensor model of dark energy with Gauss–Bonnet and non-minimal couplings. The critical points describe important stable asymptotic scenarios including quintessence, phantom and de Sitter attractor solutions. Two functional forms for the coupling functions and the scalar potential are considered: power-law and exponential functions of the scalar field. For the exponential functions the existence of stable quintessence, phantom or de Sitter solutions, allows for an asymptotic behavior where the effective Newtonian coupling becomes constant. The phantom solutions could be realized without appealing to ghost degrees of freedom. Transient inflationary and radiation-dominated phases can also be described.

Strongly coupled dark energy with warm dark matter vs. LCDM

Cosmologies including strongly Coupled (SC) Dark Energy (DE) and Warm dark matter (SCDEW) are based on a conformally invariant (CI) attractor solution modifying the early radiative expansion. Then, aside of radiation, a kinetic field Φ and a DM component account for a stationary fraction, ∼ 1 %, of the total energy. Most SCDEW predictions are hardly distinguishable from LCDM, while SCDEW alleviates quite a few LCDM conceptual problems, as well as its difficulties to meet data below the average galaxy scale. The CI expansion begins at the end of inflation, when Φ (future DE) possibly plays a role in reheating, and ends at the Higgs scale. Afterwards, a number of viable options is open, allowing for the transition from the CI expansion to the present Universe. In this paper: (i) We show how the attractor is recovered when the spin degrees of freedom decreases. (ii) We perform a detailed comparison of CMB anisotropy and polarization spectra for SCDEW and LCDM, including tensor components, finding negligible discrepancies. (iii) Linear spectra exhibit a greater parameter dependence at large k's, but are still consistent with data for suitable parameter choices. (iv) We also compare previous simulation results with fresh data on galaxy concentration. Finally, (v) we outline numerical difficulties at high k. This motivates a second related paper [1], where such problems are treated in a quantitative way.