Exact Power-law Research Articles

Anisotropic power-law inflation for models of non-canonical scalar fields non-minimally coupled to a two-form field

In this paper, we investigate the validity of the so-called cosmic no-hair conjecture in the framework of anisotropic inflation models of non-canonical scalar fields non-minimally coupled to a two-form field. In particular, we focus on two typical k-inflation and Dirac–Born–Infeld inflation models, in which we find a set of exact anisotropic power-law inflationary solutions. Interestingly, these solutions are shown to be stable and attractive during an inflationary phase using the dynamical system analysis. The obtained results indicate that the non-minimal coupling between the scalar and two-form fields acts as a non-trivial source of generating stable spatial anisotropies during the inflationary phase and therefore violates the prediction of the cosmic no-hair conjecture, even when the scalar field is of non-canonical forms. In connection with the Planck 2018 data, tensor-to-scalar ratios of these anisotropic solutions are investigated. As a result, it appears that the tensor-to-scalar ratio of the anisotropic power-law inflationary solution of k-inflation model turns out to be more highly consistent with the Planck 2018 data than that of Dirac-Born-Infeld model.

Eccentricity dynamics of wide binaries – I. The effect of Galactic tides

ABSTRACT A major puzzle concerning the wide stellar binaries (semimajor axes a ≳ 103 au) in the Solar neighbourhood is the origin of their observed superthermal eccentricity distribution function (DF), which is well approximated by P(e) ∝ eα with α ≈ 1.3. This DF evolves under the combined influence of (i) tidal torques from the Galactic disc and (ii) scattering by passing stars, molecular clouds, and substructure. Recently, it was demonstrated that Galactic tides alone cannot produce a superthermal eccentricity DF from an initially isotropic, non-superthermal one, under the restrictive assumptions that the eccentricity DF was initially of power-law form and then was rapidly phase-mixed toward a steady state by the tidal perturbation. In this paper, we first prove analytically that this conclusion is valid at all times, regardless of these assumptions. We then adopt a thin Galactic disc model and numerically integrate the equations of motion for several ensembles of tidally perturbed wide binaries to study the time evolution in detail. We find that even non-power-law DFs can be described by an effective power-law index αeff which accurately characterizes both their initial and final states, and that a DF with initial (effective or exact) power-law index αi is transformed by Galactic tides into another power law with index αf ≈ (1 + αi)/2 on a time-scale $\sim 4\, \mathrm{Gyr}\, (a/10^4\mathrm{AU})^{-3/2}$. In a companion paper, we investigate separately the effect of stellar scattering. As the GAIA data continues to improve, these results will place strong constraints on wide binary formation channels.

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.

Short length scale fluctuations in lattice growth models

Fluctuations of interfaces produced by lattice growth models scale as those of stochastic equations at distances r much larger than the lattice constant a. However, those equations may be derived from the short range interactions through renormalization, which suggests that universal properties may also be observed in short scale fluctuations of the lattice models. We first investigate this question in interfaces with preset exact power law structure factors by expanding the autocovariance function, which is shown to scale as r2α+constant at distances as small as r∼5a (α is the roughness exponent). Next we perform numerical simulations of lattice models in five universality classes, in one and two dimensions, and calculate the autocovariance function and the fluctuation of a local average height in their growth regimes, where finite-size effects are negligible. In cases of normal roughening with α>0, those quantities also scale as affine functions of r2α in distances from a to ∼10a, in contrast with the usual expectation that such relation is applicable only in the hydrodynamic limit. In a model with super-roughening in the Mullins–Herring class, similar relation is applicable with the local roughness exponent in one and two dimensions. In cases with α≤0, the fits of those functions diverge in the zero distance limit, in the same form as the one-point fluctuations of the corresponding stochastic equations. Finally, we study competitive models with crossovers in the roughness evolution and show that, at any given time, short and long range fluctuations scale with the exponent α of the dominant universality class at that time. Thus, short range correlations at long times do not keep a memory on the short time kinetics. These results reinforce the connection between discrete and continuous growth models by showing that their short range fluctuations have related properties.

EFFICIENT INCREMENTAL SIMULATION OF “1/f” NOISES

Large-scale simulations and analysis of the original powers of 2 geometric source sum algorithm for simulating [Formula: see text] noise confirm that it provides a reasonably accurate approximation to an exact [Formula: see text] spectral density with a Gaussian amplitude distribution over any arbitrarily large frequency range. This incremental algorithm is computationally efficient with a computation time, after initialization, that varies linearly with the number of samples generated. A new variation allows non-integer and random ratios for the geometric sequence that reduces variations about the exact power-law spectral density and, by varying individual source amplitudes, produces generalized [Formula: see text] noises.

Modified black hole solution with a background Kalb\u2013Ramond field

We study the gravitation effects on a static and spherically symmetric spacetime due to the vacuum expectation value (VEV) of a Kalb–Ramond field. The Kalb–Ramond VEV is a background tensor field which produces a local Lorentz symmetry breaking (LSB) of spacetime. Considering a non-minimal coupling between the Kalb–Ramond (VEV) and the Ricci tensor, we obtain an exact parameterdependent power-law modified black hole. For a particular choice of the LSB parameter, the Lorentz violation produces a solution similar to the Reissner–Nordstrom, despite the absence of charge. The near-horizon geometry is modified by including a new inner horizon and shifting the Schwarzschild horizon. Asymptotically, the usual Minkowski spacetime with a background tensor field is recovered. The vacuum configurations are studied considering the energy conditions and the Lorentz violating source properties. By means of the mercury perihelion test, an upper bound to the local Lorentz violation (LV) is obtained, and its corresponding effects on the black hole temperature is investigated.

Explicit power laws in analytic continuation problems via reproducing kernel Hilbert spaces

The need for analytic continuation arises frequently in the context of inverse problems. Notwithstanding the uniqueness theorems, such problems are notoriously ill-posed without additional regularizing constraints. We consider several analytic continuation problems with typical global boundedness constraints that restore well-posedness. We show that all such problems exhibit a power law precision deterioration as one moves away from the source of data. In this paper we demonstrate the effectiveness of our general Hilbert space-based approach for determining these exponents. The method identifies the ‘worst case’ function as a solution of a linear integral equation of Fredholm type. In special geometries, such as the circular annulus or upper half-plane this equation can be solved explicitly. The obtained solution in the annulus is then used to determine the exact power law exponent for the analytic continuation from an interval between the foci of an ellipse to an arbitrary point inside the ellipse. Our formulas are consistent with results obtained in prior work in those special cases when such exponents have been determined.

The exact power law for Buffon’s needle landing near some random Cantor sets

In this paper, we study the Favard length of some random Cantor sets of Hausdorff dimension 1. We start with a unit disk in the plane and replace the unit disk by 4 disjoint subdisks (with equal distance to each other) of radius 1/4 inside and tangent to the unit disk. By repeating this operation in a self-similar manner and adding a random rotation in each step, we can generate a random Cantor set {\cal D}(\omega) . Let {\cal D}_n be the n -th generation in the construction, which is comparable to the 4^{-n} -neighborhood of {\cal D} . We are interested in the decay rate of the Favard length of these sets {\cal D}_n as n\to\infty , which is the likelihood (up to a constant) that "Buffon's needle'' dropped randomly will fall into the 4^{-n} -neighborhood of {\cal D} . It is well known that the lower bound of the Favard length of {\cal D}_n(\omega) is a constant multiple of n^{-1} . We show that the upper bound of the Favard length of {\cal D}_n(\omega) is C n^{-1} for some C > 0 in the average sense. We also prove the a similar linear decay for the Favard length of {\cal D}^d_n(\omega) , which is the d^{-n} -neighborhood of a self-similar random Cantor set with degree d greater than 4 . Notice that in the non-random case where the self-similar set has degree greater than 4 , the best known result for the decay rate of the Favard length is e^{-c\sqrt {\log n}} .

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.

Validity of the effective potential method to study PT-symmetric field theories

Abstract We calculate the one loop effective potential for the class ( - ( i ϕ ) α ) of PT -symmetric and non-Hermitian field theories in 0 + 1 space-time dimensions. To test the method, we showed that for the massless Hermitian ϕ 4 theory, the method gives the exact power law behavior known from the literature. We show that this order of calculations goes beyond the truncation of the Schwinger-Dyson equations at the two-point green functions applied to the PT -symmetric ( - ϕ 4 ) theory in the literature. We found that the effective potential calculation represents good approximations of the vacuum energies of the class ( - ( i ϕ ) α ) compared to the numerical results. For the vacuum condensate, the method gives also accurate results for the absolute values but gives both positive as well as negative imaginary condensates for even α which again agrees with the prediction of the Schwinger-Dyson equations. Unlike other methods, the effective potential can be directly extended to higher dimensions as it offers a way to implement the PT -symmetric boundary conditions as well as there exist well known methods to regularize the theory at higher dimensions.

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.

Exploring complexity matching and asynchrony dynamics in synchronized and syncopated task performances

When two people synchronize their rhythmic behaviors (e.g., finger tapping; walking) they match one another not only at a local scale of beat-to-beat intervals, but also at a global scale of the complex (fractal) patterns of variation in their interval series. This “complexity matching” had been demonstrated in a variety of timing behaviors, but the current study was designed to address two important gaps in previous research. First, very little was known about complexity matching outside of synchronization tasks. This was important because different modes are associated with differences in the strength of coordination and the fractal scaling of the task performance. Second, very little was known about the dynamics of the asynchrony series. This was important because asynchrony is a variable directly quantifying the coordination between the two timing behaviors and the task goal. So, the current study explored complexity matching in both synchronized and syncopated finger tapping tasks, and included analyses of the fractal scaling of the asynchrony series. Participants completed an interpersonal finger tapping task, in both synchronization and syncopation conditions. The magnitude of variation and the exact power law scaling of the tapping intervals were manipulated by having one participant tap in time with a metronome. Complexity matching was most stable when there was sufficient variation in the task behavior and when a persistent scaling dynamic was presented. There were, however, several interesting differences between the two coordination modes, in terms of the heterogeneity of the complexity matching effect and the scaling of the asynchronies. These findings raised a number of important points concerning how to approach and understand the interaction of inherently complex systems.

How driving rates determine the statistics of driven non-equilibrium systems with stationary distributions

Sample space reducing (SSR) processes offer a simple analytical way to understand the origin and ubiquity of power-laws in many path-dependent complex systems. SRR processes show a wide range of applications that range from fragmentation processes, language formation to search and cascading processes. Here we argue that they also offer a natural framework to understand stationary distributions of generic driven non-equilibrium systems that are composed of a driving- and a relaxing process. We show that the statistics of driven non-equilibrium systems can be derived from the understanding of the nature of the underlying driving process. For constant driving rates exact power-laws emerge with exponents that are related to the driving rate. If driving rates become state-dependent, or if they vary across the life-span of the process, the functional form of the state-dependence determines the statistics. Constant driving rates lead to exact power-laws, a linear state-dependence function yields exponential or Gamma distributions, a quadratic function produces the normal distribution. Logarithmic and power-law state dependence leads to log-normal and stretched exponential distribution functions, respectively. Also Weibull, Gompertz and Tsallis-Pareto distributions arise naturally from simple state-dependent driving rates. We discuss a simple physical example of consecutive elastic collisions that exactly represents a SSR process.

Elastic interaction of hydrogen atoms on graphene: A multiscale approach from first principles to continuum elasticity

The deformation of graphene due to the chemisorption of hydrogen atoms on its surface and the long-range elastic interaction between hydrogen atoms induced by these deformations are investigated using a multiscale approach based on first principles, empirical interactions, and continuum modeling. Focus is given to the intrinsic low-temperature structure and interactions. Therefore, all calculations are performed at $T=0$, neglecting possible temperature or thermal fluctuation effects. Results from different methods agree well and consistently describe the local deformation of graphene on multiple length scales reaching 500 $\AA{}$. The results indicate that the elastic interaction mediated by this deformation is significant and depends on the deformation of the graphene sheet both in and out of plane. Surprisingly, despite the isotropic elasticity of graphene, within the linear elastic regime, atoms elastically attract or repel each other depending on (i) the specific site they are chemisorbed; (ii) the relative position of the sites; (iii) and if they are on the same or on opposite surface sides. The interaction energy sign and power-law decay calculated from molecular statics agree well with theoretical predictions from linear elasticity theory, considering in-plane or out-of-plane deformations as a superposition or in a coupled nonlinear approach. Deviations on the exact power law between molecular statics and the linear elastic analysis are evidence of the importance of nonlinear effects on the elasticity of monolayer graphene. These results have implications for the understanding of the generation of clusters and regular formations of hydrogen and other chemisorbed atoms on graphene.

Time-Domain Simulation of Ultrasound Propagation in a Tissue-Like Medium Based on the Resolution of the Nonlinear Acoustic Constitutive Relations

A time-domain numerical code based on the constitutive relations of nonlinear acoustics for simulating ultrasound propagation is presented. To model frequency power law attenuation, such as observed in biological media, multiple relaxation processes are included and relaxation parameters are fitted to both exact frequency power law attenuation and empirically measured attenuation of a variety of tissues that does not fit an exact power law. A computational technique based on artificial relaxation is included to correct the non-negligible numerical dispersion of the numerical method and to improve stability when shock waves are present. This technique avoids the use of high order finite difference schemes, leading to fast calculations. The numerical code is especially suitable to study high intensity and focused axisymmetric acoustic beams in tissue-like medium, as it is based on the full constitutive relations that overcomes the limitations of the parabolic approximations, while some specific effects not contemplated by the Westervelt equation can be also studied. The accuracy of the method is discussed by comparing the proposed simulation solutions to one-dimensional analytical ones, to $k$-space numerical solutions and also to experimental data from a focused beam propagating in a frequency power law attenuation media.

Anisotropic inflation in Gauss-Bonnet gravity

We study anisotropic inflation with Gauss-Bonnet correction in presence of a massless vector field. In this scenario, exact anisotropic power-law inflation is realized when the inflaton potential, gauge coupling function and the Gauss-Bonnet coupling are exponential functions. We show that anisotropy becomes proportional to two slow-roll parameters of the theory and hence gets enhanced in presence of quadratic curvature corrections. The stability analysis reveals that anisotropic power-law solutions remain stable over a substantially large parameter region.

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.

Understanding scaling through history-dependent processes with collapsing sample space

History-dependent processes are ubiquitous in natural and social systems. Many such stochastic processes, especially those that are associated with complex systems, become more constrained as they unfold, meaning that their sample space, or their set of possible outcomes, reduces as they age. We demonstrate that these sample-space-reducing (SSR) processes necessarily lead to Zipf's law in the rank distributions of their outcomes. We show that by adding noise to SSR processes the corresponding rank distributions remain exact power laws, p(x) ~ x(-λ), where the exponent directly corresponds to the mixing ratio of the SSR process and noise. This allows us to give a precise meaning to the scaling exponent in terms of the degree to which a given process reduces its sample space as it unfolds. Noisy SSR processes further allow us to explain a wide range of scaling exponents in frequency distributions ranging from α = 2 to ∞. We discuss several applications showing how SSR processes can be used to understand Zipf's law in word frequencies, and how they are related to diffusion processes in directed networks, or aging processes such as in fragmentation processes. SSR processes provide a new alternative to understand the origin of scaling in complex systems without the recourse to multiplicative, preferential, or self-organized critical processes.