Perturbation Equations Research Articles

Gravito-electromagnetic perturbations and QNMs of regular black holes

In the framework of Einstein’s gravity coupled to nonlinear electromagnetic fields, we study gravito-electromagnetic perturbations of magnetic regular black holes (BHs). The master equations of perturbations are obtained through Chandrasekhar’s formulation, from which it can be seen, different from the electric counterparts, for magnetic BHs gravitational perturbations with odd-parity coupled only to the electromagnetic perturbations with even-parity. We solve the master equations numerically and obtain quasinormal modes (QNMs) for three typical regular BHs. Results show that QNMs of distinct regular BHs differ significantly, and they differ from that of the Reissner–Nordström BH as well. Indications of these results on the stability of these regular BHs are discussed in detail.

Ballistic Coefficient Calculation Based on Optical Angle Measurements of Space Debris.

Atmospheric drag is an important factor affecting orbit determination and prediction of low-orbit space debris. To obtain accurate ballistic coefficients of space debris, we propose a calculation method based on measured optical angles. Angle measurements of space debris with a perigee height below 1400 km acquired from a photoelectric array were used for orbit determination. Perturbation equations of atmospheric drag were used to calculate the semi-major-axis variation. The ballistic coefficients of space debris were estimated and compared with those published by the North American Aerospace Defense Command in terms of orbit prediction error. The 48 h orbit prediction error of the ballistic coefficients obtained from the proposed method is reduced by 18.65% compared with the published error. Hence, our method seems suitable for calculating space debris ballistic coefficients and supporting related practical applications.

IDECAMB: an implementation of interacting dark energy cosmology in CAMB

Interacting dark energy (IDE) scenario is a natural and important extension to the standard ΛCDM cosmology. We develop a full numerical routine, called IDECAMB, as a patch to the public Einstein-Boltzmann solver CAMB, to solve the background and perturbation equations of the IDE models. The IDECAMB solver provides a unified interface for the widely studied IDE models by employing a parametrization model with five free functions. By configuring these five functions, one can easily map the coupled quintessence (CQ) and coupled fluid (CF) models into the parametrization. We handle the perturbation evolutions of the CF models with the parametrized post-Friedmann (PPF) approach to avoid the possible large-scale instability. Compared with the previous established PPF approach whose form depends on a specific IDE model, the PPF approach in this work are model-independent, making it easy to use. We constrain a specific CQ model with the IDECAMB package. The fitting results are consistent with those obtained by Planck Collaboration, which confirms the validity of the package.

Creep of Concrete in Shell Structures: Nonlinear Theory.

The creep of concrete is one of the main problems threatening concrete structural development and the stability and safety of structures. However, the nonlinear theory is the key to solving the problem of taking into account the physical and mechanical properties of concrete creep in shell structures. To create such a theory, the original shell is replaced by a continuous equivalent elastic shell. To determine the stress-strain state of the structure, the equations of nonlinear creep and crack growth are derived, and a deformation model of the section is created. The behavior of the structure at all stages of the life cycle is investigated by solving the solving systems of differential equations of equilibrium, motion, and perturbation of the equivalent shell. The values of the ratios of dependence of long-term and short-term critical loads on deformations, forces, cracks, etc., are given. The accuracy of the solution of the developed nonlinear theory is compared with the linear theory of concrete creep as well as experimental data. The results show that, according to the linear theory, for the values for the short term and long term, up to 56% and up to 39% of critical loads are overestimated, respectively. The creep process in practical engineering can be effectively controlled by the results of the proposed theory.

Tracking the validity of the quasi-static and sub-horizon approximations in modified gravity

Within the framework of modified gravity, the quasi-static and sub-horizon approximations are widely used in analyses aiming to identify departures from the concordance model at late-times. In general, it is assumed that time derivatives are subdominant with respect to spatial derivatives given that the relevant physical modes are those well inside the Hubble radius. In practice, the perturbation equations under these approximations are reduced to a tractable algebraic system in terms of the gravitational potentials and the perturbations of involved matter fields. Here, in the framework of f(R) theories, we revisit standard results when these approximations are invoked using a new parameterization scheme that allows us to track the relevance of each time-derivative term in the perturbation equations. This new approach unveils correction terms which are neglected in the standard procedure. We assess the relevance of these differences by comparing results from both approaches against full numerical solutions for two well-known toy-models: the designer f(R) model and the Hu-Sawicki model. We find that: i) the sub-horizon approximation can be safely applied to linear perturbation equations for scales 0.06 h/Mpc ≲ k ≲ 0.2 h/Mpc, ii) in this “safety region”, the quasi-static approximation provides a very accurate description of the late-time cosmological dynamics even when dark energy significantly contribute to the cosmic budget, and iii) our new methodology performs better than the standard procedure, even for several orders of magnitude in some cases. Although, the impact of this major improvement on the linear observables is minimal for the studied cases, this does not represent an invalidation for our approach. Instead, our findings indicate that the perturbation expressions derived under these approximations in more general modified gravity theories, such as Horndeski, should be also revisited.

Stability analysis of f(Q) gravity models using dynamical systems

In recent years, the modified theory of gravity known as [Formula: see text] gravity has drawn interest as a potential alternative to general relativity. According to this theory, the gravitational force is determined by a function of the so-called “non-metricity” tensor [Formula: see text], which expresses how far a particle space-time is from the metric geometry. In contrast to general relativity, which describes the gravitational field using the curvature tensor, [Formula: see text] gravity builds a theory of gravity using the non-metricity tensor. For this class of theories, dynamical system analysis of the background and perturbation equations has been carried out in this work to determine how various models behave cosmologically. Here, the critical points are determined for two [Formula: see text] models from the literature: the power law, [Formula: see text], and the logarithmic, [Formula: see text] models. The stability behavior and corresponding cosmology are displayed for each critical point. For the power law model, we achieve a matter-dominated saddle point with the right matter perturbation growth rate. For the logarithmic model, we get a saddle point dominated by the geometric component of the [Formula: see text] model with perturbations in the decomposition of matter. For both models, we later achieved a stable and accelerating Universe with constant matter perturbations.

Response Studies of a Tethered Aerostat Balloon Subjected to Initial Disturbances

This paper presents response studies for a tethered aerostat balloon subjected to initial disturbances. The linearized small perturbations equations of motion have been decoupled into longitudinal and lateral-directional movements. First of all, trim conditions are estimated for a typical operating wind speed and then equations of motion are converted to state-space form. Eigenvalues and eigenvectors are then evaluated for the characteristic matrix. The differential equations are then solved using eigenvalues and eigenvectors approach. The initial disturbances are then applied first in isolation and then in combination. The results are presented for a reference aerostat balloon in the form of plots of responses of longitudinal and lateral motion variables with time. The plots reveal useful information which may be used for finalization of operating parameters for aerostat balloon to generate required response characteristics.

On Chaplygin models in f(G) gravity

This work treats cosmological perturbation in a mixture of standard matter, Chaplygin gas as well as Gauss–Bonnet fluids using a 1+3 covariant approach in the context of modified [Formula: see text] gravity. We define the gradient variables to obtain linear perturbation equations. After scalar and redshift transformations, we consider both an original Chaplygin and generalized Chaplygin gas models under Gauss–Bonnet gravity. For pedagogical purposes, the consideration of polynomial [Formula: see text] gravity model was used to solve the perturbation equations for short- and long-wavelength modes and investigate the late-time evolution. The numerical solutions were obtained. The results show that the energy overdensity perturbations decay with an increase in redshift. The treatment recovers GR results under limiting cases.

Acoustic characteristics of supersonic planar impinging jets

Acoustic characteristics of supersonic planar impinging jets

Quasinormal modes of nonseparable perturbation equations: The scalar non-Kerr case

Quasinormal modes of nonseparable perturbation equations: The scalar non-Kerr case

Perturbation in an interacting dark Universe

In this paper we have considered an interacting model of dark energy where the dark sectors are allowed to interact among themselves through an assumed relation. By solving the perturbation equations numerically, we have studied the imprints on the growth of matter as well as dark energy fluctuations. It has been found that for higher rate of interaction strength for the coupling term, visible imprints on the dark energy density fluctuations are observed at the early epochs of evolution. In order to check the viability of the model, we have tested the interacting model with different observational datasets and have looked into the evolution of the density contrast, and the effects on the CMB temperature fluctuation as well as matter power spectrum.

Self-balancing characteristics of three self-synchronous linear motion mechanical oscillators

How to restrain the unbalance caused by the manufacturing and installation of parts during the working process of the rotating oscillators is rarely addressed, yet it is very important for evaluating the performance of rotating machines. This paper treats a system comprising three linear motion mechanical oscillators (LMMOs) fixed to a rigid body and clarifies the self-balancing characteristics of the system in the over-resonance state. An electromechanically coupled dynamic model of LMMO is constructed using Lagrange’s energy approach. The small parameter average method is used to derive the synchronization conditions among the LMMOs, and the reasonable coupling torque parameter range that the system should be satisfied. The motion stability is analyzed using a linearized matrix of the torque perturbation equation describing the system, and the stability criteria of synchronous linear motion are provided. Subsequently, the necessary conditions are found for implementing self-balancing among the three LMMOs to eliminate the unbalanced vibration in the system, and the influences of the structural parameters on self-balancing behavior are investigated. The results demonstrate that the structural parameters and phase differences among the three LMMOs can positively reduce the unbalanced vibration of the system, when the necessary condition for the vibrating displacement response nearing to be zero is fully satisfied. This research can provide theoretical guidance for the vibrating isolation of rotating machinery body with several linear motion vibrating sources installed on a common oscillating platform.

Dynamics of a binary system around a supermassive black hole

We discuss the motion of a binary system around a supermassive black hole. Using Fermi-Walker transport, we construct a local inertial reference frame and set up a Newtonian binary system. Assuming a circular geodesic observer around a Schwarzschild black hole, we write down the equations of motion of a binary. Introducing a small acceleration of the observer, we remove the interaction terms between the center of mass (CM) of a binary and its relative coordinates. The CM follows the observer's orbit, but its motion deviates from an exact circular geodesic. We first solve the relative motion of a binary system, and then find the motion of the CM by the perturbation equations with the small acceleration. We show that von Zeipel-Lidov-Kozai (vZLK) oscillations appear when a binary is compact and the initial inclination is larger than a critical angle. In a hard binary system, vZLK oscillations are regular, whereas in a soft binary system, oscillations are irregular both in period and in amplitude, although stable. We find an orbital flip when the initial inclination is large. As for the motion of the CM, the radial deviations from a circular orbit become stable oscillations with very small amplitude.

On 1 + 3 covariant perturbations of the quasi-Newtonian spacetime in modified Gauss–Bonnet gravity

The consideration of a [Formula: see text] covariant approach to cold dark matter universe with no shear cosmological dust model with irrotational flows is developed in the context of [Formula: see text] gravity theory in this study. This approach reveals the existence of integrability conditions which do not appear in noncovariant treatments. We constructed the integrability conditions in modified Gauss–Bonnet [Formula: see text] gravity basing on the constraints and propagation equations. These integrability conditions reveal the linearized silent nature of quasi-Newtonian models in [Formula: see text] gravity. Finally, the linear equations for the overdensity and velocity perturbations of the quasi-Newtonian spacetime were constructed in the context of modified [Formula: see text] gravity. The application of harmonic decomposition and redshift transformation techniques to explore the behavior of the overdensity and velocity perturbations using [Formula: see text] model was made. On the other hand, we applied the quasi-static approximation to study the approximated solutions on small scales which helps to get both analytical and numerical results of the perturbation equations. The analysis of the energy overdensity and velocity perturbations for both short- and long-wavelength modes in a dust-Gauss–Bonnet fluids was done and we see that both energy overdensity and velocity perturbations decay with redshift for both modes. In the limits to [Formula: see text]CDM, it means [Formula: see text] the considered [Formula: see text] model results coincide with [Formula: see text]CDM.

An update on adiabatic modes in cosmology and δN formalism

In this paper, we generalize the Weinberg's procedure to determine the comoving curvature perturbation ℛ to non-attractor inflationary regimes. We show that both modes of ℛ are related to a symmetry of the perturbative equations in the Newtonian gauge. As a byproduct, we clarify that adiabaticity does not generally imply constancy of ℛ, not even in the k ⟶ 0 limit. We then show that there exist non-equivalent definitions of δN that would reproduce ℛ or the uniform density curvature perturbation ζ at linear order. We have then shown that the perturbative δN definition in terms of difference between the number of e-foldings of different gauges, can be extended non-perturbatively at leading order in gradient expansion. Nevertheless, the computer friendly definition in terms of the difference of e-foldings obtained from the evolution of a local FRW Universe, respectively with perturbed and un-perturbed initial conditions, might only give information about the linear order curvature perturbations, contrary to what is stated in the literature.

An Efficient Hybrid Computational Process for Interior Noise Prediction in Aeroacoustic Vehicle Development

Numerical methodologies for aeroacoustic analyses are increasingly crucial for car manufacturers to optimize the effectiveness of vehicle development. In the present work, a hybrid numerical tool based on the combination of a delayed detached-eddy simulation and a finite element model, which relies on the Lighthill's acoustic analogy and the acoustic perturbation equations, is presented. The computational aeroacoustics is performed by the software OpenFOAM and Actran, concerning respectively the CFD and the FEM. The aeroacoustic behavior of the SUV Lamborghini Urus at a cruising speed of 140 km/h has been investigated. The main aerodynamic noise phenomena occurring in the side mirror region in a frequency range up to 5 kHz are discussed. The numerical simulations have been verified against the measurements performed in the aeroacoustic wind tunnel of the University of Stuttgart, operated by FKFS. The predicted exterior noise propagation into the far field has been validated by comparing the sound pressure level with the experimental data measured by exterior microphones, which were located outside the turbulent region beside the wake of the side mirror. Furthermore, the noise transmission into the cabin through the side window has been modeled. Simulation results have been validated by means of interior microphones installed on the driver seat. Both the exterior and the interior noise predictions show very good correlations with experiments. Lastly, a comprehensive investigation of the most critical aeroacoustic sources has been carried out. The numerical tool has been proven to be in good accordance with the microphone array with respect to the distribution of the sound pressure level in the proximity of the side mirror. Besides, the main vortex structures involved in the generation mechanisms of wind noise have been investigated by a CFD analysis. The entire CAA process has been proven to be accurate and suitable for combined analysis between the generation mechanisms of wind noise and the resulting transfer into the interior cabin to the driver's ear as well.

Slowly rotating Kerr metric derived from the Einstein equations in affine-null coordinates

Using a quasispherical approximation of an affine-null metric adapted to an asymptotic Bondi inertial frame, we present high order approximations of the metric functions in terms of the specific angular momentum for a slowly rotating stationary and axisymmetric vacuum spacetime. The metric is obtained by following the procedure of integrating the hierarchy of Einstein equations in a characteristic formulation utilizing master functions for the perturbations. It further verified its equivalence with the Kerr metric in the slowly rotating approximation by carrying out an explicit transformation between the Boyer-Lindquist coordinates to the employed affine-null coordinates. A peculiar feature of the derivation is that in the solution of the perturbation equations for every order a new integration constant appears which cannot be set to zero using asymptotical flatness or regularity arguments. However, these additional integration constants can be absorbed into the overall Komar mass and Komar angular momentum of a slowly rotating black hole.

Continuum perturbation field in quiescent ambience: Common foundation of flows and acoustics

Here, the perturbation equation for a dissipative medium is derived from the first principles for the linearized compressible Navier–Stokes equation without Stokes' hypothesis. Dispersion relations of this generic governing equation are obtained, which exhibits both the dispersive and dissipative nature of perturbations traveling in a dissipative medium, depending upon the length scale. We specifically provide a theoretical cutoff wave number above which the perturbation equation represents diffusive and dissipative nature of the quiescent flow. It is shown that perturbation equations for pressure and velocity retain the same form in one-dimension, but it is not the same for multi-dimensional perturbation fields. Such behavior has not been reported before, as per the knowledge of the authors.

Prediction model of GEO spacecraft position after maneuver based on causal Bayesian network

Predicting the position of Geosynchronous (GEO) spacecraft under maneuver is a crucial work for space domain awareness (SDA) as it can help to improve the flexibility and operational efficiency of space surveillance network (SSN). The longitude is a unique freely assigned parameter for GEO spacecraft, in this paper, a predictive model of GEO spacecraft longitude based on causal Bayesian network is proposed. Firstly, the causal parameters of longitude is found by Gaussian perturbation equation. Secondly, the Markov order of the causal parameters is obtained by transfer entropy. Finally, the linear expressions for Gaussian functions is proved and a causal Bayesian network (CBN) prediction model for longitude is constructed. After experimental analysis, the mean absolute error (MAE) and mean square error (MSE) of the proposed method are decreased by 71.88% and 72.18%, respectively compared with the traditional Long short-term memory (LSTM) method.

Inflation from a chaotic potential with a step

In this work, we study the effects on the relevant observational parameters of an inflationary universe from a chaotic potential with a step. We numerically evolve the perturbation equations within both cold inflation and warm inflation. On the one hand, in a cold inflation scenario we analyse the scalar power spectrum $P_{\mathcal{R}}$ in terms of the number of e-folds $N_{e}$, and in terms of the ratio $k/k_{0}$, where $k_{0}$ is our pivot scale. We show how $P_{\mathcal{R}}$ oscillates around $0.2< k/k_{0} < 20$. Additionally, we present the evolution of two relevant parameters: the scalar spectral index $n_\mathrm{s}$ and the tensor-to-scalar ratio $r$. In fact, more than one region of $(n_\mathrm{s},r)$ lies within the observable window (Planck 2018). On the other hand, in the warm inflationary case, we also examine the evolution of $P_{\mathcal{R}}$ in terms of $N_{e}$ and $k/k_{0}$. Perturbations are amplified in WI; in fact, $P_{\mathcal{R}}$ can be much larger than the CMB value $P_{\mathcal{R}}> 2.22\times 10^{-9}$. This time, the spectral index $n_\mathrm{s}$ is clearly blue-tilted, at smaller scales, and the tensor-to-scalar ratio $r$ becomes too low. However, $n_\mathrm{s}$ can change from blue-tilted towards red-tilted, since $P_{\mathcal{R}}$ starts oscillating around $k/k_{0}\sim 40$. Indeed, the result from the step potential skims the Planck contours. Finally, one key aspect of this research was to contrast the features of an inflationary potential between both paradigms, and, in fact, they show similarities and differences. Due to a featured background and a combined effect of entropy fluctuations (only in warm inflation), in both scenarios certain fluctuation scales are not longer ``freeze in'' on super-horizon scales.