Perturbation Equations Research Articles

Cosmology in f(Q) gravity: A unified dynamical systems analysis of the background and perturbations

Motivated by the fact that cosmological models based on $f(Q)$ gravity are very efficient in fitting observational datasets at both background and perturbation levels, we perform a combined dynamical system analysis of both background and perturbation equations in order to examine the validity of this result through an independent method. We examine two studied $f(Q)$ models of the literature, namely the power-law and the exponential ones. For both cases, we obtain a matter-dominated saddle point characterized by the correct growth rate of matter perturbations, followed by the transition to a stable dark-energy dominated accelerated universe in which matter perturbations remain constant. Furthermore, analyzing the behavior of $f \sigma_8$, we find that the models fit observational data successfully, obtaining a behavior similar to that of $\Lambda$CDM scenario, although the exponential model does not possess the latter as a particular limit. Hence, through the independent approach of dynamical systems, we do verify the results of observational confrontation, namely that $f(Q)$ gravity can be considered as a very promising alternative to the $\Lambda$CDM concordance model.

Density structure beneath the Rungwe volcanic province and surroundings, East Africa from shear-wave velocity perturbations constrained inversion of gravity data

Density perturbations in the subsurface are the main driver of mantle convection and can contribute to lithospheric deformation. However, in many places the density structure in the subsurface is poorly constrained. Most geodynamic models rely on simplified equations of state or use linear seismic velocity perturbations to density conversions. In this study, we investigate the density structure beneath the Rungwe Volcanic Province (RVP), which is the southernmost volcanic center in the Western Branch of the East African Rift (EAR). We use shear-wave velocity perturbations (dlnvs) as a reference model to perform constrained inversions of satellite gravity data centered on the RVP. We use the code jif3D with a dlnvs-density coupling criterion based on mutual information to generate a 3D density model beneath the RVP up to a depth of 660 km. Our results reveal a conspicuous negative density anomaly (∼−200 kg/m3) in the sublithospheric mantle (at depths ranging from ∼100 km to ∼250 km) beneath the central part of the Malawi Rift extending to the west, beneath the Niassa Craton, coincident with locations with positive shear-wave velocity perturbations (+7%). We calculate a 3D model of the velocity-to-density conversion factor (f) and find negative f-values beneath the Niassa Craton which suggests the observed negative density anomaly is mostly due to compositional variations. Apart from the Niassa Craton, there are generally positive f-values in the study area, which suggest dominance of temperature control on the density structure. Although the RVP generally shows negative density anomalies and positive f-values, at shallow depths (<120 km), f≈ 0, which suggests important contributions of both temperature and composition on the density structure possibly due to the presence of plume material. The negative buoyancy of the Niassa Craton contributes to its long stability, while constituting a barrier to the southward flow of plume material, thus restricting the southward continuation of magmatism in the Western Branch of the EAR. The presence of a negative-density anomaly where dlnvs are positive is incompatible with models based on the use of simple dlnvs to density conversion factors. These results have implications on how dlnvs models are converted to density perturbations.

Stability of ion-acoustic shock profiles to one-dimensional compressible Navier-Stokes-Poisson equations with large density oscillation

It is shown in [3] that the one-dimensional compressible Navier-Stokes-Poisson system admits a unique (up to a shift) small-amplitude smooth traveling wave and this traveling wave is time-asymptotically nonlinear stable under suitably small smooth zero-mass perturbations. This paper will show that similar results still hold even for certain large initial data. Our main strategy is to use the smallness of the strength of traveling wave solution to regulate the possible growth of the solutions of the perturbation equation, and then obtain the desired uniform-in-time upper and lower bounds on the specific volume.

Polarisability and magnetisation of electrically K-mouflaged objects: the Born-Infeld ModMax case study

We consider a family of non-linear theories of electromagnetism that interpolate between Born-Infeld at small distances and the recently introduced ModMax at large distances. These models are duality invariant and feature a K-mouflage screening in the Born-Infeld regime. We focus on computing the static perturbations around a point-like screened charge in terms of two decoupled scalar potentials describing the polar and the axial sectors respectively. Duality invariance imposes that the propagation speed of the odd perturbations goes to zero as fast as the effective screened charge of the object, potentially leading to strong coupling and an obstruction to the viability of the EFT below the screened radius. We then consider the linear response to external fields and compute the electric polarisability and the magnetic susceptibility. Imposing regularity of the perturbations at the position of the particle, we find that the polarisability for the odd multipoles vanishes whilst for the magnetisation Born-Infeld emerges as the only theory with vanishing susceptibility for even multipoles. The perturbation equations factorise in terms of ladder operators connecting different multipoles. There are two such ladder structures for the even sector: one that acts as an automorphism between the first four multipoles and another one that connects multipoles separated by four units. When requiring a similar ladder structure for the odd sector, Born-Infeld arises again as the unique theory. We use this ladder structure to relate the vanishing of the polarisability and the susceptibility to the values of conserved charges. Finally the perturbation equations correspond to a supersymmetric quantum mechanical system such that the polar sector can be described in terms of Schrödinger’s equations with four generalised hyperbolic Pösch-Teller potentials whose eigenfunctions are in correspondence with the multipoles.

A CLASS OF SINGULARLY PERTURBED EQUATIONS WITH DISCONTINUOUS RIGHT-HAND SIDE IN THE CRITICAL CASE

In this paper, we investigate a class of singularly perturbed equations with discontinuous right-hand side in the critical case. An asymptotic expansion of a contrast structure solution for the system is contrasted. Moreover, results for existence of the solution and estimations of remainders are presented. Finally, an example is provided to verify the theoretical results.

Analytic fluid approximation for warm dark matter

We present the cosmological evolution of the velocity of a massive particle, along with its equation of state. Both quantities are expressed in terms of anr, the moment when a massive particle becomes nonrelativistic. The expressions for the energy density and pressure for the background evolution are also in terms of anr; therefore, the perturbation equations for any massively decoupled particle, i.e., warm dark matter (WDM) or neutrinos, can be computed in the fluid approximation. A relation between the mass of the WDM particle, mwdm, with the moment of the nonrelativistic transition, anr, and the temperature ratio with the neutrinos, Twdm/Tν, are obtained and compared to results from Boltzmann solvers as CLASS with a non-cold relic as WDM. We found that using the analytic fluid approximation, the value of anr is 2.1% different on average in a wide range of WDM masses, and the temperature ratio is 7.1% lower than previous parametrizations. A smooth velocity dispersion for the WDM allows us to compute the cut-off scale in structure formation due to the free-streaming (λfs), which is a feature for a WDM particle to explain the satellite problem. The cut-off in the matter power spectrum and halo mass function using the analytic fluid approximation is similar to the Boltzmann solvers with a non-cold relic and the transfer function from numerical simulations. This approach provides a more detailed description and a deeper understanding of the WDM cosmological evolution by understanding the velocity dispersion of a WDM particle. Comprehensive numerical modeling can incorporate the analytical fluid formulation, potentially improving calculation performance, for instance, running MCMC for a ΛWDM scenario using CMB Planck and WiggleZ data, we obtained a lower bound for the WDM mass mwdm>70.3eV at 1σ confidence. Still, more data at small scales or a combination with other observations are needed to constrain the mass value of the WDM particle.

Effect of Finite Ion Larmor Radius (FLR) Corrections on Thermal Instability of Rotating Radiative Porous Astrophysical Plasma in Interstellar Medium (ISM)

Infinite homogeneous plasma’s thermal instability has been studied in relation to finite ion Larmor radius (FLR) corrections, rotation, and porosity, as well as the impacts of radiative heat-loss function and thermal conductivity. With the aid of a normal mode analysis framework and the necessary difficulty-appropriate linearized perturbation equations, a universal dispersion relation is investigated. For the propagation of transverse waves, this dispersion relationship further condenses for rotation axes parallel to and at right angles to the magnetic field. It is proven that the presence of rotation, porosity, thermal conductivity, and radiative heat-loss function altered the thermal instability criterion. To show how different parameters affect the rate at which the thermal instability grows, numerical calculations have been carried out. We discover that rotation, FLR corrections, and medium porosity stabilised the growth rate of the thermal system in the transverse mode of propagation. The conclusion of this research states that the rotation, porosity, and FLR corrections have an impact on the configuration of dense molecular clouds and star formation.

The integrated Sachs–Wolfe effect in interacting dark matter–dark energy models

Interacting dark matter-dark energy (IDMDE) models can be considered as one of the present challenges that may affect cosmic structures. The main idea of such models is that the mass of dark matter particles is specified by their interactions with a scalar field whose energy density is characterized by dark energy. In this work, we propose to study the integrated Sachs–Wolfe (ISW) effect in IDMDE models. Regarding this, we initially introduce a theoretical framework for IDMDE models. Moreover, we briefly discuss the stability conditions of IDMDE models, and by specifying a simple functional form for the energy density transfer rate, we calculate the perturbation equations. In the following, we specify the amplitude of the matter power spectrum for IDMDE models and compare it with the corresponding result obtained from the ΛCDM model. Furthermore, we calculate the amplitude of the ISW auto-power spectrum as a function of multipole order l for IDMDE models. The results indicate that the amplitude of the ISW auto-power spectrum in IDMDE models behaves like the one for the ΛCDM model for different values of the phantom dark energy equation of state. While the amplitude of the ISW-auto power spectrum for the IDMDE model should be higher than the one for the ΛCDM model in the case of quintessence dark energy equation of state. Also, it turns out that the corresponding results by different values of the coupling parameter demonstrate that ξ is inversely proportional to the amplitude of the ISW-auto power spectrum in IDMDE models. Finally, by employing four different surveys, we calculate the amplitude of the ISW-cross power spectrum as a function of multipole order l for IDMDE models. The results exhibit that the amplitude of the ISW-cross power spectrum in IDMDE models for all values of ωx is higher than that obtained from the ΛCDM model. Also, it turns out that the amplitude of the ISW-cross power spectrum in IDMDE models changes inversely with the value of coupling parameter ξ.

Application of wavy geometries for the reduction of trailing edge instability noise

Trailing edge instability noise can occur in a wide range of applications, including wind turbines and small aerial vehicles. It can be characterised by a high amplitude tonal or narrowband sound resulting from instability waves in the boundary layer scattering as they are convected over the trailing edge. In this study, a hybrid aero-acoustic model is used to demonstrate how wavy geometries can be used to successfully reduce trailing edge instability noise. The hybrid model uses incompressible large eddy simulations to compute acoustic source terms, which are then mapped onto a larger domain using radial basis functions. The acoustic perturbation equations are then solved on this larger domain to obtain the 3-dimensional acoustic field. The study uses a modified NACA0012 wing, where the surface contains spanwise waves of different wavelengths. The results show that there is an optimal wavelength that produces the greatest noise reduction for a given flow speed and Reynolds number, and that this relates to wavelengths of the dominant structures within the boundary layer. This shows that by understanding the spectral character of the boundary layer, one can design optimal geometries for reducing this important source of fluid-induced noise.

QNMs of slowly rotating Einstein–Bumblebee black hole

We have studied the quasinormal modes (QNMs) of a slowly rotating black hole with Lorentz-violating parameter in Einstein–Bumblebee gravity. We analyse the slow rotation approximation of the rotating black hole in the Einstein–Bumblebee gravity, and obtain the master equations for scalar perturbation, vector perturbation and axial gravitational perturbation, respectively. Using the matrix method and the continuous fraction method, we numerically calculate the QNM frequencies. In particular, for scalar field, it shows that the QNMs up to the second order of rotation parameter have higher accuracy. The numerical results show that, for both scalar and vector fields, the Lorentz-violating parameter has a significant effect on the imaginary part of the QNM frequencies, while having a relatively smaller impact on the real part of the QNM frequencies. But for axial gravitational perturbation, the effect of increasing the Lorentz-violating parameter ell is similar to that of increasing the rotation parameter {tilde{a}}.

Stability analysis of water-alumina nanofluid film at the spherical interface

The interface of a viscous fluid and [Formula: see text]-water nanofluid is analyzed through linear instability analysis in a spherical configuration. The viscous fluid lies inside the sphere while the outside region contains nanofluid. In this model, the viscosity of the nanofluid is considered a function of the base fluid viscosity, nanoparticles volume fraction, fractal aggregates, and nanoparticles shape. The perturbed flow is taken as irrotational, and the linear perturbation equations are solved through viscous irrotational theory. The perturbations growth rate can be computed through a 2-degree polynomial and the coefficients of this polynomial are the functions of the Atwood number, Richardson number, Weber number, Reynolds number, etc. The nanofluid interface is found more stable than the viscous fluid interface. The density of nanofluids raises the amplitude of disturbances while the nanofluid's viscosity has a reverse effect.

Impact of interfacial rheology on finger tip splitting.

Fluid-fluid interfaces, laden with polymers, surfactants, lipid bilayers, proteins, solid particles, or other surface-active agents, often exhibit a rheologically complex response to deformations. Despite its academic and practical relevance to fluid dynamics and various other fields of research, the role of interfacial rheology in viscous fingering remains fairly underexplored. A noteworthy exception is the work by Li and Manikantan [Phys. Rev. Fluids 6, 074001 (2021)2469-990X10.1103/PhysRevFluids.6.074001], who used linear stability analysis to show that surface rheological stresses act to stabilize the development of radial viscous fingering at the linear regime. In this paper, we perform a perturbative, second-order mode-coupling analysis of the system and investigate the influence of interfacial rheology on the morphology of the fingering structures at early nonlinear stages of the dynamics. In particular, we focus on understanding how interfacial rheology impacts the emblematic finger tip-widening and finger tip-splitting phenomena that take place in radial viscous fingering in Hele-Shaw cells. We describe the viscous Newtonian fluid-fluid interface by using a Boussinesq-Scriven model, and derive a generalized Young-Laplace pressure jump condition at the fluid-fluid interface. In this framing, we go beyond the purely linear description and use Darcy's law to obtain a perturbative mode-coupling differential equationwhich describes the time evolution of the perturbation amplitudes, accurate to second order. Our early nonlinear mode-coupling results indicate that regardless of their stabilizing action at the linear regime, interfacial rheology effects favor finger tip widening, leading to the occurrence of enhanced finger tip-splitting events.

Equilibrium points of heterogeneous small body in finite element method

ABSTRACT This paper presents a finite element method to search for equilibrium points around a heterogeneous small body, which degenerates from the full two-body problem in finite element form. The gravitational potential, acceleration, gravitational gradient matrix, and the linearized perturbation equation, which are key formulas for solving the equilibrium points and discriminating their stabilities, are interpolated by the tetrahedral model nodes of the small body. The finite element method could capture the complex internal structures of small bodies and provide a uniform and simple formula for various configurations. The method is applied to the Kuiper Belt Object (486958) Arrokoth, which is a contact binary asteroid. Three types of heterogeneous structures are hypothesized to investigate the evolutions of equilibrium points, namely density disparity of two sub-lobes, hardcore structure, and cavity structure. Under the condition of constant Arrokoth mass, the density disparity of the two sub-lobes greatly influences the local gravitational field. It thus has a significant influence on the positions of equilibrium points. The hardcore and cavity structures have less impact on the equilibrium points. These three simulation groups verified that the equilibrium points of a heterogeneous small body are quite different from those of a homogeneous small body. Such investigation of the equilibrium points may give an in-depth understanding of the dynamical environment around the heterogeneous small body, which is significant for future deep-space missions.

Gravitational-Scalar Instability of Cosmological Model Based on Two-Component System of Degenerate Scalarly Charged Fermions with Asymmetric Higgs Interaction. I. Equations for Perturbations

A mathematical model is formulated for the evolution of plane perturbations in a cosmological two-component statistical system of completely degenerate scalarly charged fermions with an asymmetric scalar Higgs interaction. A complete closed system of differential equations describing the unperturbed state of a homogeneous and isotropic system and a system of self-consistent evolution equations of small perturbations are constructed.

Transverse thermal instability of partially ionized radiative plasma with impact of neutral collisions and finite electron inertia effect in ISM

The impact of neutral friction, finite electron inertia and radiative heat-loss function on the thermal instability of viscous, partially-ionized plasma has been explored integrating the impacts of finite electrical resistivity and thermal conductivity. Normal mode analysis scheme is used to derive universal dispersion relation by means of the related linearized perturbation equations of the difficulty and a tailored thermal situation of instability is explored. It is finding that the thermal condition of instability is modified owing the occurrence of radiative heat-loss function, thermal conductivity, finite electron inertia and neutral particle. For transverse mode of propagation, we discover that the condition of thermal instability depends on the finite electron inertia, magnetic field strength, radiative heat-loss function, thermal conductivity, neutral particle and finite electrical resistivity, but independent viscosity of two-components. From the graphs, we explored that the temperature dependent heat-loss function, thermal conductivity, neutral collision frequency and viscosity of two-components demonstrates stabilizing effect, while finite electron inertia and finite electrical resistivity demonstrates destabilizing effect. Our results help’s to understand the process of molecular cloud formation in ISM.

Numerical investigation of wall effects on combustion noise from a lean-premixed hydrogen/air low-swirl flame

A hybrid computational fluid dynamics (CFD)/computational aero-acoustics (CAA) approach, in which large-eddy simulation (LES) and APE-RF (solution of the acoustic perturbation equations for reacting flows) are employed for the CFD and CAA, respectively, calling it the hybrid LES/APE-RF approach, is used to analyze the influence of a wall on the combustion noise from a lean-premixed gaseous hydrogen/air low-swirl turbulent jet flame. The wall boundary conditions pertaining to the APE-RF system are formulated to account for acoustic reflection from the wall. The results show that the sound pressure level (SPL) spectrum obtained from the LES/APE-RF is in good agreement with that measured in the experiment. In the LES/APE-RF, the SPL spectrum of combustion noise with the wall plate explicitly changes compared to that without the wall plate. Specifically, the presence of the wall plate tends to ease the peaks that appeared in the case without the wall plate and create a nearly constant SPL within a specific frequency band. The analysis of the heat release rate fluctuation reveals that these phenomena are caused by the absence of a single periodic oscillation of heat release rate. The presence of the wall plate creates an asymmetric flow around the flame and distorts the flame structure, thereby altering the flame fluctuation phenomena.

A direct-hybrid CFD/CAA method based on lattice Boltzmann and acoustic perturbation equations

The accuracy of two direct coupled two-step CFD/CAA methods is discussed. For the flow field either a finite-volume (FV) method for the solution of the Navier–Stokes equations or a lattice Boltzmann (LB) method is coupled to a discontinuous Galerkin (DG) method for the solution of the acoustic perturbation equations. The coupling takes advantage of a joint Cartesian mesh allowing for the exchange of the acoustic sources without MPI communication. An immersed boundary treatment of the acoustic scattering from solid bodies by a novel solid wall formulation is implemented and validated in the DG method. Results for the case of a spinning vortex pair and the low Reynolds number unsteady flow around a circular cylinder show that a solution with comparable accuracy is obtained for the two direct-hybrid methods when using identical mesh resolution.

Bounce corrections to gravitational lensing, quasinormal spectral stability, and gray-body factors of Reissner-Nordström black holes

Gravitational lensing in the weak field limit, quasinormal spectra, and gray-body factors are investigated in the Reissner-Nordstr\"om spacetime corrected by bounce parameters. Using the Gauss-Bonnet theorem, we analyze the effects of bounce corrections to the weak gravitational deflection angle and find that the divergence of the deflection angle can be suppressed by a bounce correction in the Reissner-Nordstr\"om spacetime. We also notice that the bounce correction plays the same role as the Morse potential in the deflection angle. Moreover, we derive the perturbation equations with spin-dependent effective potentials and discuss the quasinormal spectral stability. We observe that the quasinormal spectra have the same behavior in large bounce parameters for different multiple numbers and they decrease significantly if the bounce parameter is much bigger than the charge. We further study the transmission probability of particles scattered by the effective potentials and reveal that the bounce correction introduced into the Reissner-Nordstr\"om spacetime increases the gray-body factors of perturbation fields.

Simulating gravitational waves passing through the spacetime of a black hole

We investigate how GWs pass through the spacetime of a Schwarzschild black hole using time-domain numerical simulations. Our work is based on the perturbed $3+1$ Einstein's equations up to the linear order. We show explicitly that our perturbation equations are covariant under infinitesimal coordinate transformations. Then we solve a symmetric second-order hyperbolic wave equation with a spatially varying wave speed. As the wave speed in our wave equation vanishes at the horizon, our formalism can naturally avoid boundary conditions at the horizon. Our formalism also does not contain coordinate singularities and, therefore, does not need regularity conditions. Then, based on our code, we simulate both finite and continuous initially plane-fronted wave trains passing through the Schwarzschild black hole. We find that for the finite wave train, the wave zone of GWs is wildly twisted by the black hole. While for the continuous wave train, unlike geometric optics, GWs cannot be sheltered by the black hole. A strong beam and an interference pattern appear behind the black hole along the optical axis. Moreover, we find that the back-scattering due to the interaction between GWs and the background curvature is strongly dependent on the direction of the propagation of the trailing wavefront relative to the black hole. Finally, for a realistic input waveform generated by binary black holes, we find that the lensed waveform in the merger and ringdown phases is much longer than that of the input waveform due to the effect of back-scattering.

The Effective Fluid Approach for Modified Gravity and Its Applications

In this review, we briefly summarize the so-called effective fluid approach, which is a compact framework that can be used to describe a plethora of different modified gravity models as general relativity (GR) and a dark energy (DE) fluid. This approach, which is complementary to the cosmological effective field theory, has several benefits, as it allows for the easier inclusion of most modified gravity models into the state-of-the-art Boltzmann codes that are typically hard-coded for GR and DE. Furthermore, it can also provide theoretical insights into their behavior since in linear perturbation theory it is easy to derive physically motivated quantities such as the DE anisotropic stress or the DE sound speed. We also present some explicit applications of the effective fluid approach with f(R), Horndeski and scalar–vector–tensor models, namely, how this approach can be used to easily solve the perturbation equations and incorporate the aforementioned modified gravity models into Boltzmann codes so as to obtain cosmological constraints using Monte Carlo analyses.