Perturbation Equations Research Articles

Gravitational odd-parity perturbation of a Horndeski hairy black hole: quasinormal mode and parameter constraint

During the binary black hole coalescence, gravitational waves emitted at the ringdown stage can be well described by black hole perturbation theory, where the quasinormal modes (QNMs) become the important ingredient in modeling the pattern waveform. In general relativity (GR), the QNMs can be obtained from solving the Regge–Wheeler (RW) equation of a non-rotating black hole. While in Horndeski gravity, the isospectrality between the odd and even parity perturbations is broken due to the scalar field, the odd perturbation equation can be simplified into a modified RW equation from the perturbed action. In this paper, we propose a new auxiliary field and tortoise coordinate to refine the modified RW equation in Horndeski gravity, and calculate the QNM frequencies of the odd perturbation of a specific hairy black hole. We find that this proposal not only cures the superluminal propagation addressed in the previous literature, but also holds the original QNM spectrum of the odd perturbation. Moreover, our results indicate that such a Horndeski hairy black hole is stable under the odd perturbation, which is also verified by the time evolution of the perturbation. In particular, in contrast to GR, the modes with ℓ=2 can decay faster than modes with ℓ>22$$\\end{document}]]> for a certain range of the Horndeski hair, and the link between the null geodesics and QNM for the odd perturbation in the current theory is violated. We then use the ringdown QNMs to preliminarily investigate the signal-to-noise ratio (SNR) rescaled measurement error of the Horndeski hair. We obtain significant effects of the angular momentum and overtone on the error bound of the hair parameter. We hope that our findings will inspire further theoretical and phenomenological work on the testing of the no-hair theorem of black holes using gravitational wave physics.

Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime Part II: Even-Mode Perturbations

This is the Part II paper of our series of papers on a gauge-invariant perturbation theory on the Schwarzschild background spacetime. After reviewing our general framework of the gauge-invariant perturbation theory and the proposal of gauge-invariant treatments for l=0,1-mode perturbations on the Schwarzschild background spacetime in the Part I paper, we examine the linearized Einstein equations for even-mode perturbations. We discuss the strategy to solve the linearized Einstein equations for these even-mode perturbations including l=0,1 modes. Furthermore, we explicitly derive the l=0,1-mode solutions to the linearized Einstein equations in both the vacuum and the non-vacuum cases. We show that the solutions for l=0-mode perturbations includes the additional Schwarzschild mass parameter perturbation, which is physically reasonable. Then, we conclude that our proposal of the resolution of the l=0,1-mode problem is physically reasonable due to the realization of the additional Schwarzschild mass parameter perturbation and the Kerr parameter perturbation in the Part I paper.

Gauge-Invariant Perturbation Theory on the Schwarzschild Background Spacetime Part I: Formulation and Odd-Mode Perturbations

This article is Part I of our series of full papers on a gauge-invariant “linear” perturbation theory on the Schwarzschild background spacetime which was briefly reported in our short papers by the present author in 2021. We first review our general framework of the gauge-invariant perturbation theory, which can be easily extended to the “higher-order” perturbation theory. When we apply this general framework to perturbations on the Schwarzschild background spacetime, gauge-invariant treatments of l=0,1 mode perturbations are required. On the other hand, in the current consensus on the perturbations of the Schwarzschild spacetime, gauge-invariant treatments for l=0,1 modes are difficult if we keep the reconstruction of the original metric perturbations in our mind. Due to this situation, we propose a strategy of a gauge-invariant treatment of l=0,1 mode perturbations through the decomposition of the metric perturbations by singular harmonic functions at once and the regularization of these singularities through the imposition of the boundary conditions to the Einstein equations. Following this proposal, we derive the linearized Einstein equations for any modes of l≥0 in a gauge-invariant manner. We discuss the solutions to the odd-mode perturbation equations in the linearized Einstein equations and show that these perturbations include the Kerr parameter perturbation in these odd-mode perturbations, which is physically reasonable. In the Part II and Part III papers of this series of papers, we will show that the even-mode solutions to the linearized Einstein equations obtained through our proposal are also physically reasonable. Then, we conclude that our proposal of a gauge-invariant treatment for l=0,1-mode perturbations is also physically reasonable.

Gravitational wave luminosity distance for Starobinsky gravity in viscous cosmological models

In this paper, in the Starobinsky gravity (SG) model, the gravitational wave (GW) luminosity distance dLGW(z) is analytically and numerically investigated by considering the dark matter as the two different kinds of shear viscous fluids in universe. Concretely, on the basis of the propagation equation of the metric perturbation derived by using transverse-traceless gauge for the linear perturbation of the metric in FRW background, we obtain the analytical expressions of the modified friction term δ(z) and the ratio dLGW(z)/dLEM(z) of GW to electromagnetic (EM) wave luminosity distance, and find that they only depend on the SG model parameter α and the shear viscous coefficient η, regardless of the bulk viscous coefficient ζ. Evidently, both of them are affected by α and η. It worth stressing that the results given by us can reduce to ones in documents [13, 22, 30]. It follows that we can extend the previous results as the special cases in our model. Furthermore, by combining the latest observational data, we more tightly constrain the values of related parameters and numerically analyze the influence of both α and η on δ(z) and dLGW(z)/dLEM(z). We observe that, in the range of 0<z≤8.2, the evolutionary curves of δ(z) and dLGW(z)/dLEM(z) exhibit that δ(z) (dLGW(z)/dLEM(z)) monotonically decrease (increase) as redshift rises, moreover there are always δ(z)<0 and dLGW(z)≥dLEM(z), specially dLGW(0)=dLEM(0) in the limit of z→0, which means that detecting GW signals might represent a more efficient tool than detecting EM signals to test modified gravity model on the given scale of cosmic distances. It is worth noting that the numerical analyses to δ(z) and dLGW(z)/dLEM(z) illustrate once again that the GW luminosity distance dLGW(z) is indeed influenced by α and η, in particular more sensitive to α than η in our model. In addition, by comparing the GW and EM luminosity distances of SG model with dL(GR) of Einstein GR, we find that the GW and EM signals in SG1 and SG2 change from stronger to weaker than the ones in GR as redshift increases and more difficultly to be detected in the same distance and detection sensitivity. Meanwhile, the EM signals change from strong to weak earlier than GW signals do. Finally, we confirm that δ(z) and dLGW(z) of our model meet the constraints imposed by the standard siren GW170817 and its electromagnetic companion GRB170817A.

Reduction of singularities in Fuchsian equations, and Heun equations for perturbations of Kerr–Newman–de Sitter and anti-de sitter spacetimes

For Fuchsian differential equations with N singularities at finite points we display two simply expressed conditions on coefficients which guarantee equivalence, under a certain inversion transformation, to a differential equation with N−1 singularities at finite points. For the radial and angular differential equations that govern perturbations of Kerr–Newman–de Sitter and anti-de Sitter spacetimes we find that these two conditions are satisfied identically for arbitrary values of all the parameters in the differential equations, including pole locations, with regular singularities and non-zero cosmological constant. From this we generalize the results of Suzuki, Tarasugi, and Umetsu, Prog. Theor. Phys. 100, 491 (1998) and find 128 Heun equations which are equivalent to the equations which govern the perturbations of the KN(A)dS spacetimes.

Stability Analysis of Gravity‐Modulated Thermohaline Convection in Viscoelastic Fluid Saturating Porous Medium

ABSTRACTThis paper investigates the modified stability analysis of thermohaline convection in a horizontal Oldroyd‐B fluid using the Darcy model, with applications in geophysics, environmental engineering, material science, and biological systems. The basic equations of motion and heat conduction in the present model are based on the Darcy model and the modified Boussinesq approximation. The perturbation equations are derived using the power series perturbation method to study two‐dimensional convective roll instability with finite amplitude disturbances. Linear stability analysis, as a first‐order stability problem, provides expressions for the Rayleigh numbers for stationary and oscillatory convection. The influence of various parameters, including the Darcy–Prandtl number, solutal Rayleigh number, stress and strain retardation parameters, and variations in the coefficient of specific heat, on the stability of the considered system, is studied numerically. It is observed that an approximate 67% increase in the coefficient of specific heat variations and a 23% increase in the solutal Rayleigh number result on average increases of 80% and 8%, respectively, in the value of the oscillatory Darcy–Rayleigh number. Conversely, an average 42% increase in the Darcy–Prandtl number leads to an average decrease of 8% in the oscillatory Darcy–Rayleigh number. In the weakly nonlinear oscillatory analysis, second‐ and third‐order stability problems are discussed, and the Landau equation is derived, which describes the amplitudes of the convection cells. The effects of the parameters on heat and mass transfer rates, as described by the Nusselt and Sherwood numbers, are studied numerically. Furthermore, the weakly nonlinear stability analysis evaluates the effects of the Darcy–Prandtl number, specific heat coefficient, stress relaxation time, salinity Rayleigh number, strain retardation time, diffusivity ratio, and gravity modulation characteristics on heat and mass transfer rates. The effects of amplitude and frequency of gravity modulation on heat and mass transport are analyzed and depicted graphically. The study establishes that heat and mass transport can be effectively controlled by a mechanism external to the system. The results are validated by comparison with previous work, which considered a special case without the velocity gradient term in the momentum equation.

Quasinormal modes and shadow of Schwarzschild black holes embedded in a Dehnen-type dark matter halo exhibiting a cloud of strings

In this paper we consider a static spherically symmetric black hole (BH) embedded in a Dehnen-(1, 4, 0)-type dark matter (DM) halo in the presence of a cloud string. We examine and present data on how the core density of the DM halo parameter and the cloud string parameter affect BH attributes such as quasinormal modes (QNMs) and shadow cast. To do this, we first look into the effective potential of perturbation equations for three types of perturbation fields with different spins: massless scalar field, electromagnetic field and gravitational field. Then, using the sixth-order Wentzel–Kramers–Brillouin approximation, we examine QNMs of the BH disturbed by the three fields and derive quasinormal frequencies. The changes in QNM versus the core density parameter and the cloud string parameter for three disturbances are explored. We also investigate how the core density and the cloud string parameter affect the photon sphere and shadow radius. Interestingly, the study shows that the influence of Dehnen-type DM and cloud strings increases both the photon sphere and the shadow radius. Finally, we employ observational data from Sgr A⋆ and M87⋆ to set limitations on the BH parameters.

Quasinormal modes of three (2+1)-dimensional black holes in string theory, conformal gravity, and Hu–Sawicki F(R) theory via the Heun function

We study the propagation of massless fermionic fields, implementing a family of special functions: Heun functions, in solving the wave equation in three three-dimensional backgrounds, including the BTZ black hole in string theory and Lifshitz black hole solutions in conformal gravity and Hu–Sawicki F(R) theory. The main properties of the selected black hole solutions is that their line elements are Weyl related to that of a homogeneous spacetime, whose spatial part possesses Lie symmetry, described by Lobachevsky-type geometry with arbitrary negative Gaussian curvature. Using the Weyl symmetry of massless Dirac action, we consider the perturbation equations of fermionic fields in relation to those of the homogeneous background, which having definite singularities, are transformed into Heun’s equation. We point out the existence of quasinormal modes labeled by the accessory parameter of the Heun function. The distribution of the quasinormal modes has been clarified to satisfy the boundary conditions that require ingoing and decaying waves at the event horizon and conformal infinity, respectively. It turned out that the procedure based on the Heun function, beside reproducing the previously known results obtained via hypergeometric function for the BTZ and Lifshitz black hole solution in conformal gravity, brings up new families of quasinormal frequencies, which can also contain purely imaginary modes. Also, the analysis of the quasinormal modes shows that with the negative imaginary part of complex frequencies ω=ωRe+iωIm, the fermionic perturbations are stable in this background.

Solitons in composite linear-nonlinear moiré lattices.

We produce families of two-dimensional gap solitons (GSs) maintained by moiré lattices (MLs) composed of linear and nonlinear sublattices, with the defocusing sign of the nonlinearity. Depending on the angle between the sublattices, the ML may be quasiperiodic or periodic, composed of mutually incommensurate or commensurate sublattices, respectively (in the latter case, the inter-lattice angle corresponds to Pythagorean triples). The GSs include fundamental, quadrupole, and octupole solitons, as well as quadrupoles and octupoles carrying unitary vorticity. Stability segments of the GS families are identified by means of the linearized equation for small perturbations, and confirmed by direct simulations of perturbed evolution.

Scalar induced gravitational waves in f(R) gravity

We investigate the first and second order cosmological perturbation equations in f(R) modified gravity theory and provide the equation of motion of second order scalar induced gravitational waves. We find that the effects of modified gravity not only change the form of the equation of motion of second order scalar induced gravitational waves but also contribute an additional anisotropic stress tensor, composed of first order scalar perturbations, to the source term of the gravitational waves. We calculate the energy density spectrum of second order scalar induced gravitational waves in the HS model. Utilizing current pulsar timing array observational data, we perform a rigorous Bayesian analysis of the parameter space of the HS model.

Mapping a dissipative quantum spin chain onto a generalized Coulomb gas

An XXZ spin chain at zero magnetization is subject to spatially correlated baths acting as dissipation. We show that the low-energy excitations of this model are described by a dissipative sine-Gordon field theory, i.e. a sine-Gordon action with an additional long-range interaction emerging from dissipation. The field theory is then exactly mapped onto a generalized Coulomb gas which, in addition to the usual integer charges, displays half-integer charges that originate from the dissipative baths. These new charges come in pairs linked by a charge independent logarithmic interaction. In the Coulomb gas picture, we identify a Berezinsky–Kosterlitz–Thouless-like phase transition corresponding to the binding of charges and derive the associated perturbative renormalization group equations. For superohmic baths, the transition is due to the binding of the integer charges, while for subohmic baths, it is due to the binding of the half-integer charges, thereby signaling a dissipation-induced transition.

Investigating tidal heating in neutron stars via gravitational Raman scattering

We present a scattering amplitude formalism to study the tidal heating effects of nonspinning neutron stars incorporating both worldline effective field theory and relativistic stellar perturbation theory. In neutron stars, tidal heating arises from fluid viscosity due to various scattering processes in the interior. It also serves as a channel for the exchange of energy and angular momentum between the neutron star and its environment. In the interior of the neutron star, we first derive two master perturbation equations that capture fluid perturbations accurate to linear order in frequency. Remarkably, these equations receive no contribution from bulk viscosity due to a peculiar adiabatic incompressibility which arises in stellar fluid for nonbarotropic perturbations. In the exterior, the metric perturbations reduce to the Regge-Wheeler equation which we solve using the analytical Mano-Suzuki-Takasugi method. We compute the amplitude for gravitational waves scattering off a neutron star, also known as gravitational Raman scattering. From the amplitude, we obtain expressions for the electric quadrupolar static Love number and the leading dissipation number to all orders in compactness. We then compute the leading dissipation number for various realistic equations of state and estimate the change in the number of gravitational-wave cycles due to tidal heating during inspiral in the LIGO-Virgo-KAGRA band. Published by the American Physical Society 2024

Spectrum of primordial gravitational waves in the presence of a cosmic string

In this paper we consider an inflating universe with long straight cosmic string along z-axis. We demonstrate that cosmic string's effect can be treated as a perturbation on the background of FRW metric. By performing cosmological perturbations on this inflating cosmic string background, we derived the linearized Einstein field equations. We show that at leading order (neglecting the mixing terms of cosmic string perturbations with gravitational tensor perturbations), the cosmic string appears as an inhomogeneous term on the right hand side of wave equation for tensor perturbations. Finally, by finding analytical solution of the wave equation for slow-roll inflation, we illustrate its impact on the spectrum of primordial gravitational waves.

Novel approach to solving Schwarzschild black hole perturbation equations via physics informed neural networks

Novel approach to solving Schwarzschild black hole perturbation equations via physics informed neural networks

Hamiltonian mapping and quantum perturbation equations in the point matter black hole and noncommutative black hole models

Hamiltonian mapping and quantum perturbation equations in the point matter black hole and noncommutative black hole models

Gravitational waves driven by holographic dark energy

In this paper, we have studied the effects of holographic dark energy on the evolution of gravitational waves. The background evolution of gravitational waves in a flat FRW universe is considered and studied in the presence of various holographic dark energy models. The perturbation equations governing the evolution of the gravitational waves have been constructed and solutions are obtained. These solutions are studied in detail to get a proper understanding of the characteristics of the gravitational waves in the presence of holographic dark energy. The work can be a significant tool in studying different dark energy models comparatively using the features of the gravitational wave evolution.

Rosen-Morse potential and gravitating kinks

We show that in a special type of two-dimensional dilaton-gravity-scalar model, where both the dilaton and the scalar matter fields have noncanonical kinetic terms, it is possible to construct kink solutions whose linear perturbation equation is a Schrödinger-like equation with Rosen-Morse potential. For this potential, eigenvalues and wave functions of the bound states, if had any, can be derived by using the standard shape invariance procedure. Depending on the values of the parameters, the stability potential can be reflective or reflectionless. There can be an arbitrary number of shape modes, but the zero mode is always absent.

Gravitational spinoptics in a curved space-time

In this paper we discuss propagation of the weak high-frequency gravitational waves in a curved spacetime background. We develop a so-called spinoptics approximation which takes into account interaction of the spin of the field with the curvature of the background metric. This is achieved by modifying the standard geometric optics approximation by including the helicity sensitive terms of the order 1/ω in the eikonal equation. The novelty of the approach developed in this paper is that instead of study of the high-frequency expansion of the equations for the gravitational field perturbations we construct the effective action for the gravitational spinoptics. The gravitational spinoptics equations derived by variation of the effective action correctly reproduce the earlier obtained results. However, the proposed effective action approach is technically more simple and transparent. It allows one to reduce the study of the high-frequency gravitational waves to study classical dynamics of massless particles with internal discrete degree of freedom (helicity). The formalism is covariant and it can be applied for arbitrary vacuum space-time background.

Comparing sharp and smooth transitions of the second slow-roll parameter in single-field inflation

In single-field inflation, violation of the slow-roll approximation can lead to growth of curvature perturbation outside the horizon. This violation is characterized by a period with a large negative value of the second slow-roll parameter. At an early time, inflation must satisfy the slow-roll approximation, so the large-scale curvature perturbation can explain the cosmic microwave background fluctuations. At intermediate time, it is viable to have a theory that violates the slow-roll approximation, which implies amplification of the curvature perturbation on small scales. Specifically, we consider ultraslow-roll inflation as the intermediate period. At late time, inflation should go back to the slow roll period so that it can end. This means that there are two transitions of the second slow-roll parameter. In this paper, we compare two different possibilities for the second transition: sharp and smooth transitions. Focusing on effects generated by the relevant cubic self-interaction of the curvature perturbation, we find that the bispectrum and one-loop correction to the power spectrum due to the change of the second slow-roll parameter vanish if and only if theMukhanov-Sasaki equation for perturbation satisfies a specific condition called Wands duality.We also find in the case of sharp transition that, even though this duality is satisfied in the ultraslow-roll and slow-roll phases, it is severely violated at the transition so that the resultant one-loop correction is extremely large inversely proportional to the duration of the transition.

A linearized hydrodynamic code for laser ablation and its application to laser pulse shaping for direct-drive fusion

One of the most harmful processes in inertial confinement fusion is Rayleigh–Taylor instability (RTI), and an efficient way to mitigate it is pulse shaping. However, because shaped laser pulses lead to unsteady ablation, it is insufficient to evaluate RTI based solely on the instability growth rate. Here, for better prediction of RTI during linear growth, hydrodynamic equations for laser ablation (including both balance and linearized perturbation equations) are solved numerically and used to optimize the laser pulse shape for direct-drive inertial confinement fusion. For given target conditions and laser energy, simulations show that a picket pulse before the main laser pulse can reduce RTI significantly, and it is clear that the reduction comes from two aspects: (i) the lower RTI seed due to rarefaction at the descending edge of the picket in the imprint stage and (ii) the smaller growth rate due to enhanced ablation velocity at the main pulse in the acceleration stage. It is found that the perturbed laser deposition in an underdense plasma also has a profound influence on RTI seeds in the imprint stage.