Gravitational Wave Asteroseismology Research Articles

Gravitational wave asteroseismology of accreting neutron stars in a steady state

Gravitational wave asteroseismology of accreting neutron stars in a steady state

Gravitational wave asteroseismology of charged strange stars in the Cowling approximation: the fluid pulsation modes

In this work we study, within the framework of Cowling approximation, the effect of the electric charge on the gravitational wave frequency of fluid oscillation modes of strange quark stars. For this purpose, the dense matter of the stellar fluid is described by the MIT bag model equation of state (EoS), while for the electric charge profile, we consider that the electric charge density is proportional to the energy density. We find that the gravitational wave frequencies change with the increment of electric charge; these effects are more noticeable at higher total mass values. We obtain that the f-mode is very sensitive to the change in the electric charge of the star. Furthermore, in the case of the p1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_1$$\\end{document} mode, the effect of the electric charge is not very significant. Our results reveal that the study of the fundamental pulsation mode of an electrically charged compact star is very important in distinguishing whether compact stars could contain electric charge. We also employ another electric charge distribution profile that follows a power law, and it is found that the f-mode change is more noticeable than the p1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p_1$$\\end{document}-mode when the electric charge is incremented.

Probing hadron–quark phase transition in twin stars using f-modes

ABSTRACT Although it is conjectured that a phase transitions from hadronic to deconfined quark matter in the ultrahigh-density environment of neutron stars (NS), the nature of phase transition remains an unresolved mystery. Furthermore, recent efforts reveal that the finite surface tension effects can lead to a mixed phase with different geometric shapes (so-called ‘pasta’ phases), leading to a smooth phase transition from hadronic to quark matter in the NS interior. Depending on whether there is a strong or a pasta-induced smooth first-order phase transition, one may expect a third family of stable, compact stars or ‘twin stars’ to appear, with the same mass but different radii compared to NSs. The possibility of identifying twin stars using astrophysical observations has been a subject of interest. This study investigates the potential of probing the nature of the hadron–quark phase transition through future gravitational wave (GW) detections from fundamental (f-) mode oscillations in NSs. Using a newly developed model that parametrizes the hadron–quark phase transition with ‘pasta phases’, we calculate f-mode characteristics within a full general relativistic framework. We then use universal relations in GW asteroseismology to derive stellar properties from the detected mode parameters. Our findings suggest that detecting GWs from f modes with third-generation GW detectors offers a promising scenario for the existence of twin stars. However, we also estimate various uncertainties in determining the mode parameters and conclude that these uncertainties make it more challenging to identify the nature of the hadron–quark phase transition.

Long-term gravitational wave asteroseismology of supernovae: From core collapse to 20 seconds postbounce

We use an asteroseismology method to calculate the frequencies of gravitational waves (GWs) in a long-term core-collapse supernova simulation, with a mass of 9.6 solar mass. The simulation, which includes neutrino radiation transport in general relativity is performed from core-collapse, bounce, explosion and cooling of protoneutron stars (PNSs) up to 20 s after the bounce self-consistently. Based on the hydrodynamics background, we calculate eigenmodes of the PNS oscillation through a perturbation analysis on fluid and metric. We classify the modes by the number of nodes and find that there are several eigenmodes. In the early phase before 1 s, there are low-frequency $g$-modes around 0.5 kHz, midfrequency $f$-modes around 1 kHz, and high-frequency $p$-modes above them. Beyond 1 second, the $g$-modes drop too low in frequency and the $p$-modes become too high to be detected by ground-based interferometers. However, the $f$-mode persists at 1 kHz. We present a novel fitting formula for the ramp-up mode, comprising a mixture of $g$-mode and $f$-mode, using postbounce time as a fitting parameter. Our approach yields improved results for the long-term simulation compared to prior quadratic formulas. We also fit frequencies using combinations of gravitational mass, $M$, and radius, $R$, of the PNS. We test three types of fitting variables: compactness $M/R$, surface gravity $M/{R}^{2}$, and average density $\sqrt{M/{R}^{3}}$. We present results of the time evolution of each mode and the fitting for three different ranges, from 0.2 s to 1 s, 4 s, and 20 s for each formula. We then compare the deviation of the formulas from the eigenmodes to determine which fitting formula is the best. In conclusion, any combination of $M$ and $R$ fits the eigenmodes well to a similar degree. Comparing three variables in detail, the fitting with compactness is slightly the best among them. We also find that the fitting using less than 1 s of simulation data cannot be extrapolated to the long-term frequency prediction.

General relativistic treatment of f -mode oscillations of hyperonic stars

We present a systematic study of $f$-mode oscillations in neutron stars containing hyperons, extending recent results obtained within the Cowling approximation to linearized general relativity. Employing a relativistic mean-field model, we find that the Cowling approximation can overestimate the quadrupolar $f$-mode frequency of neutron stars by up to 30% compared to the frequency obtained in the linearized general relativistic formalism. Imposing current astrophysical constraints, we derive updated empirical relations for gravitational wave asteroseismology. The frequency and damping time of quadrupole $f$-mode oscillations of hyperonic stars are found to be in the range of 1.47--2.45 kHz and 0.13--0.51 s, respectively. Our correlation studies demonstrate that among the various parameters of the nucleonic and hyperonic sectors of the model, the nucleon effective mass shows the strongest correlation with mode characteristics and neutron star observables. Estimates for the detectability of $f$ modes in a transient burst of gravitational waves from isolated hyperonic stars is also provided.

Fast Rotating Neutron Stars: Oscillations and Instabilities

In this review article, we present the main results from our most recent research concerning the oscillations of fast rotating neutron stars. We derive a set of time evolution equations for the investigation of non-axisymmetric oscillations of rapidly rotating compact objects in full general relativity, taking into account the contribution of a dynamic spacetime. Using our code, which features high accuracy at comparably low computational expense, we are able to extract the frequencies of non-axisymmetric modes of compact objects with rotation rates up to the Kepler limit. We propose various universal relations combining bulk properties of isolated neutron stars as well as of binary systems before and after merger; these relations are independent of the true equation of state and may serve as a valuable tool for gravitational wave asteroseismology. We also present an introductory example using a Bayesian analysis.

Frequency deviations in universal relations of isolated neutron stars and postmerger remnants

We relate the fundamental quadrupolar fluid mode of isolated nonrotating neutron stars (NSs) and the dominant oscillation frequency of neutron star merger remnants. Both frequencies individually are known to correlate with certain stellar parameters like radii or the tidal deformability, which we further investigate by constructing fit formulas and quantifying the scatter of the data points from those relations. Furthermore, we compare how individual data points deviate from the corresponding fit to all data points. Considering this point-to-point scatter we uncover a striking similarity between the frequency deviations of perturbative data for isolated NSs and of oscillation frequencies of rapidly rotating, hot, massive merger remnants. The correspondence of frequency deviations in these very different stellar systems points to an underlying mechanism and equation of state information being encoded in the frequency deviation. We trace the frequency scatter back to deviations of the tidal Love number from an average tidal Love number for a given stellar compactness. Our results thus indicate a possibility to break the degeneracy between NS radii, tidal deformability, and tidal Love number. We also relate frequency deviations to the derivative of the tidal deformability with respect to mass. Our findings generally highlight a possibility to improve gravitational wave (GW) asteroseismology relations where the systematic behavior of frequency deviations is employed to reduce the scatter in such relationships and consequently increase the measurement accuracy. In addition, we relate the $f$-mode frequency of static stars and the dominant GW frequency of merger remnants. We find an analytic mapping to connect the masses of both stellar systems, which yields particularly accurate mass-independent relations between both frequencies and between the postmerger frequency and the tidal deformability.

Gravitational-wave Asteroseismology with f-modes from Neutron Star Binaries at the Merger Phase

Abstract Gravitational-wave signals from coalescing binary neutron stars (BNSs) can yield important information about the properties of nuclear-matter equation of state. We investigate a direct link between the frequency of the quadrupolar 2 f-mode oscillation (f 2f ) of nonrotating and rotating neutron stars calculated by a nonlinear hydrodynamics code in the conformally flat approximation and the gravitational-wave frequency associated with the peak amplitude ( f max ) of binary neutron stars from a set of publicly available simulations. We find that f max and f 2f differ by about 1%, on average, across 45 equal-mass systems with different total mass and equations of state. Interestingly, assuming that the gravitational-wave frequency is still approximately equal to twice the orbital frequency f orb near the merger, the result indicates that the condition for tidal resonance ( ∣ m ∣ f orb ) m = 2 = f 2 f is satisfied to high accuracy near the merger. Moreover, the well-established universal relation between f max and the tidal deformability of equal-mass binary systems can now be explained by a similar relation between f 2f and the tidal deformability of isolated neutron stars. For unequal-mass binaries, f max increasingly deviates from f 2f of the two stars as the mass ratio decreases from unity. Therefore, it is possible to relate the gravitational-wave signal at the merger of a BNS system directly to the fundamental oscillation modes and the mass ratio. This work potentially brings gravitational-wave asteroseismology to the late-inspiral and merger phases of BNSs.

Fast Rotating Relativistic Stars: Spectra and Stability without Approximation.

We study oscillations and instabilities of relativistic stars using perturbation theory in general relativity and take into account the contribution of a dynamic spacetime. We present the oscillation spectrum as well as the critical values for the onset of the secular CFS instability of neutron stars, and propose universal relations for gravitational wave asteroseismology, which may help constrain the neutron star radius and/or the nuclear equation of state. The results are relevant for all stages during a neutron star's life but especially to nascent or remnant objects following a binary merger.

Gravitational-wave asteroseismology with fundamental modes from compact binary inspirals

Gravitational waves (GWs) from binary neutron stars encode unique information about ultra-dense matter through characterisic signatures associated with a variety of phenomena including tidal effects during the inspiral. The main tidal signature depends predominantly on the equation of state (EoS)-related tidal deformability parameter Λ, but at late times is also characterised by the frequency of the star’s fundamental oscillation mode (f-mode). In General Relativity and for nuclear matter, Λ and the f-modes are related by universal relations which may not hold for alternative theories of gravity or exotic matter. Independently measuring Λ and the f-mode frequency enables tests of gravity and the nature of compact binaries. Here we present directly measured constraints on the f-mode frequencies of the companions of GW170817. We also show that future GW detector networks will measure f-mode frequencies to within tens of Hz, enabling precision GW asteroseismology with binary inspiral signals alone.

Empirical relations for gravitational-wave asteroseismology of binary neutron star mergers

We construct new, multivariate empirical relations for measuring neutron star radii and tidal deformabilities from the dominant gravitational wave frequency in the post-merger phase of binary neutron star mergers. The relations determine neutron star radii and tidal deformabilities for specific neutron star masses with consistent accuracy and depend only on two observables: the post-merger peak frequency $f_{\rm peak}$ and the chirp mass $M_{\rm chirp}$. The former could be measured with good accuracy from gravitational waves emitted in the post-merger phase using next-generation detectors, whereas the latter is already obtained with good accuracy from the inspiral phase with present-day detectors. Our main data set consists of a gravitational wave catalogue obtained with CFC/SPH simulations. We also extract the $f_{\rm peak}$ frequency from the publicly available CoRe data set, obtained through grid-based GRHD simulations and find good agreement between the extracted frequencies of the two data sets. As a result, we can construct empirical relations for the combined data sets. Furthermore, we investigate empirical relations for two secondary peaks, $f_{2-0}$ and $f_{\rm spiral}$, and show that these relations are distinct in the whole parameter space, in agreement with a previously introduced spectral classification scheme. Finally, we show that the spectral classification scheme can be reproduced using machine-learning techniques.

Axial quasinormal modes of neutron stars in R2 gravity

In the present paper the axial quasi-normal modes of neutron stars in $f(R)$ gravity are examined using a large set of equations of state. The numerical calculations are made using two different approaches -- performing time evolution of the perturbation equations and solving the time-independent representation of the equations as a boundary value problem. According to the results the mode frequencies and the damping times decrease with the increase of the free parameter of the theory in comparison to the pure general relativistic case. While the frequencies deviate significantly from Einstein's theory for all realistic neutron star masses (say above $1M_\odot$), the damping times reach non-negligible differences only for the more massive models. We have constructed as well universal (equation of state independent) gravitational wave asteroseismology relations involving the frequencies and the damping times. It turns out that the equation of state independence is preserved using the same normalization as in pure general relativity and the qualitative differences of the phenomenological relations with respect to Einstein's theory of gravity can be large for large values of the free parameter in $f(R)$ gravity.

Gravitational wave asteroseismology limits from low density nuclear matter and perturbative QCD

We investigate the fundamental mode of non-radial oscillations of non-rotating compact stars in general relativity using a set of equations of state (EOS) connecting state-of-the-art calculations at low and high densities. Specifically, a low density model based on the chiral effective field theory (EFT) and high density results based on perturbative Quantum Chromodynamics (QCD) are matched through different interpolating polytropes fulfilling thermodynamic stability and subluminality of the speed of sound, together with the additional requirement that the equations of state support a two solar mass star. We employ three representative models (EOS I, II and III) presented in ref. [1] such that EOS I gives the minimum stellar radius, EOS II the maximum stellar mass, and EOS III the maximum stellar radius. Using this family of equations of state, we find that the frequency and the damping time of the f-mode are constrained within narrow quite model-independent windows. We also analyze some proposed empirical relations that describe the f-mode properties in terms of the average density and the compactness of the neutron star. We discuss the stringency of these constrains and the possible role of physical effects that cannot be encoded in a mere interpolation between low and high density EOSs.

Probing mass-radius relation of protoneutron stars from gravitational-wave asteroseismology

The gravitational-wave (GW) asteroseismology is a powerful technique for extracting interior information of compact objects. In this work, we focus on spacetime modes, the so-called $w$-modes, of GWs emitted from a proto-neutron star (PNS) in the postbounce phase of core-collapse supernovae. Using results from recent three-dimensional supernova models, we study how to infer the properties of the PNS based on a quasi-normal mode analysis in the context of the GW asteroseismology. We find that the $w_1$-mode frequency multiplied by the PNS radius is expressed as a linear function with respect to the ratio of the PNS mass to the PNS radius. This relation is insensitive to the nuclear equation of state (EOS) employed in this work. Combining with another universal relation of the $f$-mode oscillations, we point out that the time dependent mass-radius relation of the PNS can be obtained by observing both the $f$- and $w_1$-mode GWs simultaneously. Our results suggest that the simultaneous detection of the two modes could provide a new probe into finite-temperature nuclear EOS that predominantly determines the PNS evolution.

Asteroseismology of rapidly rotating neutron stars: An alternative approach

We examine gravitational wave asteroseismology relations for $f$-modes of rapidly rotating neutron stars. A different approach than previous studies is employed. First, the moment of inertia is used instead of the stellar radius and, second, the normalization of the oscillation frequencies and damping times is different. It is shown that in the nonrotating case this can lead to a much stronger equation of state independence, and our goal is to generalize the static relations to the rapidly rotating case and values of the spherical mode number $l\ensuremath{\ge}2$. We employ realistic equations of state that cover a very large range of stiffness in order to better check the universality of the relations. We later explore the inverse problem; i.e., we obtain the neutron star parameters from the observed gravitational frequencies and damping times. It turns out that, with this new set of relations, we can solve the inverse problem with very good accuracy using three frequencies. This was not possible in the previous studies, where one also needed damping times. The asteroseismology relations are also quite accurate for the massive rapidly rotating models that are subject to secular instabilities.

Gravitational wave asteroseismology of neutron and strange stars inR2gravity

We study the oscillations of neutron and strange stars in ${R}^{2}$ gravity. More precisely, the nonradial $f$ modes are examined and the differences with pure general relativity are investigated. Using these results, we build several gravitational wave asteroseismology relations. Our goal is to determine up to what extent these relations are equation of state independent and whether they deviate enough from general relativity in order to produce an observable effect. The results show that the differences coming from ${R}^{2}$ gravity are up to 10%, which will be difficult to observe in the near future. On the other hand, the small deviations in some of the asteroseismology relations show that they are not only equation of state independent, but they are also quite insensitive to the gravitational theory. That is why solving the inverse problem can give us quite robust estimates of the neutron star parameters.

Gravitational wave asteroseismology of fast rotating neutron stars with realistic equations of state

In the present paper we study the oscillations of fast rotating neutron stars with realistic equations of state (EoS) within the Cowling approximation. We derive improved empirical relations for gravitational wave asteroseismology with f-modes and for the first time we consider not only quadrupolar oscillations but also modes with higher spherical order (l=|m|=3,4). After performing a systematic comparison with polytropic EoS, it is shown that the empirical relations found in this case approximately also hold for realistic EoS. Even more, we show that these relations will not change significantly even if the Cowling approximation is dropped and the full general relativistic case is considered, although the normalization used here(frequencies and damping times in the nonrotating limit) could differ considerably. We also address the inverse problem, i.e. we investigate in detail what kind of observational data is required in order to determine characteristical neutron star parameters. It is shown that masses, radii and rotation rates can be estimated quite accurately using the derived asteroseismology relations. We also compute the instability window for certain models, i.e. the limiting curve in a T-\Omega-plane where the secular Chandrasekhar-Friedman-Schutz(CFS) instability overcomes dissipative effects, and show that some of the modern realistic EoS will lead to a larger instability window compared to all of the polytropic ones presented so far in the literature. Additionally, we calculate the r-mode instability window and compare it with the f-mode-case. The overall results for the instability window suggest that it is vital to take into account oscillations with l=3,4 when considering gravitational wave asteroseismology using the f-mode in rapidly rotating neutron stars, as these modes can become CFS unstable for a much larger range of parameters than pure quadrupolar oscillations.

Gravitational waves and non-axisymmetric oscillation modes in mergers of compact object binaries

We study the excitation of nonaxisymmetric modes in the post-merger phase of binary compact object mergers and the associated gravitational wave emission. Our analysis is based on general-relativistic simulations, in the spatial conformal flatness approximation, using smoothed-particle-hydrodynamics for the evolution of matter, and we use a set of equal and unequal mass models, described by two nonzero-temperature hadronic equations of state and by one strange star equation of state. Through Fourier transforms of the evolution of matter variables, we can identify a number of oscillation modes, as well as several nonlinear components (combination frequencies). We focus on the dominant m=2 mode, which forms a triplet with two nonlinear components that are the result of coupling to the quasiradial mode. A corresponding triplet of frequencies is identified in the gravitational wave spectrum, when the individual masses of the compact objects are in the most likely range of 1.2 to 1.35 $M_\odot$. We can thus associate, through direct analysis of the dynamics of the fluid, a specific frequency peak in the gravitational wave spectrum with the nonlinear component resulting from the difference between the m=2 mode and the quasiradial mode. Once such observations become available, both the m=2 and quasiradial mode frequencies could be extracted, allowing for the application of gravitational-wave asteroseismology to the post-merger remnant and leading to tight constraints on the equation of state of high-density matter.

Magnetar asteroseismology with long-term gravitational waves

Magnetic flares and induced oscillations of magnetars (super-magnetized neutron stars) are promising sources of gravitational waves (GWs). We suggest that the GW emission, if any, would last longer than the observed X-ray quasi-periodic oscillations (X-QPOs), calling for the longer-term GW analyses from a day to months than the current searches. Like the pulsar timing, the oscillation frequency would also evolve with time because of the decay or reconfiguration of magnetic field, which is crucial for the GW detection. With the observed GW frequency and its time-derivatives, we can probe the interior magnetic field strength and its evolution to open a new GW asteroseismology with the next generation interferometers like advanced LIGO, advanced VIRGO, LCGT and ET.

Gravitational wave asteroseismology with fast rotating neutron stars

We investigate damping and growth times of the f-mode for rapidly rotating stars and a variety of different polytropic equations of state in the Cowling approximation. We discuss the differences in the eigenfunctions of co- and counterrotating modes and compute the damping times of the f-mode for several EoS and all rotation rates up to the Kepler-limit. This is the first study of the damping/growth time of this type of oscillations for fast rotating neutron stars in a general relativistic framework. We use these frequencies and damping/growth times to create robust empirical formulae which can be used for gravitational wave asteroseismology. The estimation of the damping/growth time is based on the quadrupole formula and our results agree very well with Newtonian ones in the appropriate limit.