Spin-induced Quadrupole Moment Research Articles

Effect of double spin-precession and higher harmonics on spin-induced quadrupole moment measurements

Effect of double spin-precession and higher harmonics on spin-induced quadrupole moment measurements

Population inference of spin-induced quadrupole moments as a probe for nonblack hole compact binaries

Gravitational wave (GW) measurements of physical effects such as spin-induced quadrupole moments can distinguish binaries consisting of black holes from nonblack hole binaries. While these effects may be poorly constrained for single-event inferences with the second-generation detectors, combining information from multiple detections can help uncover features of nonblack hole binaries. The spin-induced quadrupole moment has specific predictions for different types of compact objects, and a generalized formalism must consider a population where different types of compact objects co-exist. In this study, we introduce a hierarchical mixture-likelihood formalism to estimate the fraction of nonbinary black holes in the population. We demonstrate the applicability of this method using simulated GW signals injected into Gaussian noise following the design sensitivities of the Advanced LIGO Advanced Virgo detectors. We compare the performance of this method with a traditionally-followed hierarchical inference approach. Both the methods are equally effective to hint at inhomogeneous populations, however, we find the mixture-likelihood approach to be more natural for mixture populations comprising compact objects of diverse classes. We also discuss the possible systematics in the mixture-likelihood approach, caused by several reasons, including the limited sensitivity of the second-generation detectors, specific features of the astrophysical population distributions, and the limitations posed by the waveform models employed. Finally, we apply this method to the LIGO-Virgo detections published in the second GW transient catalog (GWTC-2) and find them consistent with a binary black hole population with ${f}_{\mathrm{nbh}}$ estimated to be $\ensuremath{\le}0.29$ at 90% credibility.

Neutron Star–Neutron Star and Neutron Star–Black Hole Mergers: Multiband Observations and Early Warnings

Abstract The detections of gravitational waves (GWs) from binary neutron star systems and neutron star–black hole systems provide new insights into dense matter properties in extreme conditions and associated high-energy astrophysical processes. However, currently, information about the neutron star equation of state (EoS) is extracted with very limited precision. Meanwhile, the fruitful results from the serendipitous discovery of the γ-ray burst alongside GW170817 show the necessity of early warning alerts. Accurate measurements of the matter effects and sky location could be achieved by joint GW detection from space and ground. In our work, based on two example cases, GW170817 and GW200105, we use the Fisher information matrix analysis to investigate the multiband synergy between the space-borne decihertz GW detectors and the ground-based Einstein Telescope (ET). We especially focus on the parameters pertaining to the spin-induced quadrupole moment, tidal deformability, and sky localization. We demonstrate that (i) only with the help of multiband observations we can constrain the quadrupole parameter; and (ii) with the inclusion of decihertz GW detectors, the errors of tidal deformability would be a few times smaller, indicating that many more EoSs could be excluded; (iii) with the inclusion of ET, the sky localization improves by about 1 order of magnitude. Furthermore, we have systematically compared the different limits from four planned decihertz detectors and adopting two widely used waveform models.

Gravitational-wave constraints on the GWTC-2 events by measuring the tidal deformability and the spin-induced quadrupole moment

Gravitational waves from compact binary coalescences provide a unique laboratory to test properties of compact objects. As alternatives to the ordinary black holes in general relativity, various exotic compact objects have been proposed. Some of them have largely different values of the tidal deformability and spin-induced quadrupole moment from those of black holes, and their binaries could be distinguished from the binary black hole by using gravitational waves emitted during their inspiral regime, excluding the highly model-dependent merger and ringdown regimes. We reanalyze gravitational waves from low-mass merger events in the GWTC-2, detected by the Advanced LIGO and Advanced Virgo. Focusing on the influence of tidal deformability and spin-induced quadrupole moment in the inspiral waveform, we provide model-independent constraints on deviations from the standard binary black hole case. We find that all events that we have analyzed are consistent with the waveform of the binary black hole in general relativity. Bayesian model selection shows that the hypothesis that the binary is composed of exotic compact objects is disfavored by all events.

Tests of general relativity with binary black holes from the second LIGO-Virgo gravitational-wave transient catalog

Gravitational waves enable tests of general relativity in the highly dynamical and strong-field regime. Using events detected by LIGO-Virgo up to 1 October 2019, we evaluate the consistency of the data with predictions from the theory. We first establish that residuals from the best-fit waveform are consistent with detector noise, and that the low- and high-frequency parts of the signals are in agreement. We then consider parametrized modifications to the waveform by varying post-Newtonian and phenomenological coefficients, improving past constraints by factors of ${\sim}2$; we also find consistency with Kerr black holes when we specifically target signatures of the spin-induced quadrupole moment. Looking for gravitational-wave dispersion, we tighten constraints on Lorentz-violating coefficients by a factor of ${\sim}2.6$ and bound the mass of the graviton to $m_g \leq 1.76 \times 10^{-23} \mathrm{eV}/c^2$ with 90% credibility. We also analyze the properties of the merger remnants by measuring ringdown frequencies and damping times, constraining fractional deviations away from the Kerr frequency to $\delta \hat{f}_{220} = 0.03^{+0.38}_{-0.35}$ for the fundamental quadrupolar mode, and $\delta \hat{f}_{221} = 0.04^{+0.27}_{-0.32}$ for the first overtone; additionally, we find no evidence for postmerger echoes. Finally, we determine that our data are consistent with tensorial polarizations through a template-independent method. When possible, we assess the validity of general relativity based on collections of events analyzed jointly. We find no evidence for new physics beyond general relativity, for black hole mimickers, or for any unaccounted systematics.

The I-Love-Q Relations for Superfluid Neutron Stars

The I-Love-Q relations are approximate equation-of-state independent relations that connect the moment of inertia, the spin-induced quadrupole moment, and the tidal deformability of neutron stars. In this paper, we study the I-Love-Q relations for superfluid neutron stars for a general relativistic two-fluid model: one fluid being the neutron superfluid and the other a conglomerate of all charged components. We study to what extent the two-fluid dynamics might affect the robustness of the I-Love-Q relations by using a simple two-component polytropic model and a relativistic mean field model with entrainment for the equation-of-state. Our results depend crucially on the spin ratio Ωn/Ωp between the angular velocities of the neutron superfluid and the normal component. We find that the I-Love-Q relations can still be satisfied to high accuracy for superfluid neutron stars as long as the two fluids are nearly co-rotating Ωn/Ωp≈1. However, the deviations from the I-Love-Q relations increase as the spin ratio deviates from unity. In particular, the deviation of the Q-Love relation can be as large as O(10%) if Ωn/Ωp differ from unity by a few tens of percent. As Ωn/Ωp≈1 is expected for realistic neutron stars, our results suggest that the two-fluid dynamics should not affect the accuracy of any gravitational waveform models for neutron star binaries that employ the relation to connect the spin-induced quadrupole moment and the tidal deformability.

Universal relationship for magnetized white dwarfs

Recently super-Chandrasekhar mass limit has been derived theoretically in presence of strong magnetic field to complement experimental observations. In the framework of Newtonian physics, we have studied the equilibrium configurations of such magnetized white dwarfs by using the relativistic Thomas-Fermi equation of state for magnetized white-dwarfs. Hartle formalism, for slowly rotating stars, has been employed to obtain the equations of equilibrium. Various physical quantities of uniformly rotating and non-rotating white dwarfs have been calculated within this formalism. Consequently, the universality relationship between the moment of inertia(I), rotational love number($\lambda$) and spin induced quadrupole moment(Q), namely the I-Love-Q relationship, has been investigated for such magnetized white dwarfs. The relationship between I, eccentricity and Q i.e. I-eccentricity-Q relationship has also been derived. Further, we have found that, the I-eccentricity-Q relationship is more universal in comparison to I-Love-Q relationship.

Quasiuniversal relations for generalized Skyrme stars

First proposed in 2013 by Yagi and Yunes, the quasi-universal \emph{I-Love-Q relations} consist of a set of relations between the moment of inertia, the spin-induced quadrupole moment and the electric quadrupolar tidal deformability of neutron stars which are independent of the Equation of State (EoS) within an accuracy of $\sim1\%$. In this work, we show that these relations hold for different Skyrme-based nuclear matter EoS and also for the star-like solutions of different Einstein-BPS-Skyrme-models, some of which do not even present a barotropic equation of state. Further, other quasi-universal relations are analyzed, and together with recent GW observations, we use them to select the generalized Skyrme model that better reproduces observations. Our results reaffirm both the universality of the \emph{I-Love-Q} relations and the suitability of generalized Skyrme models to describe nuclear matter inside neutron stars.

Testing the Kerr nature of supermassive and intermediate-mass black hole binaries using spin-induced multipole moment measurements

The gravitational wave measurements of spin-induced multipole moment coefficients of a binary black hole system can be used to distinguish black holes from other compact objects []. Here, we apply the idea proposed in reference [] to binary systems composed of supermassive and intermediate-mass black holes and derive the expected bounds on their Kerr nature using future space-based gravitational wave detectors. Using astrophysical models of binary black hole population, we study the measurability of the spin-induced quadrupole and octupole moment coefficients using LISA and DECIGO. The errors on spin-induced quadrupole moment parameter of the binary system are found to be ⩽0.1 for almost 3% of the total supermassive binary black hole population which is detectable by LISA whereas it is ∼46% for the intermediate-mass black hole binaries observable by DECIGO at its design sensitivity. We find that errors on both the quadrupole and octupole moment parameters can be estimated to be ⩽1 for ∼2% and ∼50% of the population respectively for LISA and DECIGO detectors. Our findings suggest that a subpopulation of binary black hole events, with the signal to noise ratio thresholds greater than 200 and 100 respectively for LISA and DECIGO detectors, would permit tests of black hole nature to 10% precision.

Detweiler’s redshift invariant for extended bodies orbiting a Schwarzschild black hole

We compute the first-order self-force contribution to Detweiler's redshift invariant for extended bodies endowed with both dipolar and quadrupolar structure (with spin-induced quadrupole moment) moving along circular orbits on a Schwarzschild background. Our analysis includes effects which are second order in spin, generalizing previous results for purely spinning particles. The perturbing body is assumed to move on the equatorial plane, the associated spin vector being orthogonal to it. The metric perturbations are obtained by using a standard gravitational self-force approach in a radiation gauge. Our results are accurate through the 6.5 post-Newtonian order, and are shown to reproduce the corresponding post-Newtonian expression for the same quantity computed by using the available Hamiltonian from an effective field theory approach for the dynamics of spinning binaries.

Constraints on the binary black hole nature of GW151226 and GW170608 from the measurement of spin-induced quadrupole moments

According to the "no-hair" conjecture, a Kerr black hole (BH) is completely described by its mass and spin. In particular, the spin-induced quadrupole moment of a Kerr BH with mass $m$ and dimensionless spin $\chi$ can be written as $Q=-\kappa\,m^3\chi^2$, where $\kappa_{\rm BH}=1$. Thus by measuring the spin-induced quadrupole parameter $\kappa$, we can test the binary black hole nature of compact binaries and distinguish them from binaries comprised of other exotic compact objects, as proposed in [N. V. Krishnendu et al., PRL 119, 091101 (2017)]. Here, we present a Bayesian framework to carry out this test where we measure the symmetric combination of individual spin-induced quadrupole moment parameters fixing the anti-symmetric combination to be zero. The analysis is restricted to the inspiral part of the signal as the spin-induced deformations are not modeled in the post-inspiral regime. We perform detailed simulations to investigate the applicability of this method for compact binaries of different masses and spins and also explore various degeneracies in the parameter space which can affect this test. We then apply this method to the gravitational wave events, GW151226 and GW170608 detected during the first and second observing runs of Advanced LIGO and Advanced Virgo detectors. We find the two events to be consistent with binary black hole mergers in general relativity. By combining information from several more of such events in future, this method can be used to set constraints on the black hole nature of the population of compact binaries that are detected by the Advanced LIGO and Advanced Virgo detectors.

Surrogate model for an aligned-spin effective-one-body waveform model of binary neutron star inspirals using Gaussian process regression

Fast and accurate waveform models are necessary for measuring the properties of inspiraling binary neutron star systems such as GW170817. We present a frequency-domain surrogate version of the aligned-spin binary neutron star waveform model using the effective-one-body formalism known as SEOBNRv4T. This model includes the quadrupolar and octopolar adiabatic and dynamical tides. The version presented here is improved by the inclusion of the spin-induced quadrupole moment effect and completed by a prescription for tapering the end of the waveform to qualitatively reproduce numerical relativity simulations. The resulting model has 14 intrinsic parameters. We reduce its dimensionality by using universal relations that approximate all matter effects in terms of the leading quadrupolar tidal parameters. The implementation of the time-domain model can take up to an hour to evaluate using a starting frequency of 20 Hz, and this is too slow for many parameter estimation codes that require $\mathcal{O}({10}^{7})$ sequential waveform evaluations. We therefore construct a fast and faithful frequency-domain surrogate of this model using Gaussian process regression. The resulting surrogate has a maximum mismatch of $4.5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$ for the Advanced LIGO detector and requires $\ensuremath{\sim}0.13\text{ }\text{ }\mathrm{s}$ to evaluate for a waveform with a starting frequency of 20 Hz. Finally, we perform an end-to-end test of the surrogate with a set of parameter estimation runs and find that the surrogate accurately recovers the parameters of injected waveforms.

Spin-induced deformations and tests of binary black hole nature using third-generation detectors

In a recent letter [N. V. Krishnendu et al., PRL 119, 091101 (2017)] we explored the possibility of probing the binary black hole nature of coalescing compact binaries, by measuring their spin-induced multipole moments, observed in advanced LIGO detectors. Coefficients characterizing the spin-induced multipole moments of Kerr black holes are predicted by the "no-hair" conjecture and appear in the gravitational waveforms through quadratic and higher order spin interactions and hence can be directly measured from gravitational wave observations. We assess the capabilities of the third-generation gravitational wave interferometers such as Cosmic Explorer and Einstein Telescope in carrying out such measurements and use them to test the binary black hole nature of observed binaries. In this paper, we extend the investigations of our previous work, by proposing to measure (a) spin-induced quadrupole effects, (b) simultaneous measurements of spin-induced quadrupole and octupole effects, in the context of the third-generation detectors. We find that, using third-generation detectors the symmetric combination of coefficients associated with spin-induced quadrupole moment of each binary component may be constrained to a value $\leq 1.1$ while a similar combination of coefficients for spin-induced octupole moment may be constrained to $\leq 2$, where both combinations take the value of 1 for a binary black hole system. Further, we consider two different binary black hole populations, as proxies of the population that will be observed by the third generation detectors, and obtain the resulting distribution of the spin-induced quadrupole coefficient. These estimates suggest that third-generation detectors can accurately constrain the first four multipole moments of the compact objects (mass, spin, quadrupole, and octupole) facilitating a thorough probe of their black hole nature.

Tidal Love numbers and moment–Love relations of polytropic stars

The physical significance of tidal deformation in astronomical systems has long been known. The recently discovered universal I-Love-Q relations, which connect moment of inertia, quadrupole tidal Love number, and spin-induced quadrupole moment of compact stars, also underscore the special role of tidal deformation in gravitational wave astronomy. Motivated by the observation that such relations also prevail in Newtonian stars and crucially depend on the stiffness of a star, we consider the tidal Love numbers of Newtonian polytropic stars whose stiffness is characterised by a polytropic index $n$. We first perturbatively solve the Lane-Emden equation governing the profile of polytropic stars through the application of the scaled delta expansion method and then formulate perturbation series for the multipolar tidal Love number about the two exactly solvable cases with $n=0$ and $n=1$, respectively. Making use of these two series to form a two-point Pad\'e approximant, we find an approximate expression of the quadrupole tidal Love number, whose error is less than $2.5 \times 10^{-5}$ per cent (0.39 per cent) for $n\in[0,1]$ ($n\in[0,3]$). Similarly, we also determine the mass moments for polytropic stars accurately. Based on these findings, we are able to show that the I-Love-Q relations are in general stationary about the incompressible limit irrespective of the equation of state (EOS) of a star. Moreover, for the I-Love-Q relations, there is a secondary stationary point near $n \approx 0.4444$, thus showing the insensitivity to $n$ for $n\in[0,1]$. Our investigation clearly tracks the universality of the I-Love-Q relations from their validity for stiff stars such as neutron stars to their breakdown for soft stars.

Testing the Binary Black Hole Nature of a Compact Binary Coalescence.

We propose a novel method to test the binary black hole nature of compact binaries detectable by gravitational wave (GW) interferometers and, hence, constrain the parameter space of other exotic compact objects. The spirit of the test lies in the "no-hair" conjecture for black holes where all properties of a Kerr black hole are characterized by its mass and spin. The method relies on observationally measuring the quadrupole moments of the compact binary constituents induced due to their spins. If the compact object is a Kerr black hole (BH), its quadrupole moment is expressible solely in terms of its mass and spin. Otherwise, the quadrupole moment can depend on additional parameters (such as the equation of state of the object). The higher order spin effects in phase and amplitude of a gravitational waveform, which explicitly contains the spin-induced quadrupole moments of compact objects, hence, uniquely encode the nature of the compact binary. Thus, we argue that an independent measurement of the spin-induced quadrupole moment of the compact binaries from GW observations can provide a unique way to distinguish binary BH systems from binaries consisting of exotic compact objects.

Tidal deformability and I-Love-Q relations for gravastars with polytropic thin shells

The moment of inertia, the spin-induced quadrupole moment, and the tidal Love number of neutron-star and quark-star models are related through some relations which depend only mildly on the stellar equation of state. These "I-Love-Q" relations have important implications for astrophysics and gravitational-wave astronomy. An interesting problem is whether similar relations hold for other compact objects and how they approach the black-hole limit. To answer these questions, here we investigate the deformation properties of a large class of thin-shell gravastars, which are exotic compact objects that do not possess an event horizon nor a spacetime singularity. Working in a small-spin and small-tidal field expansion, we calculate the moment of inertia, the quadrupole moment, and the (quadrupolar electric) tidal Love number of gravastars with a polytropic thin shell. The I-Love-Q relations of a thin-shell gravastar are drastically different from those of an ordinary neutron star. The Love number and quadrupole moment are negative for less compact models and the I-Love-Q relations continuously approach the black-hole limit. We consider a variety of polytropic equations of state for the matter shell, and find no universality in the I-Love-Q relations. However, we cannot deny the possibility that, similarly to the neutron-star case, an approximate universality might emerge for a limited class of equations of state. Finally, we discuss how a measurement of the tidal deformability from the gravitational-wave detection of a compact-binary inspiral can be used to constrain exotic compact objects like gravastars.

Universality and stationarity of the I-Love relation for self-bound stars

The emergence of the I-Love-Q relations, revealing that the moment of inertia, the tidal Love number (deformability) and the spin-induced quadrupole moment of compact stars are, to high accuracy, interconnected in a universal way disregarding the wide variety of equations of state (EOSs) of dense matter, has attracted much interest recently. However, the physical origin of these relations is still a debatable issue. In the present paper, we focus on the I-Love relation for self-bound stars (SBSs) such as incompressible stars and quark stars. We formulate perturbative expansions for the moment of inertia, the tidal Love number (deformability) and the I-Love relation of SBSs. By comparing the respective I-Love relations of incompressible stars and a specific kind of SBSs, we show analytically that the I-Love relation is, to relevant leading orders in stellar compactness, stationary with respect to changes in the EOS about the incompressible limit. Hence, the universality of the I-Love relation is indeed attributable to the proximity of compact stars to incompressible stars, and the stationarity of the relation as unveiled here. We also discover that the moment of inertia and the tidal deformability of a SBS with finite compressibility are, to leading order in compactness, equal to their counterparts of an incompressible star with an adjusted compactness, thus leading to a novel explanation for the I-Love universal relation.

I-Love relations for incompressible stars and realistic stars

In spite of the diversity in the equations of state of nuclear matter, the recently discovered I-Love-Q relations [Yagi and Yunes, Science {\bf 341}, 365 (2013)], which relate the moment of inertia, tidal Love number (deformability) and the spin-induced quadrupole moment of compact stars, hold for various kinds of realistic neutron stars and quark stars. While the physical origin of such universality is still a current issue, the observation that the I-Love-Q relations of incompressible stars can well approximate those of realistic compact stars hints at a new direction to approach the problem. In this paper, by establishing recursive post-Minkowskian expansion for the moment of inertia and the tidal deformability of incompressible stars, we analytically derive the I-Love relation for incompressible stars and show that the so obtained formula can be used to accurately predict the behavior of realistic compact stars from the Newtonian limit to the maximum mass limit.

UNVEILING THE UNIVERSALITY OF I-LOVE-Q RELATIONS

The recent discovery of the universal I-Love-Q relations connecting the moment of inertia, tidal deformability, and the spin-induced quadrupole moment of compact stars is intriguing and totally unexpected. In this paper, we provide numerical evidence showing that the universality can be attributed to the incompressible limit of the I-Love-Q relations. The fact that modern equations of state are stiff, with an effective adiabatic index larger than about two, above the nuclear density range is the key to establishing the universality for neutron stars and quark stars with typical compactness from about 0.1 to 0.3. On the other hand, the I-Love-Q relations of low-mass neutron stars near the minimum mass limit depend more sensitively on the underlying equation of state because these stars are composed mainly of softer matter at low densities. However, the I-Love-Q relations for low-mass quark stars can still be represented accurately by the incompressible limit. We also study the I-Love relation connecting the moment of inertia and tidal deformability analytically in Newtonian gravity and show why the I-Love-Q relation is weakly dependent on the underlying equation of state and can be attributed to its incompressible limit.

Slowly rotating neutron stars in scalar-tensor theories

We construct models of slowly rotating, perfect-fluid neutron stars by extending the classical Hartle-Thorne formalism to generic scalar-tensor theories of gravity. Working at second order in the dimensionless angular momentum, we compute the mass M, radius R, scalar charge q, moment of inertia I and spin-induced quadrupole moment Q, as well as the tidal and rotational Love numbers. Our formalism applies to generic scalar-tensor theories, but we focus in particular on theories that allow for spontaneous scalarization. It was recently discovered that the moment of inertia, quadrupole moment and Love numbers are connected by approximately universal (i.e., equation-of-state independent) "I-Love-Q" relations. We find that similar relations hold also for spontaneously scalarized stars. More interestingly, the I-Love-Q relations in scalar-tensor theories coincide with the general relativistic ones within less than a few percent, even for spontaneously scalarized stars with the largest couplings allowed by current binary-pulsar constraints. This implies that astrophysical measurements of these parameters cannot be used to discriminate between general relativity and scalar-tensor theories, even if spontaneous scalarization occurs in nature. Because of the well known equivalence between f(R) theories and scalar-tensor theories, the theoretical framework developed in this paper can be used to construct rotating compact stellar models in f(R) gravity. Our slow-rotation expansion can also be used as a benchmark for numerical calculations of rapidly spinning neutron stars in generic scalar-tensor theories.