Recent observations of neutron stars (NS) with radio, X-rays and gravitational waves have begun to constrain the equation of state (EoS) for nuclear matter beyond the nuclear saturation density. To one's surprise, there exist approximate universal relations connecting certain bulk properties of NSs that are insensitive to the underlying EoS and having important applications on probing fundamental physics including nuclear and gravitational physics. To date, analytic works on universal relations for realistic NSs are lacking, which may lead to a better understanding of the origin of the universality. Here, we focus on the universal relations between the compactness(C), moment of inertia (I), and tidal deformability(related to the Love number), and derive analytic, approximate I-Love-C relations. To achieve this, we construct slowly-rotating/ tidally-deformed NS solutions analytically starting from an extended Tolman VII model that accurately describes non-rotating realistic NSs, allowing us to extract the moment of inertia and the tidal deformability on top of the compactness. We solve the field equations analytically by expanding them about the Newtonian limit and keeping up to 6th order in the stellar compactness. Based on these analytic solutions, we can mathematically demonstrate the O(10%) EoS variation in the I-C and Love-C relations and the O(1%) variation in the I-Love relation that have previously been found numerically. Our new analytic relations agree more accurately with numerical results for realistic NSs(especially the I-C and Love-C ones) than the analytic relations for constant density stars derived in previous work. Based on these analytic findings, we attribute a possible origin of the universality for the I-C and Love-C relations to the fact that the energy density of realistic NSs can be approximated as a quadratic function, as is the case for the Tolman VII solution.
Observations of gravitational waves from inspiralling neutron star binaries---such as GW170817---can be used to constrain the nuclear equation of state by placing bounds on stellar tidal deformability. For slowly rotating neutron stars, the response to a weak quadrupolar tidal field is characterized by four internal-structure-dependent constants called "Love numbers." The tidal Love numbers $k_2^\text{el}$ and $k_2^\text{mag}$ measure the tides raised by the gravitoelectric and gravitomagnetic components of the applied field, and the rotational-tidal Love numbers $\mathfrak{f}^\text{o}$ and $\…
Gravitational-wave measurements of the tidal deformability of neutron stars could reveal important information regarding their internal structure, the equation of state of high-dense nuclear matter and gravity in strong field regime. In this work, we extend the relativistic theory of the tidal deformability of neutron stars to Rastall gravity. Both the electric-type and magnetic-type quadrupole tidal Love numbers are calculated for neutron stars in the polytrope model. It is found that neutron star's tidal Love numbers in Rastall gravity is significantly smaller than those in general relativit…
In this work, we study neutron stars (NSs) in an ungravity (UG) inspired model. We examine the UG effects on the static properties of the selected NSs, in different mass and radius regimes, i.e., mini, moderate, and heavy NSs, using a polytropic equation of state approach. Based on the observational data, we obtain bounds on the characteristic length and scaling dimension of the UG model. Furthermore, we obtain dynamic properties, such as inertial moment (I), Love number (Love), and quadrupole moment (Q) of a slowly rotating NS in the presence of the exterior gravity and ungravity fields. The …
Neutron star (NS) equation of state (EoS) insensitive relations or universal relations (UR) involving neutron star bulk properties play a crucial role in gravitational-wave astronomy. Considering a wide range of equations of state originating from (i) phenomenological relativistic mean field models, (ii) realistic EoS models based on different physical motivations, and (iii) polytropic EoSs described by spectral decomposition method, we update the EoS-insensitive relations involving NS tidal deformability (Multipole Love relation) and the UR between f-mode frequency and tidal deformability (f-…
The exterior gravitational field of a slowly-rotating neutron star can be characterized by its multipole moments, the first few being the neutron star mass, moment of inertia, and quadrupole moment to quadratic order in spin. In principle, all of these quantities depend on the neutron star's internal structure, and thus, on unknown nuclear physics at supra-nuclear energy densities. We here find relations between the moment of inertia, the Love numbers and the quadrupole moment (I-Love-Q relations) that do not depend sensitively on the neutron star's internal structure. Three important conseque…
Inferring neutron-star Love-Q relations from gravitational waves in the hierarchical Bayesian framework
Peking University（北京大学）；Kavli Institute for Astronomy and Astrophysics, Peking University（北京大学科维理天文与天体物理研究所）；National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台）
By extending our recent framework to describe the tidal deformations of a spinning compact object, we compute for the first time the tidal Love numbers of a spinning neutron star to linear order in the angular momentum. The spin of the object introduces couplings between electric and magnetic distortions and new classes of spin-induced ("rotational") tidal Love numbers emerge. We focus on stationary tidal fields, which induce axisymmetric perturbations. We present the perturbation equations for both electric-led and magnetic-led rotational Love numbers for generic multipoles and explicitly sol…
We comment on the paper of S. Postnikov et al. in Phys. Rev. D 82, 024016 (2010) and give a modified formula that needs to be taken into account when calculating the tidal Love number of neutron stars in case a first order phase-transition occurs at non-zero pressure. We show that the error made when using the original formula tends to zero as $p \rightarrow 0$ and we estimate the maximum relative error to be $\sim 5\%$ if the density discontinuity is at larger densities.
The recent detection of gravitational waves from a neutron star merger was a significant step towards constraining the nuclear matter equation of state by using the tidal Love numbers (TLNs) of the merging neutron stars. Measuring or constraining the neutron star TLNs allows us in principle to exclude or constraint many equations of state. This approach, however, has the drawback that many modified theories of gravity could produce deviations from General Relativity similar to the deviations coming from the uncertainties in the equation of state. The first and the most natural step in resolvin…
Rossby modes (r-modes) of rotating neutron stars can be excited by the gravitomagnetic forces in coalescing binary systems. The previous study by Flanagan and Racine [Phys. Rev. D 75, 044001 (2007)] showed that this kind of dynamical tide (DT) can induce phase shifts of 0.1 rad on gravitational waveforms, which is detectable by third-generation (3G) detectors. In this paper, we study the impact of this DT on measuring neutron-star parameters in the era of 3G detectors. We incorporate two universal relations among neutron star properties predicted by different equations of state: (i) the well-k…
The coupling between the angular momentum of a compact object and an external tidal field gives rise to the "rotational" tidal Love numbers, which affect the tidal deformability of a spinning self-gravitating body and enter the gravitational waveform of a binary inspiral at high post-Newtonian order. We provide numerical evidence for a surprising "hidden" symmetry among the rotational tidal Love numbers with opposite parities, which are associated to perturbations belonging to separate sectors. This symmetry, whose existence had been suggested on the basis of a Lagrangian description of the ti…
Neutron stars are ideal to probe, not only nuclear physics, but also strong-field gravity. Approximate universal relations insensitive to the star's internal structure exist among certain observables and are useful in testing General Relativity, as they project out the uncertainties in the equation of state. One such set of universal relations between the moment of inertia $(I)$, the tidal Love number and the quadrupole moment $(Q)$ has been studied both in General Relativity and in modified theories. In this paper, we study the relations in dynamical Chern-Simons gravity, a well-motivated, pa…
The space-borne gravitational wave interferometer, LISA, is expected to detect signals from numerous binary white dwarfs. At small orbital separation, rapid rotation and large tidal bulges may allow for the stellar internal structure to be probed through such observations. Finite-size effects are encoded in quantities like the moment of inertia ($I$), tidal Love number (Love), and quadrupole moment ($Q$). The universal relations among them (I-Love-Q relations) can be used to reduce the number of parameters in the gravitational-wave templates. We here study I-Love-Q relations for more realistic…
Finite size effects come into play during the late stages of neutron star binary inspiral, with the tidal deformability of the supranuclear density matter leaving an imprint on the gravitational-wave signal. As demonstrated in the case of GW170817, this leads to a constraint on the neutron star radius (and hence the equation of state). A deeper understanding of the tidal response requires an analysis of both the state and composition of matter. While these aspects may not have dramatic impact, they could lead to systematic effects that need to be kept in mind as the observational data become m…
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 wi…
Moment of inertia ($I$), rotational~(tidal) Love number ($λ^{\rm (rot)}$) and quadrupole moment ($Q$) of slowly rotating massive neutron star~(NS) with holographic multiquark~(MQ) core are computed in comparison to pure MQ star. The Chiral Effective Theory~(CET) stiff equation of state~(EoS) is used in the crust of the neutron star. The dimensionless multipole moments $\bar{I},\barλ^{\rm (rot)}, \bar{Q}$ are found to be independent of the rotation parameters and determined completely by the zeroth-order star profile. Universal ``I-Love-Q'' relations found by Yagi and Yunes [1,2] are mostly pre…
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 …
Gravitational-wave measurements of the tidal deformability in neutron-star binary coalescences can be used to infer the still unknown equation of state (EoS) of dense matter above the nuclear saturation density. By employing a Bayesian-ranking test we quantify the ability of current and future gravitational-wave observations to discriminate among families of nuclear-physics based EoS which differ in particle content and ab-initio microscopic calculations. While the constraining power of GW170817 is limited, we show that even twenty coalescences detected by LIGO-Virgo at design sensitivity are …
In this present work, we suggest studying neutron star with a conformal coupling by using the analytical expressions of realistic equations of state. The equations of perturbed and nonperturbed metrics, scalar field and the pressure of the matter are derived in order to observe the effects of the conformal coupling on the mass, radius and tidal love number of polar and axial types. The numerical and the analytical analysis show a different comportment of our model from GR by varying the value of the parameters in the model. We showed also for a particular case that the scalar field becomes van…
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 fro…
Linear analysis of I-C-Love universal relations for neutron stars
Peking University（北京大学）；Kavli Institute for Astronomy and Astrophysics, Peking University（北京大学科维理天文与天体物理研究所）；Eberhard Karls University of Tübingen（蒂宾根大学）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））；National Astronomical Observatories, Chinese Academy of Sciences（中国科学院国家天文台）
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…
Analyses that connect astrophysical observations of neutron stars with nuclear matter properties sometimes rely on equation-of-state insensitive relations. We show that the slope of the binary Love relations (i.e.~between the tidal deformabilities of binary neutron stars) encodes the rate of change of the nuclear matter speed of sound below three times nuclear saturation density. Twin stars lead to relations that present a signature ''hill'', ''drop'', and ''swoosh'' due to the second (mass-radius) stable branch, requiring a new description of the binary love relations.
The distance-inclination degeneracy limits gravitational-wave parameter estimation of compact binary mergers. Although the degeneracy can be partially broken by including higher-order modes or precession, these effects are suppressed in binary neutron stars. In this work, we implement a new parametrization of the tidal effects in the binary neutron-star waveform, exploiting the binary Love relations, that breaks the distance-inclination degeneracy. The binary Love relations prescribe the tidal deformability of a neutron star as a function of its source-frame mass in an equation-of-state insens…
We calculate the tidal Love numbers of black holes and neutron stars in the presence of higher dimensions. The perturbation equations around an arbitrary static and spherically symmetric metric for the even parity modes are presented in the context of an effective four-dimensional theory on the brane. This subsequently leads to the sought expression for the tidal Love number for black holes in the presence of extra spatial dimensions. Surprisingly, these numbers are non-zero and, more importantly, negative. We extend our method to determine the tidal Love number of neutron stars in a spacetime…
The gravitational wave signal from a binary neutron star merger carries the imprint of the deformability properties of the coalescing bodies, and then of the equation of state of neutron stars. In current models of the waveforms emitted in these events, the contribution of tidal deformation is encoded in a set of parameters, the tidal Love numbers. More refined models include tidal-rotation couplings, described by an additional set of parameters, the rotational tidal Love numbers, which appear in the waveform at $6.5$ post-Newtonian order. For neutron stars with spins as large as $\sim0.1$, we…
Certain bulk properties of neutron stars, in particular their moment of inertia, rotational quadrupole moment and tidal Love number, when properly normalized, are related to one another in a nearly equation of state independent way. The goal of this paper is to test these relations with extreme equations of state at supranuclear densities constrained to satisfy only a handful of generic, physically sensible conditions. By requiring that the equation of state be (i) barotropic and (ii) its associated speed of sound be real, we construct a piecewise function that matches a tabulated equation of …
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…
Gravitational waves from the final stages of inspiralling binary neutron stars are expected to be one of the most important sources for ground-based gravitational wave detectors. The masses of the components are determinable from the orbital and chirp frequencies during the early part of the evolution during which tidal effects provide small correction; however, during this phase the signal is relatively clean. The accumulated phase shift due to tidal corrections is characterized by a single quantity, the Love number, which is sensitive to the compactness parameter M/R and the star's internal …
Hořava-Lifshitz gravity is an alternative theory to general relativity which breaks Lorentz invariance in order to achieve an ultraviolet complete and power-counting renormalizable theory of gravity. In the low energy limit, Hořava-Lifshitz gravity coincides with a vector-tensor theory known as khronometric gravity. The deviation of khronometric gravity from general relativity can be parametrized by three coupling constants: $α$, $β$, and $λ$. Solar system experiments and gravitational wave observations impose stringent bounds on $α$ and $β$, while $λ$ is still relatively unconstrained ($λ\les…
Shortly after its birth in a gravitational collapse, a proto-neutron star enters in a phase of quasi-stationary evolution characterized by large gradients of the thermodynamical variables and intense neutrino emission. In few tens of seconds the gradients smooth out while the star contracts and cools down, until it becomes a neutron star. In this paper we study this phase of the proto-neutron star life including rotation, and employing finite temperature equations of state. We model the evolution of the rotation rate, and determine the relevant quantities characterizing the star. Our results s…
Recently, general relations among the quadrupole moment (Q), the moment of inertia (I), and the tidal deformability (Love number) of a neutron star were shown to exist. They are nearly independent of the nuclear matter equation of state and would be of great aid in extracting parameters from observed gravitational-waves and in testing general relativity. These relations, however, do not account for strong magnetic fields. We consider this problem by studying the effect of a strong magnetic field on slowly rotating relativistic neutron stars and show that, for simple magnetic field configuratio…
Neutron stars and quark stars are ideal laboratories to study fundamental physics at supra nuclear densities and strong gravitational fields. Astrophysical observables, however, depend strongly on the star's internal structure, which is currently unknown due to uncertainties in the equation of state. Universal relations, however, exist among certain stellar observables that do not depend sensitively on the star's internal structure. One such set of relations is between the star's moment of inertia ($I$), its tidal Love number (Love) and its quadrupole moment ($Q$), the so-called I-Love-Q relat…
Neutron stars and quark stars are not only characterized by their mass and radius, but also by how fast they spin, through their moment of inertia, and how much they can be deformed, through their Love number and quadrupole moment. These depend sensitively on the star's internal structure, and thus on unknown nuclear physics. We find universal relations between the moment of inertia, the Love number and the quadrupole moment that are independent of the neutron star's and quark star's internal structure. These can be used to learn about the deformability of these compact objects through observa…
One of the most common assumptions has been made that the pressure inside the star is isotropic in nature. However, the pressure is locally anisotropic in nature which is a more realistic case. In this study, we investigate certain properties of anisotropic neutron stars with the scalar pressure anisotropy model. Different perfect fluid conditions are tested within the star with the relativistic mean-field model equation of states (EOSs). The anisotropic neutron star properties such as mass ($M$), radius ($R$), compactness ($C$), Love number ($k_2$), dimensionless tidal deformability ($Λ$), an…
Gravitational-wave cosmology began in 2017 with the observation of the gravitational waves emitted in the merger of two neutron stars, and the coincident observation of the electromagnetic emission that followed. Although only a $30\%$ measurement of the Hubble constant was achieved, future observations may yield more precise measurements either through other coincident events or through cross correlation of gravitational-wave events with galaxy catalogs. Here, we implement a new way to measure the Hubble constant without an electromagnetic counterpart and through the use of the binary Love re…
The role of dense-matter properties in the tidal deformability of a cold nonaccreted neutron star is further investigated. Using the set of Brussels-Montreal unified equations of state, we have computed the gravitoelectric Love numbers $k_\ell$ and the gravitomagnetic Love numbers $j_\ell$ up to $\ell=5$. Their relative importance and their sensitivity to the symmetry energy and the neutron-matter stiffness are numerically assessed. Their impact on the phase of the gravitational-wave signal emitted by binary neutron star inspirals is also discussed.
Quasi-universal relations connecting the tidal deformability and the quadrupole moment of individual neutron stars are predicted by theoretical computations, but have not been measured experimentally. However, such relations are employed during the interpretation of gravitational waves and, therefore, have a direct impact on the interpretation of real data. In this work, we study how quasi-universal relations can be tested and measured from gravitational wave signals connected to binary neutron star coalescences. We study a population of $120$ binary neutron star systems and find that Advanced…
In recent years there has been a surge of interest in what has come to be known as the `universal relations' between various global properties of neutron stars. These universal relations are equation of state independent relations between quantities such as the moment of inertia $I$, the tidal deformability or Love number $λ$, and the quadrupole $Q$ (I-Love-Q relations), or the relativistic multipole moments (3-hair relations). While I-Love-Q relations have been studied extensively in both general relativity and various alternatives, 3-hair relations have been studied only in general relativit…
Utah State University（犹他州立大学）；Instituto de Física Teórica UAM-CSIC, Universidad Autónoma de Madrid（马德里自治大学UAM-CSIC理论物理研究所）；Université Paris Cité, CNRS, Astroparticule et Cosmologie（巴黎西岱大学CNRS天体粒子与宇宙学）；University of Virginia（弗吉尼亚大学）
Recently it was shown that slowly rotating neutron stars exhibit an interesting correlation between their moment of inertia $I$, their quadrupole moment $Q$, and their tidal deformation Love number $λ$ (the I-Love-Q relations), independently of the equation of state of the compact object. In the present work a similar, more general, universality is shown to hold true for all rotating neutron stars within General Relativity; the first four multipole moments of the neutron star are related in a way independent of the nuclear matter equation of state we assume. By exploiting this relation, we can…
Tidal deformability of a coalescing neutron star subjected to an external tidal field plays an important role in our probe for the structure and properties of compact stars. In particular, the tidal love number provides valuable information about the external gravitational field responsible for deforming the star. In this article, we compute the tidal love number of a particular class of anisotropic stars and analyze the impacts of anisotropy and compactness on the tidal love number.
The coalescence of two neutron stars is an important gravitational wave source for LIGO and other detectors. Numerous studies have considered the precision with which binary parameters (masses, spins, Love numbers) can be measured. Here I consider the accuracy with which these parameters can be determined in the presence of systematic errors due to waveform approximations. These approximations include truncation of the post-Newtonian (PN) series and neglect of neutron star (NS) spin, tidal deformation, or orbital eccentricity. All of these effects can yield systematic errors that exceed statis…
Gravitational-wave observations in the near future may allow us to measure tidal deformabilities of neutron stars, which leads us to the understanding of physics at nuclear density. In principle, the gravitational waveform depends on various tidal parameters, which correlate strongly. Therefore, it would be useful if one can express such tidal parameters with a single parameter. Here, we report on universal relations among various $\ell$-th (dimensionless) electric, magnetic and shape tidal deformabilities in neutron stars and quark stars that do not depend sensitively on the equation of state…
In this short note, we investigate the existence of universal relations between the gravimagnetic Love number of irrotational stars and the dimensionless moment of inertia. These Love numbers take into account the internal motion of the fluid, while the star is globally irrotational. The goal is to extend the so-called I-Love-Q relations - providing a strong correlation between the gravitoelectric Love number, the dimensionless moment of inertia and the dimensionless rotation-induced quadrupole - to the gravitomagnetic sector, where internal motion is taken into account. As a byproduct, we pre…
In this brief note, we clarify certain aspects related to the magnetic (i.e., odd parity or axial) tidal Love numbers of a star in general relativity. Magnetic tidal deformations of a compact star had been computed in 2009 independently by Damour and Nagar and by Binnington and Poisson. More recently, Landry and Poisson showed that the magnetic tidal Love numbers depend on the assumptions made on the fluid, in particular they are different (and of opposite sign) if the fluid is assumed to be in static equilibrium or if it is irrotational. We show that the zero-frequency limit of the Regge-Whee…
Investigating the impact of quasi-universal relations on neutron star constraints in third-generation detectors
Universität Potsdam（波茨坦大学）；Universidade Estadual da Região Tocantina do Maranhão, UEMASUL（马拉尼昂州托坎廷斯地区州立大学）；Universidade Federal do Maranhão（马拉尼昂联邦大学）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理研究所（爱因斯坦研究所））
The neutron star properties are generally determined by the equation of state of $β$-equilibrated dense matter. In this work, we consider the interaction of fermionic dark matter (DM) particles with the nucleons via Higgs exchange and investigate its effect on the neutron star properties with the relativistic mean-field model equation of state coupled with DM. We deduce that DM significantly affects the neutron star properties, such as considerably reduce the maximum mass of the star, which depends on the percentage of the DM considered inside the neutron star. The tidal Love numbers both for …
