g-mode oscillations in hybrid stars:
A tale of two sounds

The core of a neutron star (NS) can, in principle, support phases of dense deconfined quark matter (QM) [1]. Confirmation of the presence of quarks in NSs, however, has not been possible either through observations or from lattice-gauge calculations of finite baryon density matter starting from the Lagrangian of Quantum Chromodynamics (QCD). Although perturbative calculations of QM have been performed [2, 3, 4], their applicability is limited to baryon densities $n_{B}\gtrsim 40n_{s}$ [5], where $n_{s}\simeq 0.16~{}\rm{fm^{-3}}$ is the nuclear matter saturation density. Such densities, however, lie well beyond the central densities $n_{c}$ in the range 3-8$n_{s}$ of observed NSs. In view of this conundrum, theoretical studies of QM in NSs have been exploratory in nature by positing either a sharp 1st-order or a smooth crossover hadron-to-quark phase transition. Depending on the treatment of the phase transition and the equations of state (EOSs) of hadronic and quark matter, either a phase of pure QM or a phase in which hadrons are admixed with quarks can be realized (for a detailed account, see Ref. [6] and an extensive list of references therein). In either case, stars with quarks are difficult to distinguish from normal NSs based on the knowledge of masses and radii alone as similar results can be obtained with both. While the long-term cooling of a NS can be affected by the presence of quarks, cooling data are relatively sparse and gathered over decades [7, 8]. Gravitational wave observations from compact binary mergers can be another probe of the EOS, but currently, constraints on tidal polarizability [9, 10, 11] from gravitational wave data [12] are consistent with both normal and quark-containing stars, depending on the theoretical assumptions made [6, 13, 14].

In this paper, we are particularly interested in how NS oscillations can shed light on the presence of QM in stars that contain an admixture of nucleons and quarks (termed hybrid stars). Andersson and Kokkotas [15] have proposed that NS oscillations (in particular, the $f,p$ modes) could be a “fingerprint” for the supra-nuclear EOS in gravitational wave data. A review of potential signatures of QM in NSs in the multi-messenger era, including the role of their oscillations, can be found in [16]. Along these lines, we offer in this work a new diagnostic of deconfined QM in NSs based on asteroseismology. We show that a steep rise in the frequency of the principal $g$-mode (gravity mode) occurs as soon as QM appears in a mixed phase in the core, exceeding the typical core $g$-mode frequency of a nucleonic star by a factor of two or more. This rise is essentially driven by a drop in the local equilibrium speed of sound at the onset of the phase transition, while the adiabatic sound speed changes only slightly. If this $g$-mode becomes resonant with the tidal force during the late stages of a binary inspiral, the resulting energy transfer from the orbital motion to the star via tidal coupling can affect the phase of the gravitational waveform, and potentially signal a hybrid star.

A $g$-mode is a specific kind of non-radial fluid oscillation initiated when a parcel of fluid is displaced against the background of a stratified environment [17, 18]. While pressure equilibrium is rapidly restored via sound waves, chemical equilibrium can take longer causing buoyancy forces to oppose the displacement. Since cold NSs are not convective, the opposing force sets up stable oscillations throughout the core, with a typical frequency, called the (local) Brunt-Väisälä  frequency [19], which depends on the difference between the equilibrium and adiabatic sound speeds as well as the local metric coefficients. Convectively stable $g$-modes exist for a wide range of models of the EOS [20]. Though the $g$-mode in NSs has been studied before [21, 22, 23], with recent works incorporating additional features like hyperonic matter and superfluidity [24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34], the novel aspect of our work is the investigation of the $g$-mode frequency in a phase transition from nuclear matter to a mixed phase of quarks and nucleons. We point out that a similar result was obtained in [30] for superfluid hyperonic stars. However, the calculations presented there were for a fixed NS mass of about 1.64$M_{\odot}$ with a radius of 13.6 km. While their chosen hyperonic EOS does support a maximum mass of $2.015M_{\odot}$ [35], the nuclear and quark EOSs chosen in our work satisfy additional observational and experimental constraints, and presents the effect for a wide range of masses up to the observed maximum.

This paper also extends the results of [36] by incorporating aspects of General Relativity in the fluid oscillation equations (while remaining within the Cowling approximation), updating the nuclear EOS to include consistency with radius constraints from tidal polarizability [37] and NICER data [38, 39] on the radius of $\simeq 1.4M_{\odot}$ NS. We also provide new analytical results for the two sound speeds in a mixed phase of quarks and nucleons. Our study can be of practical interest to investigations of the sound speed in NSs, which is attracting renewed attention [40, 41, 42, 43]. The matter of detecting $g$-modes from the pre- or post-coalescence phase of binary NS mergers is not addressed in detail here, but we present an estimate of its impact on the accumulated tidal phase up to the merger.

The organization of this paper is as follows. In Sec. II, we introduce the governing equations of the $g$-mode and outline its relation to the two sound speeds. In Sec. III, we present the EOS models in the nucleonic and quark sectors chosen for our study of the $g$-mode spectrum. The rationale for our parameter choices and the basic features of these models are highlighted here for orientation. We stress that our choices are representative, but not exhaustive. In Sec. IV, we derive expressions for the two sound speeds in the mixed phase of quarks and nucleons. Results for the sound speeds, the Brunt-Väisälä  frequency and the $g$-mode frequency in nucleonic, quark and hybrid stars are gathered and interpreted in Sec. V. The tidal forcing and phase error due to $g$-mode excitation are estimated in Sec. VI. Our summary, conclusions and outlook are contained in Sec. VII. The appendices contain details about the determination of parameters in the nuclear EOS model, the resulting NS structural properties and the two sound speeds.

In this section, we outline the equations for fluid oscillations and non-rotating NS structure that were used to determine the eigenfrequencies of the $g$-mode. In general, the oscillatory displacement of a fluid element in a spherically symmetric star is represented by a vector field ${\vec{\xi}}^{nlm}(\vec{r}){\rm e}^{-i\omega t}$ with $n,l$ and $m$ denoting the radial, azimuthal and magnetic mode indices. To be precise, the frequencies also carry subscripts $nlm$ implicitly understood, with degeneracies that are broken in more realistic cases such as with rotation or magnetic fields. For even-parity or spheroidal modes, separation into radial and tangential components yields $\xi_{r}^{nlm}(\vec{r})$ = $\eta_{r}^{nl}(r)Y_{lm}(\theta,\phi)$ and $\xi_{\perp}^{nlm}(\vec{r})$ = $r\eta_{\perp}^{nl}(r)\nabla_{\perp}Y_{lm}(\theta,\phi)$, respectively, where $Y_{lm}(\theta,\phi)$ are the spherical harmonics. From the perturbed continuity equation for the fluid, the tangential function $\eta_{\perp}$ can be traded for fluid variables as $\delta p/\epsilon$ = $\omega^{2}r\eta_{\perp}(r)Y_{lm}(\theta,\phi){\rm e}^{-i\omega t}$, where $\delta p$ is the corresponding local (Eulerian) pressure perturbation and $\epsilon$ the local energy density. Within the relativistic Cowling approximation²²2The Cowling approximation neglects the back reaction of the gravitational potential and reduces the number of equations we have to solve. While this approximation is not strictly consistent with our fully general relativistic (GR) treatment of the equilibrium structure of the star, it does not change our conclusions qualitatively or even quantitatively, since this approximation is accurate for $g$-mode frequencies at the few % level [54]., the equations of motion to be solved to determine the frequency of a particular mode are [55, 17, 29]

where $h=p+\epsilon$, and we have suppressed the indices on $\omega$ and $\xi$. Equation (1) involves thermodynamic quantities that follow from the specific EOS. Specifically, $p$ denotes pressure, $\epsilon$ energy density, and $\gamma$ the adiabatic index of the fluid. The Lagrangian variation of the pressure enters as $\Delta p$, and is related to the Eulerian variation $\delta p$ through the operator relation $\Delta\equiv\delta+\xi\cdot\nabla$. The symbol $c_{s}$ denotes the adiabatic sound speed, which is related to the adiabatic index as $c_{s}^{2}=\gamma p/(\mu_{n}n_{B})$ where $\mu_{n}$ is the neutron chemical potential³³3In beta-equilibrated charge neutral matter, the neutron chemical potential is sufficient to determine all other chemical potentials. and $n_{B}$ the local baryon density. The equilibrium sound speed enters through the Brunt-Väisälä  frequency ($N$) which is given by

where $g=-\nabla\phi=-\nabla p/h$ with $h=\epsilon+p$ the enthalpy of the fluid. Finally, $\nu(r)$ and $\lambda(r)$ are metric functions of the unperturbed star which features in the Schwarzschild interior metric ($r<R$):

Explicitly,

and

where $m(r^{\prime})$ is the enclosed mass of the star at $r^{\prime}$. These metric functions must match to their exterior values at the surface $r=R$, hence the constant factor ${\rm e}^{\nu_{0}}$ [56].

In this work, we study the fundamental $g$-mode with $n$ = 1 and fix the mode’s multipolarity at $l$ = 2. For the non-rotating stars we consider here, solutions are degenerate in $m$. Note that our definition of the “fundamental” mode refers to the lowest radial order of the $g$-mode which also has the highest frequency. This should not be confused with the qualitatively different $f$-mode which is also referred to sometimes as the fundamental mode. Furthermore, overtones with lower frequency exist, but we do not perform any computations with them here, since the fundamental $g$-mode has the highest frequency and will be excited during the final stage of the pre-merger phase when tidal forces are strongest. The system of equations in Eq. (1) cannot be solved analytically even with a simple model of a neutron star. Our aim will be to solve this numerically as an eigenvalue system for the $g$-mode frequency $\omega$. Physically, the solution to this system of equations, under the boundary conditions $\Delta p=0$ at the surface and $\xi_{r},\,\delta p/\epsilon$ regular at the center, only exists for discrete values of the mode frequency $\omega$. These values represent the $g$-mode spectrum for a chosen stellar model. Because we have employed the Cowling approximation and ignored the perturbations of the metric that must accompany fluid perturbations, we cannot compute the imaginary part of the eigenfrequency (damping time) of the $g$-mode⁴⁴4The damping time of $g$-modes due to viscosity and gravitational wave emission, crudely estimated in [57, 36], suggests that the $g$-mode can become secularly unstable for temperatures $10^{8}~{}{\rm K}<T<10^{9}~{}{\rm K}$ for rotational speeds exceeding twice the $g$-mode frequency of a static star.. We turn now to discuss the EOS models for nucleonic and quark matter employed in this work.

The EOS models chosen in this work were predicated on the requirement that the squared sound speeds $c_{e}^{2}$ (see Eq.(25)) and $c_{s}^{2}$ could be calculated straightforwardly. In the nucleonic sector, we employ the model of Zhao and Lattimer (ZL) [43] which is consistent with nuclear systematics at and below the nuclear saturation density $n_{s}$. With suitable choices of the slope of the nuclear symmetry energy at $n_{s}$ (see below), this EOS is also consistent with the recent chiral effective theory calculations of Drischler et al. [58] in which uncertainty estimates of the EOS up to $2n_{s}$ were provided (see Fig. 2 of this reference). In addition, the ZL EOS is able to support $\simeq 2M_{\odot}$ stars required by mass measurements of heavy NSs [59], and is consistent with the recent radius measurements of $\sim 1.4M_{\odot}$ stars [38, 39] and the tidal deformability estimates from the binary neutron star merger GW170817 [37].

Among the many models and treatments available in the quark sector [6], we utilize the vMIT model of Gomes et al. [60] as a caricature of strongly interacting quarks at the densities attained within NSs. Such interactions between quarks are required to satisfy astrophysical data, particularly those of heavy mass NSs. For the treatment of the nucleon-to-quark transition at supra-nuclear densities, we employ the Gibbs construction [61] which renders the transition to be smooth. Alternative models and treatments that feature strong first- or second-order phase transitions will be undertaken in subsequent work.

As the difference of the adiabatic and equilibrium sound speeds drives the restoring force for $g$-modes, it is instructive to collect some general expressions for these two sound speeds in the pure and mixed phases. For the pure phase of $npe$ matter, these expressions are derived and applied in [20], but given their central role in this work, and the fact that we also extend the application to $npe\mu$ and quark matter, we detail their derivation below for completeness. For the mixed phase, we derive expressions that have not, to our knowledge, been previously reported in the literature. First, a point of notation: the equilibrium squared sound speed is commonly defined in the literature [20, 64, 6] by the symbol $c_{s}^{2}$, which we reserve here for the squared adiabatic sound speed, as in [29]. The equilibrium sound speed is defined by

where $p$ and $\epsilon$ are the total pressure and energy density in matter

In Eq.(26), the leptonic energies are the $T$=0 degenerate Fermi gas expressions for massive leptons. Being a total derivative, the derivative is taken along the curve satisfying both mechanical and chemical equilibrium, i.e., $\beta$-equilibrium conditions hold. In NSM, when only $npe$ are present in equilibrium, the composition at fixed baryon density ($n_{n}+n_{p}$) is completely fixed once the proton fraction $x_{p}$ (=$x_{e}$ by charge neutrality) is determined. In this case, the squared adiabatic sound speed is defined as

where $x=x_{p}=x_{e}$. In the partial derivative, the composition is held fixed, i.e., $\beta$-equilibrium conditions are imposed only after all derivatives have been evaluated. The resulting distinction between these two speeds plays an important role in determining the oscillation frequencies of non-radial oscillations such as $g$-modes:

Note that both the above speeds are dependent on density which varies over a large range in NSM. Furthermore, an individual knowledge of both speeds is required. In what follows, we apply Eqs. (25) and (27) to the case of a pure and mixed phase.