@unpublished{constantinou2023framework,
  title = {A framework for phase transitions between the Maxwell and Gibbs constructions},
  author = {Constantinou, Constantinos and Zhao, Tianqi and Han, Sophia and Prakash, Madappa},
  note = {arXiv:2302.04289},
  doi = {2302.04289},
  file = {2302.04289.pdf}
}

By taking the nucleon-to-quark phase transition within a neutron star as an example, we present a thermodynamically-consistent method to calculate the equation of state of ambient matter so that transitions that are intermediate to those of the familiar Maxwell and Gibbs constructions can be described. This method does not address the poorly known surface tension between the two phases microscopically (as, for example, in the calculation of the core pasta phases via the Wigner-Seitz approximation) but instead combines the local and global charge neutrality conditions characteristic of the Maxwell and Gibbs constructions, respectively. Overall charge neutrality is achieved by dividing the leptons to those that obey local charge neutrality (Maxwell) and those that maintain global charge neutrality (Gibbs). The equation of state is obtained by using equilibrium constraints derived from minimizing the total energy density results of which are then used to calculate neutron star mass-radius curves, tidal deformabilities, equilibrium and adiabatic sound speeds and non-radial g-mode oscillation frequencies for several intermediate constructions. Various quantities of interest transform smoothly from their Gibbs structures to those of Maxwell as the local-to-total electron ratio g, introduced to mimic the hadron-to-quark interface tension from 0 (Gibbs) to infinity (Maxwell), is raised from 0 to1, A notable exception is the g-mode frequency for the specific case of g=1 for which a gap appears between the quark and hadronic branches.

@unpublished{sen2022radial,
  title = {Radial oscillations in neutron stars from unified hadronic and quarkyonic equation of states},
  author = {Sen, Souhardya and Kumar, Shubham and Kunjipurayi, Athul and Routaray, Pinku and Zhao, Tianqi and Kumar, Bharat},
  note = {arXiv:2205.02076},
  doi = {2205.02076},
  file = {2205.02076.pdf}
}

We study radial oscillations in non-rotating neutron stars by considering the unified equation of states (EoSs), which support the 2 M⊙ star criterion. We solve the Sturm-Liouville problem to compute 20 lowest radial oscillation modes and their eigenfunctions for neutron star modelled with eight selected unified EoSs from distinct Skyrme-Hartree Fock, Relativistic Mean-Field and quarkyonic models. We compare the behavior of the computed eigenfrequency for NS modelled with hadronic to that with quarkyonic EoSs while varying central densities. The lowest order, f-mode frequency varies substantially between the two classes of the of EoS at 1.4 M⊙ but vanishes at their respective maximum masses, consistent with the stability criterion ∂M/∂ρc>0. Moreover, we also computed large frequency separation and discovered that higher-order mode frequencies are significantly reduced by incorporating crust in the EoS.

@unpublished{routray2022radial,
  title = {Radial Oscillations of Dark Matter admixed Neutron Stars},
  author = {Routray, Pinku and Das, HC and Sen, Souhardya and Kumar, Bharat and Panotopoulos, Grigoris and Zhao, Tianqi},
  note = {arXiv:2211.12808},
  doi = {2211.12808},
  file = {2211.12808.pdf}
}

Within the relativistic mean-field model, we investigate the properties of dark matter (DM) admixed neutron stars, considering non-rotating objects made of isotropic matter. We adopt the IOPB-I hadronic equation of state (EOS) by assuming that the fermionic DM within super-symmetric models has already been accreted inside the neutron star (NS). The impact of DM on the mass-radius relationships and the radial oscillations of pulsating DM admixed neutron stars (with and without the crust) are explored. It is observed that the presence of DM softens the EOS, which in turn lowers the maximum mass and its corresponding radius. Moreover, adding DM results in higher frequencies of pulsating objects and hence we show the linearity of fundamental mode frequency of canonical NS with DM Fermi momentum. We also investigate the profile of eigenfunctions solving the Sturm-Liouville boundary value problem, and verify its validity. Further, we study the stability of NSs considering the fundamental mode frequency variation with the mass of the star, and verify the stability criterion ∂M/∂ρc>0. Finally, the effect of the crust on the large frequency separation for different DM Fermi momenta is shown as well.

@article{zhao22universal,
  title = {Universal relations for neutron star $f$-mode and $g$-mode oscillations},
  author = {Zhao, Tianqi and Lattimer, James M.},
  journal = {Phys. Rev. D},
  volume = {106},
  issue = {12},
  pages = {123002},
  numpages = {23},
  year = {2022},
  month = dec,
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevD.106.123002},
  url = {https://link.aps.org/doi/10.1103/PhysRevD.106.123002},
  file = {zhao22universal.pdf}
}

Among the various oscillation modes of neutron stars, f- and g- modes are the most likely to be ultimately observed in binary neutron star mergers. The f-mode is known to correlate in normal neutron stars with their tidal deformability, moment of inertia and quadrupole moment. Using a piecewise polytropic parameterization scheme to model the uncertain hadronic high-density EOS and a constant sound-speed scheme to model pure quark matter, we refine this correlation and show that these universal relations also apply to both self-bound stars and hybrid stars containing phase transitions. We identify a novel 1-node branch of the f-mode that occurs in low-mass hybrid stars in a narrow mass range just beyond the critical mass necessary for a phase transition to appear. This 1-node branch shows the largest, but still small, deviations from the universal correlation we have found. The g-mode frequency only exists in matter with a non-barotropic equation of state involving temperature, chemical potential or composition, or a phase transition in barotropic matter. The g-mode therefore could serve as a probe for studying phase transitions in hybrid stars. In contrast with the f-mode, discontinuity g-mode frequencies depend strongly on properties of the transition (the density and the magnitude of the discontinuity) at the transition. Imposing causality and maximum mass constraints, the g-mode frequency in hybrid stars is found to have an upper bound of about 1.25 kHz. However, if the sound speed c_s in the inner core at densities above the phase transition density is restricted to c_s^2 < c^2/3, the g-mode frequencies can only reach about 0.8 kHz, which are significantly lower than f-mode frequencies, 1.3-2.8 kHz. Also, g-mode gravitational wave damping times are extremely long, >10^4 s (10^2 s) in the inner core with c_s^2< c^2/3 (c^2), in comparison with the f-mode damping time, 0.1-1 s.

@article{kunjipurayil22impact,
  title = {Impact of the equation of state on $f$- and $p$- mode oscillations of neutron stars},
  author = {Kunjipurayil, Athul and Zhao, Tianqi and Kumar, Bharat and Agrawal, Bijay K. and Prakash, Madappa},
  journal = {Phys. Rev. D},
  volume = {106},
  issue = {6},
  pages = {063005},
  numpages = {16},
  year = {2022},
  month = sep,
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevD.106.063005},
  url = {https://link.aps.org/doi/10.1103/PhysRevD.106.063005},
  file = {kunjipurayil22impact.pdf}
}

We investigate the impact of the neutron-star matter equation of state on the f- and p1-mode oscillations of neutron stars obtained within the Cowling approximation and linearized general relativity. The f- and p1-mode oscillation frequencies, and their damping times are calculated using representative sets of Skyrme Hartree-Fock and relativistic mean-field models, all of which reproduce nuclear systematics and support 2M⊙ neutron stars. Our study shows strong correlations between the frequencies of f- and p1-modes and their damping times with the pressure of β-equilibrated matter at densities equal to or slightly higher than the nuclear saturation density ρ0. Such correlations are found to be almost independent of the composition of the stars. The frequency of the p1-mode of 1.4M⊙ star is strongly correlated with the slope of the symmetry energy L0 and β-equilibrated pressure at density ρ0. Compared to GR calculations, the error in the Cowling approximation for the f-mode is about 30% for neutron stars of low mass, whereas it decreases with increasing mass. The accuracy of the p1-mode is better than 15% for neutron stars of maximum mass, and improves for lower masses and higher number of radial nodes.

@article{zhao22qusinormal,
  title = {Quasinormal $g$ modes of neutron stars with quarks},
  author = {Zhao, Tianqi and Constantinou, Constantinos and Jaikumar, Prashanth and Prakash, Madappa},
  journal = {Phys. Rev. D},
  volume = {105},
  issue = {10},
  pages = {103025},
  numpages = {15},
  year = {2022},
  month = may,
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevD.105.103025},
  url = {https://link.aps.org/doi/10.1103/PhysRevD.105.103025},
  file = {zhao22qusinormal.pdf}
}

Quasi-normal oscillation modes of neutron stars provide a means to probe their interior composition using gravitational wave astronomy. We compute the frequencies and damping times of composition-dependent core g-modes of neutron stars containing quark matter employing linearized perturbative equations of general relativity. We find that ignoring background metric perturbations due to the oscillating fluid, as in the Cowling approximation, underestimates the g-mode frequency by up to 10% for higher mass stars, depending on the parameters of the nuclear equation of state and how the mixed phase is constructed. The g-mode frequencies are well-described by a linear scaling with the central lepton (or combined lepton and quark) fraction for nucleonic (hybrid) stars. Our findings suggest that neutron stars with and without quarks are manifestly different with regards to their quasi-normal g-mode spectrum, and may thus be distinguished from one another in future observations of gravitational waves from merging neutron stars.

@article{limiting21drischler,
  title = {Limiting masses and radii of neutron stars and their implications},
  author = {Drischler, Christian and Han, Sophia and Lattimer, James M. and Prakash, Madappa and Reddy, Sanjay and Zhao, Tianqi},
  journal = {Phys. Rev. C},
  volume = {103},
  issue = {4},
  pages = {045808},
  numpages = {23},
  year = {2021},
  month = apr,
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevC.103.045808},
  url = {https://link.aps.org/doi/10.1103/PhysRevC.103.045808},
  file = {limiting21drischler.pdf}
}

We combine equation of state of dense matter up to twice nuclear saturation density (nsat=0.16fm−3) obtained using chiral effective field theory (χEFT), and recent observations of neutron stars to gain insights about the high-density matter encountered in their cores. A key element in our study is the recent Bayesian analysis of correlated EFT truncation errors based on order-by-order calculations up to next-to-next-to-next-to-leading order in the χEFT expansion. We refine the bounds on the maximum mass imposed by causality at high densities, and provide stringent limits on the maximum and minimum radii of ∼1.4M⊙ and ∼2.0M⊙ stars. Including χEFT predictions from nsat to 2nsat reduces the permitted ranges of the radius of a 1.4M⊙ star, R1.4, by ∼3.5km. If observations indicate R1.4<11.2km, our study implies that either the squared speed of sound c2s>1/2 for densities above 2nsat, or that χEFT breaks down below 2nsat. We also comment on the nature of the secondary compact object in GW190814 with mass ≃2.6M⊙, and discuss the implications of massive neutron stars >2.1M⊙(2.6M⊙) in future radio and gravitational-wave searches. Some form of strongly interacting matter with c2s>0.35(0.55) must be realized in the cores of such massive neutron stars. In the absence of phase transitions below 2nsat, the small tidal deformability inferred from GW170817 lends support for the relatively small pressure predicted by χEFT for the baryon density nB in the range 1−2nsat. Together they imply that the rapid stiffening required to support a high maximum mass should occur only when nB≳1.5−1.8nsat.

@article{zhao20quarkyonic,
  title = {Quarkyonic matter equation of state in beta-equilibrium},
  author = {Zhao, Tianqi and Lattimer, James M.},
  journal = {Phys. Rev. D},
  volume = {102},
  issue = {2},
  pages = {023021},
  numpages = {12},
  year = {2020},
  month = jul,
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevD.102.023021},
  url = {https://link.aps.org/doi/10.1103/PhysRevD.102.023021},
  file = {zhao20quarkyonic.pdf}
}

Quasi-normal oscillation modes of neutron stars provide a means to probe their interior composition using gravitational wave astronomy. We compute the frequencies and damping times of composition-dependent core g-modes of neutron stars containing quark matter employing linearized perturbative equations of general relativity. We find that ignoring background metric perturbations due to the oscillating fluid, as in the Cowling approximation, underestimates the g-mode frequency by up to 10% for higher mass stars, depending on the parameters of the nuclear equation of state and how the mixed phase is constructed. The g-mode frequencies are well-described by a linear scaling with the central lepton (or combined lepton and quark) fraction for nucleonic (hybrid) stars. Our findings suggest that neutron stars with and without quarks are manifestly different with regards to their quasi-normal g-mode spectrum, and may thus be distinguished from one another in future observations of gravitational waves from merging neutron stars.

@article{zhao18tidal,
  title = {Tidal deformabilities and neutron star mergers},
  author = {Zhao, Tianqi and Lattimer, James M.},
  journal = {Phys. Rev. D},
  volume = {98},
  issue = {6},
  pages = {063020},
  numpages = {15},
  year = {2018},
  month = sep,
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevD.98.063020},
  url = {https://link.aps.org/doi/10.1103/PhysRevD.98.063020},
  file = {zhao18tidal.pdf}
}

Finite size effects in a neutron star merger are manifested, at leading order, through the tidal deformabilities (Lambdas) of the stars. If strong first-order phase transitions do not exist within neutron stars, both neutron stars are described by the same equation of state, and their Lambdas are highly correlated through their masses even if the equation of state is unknown. If, however, a strong phase transition exists between the central densities of the two stars, so that the more massive star has a phase transition and the least massive star does not, this correlation will be weakened. In all cases, a minimum Lambda for each neutron star mass is imposed by causality, and a less conservative limit is imposed by the unitary gas constraint, both of which we compute. In order to make the best use of gravitational wave data from mergers, it is important to include the correlations relating the Lambdas and the masses as well as lower limits to the Lambdas as a function of mass. Focusing on the case without strong phase transitions, and for mergers where the chirp mass M_chirp<1.4M_sun, which is the case for all observed double neutron star systems where a total mass has been accurately measured, we show that the dimensionless Lambdas satisfy Lambda_1/Lambda_2= q^6, where q=M_2/M_1 is the binary mass ratio; M is mass of each star, respectively. Moreover, they are bounded by q^n_->Lambda_1/Lambda_2> q^n_0++qn_1+, where n_-<n_0++qn_1+; the parameters depend only on M_chirp, which is accurately determined from the gravitational-wave signal. We also provide analytic expressions for the wider bounds that exist in the case of a strong phase transition. We argue that bounded ranges for Lambda_1/Lambda_2, tuned to M_chirp, together with lower bounds to Lambda(M), will be more useful in gravitational waveform modeling than other suggested approaches.

@inproceedings{kunjipurayil2022f,
  title = {The f-mode and density dependence of symmetry energy},
  author = {Kunjipurayil, Athul and Zhao, Tianqi and Kumar, Bharat and Agrawal, Bijay K and Prakash, Madappa},
  booktitle = {Proceedings of the DAE Symp. on Nucl. Phys},
  volume = {66},
  pages = {728},
  year = {2022},
  file = {kunjipurayil2022f.pdf}
}

Any external or internal influ- ence/perturbation can cause the fluid in a neutron star (NS) to oscillate resulting in a loss of equilibrium. However, restoring forces act to return the star to a state of equilibrium causing different modes of oscillation. The complex eigen frequencies corresponding to these oscillation modes, termed “quasi-normal modes” can be calculated in general relativ- ity. Here, we solve the fluid perturbation equations to calculate the fundamental mode, called the f-mode, of quadrupolar oscillation. The structure and bulk properties of NSs are determined by the pressure vs energy den- sity relation or the equation of state (EOS) of beta-stable and charge neutral matter within these stars. Usually, the EOSs are charac- terised by a set of nuclear matter parame- ters (NMP) evaluated at the saturation den- sity, ρ0, that describes the symmetric nuclear matter part and the density dependence of the symmetry energy. The NMPs considered are the binding energy e0 , the incompressibility K0, the skewness coefficient Q0, the symme- try energy coefficient J0, its slope L0, and the curvature Ksym,0. A central problem in nu- clear physics and nuclear astrophysics is to constrain the EOS at high densities. In this contribution, we present correlations of the f-mode frequencies with various NMPs. We have considered 35 different EOS’s of which 23 correspond to non-relativistic mean field models based on the Skyrme interaction. The remaining 12 represent the two different.