Equation-of-state Table with Hyperon and Antikaon for Supernova and Neutron Star Merger

The study of the equation of state (EOS) of matter in neutron stars has reached a high level of sophistication observationally as well as theoretically in recent years. This journey had started with the discovery of the first pulsar (Hewish et al. 1968) in the year 1967. In recent years, it has gotten a huge boost with the detection of gravitational waves in binary neutron star (BNS) merger event GW170817 (Abbott et al. 2017a, 2017b, 2018, 2019; The LIGO Scientific Collaboration et al. 2019, 2020). Parallelly, the observations of heavy NSs led to the determination of masses ≥2 M_⊙ from post Keplerian parameters using the pulsar timing technique (Antoniadis et al. 2013; Cromartie et al. 2019). The simultaneous measurement of mass and radius of the X-ray-powered pulsar PSR J0030+0451 is another important step forward by the Neutron Star Composition Explorer (NICER; Miller et al. 2019; Raaijmakers et al. 2019; Riley et al. 2019). All these observations are providing valuable inputs to constrain the EOS from low to very high density, which could not otherwise be probed using the knowledge of laboratory experiments on nuclei and relativistic heavy ion collisions.

Observations of galactic massive pulsars PSR J0348+0432 of 2.01 ± 0.04 M_⊙, PSR J0740+6620 of ${2.14}_{-0.09}^{+0.10}$ M_⊙, and PSR J1810+1744 of 2.13 ± 0.04 M_⊙ set a lower limit on the maximum mass of NSs (Antoniadis et al. 2013; Cromartie et al. 2019; Romani et al. 2021). On the other hand, the matter ejected in the BNS merger GW170817 was also observed across the wide electromagnetic spectrum. An upper bound on the maximum mass of NSs might be obtained from the electromagnetic observation of GW170817 if the remnant collapsed to a black hole. Different groups estimated the upper bound to be in the range of ∼2.17 − 2.3 M_⊙ (Margalit & Metzger 2017; Shibata et al. 2017, 2019; Rezzolla et al. 2018; Ruiz et al. 2018). Furthermore, it was possible for the first time to extract the value of tidal deformability ($\widetilde{{\rm{\Lambda }}}$) from the gravitational wave signal of GW170817. Its range of $70\leqslant \widetilde{{\rm{\Lambda }}}\leqslant 720$ gives an estimate for the radius of 9–14 km for a 1.4 M_⊙ NS (Abbott et al. 2018; De et al. 2018; Fattoyev et al. 2018; Most et al. 2018; Raithel et al. 2018; Zhao & Lattimer 2018; Soma & Bandyopadhyay 2020). The values of the mass–radius relation of PSR J0030+0451 from two different analyses of NICER results are ${1.44}_{-0.14}^{+0.15}$ M_⊙ − 13.02${}_{-1.06}^{+1.24}$ km and ${1.34}_{-0.16}^{+0.15}$ M_⊙ − 12.71${}_{-1.19}^{+1.14}$ km (Miller et al. 2019; Riley et al. 2019). Another compact binary coalescence event, GW190814, involving a black hole of ${23.2}_{-1.0}^{+1.1}{M}_{\odot }$ and a compact object of ${2.59}_{-0.9}^{+0.8}{M}_{\odot }$, was reported recently (Abbott et al. 2020). This event has generated a debate regarding whether the secondary mass makes the compact object the heaviest NS or the lightest black hole. The observed values of masses and radii are stringent probes of the EOS of NS matter. Theoretical models of the EOS should be compatible with these observations.

The theoretical modeling of the EOS has undergone a sea of changes over the past several years. Traditional EOS models are based on two-body plus three-body interactions in nonrelativistic approaches and the strong-interaction Lagrangian in relativistic field theoretic approaches (Lattimer & Swesty 1991; Shen et al. 1998). Furthermore, compositions of matter are assumed in both types of models. Recently, large numbers of EOSs were constructed by adopting the phenomenological Skyrme interaction or nucleon–nucleon chiral potentials for the low density and perturbative quantum chromodynamics for asymptotically high-density regimes. The EOS in the intermediate-density regime is connected to EOSs at two extreme densities either by polytropes or just demanding the causality and monotonicity (Hebeler et al. 2013; Kurkela et al. 2014; Most et al. 2018; Lope Oter et al. 2019). However, such EOS models do not assume any kind of composition of matter in the intermediate-density regime, which plays the most important role for masses and radii of NSs.

Traditional models of the EOS are widely used in numerical relativity simulations of compact astrophysical objects. Our motivation in this work is to compute an EOS including strange matter within the framework of relativistic field theoretical models to be used as input in core-collapse supernova (CCSN) and NS merger simulations.

A large set of EOSs with and without strange matter, such as hyperon, antikaon condensed, or quark matter, are already available for CCSN and NS merger simulations (Hempel & Schaffner-Bielich 2010; Raduta & Gulminelli 2010; Shen et al. 2010, 2011a, 2011b; Blinnikov et al. 2011; Fischer et al. 2011, 2014; Hempel et al. 2012; Steiner et al. 2013; Typel et al. 2013; Banik et al. 2014; Buyukcizmeci et al. 2014; Constantinou et al. 2014; Togashi et al. 2014; Oertel et al. 2017; Malik et al. 2021). The EOS tables with nucleons-only matter satisfy the 2 M_⊙ lower bound on the maximum mass of NSs. However, many EOSs with hyperons or quarks used in CCSN simulations do not conform to the lower bound on the maximum mass (Ishizuka et al. 2008; Nakazato et al. 2008; Sagert et al. 2009; Sumiyoshi et al. 2009; Shen et al. 2011c; Nakazato et al. 2012; Oertel et al. 2012; Peres et al. 2013; Banik 2014). Keeping this in mind, we constructed an EOS table including Λ-hyperons, which is compatible with 2 M_⊙ NSs (Banik et al. 2014). This EOS, known as BHBΛϕ, is being widely used in CCSN (Char et al. 2015) and BNS merger simulations (Radice et al. 2018). It has long been debated that the Bose–Einstein condensate of negatively charged kaons might appear in dense matter (Kaplan & Nelson 1986; Knorren et al. 1995). This idea was extended to understand the nonobservation of an NS in SN 1987A (Bethe & Brown 1995). There was no EOS table involving an antikaon condensate in nuclear matter before our very recent work on this problem (Malik et al. 2021). We did not consider the appearance of hyperons along with the antikaon condensate in the previous work. The early appearance of hyperons would delay the onset of the Bose–Einstein condensate of antikaons in dense matter or vice versa. It is worth investigating how one form of strange matter behaves in the presence of another. This motivates us to investigate this issue and extend our hyperon EOS BHBΛϕ to include the Bose–Einstein condensate of K⁻ mesons fulfilling the most updated information on NSs.

The paper is organized as follows. In Section 2, the hadronic field theory models of EOSs at zero and finite temperatures are described. The results of our calculation are discussed in Section 3. Section 4 contains the summary and conclusions.

2.1. Hadronic Model

The baryonic matter is described within the framework of the density-dependent model adopting the relativistic mean field (RMF) approximation. The baryon–baryon interaction is mediated by σ, ω, ρ, and ϕ mesons. However, nucleons do not couple to ϕ mesons; they account for the repulsive hyperon–hyperon interaction. The Lagrangian density is given by (Typel et al. 2010; Banik et al. 2014)

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{ \mathcal L } & = & \displaystyle \sum _{B=N,{\rm{\Lambda }}}{\bar{{\rm{\Psi }}}}_{B}\left(i{\gamma }_{\mu }{{\rm{\partial }}}^{\mu }-{m}_{B}+{g}_{\sigma B}\sigma \right.\\ & & -\,\left.{g}_{\omega B}{\gamma }_{\mu }{\omega }^{\mu }-{g}_{\rho B}{\gamma }_{\mu }{{\boldsymbol{\tau }}}_{{\boldsymbol{B}}}\cdot {{\boldsymbol{\rho }}}^{{\boldsymbol{\mu }}}-{g}_{\phi B}{\gamma }_{\mu }{\phi }^{\mu }\right){{\rm{\Psi }}}_{B}\\ & & +\,\displaystyle \frac{1}{2}\left({{\rm{\partial }}}_{\mu }\sigma {{\rm{\partial }}}^{\mu }\sigma -{m}_{\sigma }^{2}{\sigma }^{2}\right)-\displaystyle \frac{1}{4}{\omega }_{\mu \nu }{\omega }^{\mu \nu }+\displaystyle \frac{1}{2}{m}_{\omega }^{2}{\omega }_{\mu }{\omega }^{\mu }\\ & & -\,\displaystyle \frac{1}{4}{\rho }_{\mu \nu }\cdot {\rho }^{\mu \nu }+\displaystyle \frac{1}{2}{m}_{\rho }^{2}{{\boldsymbol{\rho }}}_{{\boldsymbol{\mu }}}\cdot {{\boldsymbol{\rho }}}^{{\boldsymbol{\mu }}}\\ & & -\,\displaystyle \frac{1}{4}{\phi }_{\mu \nu }{\phi }^{\mu \nu }+\displaystyle \frac{1}{2}{m}_{\phi }^{2}{\phi }_{\mu }{\phi }^{\mu },\end{array}\end{array}\end{eqnarray} \tag{ 1 }$$

where Ψ_B denotes the isospin multiplets for baryons B, with m_B being their bare masses and τ_B the isospin operator. The density-dependent meson–nucleon couplings are denoted by g_xB with x = σ, ω, and ρ meson fields. The vector meson field strength tensors are particularly represented by x^{μ ν} = ∂^μ x^ν − ∂^ν x^μ . In the mean field approximations, meson fields are replaced by their expectation values $\left\langle x\right\rangle$. Only the time-like components of vector fields and the third isospin component of the ρ field survive in a uniform and static matter. They are denoted by σ, ω₀, ρ₀₃, and ϕ₀ and are obtained by solving the meson field equations in the RMF approximation,

$$\begin{eqnarray*}\begin{array}{rcl}{m}_{\sigma }^{2}\sigma & = & \displaystyle \sum _{B}{g}_{\sigma B}{\bar{\psi }}_{\sigma B}{\psi }_{B},\quad {m}_{\omega }^{2}{\omega }_{0}=\displaystyle \sum _{B}{g}_{\omega B}{\bar{\psi }}_{B}{\gamma }_{0}{\psi }_{B},\\ {{m}_{\rho }}^{2}{\rho }_{03} & = & \displaystyle \frac{1}{2}\displaystyle \sum _{B}{g}_{\rho B}{\bar{\psi }}_{B}{\gamma }_{0}{\tau }_{3B}{\psi }_{B},\quad {m}_{\phi }^{2}{\phi }_{0}=\displaystyle \sum _{B}{g}_{\phi B}{\bar{\psi }}_{B}{\gamma }_{0}{\psi }_{B}.\end{array}\end{eqnarray*}$$

The grand-canonical thermodynamic potential per unit volume of the hadronic phase is given by (Banik et al. 2014; Soma & Bandyopadhyay 2020)

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{\Omega }}}{V} & = & \displaystyle \frac{1}{2}{m}_{\sigma }^{2}{\sigma }^{2}-\displaystyle \frac{1}{2}{m}_{\omega }^{2}{\omega }_{0}^{2}-\displaystyle \frac{1}{2}{m}_{\rho }^{2}{\rho }_{03}^{2}\\ & & -\displaystyle \frac{1}{2}{m}_{\phi }^{2}{\phi }_{0}^{2}-{{\rm{\Sigma }}}^{r}\displaystyle \sum _{B=n,p,{\rm{\Lambda }}}{n}_{B}\\ & & -2T\displaystyle \sum _{B=n,p,{\rm{\Lambda }}}\int \displaystyle \frac{{d}^{3}k}{{\left(2\pi \right)}^{3}}\left[\mathrm{ln}\left(1+{e}^{-\beta ({E}^{* }-{\nu }_{B})}\right)\right.\\ & & \left.+\mathrm{ln}\left(1+{e}^{-\beta ({E}^{* }+{\nu }_{B})}\right)\right],\end{array}\end{eqnarray} \tag{ 2 }$$

where the temperature is defined as β = 1/T and ${E}^{* }\,=\sqrt{({k}^{2}+{m}_{B}^{* 2})}$.

The number densities and scalar number densities of baryon B are ${n}_{B}= \langle {\bar{\psi }}_{B}{\gamma }_{0}{\psi }_{B} \rangle$ and ${n}_{B}^{S}= \langle {\bar{\psi }}_{\sigma B}{\psi }_{B} \rangle$, respectively, and at finite temperature (T) are given by

$$\begin{eqnarray}&&{n}_{B}=2\int \displaystyle \frac{{d}^{3}k}{{\left(2\pi \right)}^{3}}\left(\displaystyle \frac{1}{{e}^{\beta ({E}^{* }-{\nu }_{B})}+1}-\displaystyle \frac{1}{{e}^{\beta ({E}^{* }+{\nu }_{B})}+1}\right),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{n}_{B}^{S}=2\int \displaystyle \frac{{d}^{3}k}{{\left(2\pi \right)}^{3}}\displaystyle \frac{{m}_{B}^{* }}{{E}^{* }}\left(\displaystyle \frac{1}{{e}^{\beta ({E}^{* }-{\nu }_{B})}+1}+\displaystyle \frac{1}{{e}^{\beta ({E}^{* }+{\nu }_{B})}+1}\right).\end{eqnarray} \tag{ 4 }$$

Here the effective nucleon mass is ${m}_{B}^{* }={m}_{B}-{g}_{\sigma B}\sigma$. The chemical potential is given by μ_B = ν_B + g_{ω B} ω₀ + g_{ρ B} τ_3B ρ₀₃ + g_{ϕ B} ϕ₀ + Σ^r , where the rearrangement term Σ^r takes care of many-body effects in nuclear interaction (Typel et al. 2010; Banik et al. 2014). It arises as a result of the density dependence of the couplings and is expressed as

$$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Sigma }}}^{r} & = & \displaystyle \sum _{B}\left[-\displaystyle \frac{\partial {g}_{\sigma B}}{\partial {n}_{B}}\sigma {n}_{B}^{s}+\displaystyle \frac{\partial {g}_{\omega B}}{\partial {n}_{B}}{\omega }_{0}{n}_{B}\right.\\ & & +\left.\displaystyle \frac{\partial {g}_{\rho B}}{\partial {n}_{B}}{\tau }_{3N}{\rho }_{03}{n}_{B}+\displaystyle \frac{\partial {g}_{\phi B}}{\partial {n}_{B}}{\phi }_{0}{n}_{B}\right].\end{array}\end{eqnarray} \tag{ 5 }$$

The pressure is calculated from the grand-canonical thermodynamic potential per unit volume as P = − Ω/V as given by Equation (2) and includes the rearrangement term.

However, the energy density does not explicitly contain the rearrangement term, as evident from its expression,

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{\epsilon }_{B} & = & \displaystyle \frac{1}{2}{m}_{\sigma }^{2}{\sigma }^{2}+\displaystyle \frac{1}{2}{m}_{\omega }^{2}{\omega }_{0}^{2}+\displaystyle \frac{1}{2}{m}_{\rho }^{2}{\rho }_{03}^{2}+\displaystyle \frac{1}{2}{m}_{\phi }^{2}{\phi }_{0}^{2}\\ & & +\,2\displaystyle \sum _{B=N,{\rm{\Lambda }}}\int \displaystyle \frac{{d}^{3}k}{{\left(2\pi \right)}^{3}}{E}^{\ast }\left(\displaystyle \frac{1}{{e}^{\beta ({E}^{\ast }-{\nu }_{B})}+1}+\displaystyle \frac{1}{{e}^{\beta ({E}^{\ast }+{\nu }_{B})}+1}\right).\end{array}\end{array}\end{eqnarray} \tag{ 6 }$$

The rearrangement term accounts for not only the energy-momentum conservation but also the thermodynamic consistency of the system. The energy density and pressure are related through the Gibbs–Duhem relation, i.e., ${{ \mathcal S }}_{B}=\beta \left({\epsilon }_{B}+{P}_{B}-{\sum }_{B}{\mu }_{B}{n}_{B}\right)$, where ${{ \mathcal S }}_{B}$ is the entropy density of baryon B.

2.2. Model for Antikaons

Kaon–nucleon interaction is considered in the same footing as the nucleon–nucleon interaction in Equation (1). The Lagrangian density for (anti)kaons in the minimal coupling scheme is (Glendenning & Schaffner-Bielich 1999; Pons et al. 2000; Banik & Bandyopadhyay 2001; Banik et al. 2008; Char & Banik 2014)

$$\begin{eqnarray}&&{{ \mathcal L }}_{K}={D}_{\mu }^{* }\bar{K}{D}^{\mu }K-{m}_{K}^{* 2}\bar{K}K,\end{eqnarray} \tag{ 7 }$$

where K and $\bar{K}$ denote kaon and (anti)kaon doublets; the covariant derivative is D_μ = ∂_μ + ig_{ω K} ω_μ + ig_{ρ K} τ_K · ρ_μ + ig_{ϕ K} ϕ_μ , and the effective mass of antikaon is ${m}_{K}^{* }={m}_{K}-{g}_{\sigma K}\sigma$. Meson fields are modified in the presence of K⁻ condensate (Glendenning & Schaffner-Bielich 1999; Banik & Bandyopadhyay 2001) and are obtained by solving the following equations:

$$\begin{eqnarray*}\begin{array}{rcl}{m}_{\sigma }^{2}\sigma & = & \displaystyle \sum _{B}{g}_{\sigma B}{n}_{B}^{S}+\displaystyle \sum _{K}{g}_{\sigma K}{n}_{K},\\ {m}_{\omega }^{2}{\omega }_{0} & = & \displaystyle \sum _{B}{g}_{\omega B}{n}_{B}-\displaystyle \sum _{K}{g}_{\sigma K}{n}_{B},\\ {{m}_{\rho }}^{2}{\rho }_{03} & = & \displaystyle \frac{1}{2}\displaystyle \sum _{B}{g}_{\rho B}{\tau }_{3B}{n}_{B}+\displaystyle \sum _{K}{g}_{\sigma K}{n}_{K}{\tau }_{3K},\\ {m}_{\phi }^{2}{\phi }_{0} & = & \displaystyle \sum _{B}{g}_{\phi B}{n}_{B}-\displaystyle \sum _{K}{g}_{\sigma K}{n}_{K}.\end{array}\end{eqnarray*}$$

The thermodynamic potential for (anti)kaons is given by (Pons et al. 2000; Banik et al. 2008)

$$\begin{eqnarray}&&\displaystyle \frac{{{\rm{\Omega }}}_{K}}{V}=T\int \displaystyle \frac{{d}^{3}k}{{\left(2\pi \right)}^{3}}[{\rm{ln}}(1-{e}^{-\beta ({\omega }_{{K}^{-}}-\mu )})+{\rm{ln}}(1-{e}^{-\beta ({\omega }_{{K}^{+}}+\mu )})],\end{eqnarray} \tag{ 8 }$$

where the in-medium energies of K^± mesons are given by

$$\begin{eqnarray}&&{\omega }_{{K}^{\pm }}=\sqrt{({k}^{2}+{m}_{K}^{* 2})}\pm \left({g}_{\omega K}{\omega }_{0}+\displaystyle \frac{1}{2}{g}_{\rho K}{\rho }_{03}+{g}_{\phi K}{\phi }_{0}\right).\end{eqnarray} \tag{ 9 }$$

The chemical potentials of nucleons are related to that of K⁻ mesons by μ = μ_n − μ_p (Pons et al. 2000). For s-wave ( k = 0) condensation, the momentum dependence vanishes in ${\omega }_{{K}^{\pm }}$. The in-medium energy of K⁻ condensate decreases from its vacuum value m_K as the meson fields build up with increasing density. The K⁻ condensate appears in the system as ${\omega }_{{K}^{-}}$ equals its chemical potential, i.e., $\mu ={\omega }_{{K}^{-}}={m}_{K}^{* }-{g}_{\omega K}{\omega }_{0}-\tfrac{1}{2}{g}_{\rho K}{\rho }_{03}-{g}_{\phi K}{\phi }_{0}$. This is the threshold condition for K⁻ condensation. Incidentally, the threshold condition for K⁺ condition $\mu ={\omega }_{{K}^{+}}$ is never attained because of the repulsive nature of K⁺−nucleon interaction. We calculate the pressure due to thermal (anti)kaons using P_K = − Ω_K /V. The K⁻ condensate does not contribute to the pressure. The energy density of (anti)kaons is due to the condensate, as well as the thermal (anti)kaons, and is given by

$$\begin{eqnarray}\begin{array}{rcl}{\epsilon }_{K} & = & {m}_{K}^{* }{n}_{K}^{C}+\left({g}_{\omega K}{\omega }_{0}+\displaystyle \frac{1}{2}{g}_{\rho K}{\rho }_{03}+{g}_{\phi K}{\phi }_{0}\right){n}_{K}^{T}\\ & & +\int \displaystyle \frac{{d}^{3}k}{{\left(2\pi \right)}^{3}}\left(\displaystyle \frac{{\omega }_{{K}^{-}}}{{e}^{\beta ({\omega }_{{K}^{-}}-\mu )}-1}+\displaystyle \frac{{\omega }_{{K}^{+}}}{{e}^{\beta ({\omega }_{{K}^{+}}+\mu )}-1}\right).\end{array}\end{eqnarray} \tag{ 10 }$$

The thermodynamic quantities—entropy density, energy density, pressure, chemical potential, and number density—are related to each other through ${{ \mathcal S }}_{K}=\beta \left({\epsilon }_{K}+{P}_{K}-\mu {n}_{K}\right)$. The entropy per baryon is given by $s={ \mathcal S }/{n}_{B}$, where n_B is the total baryon density. The total (anti)kaon number density (n_K ) is given by ${n}_{K}={n}_{K}^{C}+{n}_{K}^{T}\,$, where n_K ^C and n_K ^T are the K⁻ condensate and thermal (anti)kaon density, respectively. They are given by

$$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{n}_{K}^{C} & = & 2\left({\omega }_{{K}^{-}}+{g}_{\omega K}{\omega }_{0}+\displaystyle \frac{1}{2}{g}_{\rho K}{\rho }_{03}+{g}_{\phi K}{\phi }_{0}\right)\bar{K}K=2{m}_{K}^{\ast }\bar{K}K,\\ {n}_{K}^{T} & = & \int \displaystyle \frac{{d}^{3}k}{{\left(2\pi \right)}^{3}}\left(\displaystyle \frac{1}{{e}^{\beta ({\omega }_{{K}^{-}}-\mu )}-1}-\displaystyle \frac{1}{{e}^{\beta ({\omega }_{{K}^{+}}+\mu )}-1}\right).\end{array}\end{array}\end{eqnarray} \tag{ 11 }$$

2.3. Parameters of the DD2 Model

The nucleon–meson couplings depend on the density through g_{α B} (n_B ) = g_{α B} (n₀)f_α (x), where x = n_B /n₀ and

$$\begin{eqnarray*}&&{f}_{\alpha }(x)={a}_{\alpha }\displaystyle \frac{1+{b}_{\alpha }{\left(x+{d}_{\alpha }\right)}^{2}}{1+{c}_{\alpha }{\left(x+{d}_{\alpha }\right)}^{2}}\end{eqnarray*}$$

for α = ω, σ (Typel & Wolter 1999; Typel et al. 2010). An exponential density dependence is assumed for the isovector meson ρ, i.e., ${f}_{\alpha }(x)=\exp [-{a}_{\alpha }(x-1)]$, as their couplings decrease at higher densities (Typel & Wolter 1999). The DD2 parameter set of nucleon–meson couplings used in the calculation results in the saturation properties of symmetric nuclear matter such as n₀ = 0.149065 fm⁻³, binding energy per nucleon 16.02 MeV, incompressibility 242.7 MeV, symmetry energy 31.67 MeV, and its slope coefficient 55.03 MeV (Typel et al. 2010; Fischer et al. 2014).

The system is populated with Λ's when the chemical equilibrium condition μ_n = μ_Λ is met. The density-dependent Λ-vector meson hyperon vertices are obtained from the SU(6) symmetry of the quark model (Dover & Gal 1985; Schaffner & Mishustin 1996),

$$\begin{eqnarray*}&&{g}_{\omega {\rm{\Lambda }}}=\displaystyle \frac{2}{3}{g}_{\omega N},\quad {g}_{\rho {\rm{\Lambda }}}=0,\quad {g}_{\phi {\rm{\Lambda }}}=-\displaystyle \frac{\sqrt{2}}{3}{g}_{\omega N}.\end{eqnarray*}$$

On the other hand, the Λ-hyperon–scalar meson couplings are obtained from the hypernuclei data. We consider Λ-hyperon potential depth ${U}_{{\rm{\Lambda }}}^{N}={g}_{\omega {\rm{\Lambda }}}{\omega }_{0}-{g}_{\sigma {\rm{\Lambda }}}{\sigma }_{0}+{{\rm{\Sigma }}}^{r}=-30\,\mathrm{MeV}$ in normal nuclear matter (Millener et al. 1988; Schaffner et al. 1992; Mares et al. 1995), and the ratio of g_σΛ to g_{σ N} is 0.62008.

However, kaon–meson couplings are not density dependent. The kaon–vector meson coupling constants are also estimated exploiting the quark model and isospin counting rule, i.e., ${g}_{\omega K}=\tfrac{1}{3}{g}_{\omega N}$ and g_{ρ K} = g_{ρ N} (Schaffner & Mishustin 1996; Banik & Bandyopadhyay 2001). The scalar coupling constant is determined from the real part of K⁻ optical potential ${U}_{{K}^{-}}=-{g}_{\sigma K}{\sigma }_{0}-{g}_{\omega K}{\omega }_{0}+{{\rm{\Sigma }}}^{r}\,$ at the saturation density, for which an appropriate value of −120 MeV is considered in this work such that the cold β-equilibrated EOS is compatible with 2 M_⊙ NSs. The K⁻ optical potential could actually range from −60 to −200 MeV as indicated by the unitary chiral model calculations and phenomenological fit to kaonic atom data (Friedman et al. 1994, 1999; Tólos et al. 2006; Tolos & Fabbietti 2020).

2.4. Matching with the Low-density EOS

2.5. Accuracy and Consistency of the EOS Table

The following consistency checks on the EOS table are performed. Thermodynamic consistency is achieved by the condition

$$\begin{eqnarray*}&&f={\mu }_{n}{n}_{n}+{\mu }_{p}{n}_{p}+{\mu }_{{\rm{\Lambda }}}{n}_{{\rm{\Lambda }}}+\mu {n}_{K}-P,\end{eqnarray*}$$

where $f=\epsilon -T{ \mathcal S }$.

The modulus of the relative thermodynamic accuracy in this EOS table is given by

$$\begin{eqnarray*}&&{\rm{\Delta }}=\displaystyle \frac{T{ \mathcal S }-P+{\mu }_{n}{n}_{n}+{\mu }_{p}{n}_{p}+{\mu }_{{\rm{\Lambda }}}{n}_{{\rm{\Lambda }}}+\mu {n}_{K}}{\epsilon }-1\sim {10}^{-7}.\end{eqnarray*}$$

The sum rule of particle fractions (X_i ) is satisfied by the EOS table given by

$$\begin{eqnarray*}&&{X}_{n}+{X}_{p}+{X}_{s}+{X}_{A}+{X}_{{\rm{\Lambda }}}=1.\end{eqnarray*}$$

Finally, the EOS table fulfills the thermodynamic stability criteria:

$$\begin{eqnarray*}&&\displaystyle \frac{d{ \mathcal S }}{{dT}}\gt 0,\quad \displaystyle \frac{{dP}}{{{dn}}_{B}}\gt 0.\,\end{eqnarray*}$$

The EOS table with thermal (anti)kaons and K⁻ condensate is generated using the DD2 parameter set and ${U}_{{K}^{-}}\,=-120\,\mathrm{MeV}$ for baryon densities (n_B = 10⁻¹² to ∼1 fm⁻³), temperatures (T = 0.1–158.48 MeV), and positive charge fractions Y_q (n_B ) = n_p − n_K (Y_q = 0.01–0.60). Grid spacing for baryon density is ${\rm{\Delta }}\mathrm{log}10({n}_{B})=0.04;$ for temperature, ${\rm{\Delta }}\mathrm{log}10(T)=0.04$; and for positive charge fraction, ΔY_q = 0.01. The EOS table consists of 301 baryon density points, 81 temperatures, and 60 positive charge fractions, i.e., a total of 1 million data points.

Before describing the thermodynamic quantities in the BHBΛK⁻ ϕ EOS table, we discuss the β-equilibrated matter relevant to cold NSs. We generate the EOS of NSs by imposing charge neutrality with the inclusion of electrons and the β-equilibrium condition without neutrinos at temperature T = 0. In Figure 1(a), various particle fractions are plotted as a function of baryon number density. The NS core contains a high neutron fraction. The proton fraction increases monotonically as baryon density increases. The positive charges of protons are balanced by negative charges of electrons. Λ-hyperons appear at ∼2.2n_B followed by the K⁻ condensate. The early appearance of Λ-hyperons makes the EOS softer, delaying the appearance of the antikaon condensate. The K⁻ fraction in the condensate abundance reaches a considerable fraction at 3.5n_B . As soon as K⁻ appears, the e⁻ fraction drops and the charge neutrality is totally taken care of by the negatively charged condensate at higher density. On the other hand, the population of neutrons is arrested owing to the accumulation of more Λ's at higher density.

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The mass–radius relationship of the sequence of NSs is shown in Figure 1(b) for HS(DD2), HS(DD2)K⁻, BHBΛϕ, and BHBΛK⁻ ϕ EOSs. The solid line (green) represents the nucleons-only NS. On the other hand, dotted (magenta), dashed–dotted (blue), and dashed (red) lines represent NSs with additional K⁻ condensate only, Λ-hyperons only, and both Λ-hyperons and K⁻ condensate, respectively. The maximum masses for the four EOSs are 2.42, 2.24, 2.1, and 2.05 M_⊙, with their corresponding radii being 11.89, 12.0, 11.58, and 11.62 km, respectively. The Λ-hyperons make the EOS softer, resulting in a smaller as well as lighter maximum mass NS compared with that of the nucleons-only HS(DD2) EOS. The maximum mass is further lowered in the presence of K⁻ condensate. It is to be noted that the masses are well within the observational benchmark of measured 2 M_⊙ NSs (Antoniadis et al. 2013; Cromartie et al. 2019).

We now move on to the results relevant to the BHBΛK⁻ ϕ EOS table for SN and NS merger simulations. Figure 2 exhibits the composition of SN matter as a function of baryon mass density for T = 1, 50, and 100 MeV and Y_q = 0.1, 0.3, and 0.5. Number densities of various particles, such as light nuclei (Z ≤ 5), heavy nuclei (Z ≥ 6), neutrons, protons, Λ-hyperons, and antikaons (both thermal and condensate) are plotted. The heavy nuclei exist only at the low-temperature and low-density nonuniform matter. The light nuclei, on the other hand, appear in the low-density region only for T = 50 MeV and higher charge fractions, as evident from the middle panel of Figure 2. At very high temperatures >48 MeV, nuclei do not appear. Furthermore, nuclei dissolve into their fundamental constituents around normal nuclear matter density and form a uniform matter of neutrons and protons. The Λ-hyperons appear with significant abundance (>10⁻⁵) at higher density at the cost of neutrons. The higher the temperature, the lower is their threshold density. However, the population fractions of baryons do not differ with temperature at relatively higher density. Next, we focus on the (anti)kaon number density. The antikaon condensate does not appear at all for a system with higher charge fraction Y_q . Even for Y_q = 0.1, its threshold is shifted toward higher density, and it fails to appear at T = 100 MeV. This may be attributed to the abundance of thermal kaons at higher temperatures, which cannot be traced at the low-temperature regime. In fact, above the critical temperature, the condensate disappears, producing thermal (anti)kaons. Nevertheless, Λ-hyperons also play a dominant role. We have reported in an earlier work with (anti)kaons, but no hyperons, that the density of K⁻ mesons in the condensate even dominates over that of thermal (anti)kaons (n^T _K ) at T = 50 MeV for Y_q = 0.1 and 0.3 (Malik et al. 2021). Here this trend is noted only for Y_q = 0.1. Thus, we conclude that the Λ-hyperons delay or do not allow the K⁻ condensate to appear in the system, and more so at higher Y_q .

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We have studied various thermodynamic observables such as free energy per baryon, entropy per baryon, and pressure as a function of baryon mass density. In all the following figures, we only show the hadronic contribution and plot them for various regimes of temperatures, T = 1, 50, and 100 MeV, and positive charge fractions, Y_q = 0.1, 0.3, and 0.5. The free energy per baryon with respect to the arbitrary value of m₀ = 938 MeV is plotted for the BHBΛK⁻ ϕ EOS in Figure 3 as a function of mass density (red dashed line). The results for the nucleons-only HS(DD2) EOS (green solid) and BHBΛϕ EOSs (black solid line) are also drawn for comparison. At lower densities, there is practically no difference between the results of nuclear and strange matter for different situations considered. A slight difference is observed at higher density for T = 1 and 50 MeV, when the strange particles appear. The difference is quite prominent at T = 100 MeV, owing to the appearance of thermal (anti)kaons at a lower density along with the Λ values. If we compare the right panels of Figure 3 at T = 100 MeV, the effect of thermal (anti)kaons becomes evident. The difference between free energy of nuclear and strange matter is noticed to disappear apparently for Y_q = 0.5. This may be attributed to the dissolution of the condensate with the production of thermal (anti)kaons at comparatively higher density. Also, its abundance is observed to be relatively small compared to the matter with lower Y_q .

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The pressure is plotted in Figure 4, as a function of baryon mass density. Just like the free energy case, we find that the nuclear and strange EOSs do not show any difference at low densities for the different values of temperature and positive charge fractions considered here. However, at the higher density the exotic particles appear, which clearly makes the BHBΛK⁻ ϕ EOS softer compared to the HS(DD2) and BHBΛϕ EOSs. The inset box shows the high-density portion zoomed in for Y_q = 0.1 and T = 1 MeV to highlight this difference. A kink in pressure is observed at ∼10¹⁰ g cm⁻³ for T = 100 MeV and Y_q = 0.1 and 0.3, which is due to significant contribution of thermal K⁻ mesons to the pressure. There is no kink or jump in pressure when Λ hyperons appear in the system, indicating a smooth transition from nuclear to hyperon matter.

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In Figure 5 the entropy per baryon as a function of baryon mass density for the same set of values of temperatures and positive charge fractions is plotted. Here also, the effect of strange particles becomes evident in the high-density region. The kinks at low densities and temperatures originate from changes in the nuclear composition that are related to nuclear shell effects, whereas the kinks at higher densities mark the appearance of exotic particles. The difference between the three EOSs is prominent only at higher density for T = 50 MeV. However, the large fraction of thermal (anti)kaons at low density does have a significant effect at T = 100 MeV for Y_q = 0.1 and 0.3. This effect is blurred at Y_q = 0.5, as the thermal (anti)kaons appear at higher density and the BHBΛK⁻ ϕ curve deviates from HS(DD2) and BHBΛϕ EOSs at the higher-density end.

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Finally, in Figure 6 we plot the phase diagram, i.e., the temperature versus baryon density for Y_q = 0.1 and 0.3. The shaded regions below the solid lines represent the kaon condensed phase in the presence of hyperons. The solid line demarcates the hyperon matter phase from the condensate phase for a particular value of Y_q and denotes the critical temperature of kaon condensation at each density point. At higher Y_q , the condensate appears at higher density. The condensate may not appear at all if the temperature of the system is high, as was seen in Figure 2. Malik et al. (2021) exhibited a similar phase diagram of nuclear matter with the antikaon condensate. Comparing this phase diagram with that of Malik et al. (2021) clearly shows the effect of hyperons. The hyperons visibly delay the kaon condensation to a higher density. Also, the critical temperature, above which the K⁻ condensate dissolves, comes down at a given density. The condensed phase is observed to shrink with higher Y_q , in nuclear as well as hyperon matter; it ceases to exist at all in the presence of hyperons at Y_q = 0.5.

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We have constructed a new EOS table including (anti)kaons and Λ hyperons for core collapse supernova and binary NS merger simulations. We call it HS(DD2)ΛK⁻ ϕ. The low-density, nonuniform matter of this EOS table is generated within an extended version of the NSE model with excluded volume (Hempel & Schaffner-Bielich 2010). The uniform matter, on the other hand, is described in a finite-temperature, density-dependent relativistic mean field model. The nucleons are described within a widely used meson exchange model using the DD2 parameter set (Typel et al. 2010). The interaction between Λ-hyperons is mediated by additional ϕ mesons. It is noted that the fraction of Λ-hyperons is significant at ∼2 times normal nuclear matter density for cold matter. At higher temperature Λ hyperons populate even at low-density matter, and their population grows in uniform matter at the cost of neutrons at high density, eventually overshooting neutron fraction at very large density.

We have also considered the role of thermal (anti)kaons and the K⁻ condensate on the EOS and other thermodynamic observables. At low temperature and low positive charge fraction, the system is populated with K⁻ condensate and a very small amount of thermal (anti)kaons. However, a high fraction of Λ hyperons at low density does not favor the onset of K⁻ condensate. At high temperature, only the thermal (anti)kaons populate the matter.

The presence of Λ-hyperons, thermal (anti)kaons, and the antikaon condensate makes the HS(DD2)ΛK⁻ ϕ EOS softest of the nucleon-only HS(DD2) EOSs, and the strange matter EOS HS(DD2)Λϕ and the HS(DD2)K⁻ EOS.

We have shown various thermodynamic observables such as the free energy per baryon, entropy per baryon, and pressure of HS(DD2)ΛK⁻ ϕ matter as a function of baryon mass density for the set of temperatures T = 1, 50, and 100 MeV and positive charge fractions Y_q = 0.1, 0.3, and 0.5. For comparison we also plot the corresponding quantities of HS(DD2) and HS(DD2) Λϕ matter. The effect of the strange particles is mainly observed at high density, which becomes visibly prominent at high temperature.

Finally, we also compute the EOS of the charge-neutral and β-equilibrated cold EOS and report the lowering of maximum mass NS in the presence of strange particles. The maximum masses are 2.42, 2.24, 2.1, and 2.05 M_⊙; all are compatible with the heavy NSs of masses ∼ 2 M_⊙, discovered in the past decade (Antoniadis et al. 2013; Cromartie et al. 2019). Also, the radii match with the range of radius calculated from the tidal deformability extracted from the BNS merger GW170817 event for a 1.4 M_⊙ NS (Abbott et al. 2017b). Recently, the NICER observation of PSR J0030+0451 has come up with a radius estimation of ${13.02}_{-1.06}^{+1.24}$ km for the 1.44 M_⊙ pulsar PSR J0030+0451 (Miller et al. 2019; Raaijmakers et al. 2019). Our result for radius corresponding to a 1.4 M_⊙ NS is slightly higher than that of GW170817 (Soma & Bandyopadhyay 2020), whereas it is consistent with the NICER observation.

We shall perform SN simulations and NS merger simulations with a new HS(DD2)ΛK⁻ ϕ EOS table and leave them for a future publication.

The authors acknowledge the DAE-BRNS grant received under the BRNS project No. 37(3)/14/12/2018-BRNS. D.B. acknowledges the hospitality at FIAS and support from the Alexander von Humboldt Foundation, Germany. The calculations are performed on the server of the Physics Department, BITS Pilani, Hyderabad Campus.

We follow the format of the widely used, existing SN EOS tables: Shen (Shen et al. 1998) and BHBΛϕ (Banik et al. 2014). We arrange the data in a parameter grid of temperature, density, and positive charge fraction. The first two parameters have a logarithmic spacing, while the charge fraction is on a linear scale. We group them in blocks of a fixed temperature, starting with the lowest value. Within each temperature block, we group the data according to the positive charge fraction, again starting with lowest values. Finally, for a given temperature and positive charge fraction, we list all the thermodynamic properties according to ascending baryon number densities.

We record only hadronic contributions to different quantities in the table. The contributions of photons, electrons, positrons, and neutrinos can be added separately. There are 23 entries of the table, representing different thermodynamic quantities corresponding to each density grid point in the table. These thermodynamic quantities are listed below.

• 1.  
  Logarithm of baryon mass density (log ₁₀(ρ_B ) [g cm⁻³]) is defined as the baryon number density multiplied by the value of the atomic mass unit m_u = 931.49432 MeV.
• 2.  
  Baryon number density (n_B [fm⁻³]).
• 3.  
  Logarithm of total positive charge fraction (log ₁₀(Y_p )).
• 4.  
  Total positive charge fraction (Y_p ).
• 5.  
  Free energy per baryon (F) relative to 938 MeV is defined by

  $$\begin{eqnarray}&&F=\displaystyle \frac{f}{{n}_{B}}-938.\end{eqnarray} \tag{ A1 }$$
• 6.  
  Internal energy per baryon (E_int) relative to m_u is defined by

  $$\begin{eqnarray}&&{E}_{\mathrm{int}}=\displaystyle \frac{\epsilon }{{n}_{B}}-{m}_{u},\end{eqnarray} \tag{ A2 }$$

  where the energy density is given by Equation (6).
• 7.  
  Entropy per baryon (S [k_B ]) related to the entropy density through $S=\tfrac{{ \mathcal S }}{{n}_{B}}$.
• 8.  
  Average mass number of heavy nuclei ($\bar{A}$) is defined as $\bar{A}={\sum }_{A,Z\geqslant 6}{{An}}_{A,Z}/{\sum }_{A,Z\geqslant 6}{n}_{A,Z}$.
• 9.  
  Average charge number of heavy nuclei ($\bar{Z}$) is defined as $\bar{Z}={\sum }_{A,Z\geqslant 6}{{Zn}}_{A,Z}/{\sum }_{A,Z\geqslant 6}{n}_{A,Z}$.
• 10.  
  Nucleon effective mass (m^* [MeV]). In the RMF calculation, we use separate values for neutron and proton masses, 939.56536 and 938.27203 MeV, respectively. However, the average value of neutron and proton effective masses is stored.
• 11.  
  Mass fraction of unbound neutrons (X_n = n_n /n_B ).
• 12.  
  Mass fraction of unbound protons (X_p = n_n /n_B ).
• 13.  
  Mass fraction of unbound Λs (X_Λ = n_Λ/n_B ).
• 14.  
  Mass fraction of light nuclei (X_a ) is defined as X_a = ∑_A,Z≤5 An_A,Z /n_B .
• 15.  
  Mass fraction of heavy nuclei (X_A ) is defined as X_A = ∑_A,Z≥6 An_A,Z /n_B .
• 16.  
  Baryon pressure (P [MeV fm⁻³]).
• 17.  
  Neutron chemical potential relative to neutron rest mass (μ_n − m_n [MeV]). Whenever Λ hyperons are present in the system, the μ_n = μ_Λ condition is satisfied.
• 18.  
  Proton chemical potential relative to proton rest mass (μ_p − m_p [MeV]).
• 19.  
  Average mass number of light nuclei ($\bar{a}$) is defined as $\bar{a}={\sum }_{A,Z\leqslant 5}{{An}}_{A,Z}/{\sum }_{A,Z\leqslant 5}{n}_{A,Z}$.
• 20.  
  Average charge number of light nuclei ($\bar{z}$) is defined as $\bar{z}={\sum }_{A,Z\leqslant 5}{{Zn}}_{A,Z}/{\sum }_{A,Z\leqslant 5}{n}_{A,Z}$.
• 21.  
  Kaon effective mass (${m}_{K}^{* }$ [MeV]).
• 22.  
  Mass fraction of thermal kaon (${X}_{K}^{T}={n}_{K}^{T}/{n}_{B}$).
• 23.  
  Mass fraction of K⁻ condensate (${X}_{K}^{C}={n}_{K}^{C}/{n}_{B}$).