1. Introduction
The equation of state (EOS) for dense matter is critical to understanding the nature of core-collapse supernovae and neutron star (NS) mergers [1]. A reliable EOS capable of describing the properties at sub-saturation density and the NS matter at a higher density becomes a shared goal of nuclear physics and astrophysics. However, it is hampered by two realities. On the one hand, a many-body EOS built at low densities has obvious flaws when used at high densities. Currently, there are two main categories for describing NSs in terms of nuclear many-body systems: (a) the ab initio method, such as the χEFT [2–6], which is based on the construction of effective two-body interactions and allows nuclear forces to be expanded systematically at a low density using the Quantum chromodynamics symmetry. (b) The phenomenological models based on self-consistent interactions, the Skyrme–Hartree–Fock method [7–9] and Gogny–Hartree–Fock method [10, 11] are typical. Unfortunately, when extended to describe high-density NS matter, these categories produce results that are inconsistent with the observations. On the other hand, a large number of astronomical observations now provide the most stringent constraints for the high-density EOS. For example, in 2010, the Shapiro delay calculation was used to verify the existence of an NS with a mass about 2 M⊙ (the solar mass) for the first time (1.97 ± 0.04 M⊙ numbered as PSR J1614-2230 [12], now more accurate calculations are 1.928 ± 0.017 M⊙ [13] and 1.908 ± 0.016 M⊙ [14]). The NS of PSR J0348 + 0432, whose mass was calculated using a pulsar-white dwarf binary system, is 2.01 ± 0.04 M⊙ [15], and a new relativistic Shapiro delay method was recently used to measure a massive millisecond pulsar MSP J0740 + 6620, whose mass is ${2.14}_{-0.09}^{+0.10}\,{M}_{\odot }$ (the 95.4% credibility interval is ${2.14}_{-0.18}^{+0.20}\,{M}_{\odot }$) [16] and currently its mass has been revised down to 2.08 ± 0.07 M⊙ by [17]. Furthermore, the gravitational wave (GW) signal GW170817 of binary NSs merger observed by the LIGO/Virgo detectors may have novel implications, as an analysis of this event yields an approximate tidal deformability value of ${300}_{-230}^{+420}$ (at 90% confidence level) [18, 19]. Furthermore, another intriguing gravitational event of GW190814 caused by the coalescence of a stellar-mass black hole and a mysterious compact star distinguishes itself because the mass of this peculiar compact object is as high as ${2.59}_{-0.09}^{+0.08}\,{M}_{\odot }$ [20], and if it is confirmed to be an NS, this object should be the heaviest one found so far, which excludes many soft EOSs to some extent. In addition to the mass observations, the radius of an NS can also be measured at the same time, since two independent teams analyzed the data of the isolated pulsar PSR J0030 + 0451 through NICER (Neutron Star Interior Composition Explorer), and simultaneously calculated the mass and radius of this pulsar as $M={1.44}_{-0.14}^{+0.15}\,{M}_{\odot },R={13.02}_{-1.06}^{+1.24}$ [21] and $M={1.34}_{-0.16}^{+0.15}\,{M}_{\odot },R={12.71}_{-1.19}^{+1.14}$ [22]. These observations involving mass–radius relations, gravitational waves, and tidal deformability have undoubtedly posed a significant challenge to the EOSs at a high density.
The binary neutron star merger [23] and the recently observed neutron star-black hole coalescence [24] demonstrate that NSs play an important role in the detection of gravitational waves. As we know, any non-axisymmetric disturbance in an NS will generate gravitational waves, which can be classified into different modes based on the restoring force, such as fundamental f-modes, gravity g-modes, pressure p-modes, pure time w-modes and rotational r-modes [25–27], in which the most interesting mode is the f-mode that can generate a lot of gravitational radiation [28–30]. With the improved sensitivity of the new generation of gravitational wave detectors [31–33], it will be possible to accurately detect the oscillation frequency, which in turn can provide us with internal information about the NSs.
In this paper, we use the relativistic mean field theory (RMFT) [34–41] including beta equilibrium, in conjunction with saturation matter characteristics, to develop relativistic parameters that satisfy both the properties at sub-saturation density and multi-messenger observations. The scalar-isovector meson is known to affect not only the symmetry energy and its slope but also the split of nucleon effective mass and the composition of asymmetric matter. As a result, it affects the properties of NSs, so we also consider the scalar-isovector channel under RMFT in this paper to decode the relationship between the isovector channel and symmetry energy as well as its slope. We then use calibrated parameter sets to describe the tidal deformabilities and f-mode oscillations of massive NSs within the constraints imposed by multi-messenger astronomical observations. In the end, the chiral effective theory (χEFT) together with the multi-messenger astronomy constraints was used to further reinforce the conclusions we have reached.
The following is how this paper is structured. The RMFT with scalar-isovector channel is introduced in section 2 . The steps for systematically constructing the nucleon coupling parameters are discussed in section 3 . The effects of saturation parameter sets on tidal deformability and f-mode oscillation of NSs are discussed in sections 4 and 5 . Section 6 concludes with a summary.
2. Relativistic mean field theory
The RMFT is an effective theory, which was originally proposed by Walecka et al [34, 35]. It describes the interaction between nucleons through the exchange of σ, ω, and ρ mesons. This model can describe the properties of the ground-state nuclei well, and can be extrapolated to study the NS matter [40–47]. In this paper, on the basis of traditional RMFT considering scalar-isoscalar meson σ, vector-isoscalar meson ω and vector-isovector meson ρ, we further consider the scalar-isovector meson δ [48–50], and the Lagrangian of the system can be written as the sum of the following seven items (here we adopt the natural units)3 . The RMFT told us that the meson field operator will be replaced by the classical field with its ground-state expectation value, i.e. ${\sigma }_{0}=\langle \sigma \rangle ,{\delta }_{a3}{\delta }_{0}=\langle \delta \rangle ,{\delta }_{\mu 0}{\omega }_{0}=\left\langle {\omega }_{\mu }\right\rangle ,{\delta }_{\mu 0}{\delta }_{a3}{\rho }_{0}=\left\langle {\rho }_{\mu }\right\rangle ,$ so the equation of motion of Lagrangian can be written as follows,
$\begin{eqnarray}L={L}_{N}+{L}_{l}+{L}_{\sigma }+{L}_{\omega }+{L}_{\rho }+{L}_{\delta }+{L}_{\rho -\omega },\end{eqnarray}$
where LN, Ll, Lσ, Lω, Lρ, Lδ, Lω−ρ are the Lagrangian of nucleons, leptons, σ, ω, ρ, δ and ω − ρ, which can be written respectively as $\begin{eqnarray}\begin{array}{l}{L}_{N}=\sum _{N=n,p}\bar{{\psi }_{N}}\left(\Space{0ex}{2.98ex}{0ex}{\rm{i}}{\partial }_{\mu }{\gamma }^{\mu }-{m}_{N}+{g}_{\sigma }\sigma -{g}_{\omega }{\omega }_{\mu }{\gamma }^{\mu }\right.\\ \left.-\displaystyle \frac{1}{2}{g}_{\rho }{\vec{\rho }}_{\mu }{\gamma }^{\mu }\vec{\tau }+{g}_{\delta }\vec{\delta }\vec{\tau }\right){\psi }_{N},\end{array}\end{eqnarray}$
$\begin{eqnarray}{L}_{l}=\displaystyle \sum _{l=e,\nu }\bar{{\psi }_{l}}\left({\rm{i}}{\partial }_{\mu }{\gamma }^{\mu }-{m}_{l}\right){\psi }_{l},\end{eqnarray}$
$\begin{eqnarray}{L}_{\sigma }=\displaystyle \frac{1}{2}{\partial }_{\mu }\sigma {\partial }^{\mu }\sigma -\displaystyle \frac{1}{2}{m}_{\sigma }^{2}{\sigma }^{2}-U(\sigma ),\end{eqnarray}$
$\begin{eqnarray}{L}_{\delta }=\displaystyle \frac{1}{2}{\partial }_{\mu }\delta {\partial }^{\mu }\delta -\displaystyle \frac{1}{2}{m}_{\delta }^{2}{\delta }^{2},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{L}_{\rho }=-\displaystyle \frac{1}{4}\left({\partial }_{\mu }{\vec{\rho }}_{v}-{\partial }_{\nu }{\vec{\rho }}_{\mu }\right)\left({\partial }^{\mu }{\vec{\rho }}^{\nu }-{\partial }^{\nu }{\vec{\rho }}^{\mu }\right)\\ \quad +\displaystyle \frac{1}{2}{m}_{\rho }^{2}{\vec{\rho }}_{\mu }{\vec{\rho }}^{\mu },\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{L}_{\omega }=-\displaystyle \frac{1}{4}\left({\partial }_{\mu }{\omega }_{v}-{\partial }_{v}{\omega }_{\mu }\right)\left({\partial }^{\mu }{\omega }^{\nu }-{\partial }^{\nu }{\omega }^{\mu }\right)\\ +\displaystyle \frac{1}{2}{m}_{\omega }^{2}{\omega }_{\mu }{\omega }^{\mu },\end{array}\end{eqnarray}$
$\begin{eqnarray}{L}_{\rho -\omega }={{\rm{\Lambda }}}_{\omega }({g}_{\rho }^{2}{\rho }^{\mu }{\rho }_{\mu })({g}_{\omega }^{2}{\omega }^{\mu }{\omega }_{\mu }),\end{eqnarray}$
where $U(\sigma )=\tfrac{1}{4}c{\left({g}_{\sigma }\sigma \right)}^{4}+\tfrac{1}{3}{{bm}}_{N}{\left({g}_{\sigma }\sigma \right)}^{3}$ is the σ nonlinear coupling terms, mN and ml are the mass of nucleon and lepton respectively, mσ, mω, mρ, mδ are the mass of corresponding meson (σ, ω, ρ, δ). The parameters such as gσ, gω, gρ, gδ, b, c, Λω need to be further determined through subsequent discussion in section $\begin{eqnarray}\begin{array}{l}\left(\Space{0ex}{2.95ex}{0ex}{\rm{i}}{\partial }_{\mu }{\gamma }^{\mu }-{m}_{N}+{g}_{\sigma }{\sigma }_{0}-{g}_{\omega }{\omega }_{0}{\gamma }^{0}\right.\\ \quad \left.-\displaystyle \frac{1}{2}{g}_{\rho }{\rho }_{0}{\gamma }^{0}{\tau }_{3}+{g}_{\delta }{\delta }_{0}{\tau }_{3}\right){\psi }_{N}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\left({\rm{i}}{\partial }_{\mu }{\gamma }^{\mu }-{m}_{l}\right){\psi }_{l}=0,\end{eqnarray}$
$\begin{eqnarray}{g}_{\sigma }{\sigma }_{0}={\left(\displaystyle \frac{{g}_{\sigma }}{{m}_{\sigma }}\right)}^{2}\left[{n}_{p}^{s}+{n}_{n}^{s}-\displaystyle \frac{1}{{g}_{\sigma }}\displaystyle \frac{{\rm{d}}{U}(\sigma )}{{\rm{d}}\sigma }\right],\end{eqnarray}$
$\begin{eqnarray}{g}_{\omega }{\omega }_{0}={\left(\displaystyle \frac{{g}_{\omega }}{{m}_{\omega }}\right)}^{2}\left[{n}_{p}^{v}+{n}_{n}^{v}-2{{\rm{\Lambda }}}_{\omega }{\left({g}_{\rho }{\rho }_{0}\right)}^{2}({g}_{\omega }{\omega }_{0})\right],\end{eqnarray}$
$\begin{eqnarray}{g}_{\rho }{\rho }_{0}={\left(\displaystyle \frac{{g}_{\rho }}{{m}_{\rho }}\right)}^{2}\left[\displaystyle \frac{{n}_{p}^{v}-{n}_{n}^{v}}{2}-2{{\rm{\Lambda }}}_{\omega }({g}_{\rho }{\rho }_{0}){\left({g}_{\omega }{\omega }_{0}\right)}^{2}\right],\end{eqnarray}$
$\begin{eqnarray}{g}_{\delta }{\delta }_{0}={\left(\displaystyle \frac{{g}_{\delta }}{{m}_{\delta }}\right)}^{2}\left({n}_{p}^{s}-{n}_{n}^{s}\right),\end{eqnarray}$
where the nucleon vector density niv expressed by the Fermi momentum ki reads $\begin{eqnarray}{n}_{i}^{v}=\displaystyle \frac{1}{{\pi }^{2}}{\int }_{0}^{{k}_{i}}{k}^{2}{\rm{d}}{k}=\displaystyle \frac{{k}_{i}^{3}}{3{\pi }^{2}},\quad i=p,n,\end{eqnarray}$
and nucleon scalar density nis can be written by Fermi momentum ki and effective mass m* $\begin{eqnarray}{n}_{i}^{s}=\displaystyle \frac{1}{{\pi }^{2}}{\int }_{0}^{{k}_{i}}\displaystyle \frac{{k}^{2}{m}_{i}^{* }}{\sqrt{{k}^{2}+{m}_{i}^{* 2}}}{\rm{d}}{k},\quad i=p,n,\end{eqnarray}$
where proton effective mass splitting is ${m}_{p}^{* }={m}_{N}-{g}_{\sigma }{\sigma }_{0}\,-{g}_{\delta }{\delta }_{0}$ and neutron is ${m}_{n}^{* }={m}_{N}-{g}_{\sigma }{\sigma }_{0}+{g}_{\delta }{\delta }_{0}$.The energy density can be obtained by the momentum energy tensor Tμν = −pgμν + (p + ε)uμuν,
$\begin{eqnarray}\begin{array}{l}\varepsilon ={T}^{00}=\sum _{i=p,n}\displaystyle \frac{1}{{\pi }^{2}}{\displaystyle \int }_{0}^{{k}_{i}}{k}^{2}\sqrt{{k}^{2}+{m}_{i}^{* 2}}{\rm{d}}{k}\\ \quad +{g}_{\omega }{\omega }_{0}({n}_{p}^{v}+{n}_{n}^{v})+\displaystyle \frac{1}{2}{g}_{\rho }{\rho }_{0}({n}_{p}^{v}-{n}_{n}^{v})\\ \quad +\displaystyle \frac{1}{2}{m}_{\sigma }^{2}{\sigma }_{0}^{2}+U(\sigma )\\ -\displaystyle \frac{1}{2}{m}_{\omega }^{2}{\omega }_{0}^{2}-\displaystyle \frac{1}{2}{m}_{\rho }^{2}{\rho }_{0}^{2}-{{\rm{\Lambda }}}_{\omega }{\left({g}_{\rho }{\rho }_{0}\right)}^{2}{\left({g}_{\omega }{\omega }_{0}\right)}^{2}\\ +\displaystyle \frac{1}{2}{m}_{\delta }^{2}{\delta }_{0}^{2}+\sum _{l=e,\mu }\displaystyle \frac{1}{{\pi }^{2}}{\displaystyle \int }_{0}^{{k}_{l}}\sqrt{{k}^{2}+{m}_{l}^{2}}{k}^{2}{\rm{d}}{k}.\end{array}\end{eqnarray}$
The pressure can be given as: $\begin{eqnarray}p=\displaystyle \frac{1}{3}{T}^{{ii}}=\sum _{i=n,p}{\mu }_{i}{n}_{i}-\varepsilon ,\quad {\mu }_{i}=\displaystyle \frac{\partial \varepsilon }{\partial {n}_{i}},\end{eqnarray}$
where ${\mu }_{p}\,=\,{g}_{\omega }{\omega }_{0}\,+\,\tfrac{{g}_{\rho }}{2}{\rho }_{0}\,+\,\sqrt{{k}_{p}^{2}\,+\,{m}_{p}^{* 2}},{\mu }_{n}={g}_{\omega }{\omega }_{0}\,-\,\tfrac{{g}_{\rho }}{2}{\rho }_{0}\,+\sqrt{{k}_{n}^{2}+{m}_{n}^{* 2}}$ are the Fermi energy of system.With the charge conservation and beta equilibrium condition, the mass and radius of a NS can be solved by the Tolman–Oppenheimer–Volkoff (TOV) equation [51]
$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{\rm{d}}p}{{\rm{d}}r} & = & -\displaystyle \frac{(p+\epsilon )\left(M+4\pi {r}^{3}p\right)}{r(r-2M)},\\ {\rm{d}}M & = & 4\pi {r}^{2}\epsilon {\rm{d}}r.\end{array}\end{eqnarray}$
3. Coupling parameters
As mentioned above, in order to calculate the properties like mass and radius of an NS, seven unknown parameters which are gσ, gω, gρ, gδ, b, c, Λω need to be determined. In this work we take the coupling parameter Λω which characterizes coupling strength between scalar-isoscalar meson ω and vector-isovector meson ρ to be 0.01 [52–54]. The remaining six parameters can be divided into two categories, one type is the isospin-independent quantity (isoscalar) like gσ, gω, b, c, and another one is the isospin-dependent (isovector) like gρ, gδ. Next, we use the physical characteristics at saturation density to construct these two types of parameters.
3.1. Isoscalar coupling parameters
The isospin-independent parameters gσ, gω, b, c can be obtained through the quantities like the binding energy per nucleon B/A, the incompressibility coefficient K, the nucleon effective mass m* and the saturation density n0. At saturation density, because protons and neutrons tend to be symmetrically distributed, δ meson and ρ meson that both characterize asymmetry matter have no effect in symmetry nuclear matter, i.e. ρ0 = δ0 = 0. Their relationships can be expressed as:
$\begin{eqnarray}{\left(\displaystyle \frac{{g}_{\omega }}{{m}_{\omega }}\right)}^{2}=\displaystyle \frac{{m}_{N}+B/A-\sqrt{{k}_{0}^{2}+{m}^{* 2}}}{{n}_{0}}.\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\left(\displaystyle \frac{{m}_{\sigma }}{{g}_{\sigma }}\right)}^{2}\left({m}_{N}-{m}^{* }\right)+{m}_{N}{\left({m}_{N}-{m}^{* }\right)}^{2}b\\ \quad +c{\left({m}_{N}-{m}^{* }\right)}^{3}={n}^{s}.\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}K={\left(\displaystyle \frac{{g}_{\omega }}{{m}_{\omega }}\right)}^{2}\displaystyle \frac{6{k}_{0}^{3}}{{\pi }^{2}}+\displaystyle \frac{3{k}_{0}^{2}}{E({k}_{0})}-\displaystyle \frac{6{k}_{0}^{3}}{{\pi }^{2}}{\left(\displaystyle \frac{{g}_{\sigma }}{{m}_{\sigma }}\right)}^{2}{\left(\displaystyle \frac{{m}^{* }}{E({k}_{0})}\right)}^{2}\\ \,{\times \,\left[1+{\left(\displaystyle \frac{{g}_{\sigma }}{{m}_{\sigma }}\right)}^{2}\displaystyle \frac{2}{{\pi }^{2}}{\displaystyle \int }_{0}^{{k}_{0}}\displaystyle \frac{{k}^{4}{\rm{d}}{k}}{{E}^{3}(k)}+{\left(\displaystyle \frac{{g}_{\sigma }}{{m}_{\sigma }}\right)}^{2}\displaystyle \frac{{\partial }^{2}U({\sigma }_{0})}{\partial {\sigma }_{0}^{2}}\displaystyle \frac{1}{{g}_{\sigma }^{2}}\right]}^{-1}.\end{array}\end{eqnarray}$
At saturation density, the energy density can be expressed by20 ), (21 ) (22 ) and (24 ) with respect to the isoscalar coupling parameters gσ, gω, b, c. In order to solve these unknown coupling parameters, we need to know the binding energy per nucleon B/A, the incompressibility coefficient K, the nucleon effective mass m* and the saturation density n0 which are still quite challenging to strictly restrict them. In this work, we adopt the symmetry nuclear matter that saturates at n0 = 0.16 fm−3 [55, 56] with B/A = −16 MeV [55, 57]. For the K, we select 220 MeV ≤ K ≤ 300 MeV in the credible interval according to the recommendations given in [1, 58–60]. For the effective mass m*, some commonly used RMF parameter sets give the range could alter from 0.55 to 0.75 [61, 62]. The [53] pointed out that the effective mass that can meet the dimensionless tidal deformability within the 90% confidence level should be greater than 0.6. According to these suggestions, we select a suitable interval with 0.61 ≤ m* ≤ 0.65 in this work, so gσ, gω, b, c will be obtained by choosing a suitable value in this range accordingly.
$\begin{eqnarray}\begin{array}{l}\varepsilon =\displaystyle \frac{1}{2}{m}_{\omega }^{2}{\omega }_{0}^{2}+\displaystyle \frac{2}{{\pi }^{2}}{\displaystyle \int }_{0}^{{k}_{0}}{k}^{2}E(k){\rm{d}}{k}\\ \quad +\displaystyle \frac{1}{2}{m}_{\sigma }^{2}{\sigma }_{0}^{2}+\displaystyle \frac{1}{3}{{bm}}_{N}{\left({g}_{\sigma }{\sigma }_{0}\right)}^{3}+\displaystyle \frac{1}{4}c{\left({g}_{\sigma }{\sigma }_{0}\right)}^{4},\end{array}\end{eqnarray}$
we also know that σ0 and ω0 equations satisfy ${m}_{\omega }^{2}{\omega }_{0}^{2}/2\,={\left({g}_{\omega }/{m}_{\omega }\right)}^{2}{n}_{0}^{2}/2$ and ${m}_{\sigma }^{2}{\sigma }_{0}^{2}/2={\left({m}_{\sigma }/{g}_{\sigma }\right)}^{2}{\left({g}_{\sigma }{\sigma }_{0}\right)}^{2}/2$ respectively, then combining with ϵ/n0 = E/A = B/A + mN, we get a useful expression as $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{2}{\left({m}_{N}-{m}^{* }\right)}^{2}{\left(\displaystyle \frac{{m}_{\sigma }}{{g}_{\sigma }}\right)}^{2}+\displaystyle \frac{1}{3}{m}_{N}{\left({m}_{N}-{m}^{* }\right)}^{3}b\\ \quad +\displaystyle \frac{1}{4}{\left({m}_{N}-{m}^{* }\right)}^{4}c=(B/A+{m}_{N}){n}_{0}-\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{g}_{\omega }}{{m}_{\omega }}\right)}^{2}{n}_{0}^{2}\\ \quad -\displaystyle \frac{2}{{\pi }^{2}}{\displaystyle \int }_{0}^{{k}_{0}}{k}^{2}E(k){\rm{d}}{k}.\\ \end{array}\end{eqnarray}$
Now, we have obtained four analytical equations (3.2. Isovector coupling parameters
In this section, we elaborate on how the isovector coupling parameters is related to the symmetry energy and its slope. In asymmetry nuclear matter one considers the energy per nucleon E/A, which is a function of the asymmetry β and total number density n. This physical quantity can be expanded into the Taylor series of β around symmetry nuclear matter (i.e. β = 0) at a given number density [63–65],
$\begin{eqnarray}\displaystyle \frac{E}{A}(n,\beta )=\displaystyle \frac{E}{A}(n)+{E}_{{\rm{s}}}(n){\beta }^{2}+......,\end{eqnarray}$
where β = (N − Z)/A with N and Z being the neutron number and proton number. The coefficient of the second term is called the symmetry energy $\begin{eqnarray}{E}_{{\rm{s}}}(n)=\displaystyle \frac{1}{2}\displaystyle \frac{{\partial }^{2}(E/A)}{\partial {\beta }^{2}}{| }_{\beta =0},\end{eqnarray}$
which can also be expanded in the Taylor series of n at saturation number density (n0 = 0.16 fm−3) $\begin{eqnarray}{E}_{{\rm{s}}}(n)={E}_{{\rm{s}}}({n}_{0})+3{n}_{0}\displaystyle \frac{\partial {E}_{{\rm{s}}}}{\partial n}{| }_{n\,=\,{n}_{0}}(\displaystyle \frac{n-{n}_{0}}{3{n}_{0}})+......,\end{eqnarray}$
and then we get the saturation symmetry energy Esym ≡ Es(n0) and its slope parameter $L\equiv 3{n}_{0}\tfrac{\partial {E}_{{\rm{s}}}}{\partial n}{| }_{n={n}_{0}}$. For convenience, we usually split them into three parts: $\begin{eqnarray*}{E}_{\mathrm{sym}}={E}_{\mathrm{sym},1}+{E}_{\mathrm{sym},2}+{E}_{\mathrm{sym},3},\,\,\,L={L}_{1}+{L}_{2}+{L}_{3},\end{eqnarray*}$
where Esym,1 is the momentum related part $\begin{eqnarray}{E}_{\mathrm{sym},1}=\displaystyle \frac{{k}_{0}^{2}}{6E({k}_{0})},\end{eqnarray}$
and $\begin{eqnarray}{E}_{\mathrm{sym},2}=\displaystyle \frac{{k}_{0}^{3}}{12{\pi }^{2}}\displaystyle \frac{{g}_{\rho }^{2}}{{m}_{\rho }^{2}+2{{\rm{\Lambda }}}_{\omega }{\left({g}_{\omega }{\omega }_{0}\right)}^{2}{g}_{\rho }^{2}},\end{eqnarray}$
is the ρ-related part. Esym,3 denotes δ-related part (see [48–50] for the full details), Here only the result is listed $\begin{eqnarray}{E}_{\mathrm{sym},3}=-\displaystyle \frac{{m}^{* 2}{n}_{0}{C}_{\delta }^{2}}{2{E}^{2}({k}_{0})}\times \displaystyle \frac{1}{\left[1+A({k}_{0},{m}^{* }){C}_{\delta }^{2}\right]},\end{eqnarray}$
where ${C}_{\delta }=\tfrac{{g}_{\delta }}{{m}_{\delta }},A({k}_{0},{m}^{* })=\tfrac{1}{{\pi }^{2}}{\int }_{0}^{{k}_{0}}\tfrac{2{k}^{4}{\rm{d}}{k}}{{E}^{3}(k)}$. With the above three symmetric energy expressions, their corresponding differential analytical read as: $\begin{eqnarray*}\begin{array}{l}{L}_{1}={E}_{\mathrm{sym},1}\left[1+\displaystyle \frac{{m}^{* 2}}{{E}^{2}({k}_{0})}\left(1+\displaystyle \frac{3{n}_{0}}{E({k}_{0})}\left[\displaystyle \frac{{m}_{\sigma }^{2}}{{g}_{\sigma }^{2}}\right.\right.\right.\\ \quad +2{{bm}}_{N}{g}_{\sigma }{\sigma }_{0}+3{{cg}}_{\sigma }^{2}{\sigma }_{0}^{2}+\displaystyle \frac{1}{{\pi }^{2}}\left[\displaystyle \frac{{k}_{0}^{3}+3{k}_{0}{m}^{* 2}}{E({k}_{0})}\right.\\ \quad \left.\left.{\left.\left.-3{m}^{* 2}\mathrm{ln}\left(\displaystyle \frac{{k}_{0}+E({k}_{0})}{{m}^{* }}\right)\right]\right]}^{-1}\right)\right],\end{array}\end{eqnarray*}$
$\begin{eqnarray}{L}_{2}=3{E}_{\mathrm{sym},2}\left(1-32\displaystyle \frac{{g}_{\omega }^{3}{\omega }_{0}{{\rm{\Lambda }}}_{\omega }}{{m}_{\omega }^{2}}{E}_{\mathrm{sym},2}\right),\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}{L}_{3}=3{E}_{\mathrm{sym},3}\left[\displaystyle \frac{{k}_{0}^{2}+3{m}^{* 2}}{3{E}^{2}({k}_{0})}-\displaystyle \frac{2{n}_{0}{C}_{\delta }^{2}{k}_{0}^{2}}{{E}^{3}({k}_{0})[1+A({k}_{0},{m}^{* }){C}_{\delta }^{2}]}\right.\\ \quad +\displaystyle \frac{2{n}_{0}{k}_{0}^{2}}{{E}^{2}({k}_{0}){m}^{* }}\displaystyle \frac{\partial {m}^{* }}{\partial n}+\displaystyle \frac{4{m}^{* }{n}_{0}{C}_{\delta }^{2}}{1+A({k}_{0},{m}^{* }){C}_{\delta }^{2}}\displaystyle \frac{1}{{\pi }^{2}}\\ \quad \left.\times {\displaystyle \int }_{0}^{{k}_{0}}\displaystyle \frac{3{k}^{4}{\rm{d}}{k}}{{E}^{5}(k)}\displaystyle \frac{\partial {m}^{* }}{\partial n}\right]\\ \end{array}.\end{eqnarray}$
The above formulas clearly show the relationship between isovector parameters and asymmetry properties, so the determination of gρ and gδ is conditional upon the values of Esym and L which is also a very difficult and challenging task to constrict their values, especially at a supersaturated density [1, 63–65]. According to recent work [1, 66], suggest Esym roughly located at 31.6 ± 2.7 MeV and [67] suggests at 25–35 MeV [68], use measurements of skin thicknesses to constrain the symmetry energy to be located at 30.2–33.7 MeV. In terms of the value of L for which different experiments and works give different ranges are still more uncertain [69–72]. So far there has been no consensus on this value. The work by [73] uses two different methods to obtain L = 64.29 ± 11.84 MeV and L = 53.85 ± 10.29 MeV respectively, other works like [66] provided a fiducial value of L = 59 ± 16 MeV, and the latest work combined a huge amount of data to analyze the most likely range of L = 58.7 ± 28.1 MeV [1]. It is worth mentioning that some well-known parameter sets are excellent in describing nuclear matter and NSs, but the values of L are very different from each other (L = 61 MeV in FSU [74], L = 47.2 MeV in IU-FSU [75], L = 110.8 MeV in TM1 [76], L = 85MeV in NLρ [48], L = 118MeV in NL3 [37] and L = 94 MeV in GM1 [77]). In view of the above facts, we select Esym with an uncertain region at 28 MeV ≤ Esym ≤ 36 MeV and choose the credible value of L at 40 MeV ≤ L ≤ 80 MeV. As a result, the values of gρ and gδ could be calculated together with the isoscalar coupling part. In this paper, in order to construct a series of credible relativistic parameters, we do not use some commonly used parameter groups, such as some well-known parameter groups TM series [76], GM series [77], FSU series [74] and so on. It is because these well-known parameter groups are based on different models. For example, the TM and FSU series consider the nonlinear σ and ω self-interacting mesons and possible meson cross terms, while the GM only considers the nonlinear coupling term. With δ meson considered in this manuscript, these model parameters cannot be used directly. As the purpose of our work, we hope to study the influence of different nuclear matter parameters (K,m*,Esym and L) on the EOS of neutron stars. In order to avoid the inconsistency caused by the direct use of the above parameters, we re-construct the parameters within the range allowed by experiment and theory. Our specific steps are: when we study the effect of one of the nuclear matter parameters, we fix the other saturation parameters at an optimal value according to the suggestions from the above literature. For example, in order to study K, we fix m*,Esym and L. Following the same step, we can construct 20 sets of parameters listed in table 1.
Table 1. The relativistic coupling parameter sets constructed by the bulk properties of saturation nuclear matter. We have adopted mN = 939 MeV, mσ = 550 MeV, mω = 783 MeV, mρ = 770 MeV, B/A = −16 MeV, n0 = 0.16 fm−3, Λω = 0.01. |
| K (MeV) | m* | Esym (MeV) | L (MeV) | gσ | gω | b | c | gρ | gδ |
|---|---|---|---|---|---|---|---|---|---|
| 220 | 0.65 | 32 | 60 | 10.1702 | 11.3512 | 0.00366354 | −0.00466868 | 9.87666 | 6.5562 |
| 240 | 0.65 | 32 | 60 | 10.1147 | 11.3512 | 0.00335058 | −0.00407256 | 9.85983 | 6.4902 |
| 260 | 0.65 | 32 | 60 | 10.0592 | 11.3512 | 0.00303235 | −0.00346641 | 9.84308 | 6.3562 |
| 280 | 0.65 | 32 | 60 | 10.0036 | 11.3512 | 0.00270872 | −0.00284998 | 9.82641 | 6.2089 |
| 300 | 0.65 | 32 | 60 | 9.94806 | 11.3512 | 0.00237956 | −0.00222301 | 9.80983 | 6.0982 |
| 260 | 0.61 | 32 | 60 | 10.5027 | 12.0815 | 0.002483 | −0.0032086 | 10.2291 | 7.5609 |
| 260 | 0.62 | 32 | 60 | 10.3927 | 11.9039 | 0.00259835 | −0.00327058 | 10.1053 | 7.4399 |
| 260 | 0.63 | 32 | 60 | 10.2822 | 11.723 | 0.00272751 | −0.0033353 | 10.0021 | 7.3209 |
| 260 | 0.64 | 32 | 60 | 10.1711 | 11.5388 | 0.00287173 | −0.0034013 | 9.91561 | 7.28532 |
| 260 | 0.65 | 32 | 60 | 10.5092 | 11.3512 | 0.00303235 | −0.00346641 | 9.84308 | 7.17003 |
| 260 | 0.65 | 28 | 60 | 10.0592 | 11.3512 | 0.00303235 | −0.00346641 | 8.17579 | 6.9568 |
| 260 | 0.65 | 30 | 60 | 10.0592 | 11.3512 | 0.00303235 | −0.00346641 | 9.04491 | 6.8563 |
| 260 | 0.65 | 32 | 60 | 10.0592 | 11.3512 | 0.00303235 | −0.00346641 | 9.84308 | 6.7908 |
| 260 | 0.65 | 34 | 60 | 10.0592 | 11.3512 | 0.00303235 | −0.00346641 | 10.5847 | 6.6005 |
| 260 | 0.65 | 36 | 60 | 10.0592 | 11.3512 | 0.00303235 | −0.00346641 | 11.2801 | 6.5567 |
| 260 | 0.65 | 32 | 40 | 10.0592 | 11.3512 | 0.00303235 | −0.00346641 | 11.6715 | 6.00567 |
| 260 | 0.65 | 32 | 50 | 10.0592 | 11.3512 | 0.00303235 | −0.00346641 | 10.2473 | 6.1028 |
| 260 | 0.65 | 32 | 60 | 10.0592 | 11.3512 | 0.00303235 | −0.00346641 | 9.84308 | 6.3562 |
| 260 | 0.65 | 32 | 70 | 10.0592 | 11.3512 | 0.00303235 | −0.00346641 | 9.76635 | 6.6475 |
| 260 | 0.65 | 32 | 80 | 10.0592 | 11.3512 | 0.00303235 | −0.00346641 | 9.83255 | 6.9983 |
4. Multi-messenger observations constraints
Now we have calculated the isoscalar and isovector coupling parameters based on the bulk properties at saturation density, it should be particularly emphasized that only the results calculated by these sets need to meet the following constraints at the same time are acceptable:
| • | Requirements for reproducing massive NSs matched with PSR J1614-2230, PSR J0348 + 0432, and MSP J0740 + 6620. |
| • | The results should fall within the radius and tidal deformability range given by NICER and gravitational wave event GW170817; the requirement that the speed of sound cannot exceed the speed of light should also be maintained as well. |
4.1. Mass–radius relationship constraint
Once the coupling parameters are determined, we can use the theoretical method provided in section III to describe NSs with standard scheme by the beta equilibrium system (npeμ) for core region, while in the NS crust region, where the density is located at 6.3 × 10−12 fm−3 ≤ ρ ≤ 2.46 × 10−4 fm−3, we use the Baym–Pethick–Sutherland EOS [78] for the outer part. For the inner region with a range of 2.46 × 10−4 fm−3 ≤ ρ ≤ ρt, we use the polytropic EOSs parameterized form given by P = a + bϵ4/3 [79–81], where constants a and b are associated with the core-crust transition ρt are determined by the thermodynamical method [82–84]. The numerical results are shown in figure 1 depicting the mass–radius relation with our parameter sets. It can be seen that the maximum masses calculated by these sets all exceed 2 M⊙. The upper right corner of figure 1 shows that when the mass exceeds 2 M⊙, the effect of m* on the NS structure is significant. The larger the effective mass will give a larger radius under the same NS mass, a similar behavior is observed by Nadine Hornick [53]. The upper left corner of figure 1 shows the influence of K, as the value of K increases, the radius of the same mass also increases due to the large K making nuclei difficult to be compressed. The lower part of figure 1 shows the influence of Esym and L on the NS structure, it can be speculated that the NS radius is sensitive to the symmetry energy, especially its slope. For the larger symmetry energy slope, the NS radius is larger, similar behaviors can be found in [62, 85, 86].
Figure 1. Relationships between mass and radius under different parameter sets. The red, green and grey regions represent the three accurately measured massive NSs of PSR J1614-2230, PSR J0348 + 0432 and MSP J0740 + 6620, the horizontal solid and dotted lines in the figure represent the radius range constrained by gravitational wave events GW170817 and NICER, respectively. |
It is worth noting that for low-mass NSs, all nuclear saturation parameters have a significant effect on the radius of 1.4 M⊙. The radius of 1.4 M⊙ affected by the K varies from 12.81 km (220 MeV) to 13.25 km (300 MeV), by the m* varies from 12.60 km (0.61) to 13.21 km (0.65), by the Esym varies from 12.36 km (32 MeV) to 12.82 km (36 MeV) and by the L varies from 12.59 km (40 MeV) to 13.79 km (80 MeV). So far, it is still a relatively difficult task to determine the radius of a NS with 1.4 M⊙ precisely, and it has a strong model dependence [1, 69, 87–90]. In addition, according to the recent analysis of gravitational wave event GW170817 by the Advanced LIGO and Advanced Virgo gravitational-wave detectors [23, 91–94], the radius of 1.4 M⊙ is roughly in the range of 12–13.5 km. Furthermore, other groups, such as the NICER [21, 22], gave a radius of 1.4 M⊙ at 11.52 km ∼14.26 km, which are shown by the horizontal solid and dashed lines respectively. Our results all fall within this limit range. In the figure, we also mark the masses of three massive NSs recently observed, namely PSR J1614-2230, PSR J0348 + 0432, and MSP J0740 + 6620. Obviously, these massive NSs can be reproduced by using our parameter sets.
4.2. Tidal deformability and sound speed constraints
Under the constraints of mass–radius relationship, all parameters we constructed meet the constraints by GW170817 and NICER. However, it is often not enough to ascertain the rationality of parameter construction from the perspective of mass–radius relationship. With the help of LIGO and Virgo Collaboration [23, 91], the most significant phenomenon during the binary inspiral is that each NS will produce an observable tidal deformability under the gravitational field of its companion star. This dimensionless tidal deformability parameter further limits current EOSs, which can be expressed in form of the second tidal Love number k2 as Λ = 2k2/(3C5) with C being the compactness parameter (M/R), where k2 can be calculated by the following expression [95–97]:19 ).
$\begin{eqnarray}\begin{array}{rcl}{k}_{2} & = & \displaystyle \frac{8{C}^{5}}{5}{\left(1-2C\right)}^{2}\left[2+2C\left({y}_{R}-1\right)-{y}_{R}\right]\\ & & \times \left\{2C\left[6-3{y}_{R}+3C\left(5{y}_{R}-8\right)\right]\right.\\ & & +4{C}^{3}\left[13-11{y}_{R}+C\left(3{y}_{R}-2\right)+2{C}^{2}\left(1+{y}_{R}\right)\right]\\ & & {\left.+3{\left(1-2C\right)}^{2}\left[2-{y}_{R}+2C\left({y}_{R}-1\right)\mathrm{ln}(1-2C)\right]\right\}}^{-1},\end{array}\end{eqnarray}$
where yR ≡ y(R) is the solution of the first-order differential equation with y(0) = 2 as the boundary condition, $\begin{eqnarray}r\displaystyle \frac{{\rm{d}}y(r)}{{\rm{d}}r}+{y}^{2}(r)+y(r)F(r)+{r}^{2}Q(r)=0,\end{eqnarray}$
where F(R) and Q(R) have the following form $\begin{eqnarray}\begin{array}{l}F(r)={\left[1-\displaystyle \frac{2M(r)}{r}\right]}^{-1}\left\{1-4\pi {r}^{2}[\varepsilon (r)-p(r)]\right\},\\ {r}^{2}Q(r)=\left\{4\pi {r}^{2}\left[5\varepsilon (r)+9p(r)+\displaystyle \frac{\varepsilon (r)+p(r)}{\tfrac{\partial p}{\partial \varepsilon }(r)}\right]-6\right\}\\ \quad \times {\left[1-\displaystyle \frac{2M(r)}{r}\right]}^{-1}-{\left[\displaystyle \frac{2M(r)}{r}+2\times 4\pi {r}^{2}p(r)\right]}^{2}\\ \quad \times {\left[1-\displaystyle \frac{2M(r)}{r}\right]}^{-2}.\end{array}\end{eqnarray}$
Then Λ can be obtained by combining with the TOV equation (The results of Λ and dimensionless tidal deformability related to the binary system of the GW170817 event calculated by parameter sets are shown in figures 2 and 3 respectively. In figure 2, we can find that the tidal deformability below 1.4 M⊙ increases rapidly as the NS mass decreases, and the closer to 2 M⊙, the less obvious the tidal deformability effect becomes. This is consistent with many other works [62, 85, 86]. For NSs with a mass around 1.4 M⊙, the isovector saturation parameters Esym and L, especially the symmetry energy slope, have a more significant influence on tidal deformability than m* and K. On the one hand, since the radius of the NS around 1.4M⊙ is very sensitive to the isovector parameter, resulting in a very obvious change in the radius. On the other hand, the tidal deformability has a strong dependence on the NS radius and there is an empirical positive correlation function between them [62, 85, 92, 98–100]. In addition, in the upper right corner of each graph, we also plot the Love number as a function of neutron star mass, which can help us better understand tidal behavior. The values of k2, which depend on the compaction parameter C and yR, show a noticeable spread under different parameter sets and increase in the low-mass region and decrease slowly in the massive mass region. One can see the values of k2 given by different parameter groups are roughly located between 0.01 and 0.12, which are consistent with the results found in [85, 101]. Furthermore, our results also need to meet the constraints given by the GW170817 event [23], in which they obtain the tidal deformability range of 1.4 M⊙ with ${{\rm{\Lambda }}}_{1.4}={190}_{-120}^{+390}$, obviously the results given by our parameters all fall into this interval. If a new generation of gravitational wave detectors such as Cosmic Explorer or Einstein Telescope [31–33] can provide tidal deformability with higher sensitivity and precision in the near future, this will provide stronger constraints to the saturation parameters.
Figure 2. The relationship between the tidal deformability and the NS mass under different parameter sets. The purple dotted line represents the tidal deformability range obtained from the analysis of the GW170817 event. Each upper right corner gives the tidal love number as a function of neutron star mass. |
Figure 3. The relationship between Λ1 and Λ2 in GW170817 event calculated with different parameter sets. The grey dash-lines represent the 90% credible interval and the 50% credible interval. |
In addition to the above constraint, in figure 3, we also calculate the respective tidal deformability of two NSs in GW170817 event. We consider the most credible value for the chirp mass with $[{\left({m}_{1}{m}_{2}\right)}^{3/5}]/[{\left({m}_{1}+{m}_{2}\right)}^{1/5}]=1.188\,{M}_{\odot }$ [91]. Λ1 and Λ2 stand for the high-mass one and the low-mass one respectively. The grey contour lines represent the 90% credible interval and the 50% credible interval, and our sets give a strong approval for these requirements and show highly reliable results.
In order to better understand the influence of saturation parameters on R1.4 and ${{\rm{\Lambda }}}_{2{M}_{\odot }}$, we plotted these data points in figure 4 and figure 5, expressing a simplified diagram of the radii and tidal deformabilities under different saturation parameters. After fitting, it is found the R1.4 and ${{\rm{\Lambda }}}_{2{M}_{\odot }}$ have a relatively good linear relationship with the saturation parameters, similar behavior is observed in [62, 85]. Figure 4 shows that R1.4 increases linearly with saturation parameters. In figure 5, we give the ${{\rm{\Lambda }}}_{2{M}_{\odot }}$ under different parameter sets and find the tidal deformation has the same trend as figure 4. These linear results can better build a bridge between the parameters of nuclear matter and the properties of NSs.
Figure 4. The data points of R1.4 under different saturation parameters. The red line represents the result of linear fitting, and C represents the Pearson's correlation coefficient. |
Figure 5. The data points of ${{\rm{\Lambda }}}_{2{M}_{\odot }}$ under different saturation parameters. The red line represents the result of linear fitting, and C represents the Pearson's correlation coefficient. |
The determination of the upper limit of the speed of sound in dense nuclear matter is still an open issue so far, and there is still no consensus. According to the law of causality, it is generally believed that the speed of dense nuclear matter should not exceed the speed of light [102, 103]. Recent work like [104] points out that if the tidal deformability of an NS with 1.4M⊙ is less than 600, then the speed of sound in the NS core should theoretically exceed the conformal limit of $c/\sqrt{3}$, Eemeli Annala's work [105] stated that at places below the saturation density, the speed of sound obtained by the χEFT calculations should be less than $c/\sqrt{3}$, while at high densities, the maximum upper limit given by the hadronic models should exceed $c/\sqrt{2}$. Fortunately, the RMFT used in this work, by design, is a relativistic theory that will not break the causality, in order to check our calculations, here we only select the case associated with L, other cases are similar. We plot the results in figure 6, which plots the relationship between the internal speed of sound and pressure, for ease of explanation, the speed of sound of entire NS region is also plotted in the lower right corner. In the NS surface area where the pressure is less than 150 MeV fm−3, the speed of sound changes rapidly with the pressure, while the value tends to be gentle as pressure goes further, at high density is almost constant. Our results also show that NS matter satisfies ${v}_{s}\leqslant c/\sqrt{3}$ near the surface, and satisfies $c/\sqrt{3}\leqslant {v}_{s}\leqslant c$ in the inner region, which is consistent with recent work. This agreement further supports the credibility of parameter sets from another perspective.
Figure 6. The relationship between the ratio of speeds of sound to light inside an NS as a function of pressure. The image in the lower right corner shows the sound speed in the entire NS region. |
5. f-mode oscillation of neutron stars
The discussion of non-radial modes under the framework of general relativity was originally proposed by Thorne and Campollataro [106]. For the non-rotating neutron star considered in our work, the interior is composed of an ideal fluid, we use the Cowling approximation approach [107–109], which ignores the space-time metric perturbation and only retains the density perturbations associated with oscillations of the fluid inside the star [27]. Some recent work shows that the difference between f-mode calculated using the Cowling approximation approach and by the complete linearized equations of general relativity is only less than 20%, p-mode is about 10% [110], and the error of g-mode is only a few percent [111]. This is enough to show the practicality of the Cowling approximation [112]. In this work, we adopt this approximation, and the fluid perturbations is composed of a spherical harmonic function Ylm(θ, φ) and a time-dependent part eiωt, the Lagrangian fluid displacements associated with infinitesimal oscillatory perturbations are expressed as:19 ).
$\begin{eqnarray}\begin{array}{rcl}{\zeta }^{{\rm{i}}} & = & \left[{{\rm{e}}}^{-{\rm{\Lambda }}(r)}W(r),-V(r){\partial }_{\theta },-V(r){\sin }^{-2}\theta {\partial }_{\phi }\right]{r}^{-2}\\ & & \times {{\rm{Y}}}_{l\ m}(\theta ,\phi ){{\rm{e}}}^{{\rm{i}}\omega t},\end{array}\end{eqnarray}$
where W(r) and V(r) satisfy the following equations: $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}W(r)}{{\rm{d}}r}=\displaystyle \frac{{\rm{d}}\varepsilon }{{\rm{d}}p}\left[{\omega }^{2}{r}^{2}{{\rm{e}}}^{{\rm{\Lambda }}(r)-2\phi (r)}V(r)+\displaystyle \frac{{\rm{d}}{\rm{\Phi }}(r)}{{\rm{d}}r}W(r)\right]\\ \quad -l(l+1){{\rm{e}}}^{{\rm{\Lambda }}(r)}V(r),\\ \displaystyle \frac{{\rm{d}}V(r)}{{\rm{d}}r}=2\displaystyle \frac{{\rm{d}}{\rm{\Phi }}(r)}{{\rm{d}}r}V(r)-\displaystyle \frac{1}{{r}^{2}}{{\rm{e}}}^{{\rm{\Lambda }}(r)}W(r),\end{array}\end{eqnarray}$
where $\begin{eqnarray}\displaystyle \frac{{\rm{d}}{\rm{\Phi }}(r)}{{\rm{d}}r}=\displaystyle \frac{-1}{\varepsilon (r)+p(r)}\displaystyle \frac{{\rm{d}}p}{{\rm{d}}r}.\end{eqnarray}$
Given appropriate boundary conditions, the above equations are the eigenvalue equations of ω. In the NS interior (r = 0), W(r) and V(r) have the following approximate behaviors $\begin{eqnarray}W(r)={{Ar}}^{l+1},\quad V(r)=-\displaystyle \frac{A}{l}{r}^{l},\end{eqnarray}$
where A is an arbitrary constant. Another boundary condition is that in the NS surface area, the pressure will disappear $\begin{eqnarray}{\omega }^{2}{{\rm{e}}}^{{\rm{\Lambda }}(R)-2{\rm{\Phi }}(R)}V(R)+{\left.\displaystyle \frac{1}{{R}^{2}}\displaystyle \frac{{\rm{d}}{\rm{\Phi }}(r)}{{\rm{d}}r}\right|}_{r=R}W(R)=0.\end{eqnarray}$
The ω that satisfies the above two boundary conditions is the eigenfrequency of the equation. In order to solve the numerical value, we also need to combine the TOV equation (We calculated the most typical non-radial bar mode instability for quadrupole oscillations (l = 2) [113] using the above parameter sets and showed the f-mode frequencies under different saturation parameters in figure 7. In all cases, the f-mode frequency increases with the mass, and this increase seems to be universal and the curve bends after reaching the maximum mass, which corresponds to an unstable NS. In the low mass region, the effect of the symmetry energy slope on the f-mode frequency is still significant due to the significant change in radius. However, it should be noted that the dependence of oscillation frequency on neutron star mass and its radius is very different from that of tidal deformability. In figure 7, we can see that for larger radius, the f-mode frequency is smaller. In fact, this is because the frequency depends on the average density of the neutron star (see the following discussion), which is completely different from the dependence of the tidal deformability on the compact parameter C. It is particularly worth noting that in the case for 1.4 M⊙, although the changes in f caused by different saturation parameters are different which are plotted in the upper left corner of figure 7, the f-mode frequencies given by all the cases are located in the range from 1.95 to 2.15 kHz. If future gravitational wave detectors could measure the f-mode frequency in this range, we may have a strong reason to speculate that the mass of this NS is approximately around 1.4 M⊙.
Figure 7. Relationships between mass and f-mode frequencies under different parameter sets. |
GW asteroseismology allows us to connect the oscillation mode frequency and damping timescale with the NS's bulk properties like mass, radius and tidal deformability, and give an empirical relationship between them [28, 113–117]. To our best knowledge, this relation was first proposed by Andersson and Kokkotas [118, 119]. They established a relationship related to the average density and f-mode frequency through the realistic EOS equation as:
$\begin{eqnarray}f(\mathrm{kHz})=a+b\sqrt{\displaystyle \frac{\bar{M}}{{\bar{R}}^{3}}},\end{eqnarray}$
where $\bar{M}/{\bar{R}}^{3}$ stands for the average density, and dimensionless parameters $\bar{M}$ and $\bar{R}$ are expressed as $\bar{M}=M/1.4{M}_{\odot }$ and $\bar{R}=R/10$ km.Other groups, such as [114, 115], investigated this relationship further by considering the rotational effect and the presence of exotic matter in the NS, but these studies were not based on the current multi-messenger astronomy results. In this manuscript, we want to look at how different nuclear saturation parameters affect the above linear relationship and try to provide an updated version of it.
We plot the results in figure 8. One can find that four saturation parameters have almost no effect on this linear relationship. If we only focus on the fitting effect, the f-mode frequencies given by different saturation parameters in the whole region have a good linear correlation with $\sqrt{\bar{M}/{\bar{R}}^{3}}$, showing a parameter-independent linear relationship. We list the fitting results in table 2. The first four rows correspond to the results in figure 8, and we also list the work of other researchers. All of the fitting results from our cases show a positive correlation between f-mode frequency and average density, with a correlation coefficient close to unity. We finally give a relationship with the 90% credible intervals for $f(\mathrm{kHz})=(1.0687\pm 0.088)+(1.10882\pm 0.066)\sqrt{\tfrac{\bar{M}}{{\bar{R}}^{3}}}$ by combining these fitting results. Once the f-mode frequency and NS's mass can be accurately measured in the future, the radius can be well constrained if the error of this universal relationship is further reduced. On the contrary, if NICER can accurately determine the mass and radius of the NS in the near future, it will be able to infer the f-mode frequency from this relationship.
Figure 8. The relationships between f-mode frequency and average density $\sqrt{\bar{M}/{\bar{R}}^{3}}$ under different saturation parameters, the red solid line represents the result of linear fitting. |
Table 2. The numerical fitting results from figure 8: the first four rows correspond to the different cases in this work. We finally get a fitting relation with f(kHz) = (1.0687 ± 0.088) $+(1.10882\pm 0.066)\sqrt{\tfrac{\bar{M}}{{\bar{R}}^{3}}}$ at the 90% credible intervals. In order to compare with other work, the results from other groups are also listed in the last four lines. |
| Cases | a (kHz) | b (kHz) |
|---|---|---|
| Case1 (K) | 1.06061 ± 0.00109 | 1.0005 ± 0.01177 |
| Case1 (m*) | 1.02569 ± 0.01152 | 1.06875 ± 0.01002 |
| Case1 (Esym) | 1.15633 ± 0.00151 | 1.0366 ± 0.00265 |
| Case1 (L) | 1.007 89 ± 0.00125 | 1.653 78 ± 0.00516 |
| 90% credible intervals | 1.0687 ± 0.088 | 1.108 82 ± 0.066 |
| Andersson and Kokkotas [118, 119] | 0.78 | 1.6350 |
| Omar Benhar et al [115] | 0.79 | 1.5000 |
| Daniela D Doneva et al [114] | 1.5620 | 1.1510 |
| Bikram Keshari Pradhan et al [117] | 1.0750 | 1.4120 |
So far, the relativistic parameter sets constructed at saturation density can meet the observational constraints of NSs from multi-messenger astronomy when extrapolated to high density. Obviously, when describing physics at low density, the parameter sets should also provide reliable physical results. Chiral effective field theory (χEFT) provides a well-organized low-energy expansion method for the interaction between nucleons, and is directly related to quantum chromodynamics [2, 120], along with the progress at low cutoff scales and renormalization group methods, χEFT has achieved great success in dealing with many-body physics, especially neutron matter [5, 6, 121].
At low density, we plot the energy per nucleon of pure neutron matter calculated by the above sets as a function of number density in figure 9, and compare it with the result given by the χEFT. In this manuscript, we adopt the constraints given in [5], which presents next-to-next-to-next-to-leading order (N3LO) in the chiral expansion based on potentials developed by Epelbaum, Glöckle and Meißner (EGM). We can see that among these sets that all are in line with astronomical observations mentioned above, also almost meet the constraint from χEFT represented by light green area of figure 9. That is to say, the parameter sets we constructed can not only meet the constraints from multi-messenger astronomy but also can better match the requirements of χEFT. This will show the reliability of these parameters and further reinforce the rationality of the conclusions we have reached above.
Figure 9. The relationship between the energy per nucleon and the number density in pure neutron matter (PNM), the result constrained by the χEFT is shown in the light green area. |
6. Conclusions and remarks
In this paper, we investigated the effects of relativistic parameter sets on the tidal deformabilities and f-mode oscillations of NSs under the framework of the traditional RMFT by considering scalar-isovector meson δ and divided the nucleon coupling parameters into an isoscalar part and an isovector part. According to the ground-state properties of nuclear physics, at the saturation density (including the experimental value of the binding energy per nucleon, incompressibility coefficient, nucleon effective mass, as well as symmetry energy and its slope), twenty sets of coupling parameters are constructed. These parameters, when extrapolated to a high density, can not only reproduce the masses of recently confirmed three massive neutron stars (PSR J1614–2230, PSR J0348 + 0432 and MSP J0740 + 6620), but the results also fall within the radius and tidal deformability range given by NICER and gravitational wave event GW170817.
The follow-up result we found is that the isovector saturation parameters have more significant influences on the radii and tidal deformabilities of neutron stars than the isoscalar saturation parameters. When fitting the relationship between the saturation parameters and radii of 1.4 M⊙ as well as tidal deformabilities of 2M⊙, we found that both of them have a relatively good linear relationship with the saturation parameters. We further investigate the effect of saturation properties on f-mode frequencies, and refit the universal relationship between f-mode frequency and average density for $f(\mathrm{kHz})=(1.0687\pm 0.088)+(1.10882\pm 0.066)\sqrt{\tfrac{\bar{M}}{{\bar{R}}^{3}}}$ with the 90% credible intervals. We also show that the f-mode frequencies with 1.4 M⊙ are roughly located in the range from 1.95 kHz to 2.15 kHz. These results may be tested by the future highly sensitive GW detectors. Moreover, together with the multi-messenger astronomy constraints, we used chiral effective theory (χEFT) to examine the behavior of these relativistic parameter sets at low densities, and further to reinforce the rationality of the conclusions we have reached.
In this paper, the EOS was discussed within the frame of a beta-equilibrium system, in which an NS is composed of baryonic matter and leptonic matter. Although the existence of hyperonic matter, condensation of (anti) kaons and pions, and quark matter in the interior of an NS are still to be further confirmed, a reasonable introduction of these assumptions undoubtedly further deepens our understanding on the internal composition of an NS. We will discuss these possibilities elsewhere.