Exotic Cores with and without Dark-Matter Admixtures in Compact Stars

Abstract

We parameterize the core of compact spherical star configurations by a mass ($m_x$) and a radius ($r_x$) and study the resulting admissible areas in the total-mass–total-radius plane. The employed fiducial equation-of-state models of the corona at radii $r>r_x$ and pressures $p\le p_x$ with $p(r=r_x)=p_x$ are that of constant sound velocity and a proxy of DY$\mathsf{\Delta }$ DD-ME2 provided by Buchdahl’s exactly solvable ansatz. The core ($r<r_x$) may contain any type of material, e.g., Standard-Model matter with unspecified equation of state or/and an unspecified Dark-Matter admixture. Employing a toy model for the cool equation of state with first-order phase transition, we also discuss the mass-radius relation of compact stars with an admixture of Dark Matter in a Mirror-World scenario.

1. Introduction

The advent of detecting gravitational waves from merging neutron stars, the related multi-messenger astrophysics [1,2,3,4,5,6,7,8] and the improving mass-radius determinations of neutron stars, in particular by NICER data [9,10,11], stimulated a wealth of activities. Besides mass and radius, moments of inertia and tidal deformabilities become experimentally accessible and can be confronted with theoretical models [1,12,13,14,15,16,17,18]. The baseline of the latter ones is provided by non-rotating, spherically symmetric cold dense matter configurations. The sequence of white dwarfs (first island of stability) and neutron stars (second island of stability) and—possibly [19]—a third island of stability [20,21,22] thereby shows up when moving to more compact objects, with details depending sensitively on the actual equation of state (EoS). The quest for a fourth island has been addressed too [23,24]. Here, “stability” means the damping of radial disturbances. Since the radii of configurations of the second (neutron stars) and third (hypothetical quark/hybrid stars [25,26,27]) islands are very similar, the notion of twin stars [28,29] has been coined for equal-mass configurations; “masquerade” was another related term [30].

The standard modeling of such static compact star configurations is based on the Tolman–Oppenheimer–Volkov (TOV) equations

$$\begin{array}{ccc}\hfill \frac{dp}{dr}& =& -G_N\frac{(e+p)(m+4\pi p{r}^3)}{r^2(1-\frac{2G_{N}m}{r})},\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill \frac{dm}{dr}& =& 4\pi e{r}^2,\hfill \end{array}$$

(2)

emerging from the energy–momentum tensor of a perfect isotropical fluid and spherical symmetry of space–time and matter as well, within the framework of Einstein gravity without a cosmological term. Newton’s constant is denoted by $G_N$, natural units with $c=1$ are used, unless relating mass and length and energy density, for which $\hslash c$ is needed. Given a unique relationship of pressure p and energy density e as equation of state (EoS) $e\left(p\right)$, in particular at zero temperature, the TOV equations are integrated with boundary conditions $p\left(0\right)=p_c$ and $m\left(0\right)=0$ (implying $p\left(r\right)=p_c-\mathcal{O}\left(r^2\right)$ and $m\left(r\right)=0+\mathcal{O}\left(r^3\right)$ at small radii r), and $p\left(R\right)=0$ and $m\left(R\right)=M$ with R as circumferential radius and M as gravitational mass (acting as a parameter in the external (vacuum) Schwarzschild solution at $r>R$). The quantity $p_c$ is the central pressure. The solutions $R\left(p_c\right)$ and $M\left(p_c\right)$ provide the mass-radius relation in parametric form $M\left(R\right)$.

A great deal of effort is presently focused on the EoS at supra-nuclear densities [31]. Figure 1 in [32] exhibits the currently admitted uncertainty: up to a factor of ten in pressure as a function of energy density. At asymptotically large energy-density, perturbative QCD constrains the EoS, though it is just the non-asymptotic supra-nuclear density region which crucially determines whether twin stars may exist or quark-matter cores appear in neutron stars. Accordingly, one can fill this gap with a big number of test EoSs to scan through the possibly resulting mass–radius curves; see [33,34,35,36]. However, the option that neutron stars can accommodate Dark Matter (or other exotic material) [37,38,39,40] obscures the safe theoretical modeling of a reliable mass–radius relation in such a manner. Of course, inverting the posed problem with sufficiently precise data of masses and radii as input offers a promising avenue towards determining the EoS [33,41,42,43,44,45,46].

Here, we pursue another perspective: we parameterize the supra-nuclear core by a radius $r_x$ and the included mass $m_x$ and integrate the above TOV equations only in the corona, i.e., from pressure $p_x$ to the surface, where $p=0$. This yields the total mass $M(r_x,m_x;p_x)$ and the total radius $R(r_x,m_x;p_x)$ by assuming that the corona EoS $e\left(p\right)$ is reliably known at $p\le p_x$. (Our notion “corona” could be substituted by “mantel” or “crust” or “envelope". It refers to the complete part of the compact star outside the core, $r_x\le r\le R$.) Clearly, without knowledge of the matter composition at $p>p_x$ (may it be Standard-Model matter with uncertain EoS or may it contain a Dark-Matter admixture, for instance, or monopoles or some other type of “exotic” matter) one does not get a simple mass–radius relation by such procedure, but admissible area(s) over the mass–radius plane, depending on the core parameters $r_x$ and $m_x$. This is the price of avoiding a special model of the core matter composition.

We are going to test two models of the corona EoS: (i) constant sound velocity with crucial parameter $e_0=e(p=0)$ and $p=0$ for $e<e_0$, and (ii) Buchdahl’s EoS which, surprisingly, represents a proxy of the DY$\mathsf{\Delta }$ DD-ME2 EoS [23] and has the property ${lim}_{p\to 0}e\left(p\right)\to 0$. One could consider this as a test of the sensitivity against variations of the neutron star crust (see [47,48] for such a crust test within the traditional approach with a given core EoS).

We emphasize the relation of ultra-relativistic heavy-ion collision physics, related to the EoS $p(T,{\mu }_B\approx 0)$, and compact star physics, related to $p(T\approx 0,{\mu }_B)$, when focusing on static compact-star properties [49,50]. (Of course, in binary or ternary compact-star merging-events, finite temperatures T and a large range of baryon-chemical potential ${\mu }_B$ are also probed [51].) Implications of the conjecture of a first-order phase transition at small temperatures and large baryo-chemical potentials or densities [52,53,54,55] can be studied by neutron-hybrid-quark stars [56,57,58,59]. It has been known for some time [20,21,22,60,61] that a cold EoS with special pressure–energy-density relation $p\left(e\right)$, e.g., a strong local softening up to first-order phase transition with a density jump, can give rise to a “third family” of compact stars, beyond white dwarfs and neutron stars. In special cases, the third-family stars appear as twins of neutron stars [30,62,63]. Various scenarios of the transition dynamics to the denser configuration as mini-supernova were discussed quite early [64,65].

Our paper is organized as follows. In Section 2, we employ the corona EoS model (i) (constant sound velocity) and also consider the instructive limiting case of an incompressible fluid. Special relations of $m_x\left(r_x\right)$ are deployed to emphasize the relation to the traditional approach, which emerges as cut in the M-R plane representing the curve $M\left(R\right)$. Section 3 uses the corona EoS model (ii) (Buchdahl’s EoS as proxy of a nuclear-physics-based EoS). The $M\left(R\right)$ relations for models (i) and (ii) are quite different, as are the admitted areas over the M-R plane. We conclude in Section 4. Appendix A considers Dark-Matter or Mirror-World admixtures in neutron (quark) stars with a phase transition with the goal of illustrating the potential impact of a first-order phase transition on the mass–radius relation and the quest for twin stars. This is another way to specify the relation $m_x\left(r_x\right)$.

2. Corona with Constant Sound Velocity EoS

The constant sound velocity EoS

$$p\left(e\right)=v_s^2(e-e_0)\mathrm{for}e>e_0,p=0\mathrm{for}e\le e_0$$

(3)

introduces a scale setting parameter $e_0$, which drops out in the representation by scaled quantities: $\overline{p}=p/e_0$, $\overline{r}=r\sqrt{G_N{e}_0}$, $\overline{m}=m{G}_N\sqrt{G_N{e}_0}$. Only the sound velocity $v_s^2=\partial p/\partial e$ remains as parameter. We use this as model (i) and—arbitrarily—take ${\overline{p}}_x=1$ as the upper limit of scaled pressure of the corona EoS.

2.1. A Limiting Case: Infinite Sound Veleocity

It is instructive to consider the limiting case $v_s^{-2}\to 0$, i.e., $\overline{e}\to 1$ for $\overline{p}\le 1$. The constant energy density in Equation (2) results in $\overline{m}={\overline{m}}_x+\frac{4\pi }{3}({\overline{r}}^3-{\overline{r}}_x^3)$, and the Riccati-type ordinary differential Equation (1) becomes, by a suitable shift of the pressure, a Bernoulli equation, allowing a quadrature to get the pressure profile $\overline{p}\left(\overline{r}\right)=-1+({\overline{p}}_c+1)exp\{\frac{1}{2}(\lambda -{\lambda }_0\}N$, $N^{-1}=1+4\pi ({\overline{p}}_c+1)exp\{-\frac{1}{2}{\lambda }_0\}{\int }_{{\overline{r}}_x}^{\overline{r}}d\tilde{r}\tilde{r}exp\left\{\frac{3}{2}\lambda \left(\tilde{r}\right)\right\}$, $e^{-\lambda }=1-2\overline{m}/\overline{r}$, ${\lambda }_0=\lambda \left({\overline{r}}_x\right)$, cf. [22,66]. Since the term with ${\int }_{{\overline{r}}_x}^{\overline{r}}d\tilde{r}\tilde{r}exp\left\{\frac{3}{2}\lambda \left(\tilde{r}\right)\right\}$ is less transparent due to the elliptic integral(s), we turn on an approximation of the case $e=const$ and numerically integrate Equations (1) and (2) with $v_s^{-2}={10}^{-4}$ to mimic the above constant energy density EoS by a proxy with large sound velocity. We then find admissible area in the $\overline{M}$-$\overline{R}$ plane displayed in Figure 1. That area is mapped out here by curves ${\overline{r}}_x=const$ with ${\overline{m}}_x$ varying from small (at the r.h.s.) to large (at the l.h.s.) values. The l.h.s. limitation is given by the black hole condition $\overline{M}=\overline{R}/2$ (red fat line). In the limit of small cores, the ${\overline{r}}_x=const$ curves $\overline{M}={\overline{m}}_x+\frac{4\pi }{3}{\overline{R}}^3-\mathcal{O}\left(r_x^3\right)$ approach the Schwarzschild mass–radius relation $\overline{M}=\frac{4\pi }{3}{\overline{R}}^3$ (dashed black curve), i.e., the gap to the ${\overline{r}}_x=const$ curve becomes larger with decreasing values of $\overline{R}$ due to increasing ${\overline{m}}_x$. The Schwarzschild curve terminates in the asterisk, since that is where ${\overline{p}}_c=1$ is reached. One may recall the pressure profile of the interior Schwarzschild solution ${\overline{p}}_{Schwarz}\left(\overline{r}\right)=-[(1+{\overline{p}}_c)-(1+3{\overline{p}}_c)W]/[3(1+{\overline{p}}_c)-(1+3{\overline{p}}_c)W]$, $W:=\sqrt{1-\frac{8\pi }{3}{\overline{r}}^2}$, yielding $\overline{R}\left({\overline{p}}_c\right)$ from $\overline{p}\left(\overline{R}\right)=0$ to see this via our proposition $\overline{p}\le {\overline{p}}_x$.

In this context, it is interesting to compare the pressure profiles of the interior Schwarzschild solution and the core–corona model at small values of ${\overline{r}}_x$. Adjusting the central pressure of the interior Schwarzschild solution to obtain the same scaled radius $\overline{R}$ as in the core–corona model with a given small value of ${\overline{r}}_x$ and a given value of ${\overline{m}}_x$, one finds (not displayed) a rapidly dropping pressure in a narrow region above ${\overline{r}}_x$ towards the Schwarzschild pressure ${\overline{p}}_{Schwarz}\left(\overline{r}\right)$, which holds until the surface. In such a way, the admissible area of masses and radii of the core–corona model are bracketed by the black-hole limit at l.h.s. and the conventional mass–radius curve in the lower part of the mass–radius plane.

Let us now consider two possible sections through the mass–radius plane. First, we assume a core-mass vs. core-radius relation as ${\overline{m}}_x\left({\overline{r}}_x\right)=\frac{4\pi }{3}{\overline{r}}_x^3$. This yields the dotted curve, which clearly coincides with the Schwarzschild relation when continuing the latter one to central pressures larger than ${\overline{p}}_x$. Note that with increasing values of ${\overline{r}}_x$, the admissible region of the core–corona model leaks increasingly into the r.h.s.

A second side condition in relating the core-mass and core-radius is given by ${\overline{m}}_x\left({\overline{r}}_x\right)=\frac{4\pi }{3}\overline{\lambda }{\overline{r}}_x^3$, meaning that the energy density in the core is $\overline{\lambda }$ times the energy density of the corona. (This is essentially the two-shell model considered in [21,22]). Selecting $\overline{\lambda }=2.2$ facilitates the cyan double curve, again a well-defined mass–radius curve. Larger (smaller) values of $\overline{\lambda }$ bend down (up) that curve. The section between the local r.h.s. mass maximum at the asterisk and the local mass minimum belongs to unstable configurations, while left to the local mass minimum, the configurations become stable again. This very construction is at the heart of twin stars, i.e., stable configurations with the same mass but (slightly) different radii. There is notable leakage into the region l.h.s. to the Buchdahl bound (cf. [67] for a corollary) given by $\overline{M}=4\overline{R}/9$. This issue is caused by the fact that a central pressure ${\overline{p}}_c<\infty$ does not drop to ${\overline{p}}_x$ at radius ${\overline{r}}_x$ when using the TOV equations with core density $\overline{\lambda }{\overline{e}}_0$. Phrased differently, the above ad hoc choice ${\overline{m}}_x\left({\overline{r}}_x\right)=\frac{4\pi }{3}\overline{\lambda }{\overline{r}}_x^3$ is compatible with the TOV equations only for a limited range of values of ${\overline{r}}_x$. The validity of the TOV equations for a consistent (always decreasing) pressure profile from $p_c$ to zero limits the extension of the curve under discussion and lets it terminate before reaching Buchdahl’s limit. Nevertheless, extensions of the TOV equations, e.g., as system describing a multi-fluid medium with Dark-Matter components (see Appendix A), make the occupancy of the region beyond Buchdahl’s limit conceivable.

2.2. Finite Sound Velocity

For a finite sound velocity, the mass–radius curve bends to the left and facilitates a mass maximum, e.g., ${\overline{M}}_{max}=0.051$ at radius $\overline{R}=0.195$ for $v_s^{-2}=3$. By the requirement of $M\approx 2M_{\odot }$ one gets $e_0\approx 231.5$ MeV/fm${}^3$ and finds, from $R=M{G}_N\overline{R}/\overline{M}$, a radius of 11.27 km, (accidentally) in the right ballpark.

The mass–radius plane is exhibited in Figure 2 with the same line style conventions as in Figure 1. One meets the features discussed in the previous Subsection. The admissible region is bracketed l.h.s. by the black hole condition and r.h.s. by the plain mass–radius curve $\overline{M}\left(\overline{R}\right)$ (dashed black curve) up to the asterisk, where ${\overline{p}}_c={\overline{p}}_x$ is reached. In the dotted section, the central pressures obey ${\overline{p}}_c>{\overline{p}}_x$. Due to non-zero matter compressibility, the analog relation ${\overline{m}}_x\left({\overline{r}}_x\right)$, discussed in Section 2.1, must be constructed numerically to see that this side condition within the core–corona model in fact coincides with the mass–radius curve obtained by the (ad hoc) continuation of the EoS (3) at $p>p_x$. For small values of ${\overline{r}}_x=const$, the permitted masses as a function of radius approach the mass–radius curve $\overline{M}\left(\overline{R}\right)$. For larger constant values of ${\overline{r}}_x$, the pattern of the curves $\overline{M}(\overline{R},{\overline{r}}_x,{\overline{m}}_x;{\overline{p}}_x)$ is as discussed in Section 2.1. They also leak beyond the Buchdahl limit.

We relegate the consideration of special cuts ${\overline{m}}_x\left({\overline{r}}_x\right)$ and variations of $p_x$ to the Appendix A, where we explicitly include an admixture of Dark Matter to be dealt within a two-fluid approach.

3. Corona with Buchdahl’s EoS and NYΔ DD-EM2

While the employed corona EoS in Section 2.2 displays a finite positive slope of the mass–radius curve for a large range, the currently preferred EoS models point to a steep increase. An example of an EoS compatible with neutron star data is NY$\mathsf{\Delta }$, based on the DD-ME2 density functional [23]. Surprisingly, Buchdahl’s EoS [68]

$$e=12\sqrt{p_{*}p}-5p,p\le p_{*}$$

(4)

with $p_{*}\approx 64$ MeV/fm${}^3$ matches fairly well NY$\mathsf{\Delta }$ within the range tabulated in [23], see Table 1. Due to causality requirement $v_s^2\le 1$, Equation (4) is limited to $p\le p_{*}$. In contrast to the EoS of model (i), Equation (4) extends smoothly to $e=0$ and $p=0$. The mass–radius relation is given by $M=\beta R/G_N$, and the radius is parameterized by $R=\frac{1-\beta }{\sqrt{1-2\beta }}\sqrt{\frac{\pi }{288p_{*}{G}_N}}$ for $\beta \in [0,1/6]$ [68,69]. In this section, we use physical units, e.g., $p_{*}$ in MeV/fm${}^3$ instead of ${\overline{p}}_x$, and display masses in units of $M_{\odot }$ and radii in units of km. With the above quoted value of $p_{*}$ one finds $R=11.33\frac{1-\beta }{\sqrt{1-2\beta }}$ km and $M=7.66\beta \frac{1-\beta }{\sqrt{1-2\beta }}{M}_{\odot }$, where the maximum mass $1.31M_{\odot }$ and radius $11.56$ km can be read off.

The resulting mass–radius plot is exhibited in Figure 3, with line style conventions as seen in Figure 1. All the features discussed in Section 2 are recovered, e.g., the convex shape of the curves $r_x=const$, the approach of these curves to the mass–radius relation based on the EoS (4) at small values of $r_x$, the occupancy of a region beyond the Buchdahl limit up to the black hole limit etc. Since $p_x=p_{*}$ is comparatively small, the resulting mass–radius curve (solid black curve) terminates at rather small mass, quoted above. However, the core–corona decomposition shows that the maximum-mass region of NY$\mathsf{\Delta }$-DD-ME2 (see gray curves) is easily uncovered too, interestingly with sizeable core radii and noticeably smaller up to larger total radii.

4. Summary

We investigate a scenario which assumes the knowledge of a trustable equation of state (EoS) for compact spherical stars at pressures $p\le p_x$. In a core–corona decomposition, the core with an unspecified composition and thus unknown (maybe exotic matter) equation(s) of state is parameterized by a mass $m_x$ and a radius $r_x$; the pressure at $r_x$ is just $p_x$. Besides the unspecified equations of state at $p>p_x$, the core may also contain any type of matter, most notably Dark Matter. By definition, outside the core—in the corona—only a Standard-Model neutron-star matter is present, described by the fiducial (“trustable”) EoS. The solution of the TOV equations of the corona pressure profile $p\left(r\right)$ for radii $r\ge r_x$ delivers a broad range of admissible masses and radii of the total core–corona configurations, bracketed by the black hole limit (thus going beyond the Buchdahl limit) and, in part, by the conventional mass–radius curve. We explore these admissible regions by using two test EoS models of the corona as simple substitutes of a state-of-the-art EoS: one with a constant sound velocity EoS and another one which refers to Buchdahl’s EoS, which in turn approximates some part of a particular density-functional approach, i.e., NY$\mathsf{\Delta }$ DD-ME2. The latter is compatible with current mass–radius observations.

One may imagine that all possible continuations of a fiducial (“trustable”) EoS beyond $p_x$ generate mass–radius curves within the admissible region, even with the Buchdahl limit as a border line. However, the admissible region is larger, since it can refer to cores which are not described by the isotropic one-fluid TOV equations. A straightforward extension of the latter is provided by the two-fluid TOV equations, dealt with in the Appendix A. In line with investigations of the impact of the cool EoS on compact star properties, including the intriguing quest for twin stars, we add this schematic consideration of compact-star mass–radius relations for an EoS with first-order phase transition. In doing so, the two-fluid model is deployed where the second fluid is aimed at describing Dark Matter leaking out of a Mirror World. This scenario also facilitates the option of twin stars for a wide range of the Dark Matter admixture.