Stability of neutron stars with dark matter core using three crustal types and the impact on mass–radius relations

doi:10.1016/j.dark.2023.101185

Physics of the Dark Universe

Volume 40, May 2023, 101185

https://doi.org/10.1016/j.dark.2023.101185 Get rights and content

Abstract

We investigate the effects of dark matter (DM) on the nuclear equation of state (EoS) and neutron star structure, in the relativistic mean field theory, both in the absence and presence of a crust. The $σ - ω$ model is modified by adding a WIMP-DM component, which interacts with nucleonic matter through the Higgs portal. This model agrees well with previous studies which utilized either a more complicated nuclear model or higher-order terms of the Higgs potential, in that DM softens the EoS, resulting in stars with lower maximum masses. However, instabilities corresponding to negative pressure values in the low-energy density regime of the DM-admixed EoS are present, and this effect becomes more prominent as we increase the DM Fermi momentum. We resolve this by confining DM in the star’s core. The regions of instability were replaced by three types of crust: first by the Friedman–Pandharipande–Skyrme (FPS), Skyrme-Lyon (SLy) and BSk19 EoS from the Brussels–Montreal Group, which can be represented by analytical approximations. For a fixed value of the DM Fermi momentum $p_{F}^{D M}$ , the DM-admixed neutron star does not have significant changes in its mass with the addition of the crusts. However, the entire mass–radius relation of the neutron star is significantly affected, with an observed increase in the radius of the star corresponding to the mass. The effect of DM is to reduce the mass of the star, while the crust does not affect the radius significantly, as the value of the $p_{F}^{D M}$ increases.

Introduction

Neutron stars are good testing grounds for predictions of theories beyond the standard model, since they are compact enough to provide conditions necessary for exotic physics to occur [1], [2], [3], [4]. Furthermore, they are a staple in the studies of nuclear physics, quantum chromodynamics (QCD), and general relativity (GR) [5], [6], [7].

One area of research that is currently very active in theoretical and observational astrophysics are neutron star interiors, especially with the advent of gravitational and electromagnetic wave observations among neutron star mergers [1], [2], [3], [4]. The description of static, nonrotating neutron stars is achieved by solving the Tolman–Oppenheimer–Volkoff (TOV) equations of GR [8], [9], [10], which are completed by an equation of state (EoS) [1], [2], [11], [12], [13], [14], [15]. This yields the mass–radius relations for neutron stars which can be analyzed [16]. Models for neutron stars utilize QCD, or phenomenologically, nuclear field theory in the context of the relativistic mean field theory (rMFT) in obtaining the EoS for nuclear structure, particularly at the core of the star [17], [18], [19]. Moreover, several semi-empirical approaches have also been developed to describe the overall structure of the neutron star, by including its outer layers, such as the crust and/or the atmosphere [11], [15].

Another factor that we can consider in the studies of neutron stars are the observations and measurements of the mass–energy density of the universe which shows that majority of its mass–energy content does not come from matter that is well-described by the standard model; about 25% is of the form now known as dark matter (DM) [20], [21]. Strong evidence for the existence of DM using galactic rotation curves was provided by Vera Rubin, Kent Ford and Ken Freeman in the 1960s and 1970s [22], [23]. A favored dark matter candidate is the weakly interacting massive particle (WIMP), which is predicted by supersymmetric extensions to the standard model, and at the same time supported by N-body cosmological simulations [24], [25]. Reviews on DM can be found in Refs. [21], [26], [27].

The effects of DM on neutron star structure, and other properties such as tidal deformability, curvature, and inspiral properties of binary neutron stars have been investigated in the literature, using different assumptions on the nature of the DM involved [28], [29], [30], [31], [32], [33], [34], [35], [36], [37]. Some of these used the relativistic mean field theory (rMFT) in quantum hadrodynamics (QHD), starting with different QHD models [30], [33], [35], [36]. In particular, the DM particle is assumed to be fermionic, captured and trapped inside the neutron star [30], [33], [35], [38]. The result of this approach is that DM softens the nuclear equation of state, yielding neutron stars of lower masses than neutron stars without DM [30], [33], [35]. This effect of reducing neutron star masses is also supported by studies assuming that there is a DM core, together with a nuclear EoS in the middle of the star [31].

A nuclear EoS, however is only dominant at the core of the neutron star, with densities greater than $ρ_{c} \sim 1 0^{14}$ g/cm $^{3}$ , while an actual neutron star can have a crust or atmosphere [11], [15]. The neutron star can then be thought of as having a crust, with density $ρ$ , surrounding the core, beginning with density $ρ_{c}$ , such that $ρ < ρ_{c}$ [11], [15]. In Ref. [33], the DM-admixed nuclear EoS was added with a Baym–Pethick–Sutherland (BPS) crust [39]. The BPS crust however only satisfies the EoS at low densities, and does not include the densities in the crust-core interface, which was approximated in Ref. [33] by a polytropic formula that connects the BPS crust with the DM-admixed core. In this paper we extend these studies by admixing DM at the nuclear core, and by adding three equations of state representing the crust on top of the core: the Friedman–Pandharipande–Skyrme (FPS) EoS, the Skyrme Lyon (SLy) EoS, and the BSk19 EoS, developed by the Brussels–Montreal group .

In this paper, we deal with the simplest QHD model, the $σ - ω$ or the Walecka model [17] and include the Higgs fields $h$ up to order $h^{2}$ . In the Standard Model, the Higgs fields are small fluctuations about the vacuum and higher orders of $h$ can be ignored. The $σ - ω$ model describes the interaction between the nucleons in matter through two meson fields, a scalar $σ$ , and a vector $ω$ , satisfying only the two minimal constraints for nuclear matter: the binding energy per nucleon, and the energy density at saturation. It does not take into account other constraints such as the compression modulus, the effective nucleon mass, isospin symmetry energy, and charge neutrality and beta equilibrium condition [5]. This model also only considers pure neutron matter. Even with the simplicity of the Walecka model, we are still able to extract the implications of putting a crust on top of the core of the star. We then extend the analysis of Ref. [30] by investigating instabilities in the DM-admixed EoS, and we fix these instabilities by replacing these unstable regions, which happen to be at the low density-end of the EoS with that of crust EoS, first with the FPS [12], then the SLy [13], and then the BSk19 [40] EoS; which can be represented by semi-analytical unified models that describe the neutron star crust realistically [11], [15]. The effects of these modifications to the DM-admixed EoS are then compared and studied.

We summarize the structure of this paper as follows. In Section 2, we discuss the modification of the Walecka model with DM. Section 3 then deals with adding the crusts to the DM-admixed EoS. The consequences of these modifications to the neutron star structure are discussed in Section 4. Finally, we conclude by giving some recommendations in Section 5. In this paper, we work with natural units $ħ = c = 1$ unless otherwise explicitly stated.

Section snippets

The Walecka model equation of state with DM

The simplest QHD model is the $σ - ω$ or Walecka model [5], [17]. It is a model describing nucleon–nucleon interaction that is mediated by exchanging $σ$ and $ω$ mesons. The fields in this model are based on four particles: the nucleons (neutrons and protons) $ψ$ , the scalar meson $σ$ , and the omega vector mesons $ω^{μ}$ , with a Lagrangian density given by $L_{had} = \bar{ψ} [i γ_{μ} (\partial^{μ} + i g_{ω} ω^{μ}) - (m_{n} - g_{σ} σ)] ψ + \frac{1}{2} (\partial_{μ} σ \partial^{μ} σ - m_{σ}^{2} σ^{2}) - \frac{1}{4} ω_{μ ν} ω^{μ ν} + \frac{1}{2} m_{ω}^{2} ω_{μ} ω^{μ},$ where $ω^{μ ν} = \partial^{μ} ω^{ν} - \partial^{ν} ω^{μ}$ , $m_{n} \approx 1$ GeV is the mass of the nucleon (or neutron), $m_{σ} = 520 MeV$ is the mass of

The DM-admixed EoS with crust

A remedy for the instability problem presented in Section 2 is to replace the unstable regions in the EoS. This can be done by replacing the low-pressure regions with another EoS that better describes it, similar to an atmosphere or crust. The underlying assumption for this is that our DM is trapped only inside the core of the neutron star; this means that the crust contains a negligible amount of DM particles. We note that this method cannot precisely determine the relative amount of DM inside

The structure equations and mass–radius relations

Using the dimensionless quantities for $ϵ$ and $P$ defined in Eq. (8) as well as the following: $\tilde{M} = \frac{M}{M_{⊙}}, \tilde{r} = \frac{r}{R_{0}}, R_{0} = G M_{⊙}, Ω = \frac{4 π ϵ_{0}}{M_{⊙}} R_{0}^{3},$ where $M$ is the mass, $r$ is the distance from the center of the star, and $M_{⊙}$ is the solar mass, we can write the TOV equations in dimensionless form as $\frac{d \tilde{P}}{d \tilde{r}} = \frac{- [\tilde{ϵ} + \tilde{P}] [\tilde{M} + Ω {\tilde{r}}^{3} \tilde{P}]}{{\tilde{r}}^{2} - 2 \tilde{M} \tilde{r}},$ $\frac{d \tilde{M}}{d \tilde{r}} = Ω {\tilde{r}}^{2} \tilde{ϵ},$ with the conditions $\tilde{P} (\tilde{r} = 0) = {\tilde{P}}_{c}, \tilde{P} (r = R_{⋆}) = 0$ $\tilde{M} (\tilde{r} = 0) = 0, \tilde{M} (r = R_{⋆}) = M_{⋆},$ where $P_{c}$ is the central pressure and $R_{⋆}$ is the stellar radius. The TOV equations describe static, spherically

Conclusions

In this work, we extended the investigation of DM-admixed neutron stars by confining the DM in the star’s core and by adding a crust on the core. This simulates a neutron star with a crust, and with a core dominated by the nuclear equation of state, which, in this case, was the $σ - ω$ model added with DM, which was obtained via relativistic mean field theory.

Three types of crust were considered: the FPS, SLy, and BSk19 crusts. These crust equations of state were used to replace the instabilities

CRediT authorship contribution statement

Adrian G. Abac: Conceptualization, Methodology, Software, Verification, Formal analysis, Investigation, Data curation, Writing – original draft, Writing – review and editing, Visualization. Christopher C. Bernido: Writing – original draft, Writing – review and editing, Supervision. Jose Perico H. Esguerra: Writing – original draft, Writing – review and editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

A. G. Abac wishes to acknowledge the support of the Department of Science of Technology (Philippines) - Accelerated Science and Technology Human Resource Development Program during the course of this study.

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View full text

Stability of neutron stars with dark matter core using three crustal types and the impact on mass–radius relations

Abstract

Introduction

Section snippets

The Walecka model equation of state with DM

The DM-admixed EoS with crust

The structure equations and mass–radius relations

Conclusions

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgments

Phys. Lett.

Internat. J. Modern Phys. D

Annu. Rev. Nucl. Part. Sci.

Annu. Rev. Nucl. Part. Sci.

Eur. Phys. J.

Phys. Rev. D

Compact Stars: Nuclear Physics, Particle Physics and General Relativity

Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects

Compact Objects in Astrophysics

Phys. Rev.

Phys. Rev.

Spacetime and Geometry: An Introduction to General Relativity

Astron. Astrophys.

Hot Nuclear Matter

Astron. Astrophys.

Neutron Stars I: Equation of State and Structure

Astron. Astrophys.

Amer. J. Phys.

Internat. J. Modern Phys. E

Phys. Rev. C

Phys. Rev. D

Internat. J. Modern Phys. A

Astrophys. J.

Astrophys. J.