# Numerical Simulations of Dark Matter Admixed Neutron Star Binaries

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. BAM Code

#### 2.2. Inclusion of Mirror Dark Matter

## 3. Single Star Simulations

#### 3.1. Solving the TOV Equations

#### 3.2. Single Star Test Runs

## 4. Binary Neutron Star Simulations

#### 4.1. Initial Configurations

#### 4.2. Time Evolutions

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Resolution Study for the Binary Neutron Star Simulations

**Figure A1.**

**Left**: Evolution of Hamiltonian norm for $1.3$ M${}_{\odot}$ simulation on the coarsest level. Overall, we find a decreasing constraint violation before the merger, due to the constraint damping properties of the Z4c evolution scheme. The jumps at late times are due to the merger and then the formation of a singularity.

**Right**: Hamiltonian norm on the coarsest level for the simulation of a 1.4 M${}_{\odot}$ and $5\%$ mirror dark matter star for the four different resolutions described in Table 3.

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**Figure 1.**

**Left**: Total mass values as a function of the total central density of the neutron star, for a pure baryonic star (blue line, magenta dots) and 50% mirror dark matter admixed star (black line, red dots). The density is expressed in units of the nuclear saturation denisty. The SLy EoS is used for both plots.

**Right**: Total mass as a function of the radius, for the same neutron star models. The lines show data obtained as described in Ciancarella et al. [26] and the overplotted dots result from our updated BAM version.

**Figure 2.**Density profiles of admixed neutron stars with a 0%, 5%, 10%, 15% and 20% mirror dark matter contribution to the total gravitational mass of 1.4 M${}_{\odot}$. Blue lines refer to the baryonic density, while red ones denote the dark matter profiles. It is noticeable that with increasing dark matter content, the star has a more compact equilibrium configuration.

**Figure 3.**

**Left**: Time evolution of the central density of baryonic matter (blue) and mirror dark matter (red) for the four different resolutions presented in this section.

**Right**: Time evolution of the L2-norm of the Hamiltonian constraint for R1, R2, R3, and R4. The convergence order for the three highest resolutions is two. In all these runs the star is 50% dark matter admixed and has a total gravitational mass of 1.4 M${}_{\odot}$ and a baryonic mass of 1.56 M${}_{\odot}$.

**Figure 4.**Waveform strain and frequency of the $\ell =\left|m\right|=2$ GW mode for all simulations. Each plot contains simulations with the neutron star mass combination listed at the top of the column. The colors represent the different mirror dark matter fractions. All simulations with the same total mass are aligned during the first few orbits of the simulation time. Black dashed lines mark the time interval where the waveform alignment occurs.

**Figure 5.**Snapshots on the equatorial plane of the 1.3 M${}_{\odot}$–1.3 M${}_{\odot}$ simulation with $5\%$ dark matter. x and y are in isotropic coordinates. The colormap represents the ${log}_{10}$ of baryonic rest mass density in cgs units. The contours represent the same quantity for mirror dark matter.

**Table 1.**Initial data, for convergence tests carried out with single mirror dark matter admixed neutron stars; cf. main text for further details.

Name | Total Mass (M${}_{\odot}$) | Mirror Dark Matter % | ${\mathit{\rho}}_{\mathit{c}}^{\mathit{b}}$ [${\mathit{\rho}}_{\mathit{nuc}}$] | ${\mathit{\rho}}_{\mathit{c}}^{\mathit{dm}}$ [${\mathit{\rho}}_{\mathit{nuc}}$] |
---|---|---|---|---|

SLy_M127_0pc | 1.27 | 0% | 3.558 | 0 |

SLy_M132_25pc | 1.32 | 25% | 5.212 | 4.045 |

SLy_M14_50pc | 1.40 | 50% | 6.518 | 6.518 |

SLy_M132_75pc | 1.32 | 75% | 4.045 | 5.212 |

SLy_M127_100pc | 1.27 | 100% | 0 | 3.558 |

**Table 2.**Initial data for the mirror dark matter admixed binary neutron star simulations. The columns refer to the gravitational mass of each star, the dark matter percentage of the gravitational mass, the initial central density values for baryonic matter and dark matter respectively, and the total radius of the single stars. For the 1.4 M${}_{\odot}$–1.4 M${}_{\odot}$ and 5% configuration, runs were performed for the four different resolution setups described in Section 3.2.

M${}_{\mathit{A},\mathit{B}}$ (M${}_{\odot}$) | Mirror Dark Matter % | ${\mathit{\rho}}_{\mathit{c}}^{\mathit{b}}$ [${\mathit{\rho}}_{\mathit{nuc}}$] | ${\mathit{\rho}}_{\mathit{c}}^{\mathit{dm}}$ [${\mathit{\rho}}_{\mathit{nuc}}$] | R${}_{\mathit{A},\mathit{B}}$ [km] | |
---|---|---|---|---|---|

SLy_M14_0 | 1.4 | 0% | 3.866 | 0 | 11.45 |

SLy_M14_5 | 1.4 | 5% | 4.360 | 2.234 | 11.00 |

SLy_M14_10 | 1.4 | 10% | 4.713 | 2.854 | 10.60 |

SLy_M13_0 | 1.3 | 0% | 3.624 | 0 | 11.46 |

SLy_M13_5 | 1.3 | 5% | 4.058 | 2.087 | 11.04 |

SLy_M13_10 | 1.3 | 10% | 4.366 | 2.679 | 10.63 |

SLy_M12_0 | 1.2 | 0% | 3.398 | 0 | 11.46 |

SLy_M12_5 | 1.2 | 5% | 3.791 | 1.960 | 11.04 |

SLy_M12_10 | 1.2 | 10% | 4.056 | 2.499 | 10.65 |

**Table 3.**Table of resolutions with columns showing the number of points per direction of the fixed levels and of the moving levels, the grid spacing of the coarsest level and the grid spacing of the finest one respectively. For all binary neutron star configurations we used 6 refinement levels.

n | ${\mathit{n}}_{\mathit{mov}}$ | ${\mathit{h}}^{\mathit{max}}$ (km) | ${\mathit{h}}^{\mathit{min}}$ (m) | |
---|---|---|---|---|

R1 | 64 | 128 | 11.81 | 369 |

R2 | 96 | 192 | 7.88 | 246 |

R3 | 128 | 256 | 5.91 | 185 |

R4 | 160 | 320 | 4.73 | 148 |

**Table 4.**First column: Name of the simulation. Second column: Total unbound matter flowed through a sphere with a radius of 450 km centered around the origin. The unbound matter is identified using the ballistic criterion, (see main text). Third column: Maximum value of the integral of unbound matter over the simulation domain reached during the simulation time. The unbound matter is again identified using the ballistic criterion. lFourth column: Mass of the disk at 5 ms after the merger was identified as the total bound mass outside a radius of 12 km from the center. Fifth column: Frequency of the $\ell =\left|m\right|=2$ mode of the gravitational waveform taken at the merger, i.e., the time of the GW strain maximum. * The disk of these simulations has been measured before the collapse of the remnant. Therefore, it can be affected by a higher uncertainty due to the ambiguity in the definition of the disk.

${\mathbf{M}}_{\mathit{ej}}$ Sphere (M${}_{\odot}$) | M${}_{\mathit{ej}}$ Integral (M${}_{\odot}$) | M${}_{\mathit{disk}}$ (M${}_{\odot}$) | ${\mathit{f}}_{\mathit{merger}}$ [Hz] | |
---|---|---|---|---|

SLy_M14_0 | - | - | 0.001 | 1770 |

SLy_M14_5 | - | - | 0.0008 | 2030 |

SLy_M14_10 | - | - | 0.0014 | 2058 |

SLy_M13_0 | 0.0168 | 4.8 × 10${}^{-3}$ | 0.062 | 1817 |

SLy_M13_5 | 0 | 0.7 × 10${}^{-3}$ | 0.001 | 1910 |

SLy_M13_10 | 0 | 0.8 × 10${}^{-3}$ | 0.0006 | 2221 |

SLy_M12_0 | 0 | 0.3 × 10${}^{-3}$ | $0.19\phantom{\rule{3.33333pt}{0ex}}*$ | 1746 |

SLy_M12_5 | 0.0016 | 2.6 × 10${}^{-3}$ | $0.16\phantom{\rule{3.33333pt}{0ex}}*$ | 1818 |

SLy_M12_10 | 0.0027 | 3.3 × 10${}^{-3}$ | 0.017 | 2198 |

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**MDPI and ACS Style**

Emma, M.; Schianchi, F.; Pannarale, F.; Sagun, V.; Dietrich, T. Numerical Simulations of Dark Matter Admixed Neutron Star Binaries. *Particles* **2022**, *5*, 273-286.
https://doi.org/10.3390/particles5030024

**AMA Style**

Emma M, Schianchi F, Pannarale F, Sagun V, Dietrich T. Numerical Simulations of Dark Matter Admixed Neutron Star Binaries. *Particles*. 2022; 5(3):273-286.
https://doi.org/10.3390/particles5030024

**Chicago/Turabian Style**

Emma, Mattia, Federico Schianchi, Francesco Pannarale, Violetta Sagun, and Tim Dietrich. 2022. "Numerical Simulations of Dark Matter Admixed Neutron Star Binaries" *Particles* 5, no. 3: 273-286.
https://doi.org/10.3390/particles5030024