# The Impact of Asymmetric Dark Matter on the Thermal Evolution of Nucleonic and Hyperonic Compact Stars

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{5}years), and a final phase dominated by photon emission (age ≳ 10

^{5}years) [36,37]. The initial phase corresponds to the duration required for the core and crust of an NS to reach a thermal equilibrium, referred to as the thermal relaxation time. During this stage, the surface temperature of the NS remains constant as neutrinos gradually diffuse through the optically thick medium, supplying enough energy to counterbalance the cooling. As neutrinos reach the stellar surface, the star’s temperature drops as a consequence of the thermal connection between the core and crust.

_{⊙}[49,50,51,52]. While providing the existence of heavy NSs with masses above $\sim 2$ M

_{⊙}, stiffer hadronic EoSs (see, e.g., Ref. [53]) are discriminated by the observational constraints on the tidal deformability [54]. These difficulties are naturally overcome within the scenario of early deconfinement of quark matter before the hyperon onset [55,56]. In this case, all the observational constraints can be fulfilled simultaneously [57].

## 2. Baryonic and Dark Matter EoSs

#### 2.1. Baryonic Matter

#### 2.2. Dark Matter

## 3. Two-Fluid Formalism

## 4. NS Thermal Evolution

^{1}S

_{0}state and the triplet

^{3}P

_{2}state in the core. Cooper pairs of protons occur in the singlet

^{1}S

_{0}state in the NS core. We examine the above-mentioned nucleon pairing in this article. For simplicity, in this work, no hyperon superfluidity was considered. Based on the previous studies, e.g., in Ref. [48], we expect that incorporation of the hyperonic superfluidity would alter the onset of the hyperonic DU processes towards higher densities. Thus, the process would not be triggered right after the onset of hyperons. This could have a significant impact on the star’s cooling whereas nucleonic DU is only permitted at densities exceeding the onset of hyperons, as observed in the three models studied here. However, another interplay between the nucleonic and hyperonic DU is discussed in [48].

`NSCool`[119], using the implementation method described by [87]. The specific heat is calculated as the sum of the contributions from its constituent particles: neutrons, protons, electrons, and hyperons (if present) using the standard

`NSCool`implementation. The impact of nucleon superfluidity is accounted for by introducing control functions ${R}_{c}$ which depend on the corresponding critical temperature [120,121].

`NSCool`. This allows us to treat a two-fluid cooling problem within the single-fluid framework.

## 5. Results

#### 5.1. Hyperonic and Nucleonic DU Onsets

_{⊙}modeled by the IST and FSU2R EoSs do not exhibit a rapid cooling due to the DU process as it is triggered in heavier stars where the central density is higher (see Table 2 for details). However, heavy DM particles of ≳ GeV scale tend to form a dense core inside a star, pulling BM inwards from the outer layers, and leading to BM redistribution. As a consequence, the baryonic density in the inner core increases, triggering the nucleonic and hyperonic DU processes as shown in Figure 3 and Figure 4. Thus, the onset of the enhanced neutrino emission occurs at the same particle fractions and central baryonic density for stars with and without DM. However, as it can be seen in Figure 1 and Table 2, with the increase in the DM fraction the total gravitational mass at which the DU processes are kinematically allowed is shifted towards lower masses. Thus, for the FSU2H EoS the mass of the star at the nucleonic DU onset, ${M}_{DU}$, drops from 1.85 M

_{⊙}to 1.75 M

_{⊙}, 1.71 M

_{⊙}, and 1.66 M

_{⊙}for 2%, 3%, and 4% of DM, respectively (see Table 3). Moreover, due to hyperonic DU processes, such as $\Lambda p$ and ${\Sigma}^{-}\Lambda $, triggered before the nucleonic DU process, $np$, an accumulation of DM allows for an NS to exhibit a fast cooling behavior at 1.24 M

_{⊙}, for 4% DM (as shown in Figure 1). The values for the central baryonic densities and total gravitational masses at which the hyperonic and nucleonic DU processes become active are presented in Table 3.

#### 5.2. Mapping the Nucleonic and Hyperonic DU Regions Inside the Star

_{⊙}described by the IST, FSU2R, and FSU2H EoSs, respectively. The DM fraction continuously changes from 0% to 4.5%. All radii of the DM-admixed configurations are normalized to the baryonic radius of each configuration. The physical values of radii and total gravitational masses are listed in Table 4.

_{⊙}have the operating nucleonic DU in their interior. However, by increasing the DM fraction the $np$ DU process is triggered at 3.87%, 4.17%, 1.96% for the IST, FSU2R, and FSU2H EoSs, respectively. A much lower value for the FSU2H EoS is related to the fact that the nucleonic DU is triggered at a lower stellar mass, i.e., $1.85$ M

_{⊙}, than for the IST and FSU2R EoSs, i.e., $1.908$ M

_{⊙}and $1.921$ M

_{⊙}(see Table 2 and Table 3). The nucleonic DU region is depicted with the light red color in Figure 3. Moreover, parts of the star in which the $\Lambda p$ and ${\Sigma}^{-}\Lambda $ DU processes take place are shown in yellow and light green colors.

_{⊙}described with the same models as in Figure 3. In the case of the FSU2H EoS, not only the hyperonic DU processes but also the nucleonic DU are activated. Moreover, as it can be seen in Figure 4, no DM-admixed configurations with a DM fraction higher than 2% exist in this model. This is related to the FSU2H EoS that has an upper limit on the baryonic density, ${n}_{B}=1$ fm

^{−3}, at which the effective nucleonic mass vanishes [44]. As higher DM fractions require higher central baryonic densities, which cannot be obtained within the FSU2H EoS, such configurations do not exist.

#### 5.3. Cooling Curves

_{⊙}(red curves), 1.6 M

_{⊙}(blue curves), and 1.9 M

_{⊙}(green curves) modeled within the IST EoS (left panel) and the FSU2R EoS (right panel). The cooling curves for the stars of 1.2 M

_{⊙}(red curves), 1.5 M

_{⊙}(blue curves), and 1.7 M

_{⊙}(green curves) modeled within the FSU2H EoS are shown in Figure 6. This choice of lower mass values in comparison to Figure 5 is related to the upper limit of the baryonic density and absence of heavy stars admixed with DM within the FSU2H model.

^{1}S

_{0}pairing, described by the SFB [122] and CCDK [123] models, as the ones that provide the best description of the observational data. Following [87], in Figure 5 and Figure 6 the color grade represents the different DM fractions, whereas the higher DM fraction corresponds to a lighter shade. Pure BM stars, i.e., with 0% of DM, are shown with the darkest shade of each color.

#### 5.4. Thermal Evolution of Cassiopeia A as a DM-Admixed NS

^{3}P

_{2}pairing of neutrons in the core [33,131], rapid cooling via the DU process [132], the impact of medium-modified one-pion exchange in dense matter [133] and beyond the Standard Model physics [134]. As the estimated mass of the CCO in Cas A is around $1.55$ M

_{⊙}[130], a realistic description should take place for a middle mass star.

_{⊙}and 1.91 M

_{⊙}stars with the inclusion of neutron and proton singlet pairings, as well as with the 1.96 M

_{⊙}star for unpaired matter [136].

^{1}S

_{0}(SFB model), p

^{1}S

_{0}(CCDK model), n

^{3}P

_{2}pairing (T72 model) with the maximum critical temperature ${T}_{c}=7.105\xb7{10}^{8}$ K. In Ref. [124], the best fit of Cas A within the FSU2H EoS is obtained for the $1.88$ M

_{⊙}star considering the proton and neutron single pairing together with the triplet neutron pairing characterized by the maximum critical temperature ${T}_{c}\sim 1.41\xb7{10}^{9}$ K. For the same EoS, we also see a better agreement with Cas A data while varying the neutron

^{3}P

_{2}pairing.

_{⊙}with a DM fraction of $3\%$. The results of the fit are presented in Figure 7 whereas the curves are obtained for the heavy-elements envelope $\eta =\Delta M/M={10}^{-16}$ and a combination of n

^{1}S

_{0}(SFB model [122]), p

^{1}S

_{0}(CCDK model [123]), and n

^{3}P

_{2}pairing (T72 model [137]) with the maximum critical temperature ${T}_{c}\sim 8.52\xb7{10}^{8}$ K.

## 6. Conclusions

_{⊙}and 1.921 M

_{⊙}, respectively, and the hyperonic FSU2H model that kinematically allows both nucleonic and hyperonic DU processes. Accumulation of the DM particles with a mass of ${m}_{\chi}=1$ GeV of ${f}_{\chi}\simeq 0.161\%$ (IST EoS) and ${f}_{\chi}=0.378\%$ (FSU2R EoS) triggers the previously energetically forbidden process.

_{⊙}described by the FSU2H EoS. In [87] the best fit of the Cas A cooling rate was obtained for the $M=1.6\phantom{\rule{3.33333pt}{0ex}}{\mathrm{M}}_{\odot}$ star modeled within the FSU2R EoS with ${f}_{DM}=4\%$ and light-elements envelope. Thus, the accrued DM helps to reconcile the star mass at which the DU process is kinematically allowed with the observational data on the mass of the CCO in Cas A by lowering the mass compared to a pure BM star. We demonstrate that an increase in the DM fraction causes a shift of the DU onset towards lower gravitational masses of the star. This effect could serve as a distinctive signature of the presence of DM in compact stars. In Ref. [87], it was shown that a similar result can be obtained by considering heavier DM particles, leaving the DM fraction to be low.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

EoS | equation of state |

NS | neutron star |

BM | baryonic matter |

DM | dark matter |

DU | direct Urca |

MU | modified Urca |

GW | gravitational wave |

PBF | pair breaking and formation |

IST | Induced Surface Tension |

TOV | Tolmann–Oppenheimer–Volkoff |

JWST | James Webb Space Telescope |

## Appendix A. Observational Data

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**Figure 1.**(

**Left panel**) The proton fraction, ${Y}_{p}$, (top) and total gravitational mass of the DM-admixed and baryonic NSs (bottom) as a function of the central baryonic density, ${n}_{B,c}$, for the IST and FSU2R EoSs. On the bottom panel, the royal blue, red, and green curves represent the relative DM fractions equal to 2%, 3%, and 4%, respectively. The vertical solid and dashed gray lines correspond to the central BM densities of the stars at which the DU process is activated for the IST and FSU2R EoSs, respectively. The intersection points depict the nucleonic DU threshold. (

**Right panel**) The particle fractions, ${Y}_{j}$, (top) and total gravitational mass of the DM-admixed and baryonic NSs (bottom) as a function of the central baryonic density, ${n}_{B,c}$, for the FSU2H EoS. The vertical solid, dashed, and dashed-dotted gray lines correspond to the central BM densities at which the DU processes are activated, i.e., nucleonic ($np$) and hyperonic ($\Lambda p$ and ${\Sigma}^{-}\Lambda $), respectively.

**Figure 2.**Total gravitational mass of DM-admixed and baryonic NSs as a function of their baryonic radius ${R}_{B}$ calculated for the DM particle mass ${m}_{DM}=1$ GeV. Solid curves correspond to pure BM stars described by the IST (blue curve), FSU2R (red curve), and FSU2H (green curve) EoSs. Dashed and dotted curves depict the M-R relations obtained for the relative DM fractions equal to 2% and 4%, respectively. Orange and dark yellow bands represent $1\sigma $ constraints on the mass of PSR J0348+0432 [50] and PSR J1810+1744 [107]. While olive green and light orange contours show the NICER measurements of PSR J0030+0451 [108,109], purple and magenta contours correspond to the PSR J0740+6620 measurement [76,110]. Observations of GW170817 [54] and GW190425 [111] binary NS mergers by LIGO-Virgo collaboration are shown in blue and green. The $2\sigma $ contour of HESS J1731-347 [112] is plotted in light red. The shaded gray region is excluded by the rotation of the fastest spinning pulsar PSR J1748-2446ad [113].

**Figure 3.**Stellar configurations with different DM fractions for the IST EoS (top right slice), FSU2R EoS (top left slice), and FSU2H EoS (bottom slice). The size of the $np$, ${\Sigma}^{-}\Lambda $, $\Lambda p$ DU regions, and DM core are depicted in light red, light green, yellow, and dark gray, respectively. For a better comparison, the radii are normalized to the baryonic radius of each configuration and are given in Table 4. All configurations correspond to NSs with a total gravitational mass of 1.75 M

_{⊙}.

**Figure 4.**The same as in Figure 3, but for the total gravitational mass of 1.9 M

_{⊙}. The empty part of the bottom slice corresponds to the non-existing configurations (see details in the text).

**Figure 5.**Cooling curves for stars with total gravitational masses of $M=1.2,1.6$, and $1.9$ M

_{⊙}described by the IST EoS (

**left**) and FSU2R EoS (

**right**) are shown. The considered DM fraction is ${f}_{DM}=0\%,\phantom{\rule{4pt}{0ex}}2\%,\phantom{\rule{4pt}{0ex}}3\%$ and $4\%$ for particle’s mass ${m}_{DM}=1$ GeV. The solid and dashed curves correspond to envelopes composed of heavy elements ($\eta =\Delta M/M={10}^{-16}$) and light elements ($\eta =\Delta M/M={10}^{-7}$), respectively. The impact of neutron superfluidity in the

^{1}S

_{0}channel, employing the SFB model [122], and proton superconductivity in the

^{1}S

_{0}channel, employing the CCDK model [123], is considered. The figure is adopted from [87]. The utilized observational data are listed in the Appendix A.

**Figure 6.**The same as in Figure 5, but for stars with the total gravitational masses of $M=1.2,1.5,1.7$ M

_{⊙}modeled within the FSU2H EoS and supplemented with n

^{3}P

_{2}pairing described by the T72 model.

**Figure 7.**The results of the fit of the surface temperature as a function of age for the CCO in Cas A measured by Chandra ACIS-S in GRADED and FAINT modes. The red and magenta data points correspond to variable and fixed absorbing hydrogen column density ${\mathrm{N}}_{\mathrm{H}}=1.656\xb7{10}^{22}$ cm

^{−2}in the FAINT mode, while the green and blue data points depict the same data, but in the GRADED mode.

**Table 1.**Parameters of the IST, FSU2R and FSU2H models. The table includes the saturation density ${n}_{0}$, energy per baryon $E/A$, incompressibility factor ${K}_{0}$, symmetry energy ${E}_{sym}$, and its slope L at saturation density, as well as the maximum gravitational mass ${M}_{max}$, and radius of the 1.4 M

_{⊙}star.

${\mathit{n}}_{0}$ [fm ^{−3}]
| $\mathit{E}/\mathit{A}$ [MeV] | ${\mathit{K}}_{0}$ [MeV] | ${\mathit{E}}_{\mathbf{sym}}$ [MeV] | L [MeV] | ${\mathit{M}}_{\mathbf{max}}$ [MeV] | ${\mathit{R}}_{1.4}$ [MeV] | |
---|---|---|---|---|---|---|---|

IST EoS | 0.16 | −16.00 | 201.0 | 30.0 | 93.19 | 2.084 | 11.4 |

FSU2R EoS | 0.1505 | −16.28 | 238.0 | 30.7 | 46.90 | 2.048 | 12.8 |

FSU2H EoS | 0.1505 | −16.28 | 238.0 | 30.5 | 44.50 | 1.992 | 12.7 |

**Table 2.**Baryonic densities and total gravitational masses at the onset of the nucleonic DU process for the IST and FSU2R EoSs.

IST EoS | n_{DU}[${\mathbf{fm}}^{-\mathbf{3}}$] | M_{DU}[${\mathbf{M}}_{\odot}$] | FSU2R EoS | n_{DU}[${\mathbf{fm}}^{-\mathbf{3}}$] | M_{DU}[${\mathbf{M}}_{\odot}$] | ||
---|---|---|---|---|---|---|---|

${\mathrm{f}}_{\mathrm{DM}}$ | 0% | 0.869 | 1.908 | ${\mathrm{f}}_{\mathrm{DM}}$ | 0% | 0.608 | 1.921 |

2% | 1.83 | 2% | 1.83 | ||||

3% | 1.80 | 3% | 1.79 | ||||

4% | 1.76 | 4% | 1.75 |

**Table 3.**The same as in Table 2, but for the FSU2H model. The columns correspond to different hyperonic and nucleonic DU onsets.

FSU2H EoS | $\mathbf{\Lambda}\mathit{p}$ | ${\mathbf{\Sigma}}^{-}\mathbf{\Lambda}$ | $\mathit{np}$ | ||||
---|---|---|---|---|---|---|---|

n_{DU}[${\mathbf{fm}}^{-\mathbf{3}}$] | M_{DU}[${\mathbf{M}}_{\odot}$] | n_{DU}[${\mathbf{fm}}^{-\mathbf{3}}$] | M_{DU}[${\mathbf{M}}_{\odot}$] | n_{DU}[${\mathbf{fm}}^{-\mathbf{3}}$] | M_{DU}[${\mathbf{M}}_{\odot}$] | ||

${f}_{DM}$ | 0% | 0.332 | 1.40 | 0.446 | 1.74 | 0.534 | 1.85 |

2% | 1.32 | 1.63 | 1.75 | ||||

3% | 1.28 | 1.59 | 1.71 | ||||

4% | 1.24 | 1.54 | 1.66 |

**Table 4.**Baryonic radii for different gravitational mass configurations for the three models used in this work and their modifications due to the addition of DM.

IST EoS | ${\mathbf{f}}_{\mathbf{DM}}$ | |||
---|---|---|---|---|

$\mathbf{0}\%$ | $\mathbf{2}\%$ | $\mathbf{3}\%$ | $\mathbf{4}\%$ | |

${M}_{\mathrm{tot}}$ [${\mathrm{M}}_{\odot}$] | ${R}_{\mathrm{B}}$ [km] | |||

1.20 | 11.29 | 11.11 | 11.03 | 10.94 |

1.60 | 11.10 | 10.91 | 10.81 | 10.70 |

1.90 | 10.58 | 10.35 | 10.20 | 10.05 |

FSU2R EoS | ${\mathbf{f}}_{\mathbf{DM}}$ | |||

$\mathbf{0}\%$ | $\mathbf{2}\%$ | $\mathbf{3}\%$ | $\mathbf{4}\%$ | |

${M}_{\mathrm{tot}}$ [${\mathrm{M}}_{\odot}$] | ${R}_{\mathrm{B}}$ [km] | |||

1.20 | 12.18 | 12.09 | 12.01 | 11.93 |

1.60 | 12.39 | 12.25 | 12.15 | 12.05 |

1.90 | 12.16 | 11.93 | 11.73 | 11.47 |

FSU2H EoS | ${\mathbf{f}}_{\mathbf{DM}}$ | |||

$\mathbf{0}\%$ | $\mathbf{2}\%$ | $\mathbf{3}\%$ | $\mathbf{4}\%$ | |

${M}_{\mathrm{tot}}$ [${\mathrm{M}}_{\odot}$] | ${R}_{\mathrm{B}}$ [km] | |||

1.20 | 12.43 | 12.30 | 12.24 | 12.16 |

1.50 | 12.77 | 12.62 | 12.53 | 12.44 |

1.70 | 12.86 | 12.63 | 12.50 | 12.38 |

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Giangrandi, E.; Ávila, A.; Sagun, V.; Ivanytskyi, O.; Providência, C.
The Impact of Asymmetric Dark Matter on the Thermal Evolution of Nucleonic and Hyperonic Compact Stars. *Particles* **2024**, *7*, 179-200.
https://doi.org/10.3390/particles7010010

**AMA Style**

Giangrandi E, Ávila A, Sagun V, Ivanytskyi O, Providência C.
The Impact of Asymmetric Dark Matter on the Thermal Evolution of Nucleonic and Hyperonic Compact Stars. *Particles*. 2024; 7(1):179-200.
https://doi.org/10.3390/particles7010010

**Chicago/Turabian Style**

Giangrandi, Edoardo, Afonso Ávila, Violetta Sagun, Oleksii Ivanytskyi, and Constança Providência.
2024. "The Impact of Asymmetric Dark Matter on the Thermal Evolution of Nucleonic and Hyperonic Compact Stars" *Particles* 7, no. 1: 179-200.
https://doi.org/10.3390/particles7010010