# New Bounds for the Mass of Warm Dark Matter Particles Using Results from Fermionic King Model

## Abstract

**:**

## 1. Introduction

## 2. Antecedents

#### 2.1. Thermodynamic Effects of Evaporation

#### 2.2. Concerning the Thermodynamic Limit

#### 2.3. Thermodynamics of Fermionic King Model at Constant Total Mass

**The region I**: the interval $0\le \theta <{\theta}_{1}\simeq 1.12\times {10}^{-7}$ (black curves). The gravitational collapse of fermionic King model represents a discontinuous microcanonical phase transition, and its thermodynamics exhibits a branch with negative heat capacities. The classical King model that appears when $\mu \to +\infty $ corresponds to the infinite mass limit $\theta \to 0$. In terms of the total mass M, this region corresponds to situations with high total masses, the interval ${M}_{1}<M<+\infty $, where ${M}_{1}={M}_{F}/{\theta}_{1}\simeq 8.9\times {10}^{6}{M}_{F}$.**The region II**: the interval ${\theta}_{1}\le \theta <{\theta}_{2}\simeq 1.10\times {10}^{-2}$ (red curves). The gravitational collapse of fermionic King model turns a continuous microcanonical phase transition, and its thermodynamics exhibits a branch with negative heat capacities. In terms of the total mass M, this region corresponds to situations with intermediate total masses, the interval ${M}_{2}<M\le {M}_{1}$, where ${M}_{2}={M}_{F}/{\theta}_{2}\simeq 90.9{M}_{F}$.**The region III**: the interval ${\theta}_{2}\le \theta <{\theta}_{m}\simeq 4.0$ (green curves). The gravitational collapse of fermionic King model is a continuous microcanonical phase transition, and its thermodynamics does not exhibit negative heat capacities. In terms of the total mass M, this region corresponds to situations with low total mass, the interval ${M}_{3}<M\le {M}_{2}$, where ${M}_{3}={M}_{F}/{\theta}_{m}\simeq $¼${M}_{F}$.

## 3. Application to Dark Matter Halos

#### 3.1. Incidence of Evaporation Effects

#### 3.2. New Bounds of WDM Particles Mass m from Fermionic King Model

#### 3.3. The keV Scale and the Masses of Large Galaxies

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Additional Notes

## References

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**Figure 1.**Behavior of $\gamma -$exponential function (2). Main panel: this function converges towards usual exponential function for large values of its argument x. Inset panel: it drops to zero when $x\to {0}^{+}$ following a power-law of the type $E\left(x,\gamma \right)\propto {x}^{\gamma}$. The $\gamma -$exponential function is just the fractional derivative of the exponential function, $E\left(x,\gamma \right)\equiv {d}^{\gamma}\left({e}^{x}\right)/d{x}^{\gamma}$. After [15].

**Figure 2.**(

**a**): Contour maps of the mass ratio parameter $\theta ={M}_{F}/M$ [see in Equation (25)] in the plane of integration parameters $\left[{\Phi}_{0},\mu \right]$ of the Poisson problem (27), which were obtained from numerical procedures designed to fulfil this purpose. Here, $\mu =ln\alpha $ is the degeneration parameter defined from the normalization constant $\alpha $ in Equation (21); ${\Phi}_{0}$ is the central value of the dimensionless potential (26). Notice that for each value of the degeneration parameter $\mu $ there exist infinite values for the mass ratio parameter $\theta $ since this quantity also depends on the dimensionless potential ${\Phi}_{0}$, $\theta =\theta \left({\Phi}_{0},\mu \right)$. This very fact implies that thermodynamics of fermionic King model at constant degeneration parameter $\mu $ differs from its thermodynamics at constant mass ratio $\theta $ (or constant total mass M). (

**b**): Corresponding caloric curves at constant mass ratio parameter $\theta $ in terms of the dimensionless inverse temperature $\eta =\beta GMm/R$ and the auxiliary variable ${u}^{*}=-G{M}^{2}/RU$ defined from the total energy U. One can observed the existence of three regions with different thermodynamic behavior. The points $(a,b,c,{c}^{\prime},d)$ are some notable configurations. Among them, it is remarkable the case of configurations $(c,{c}^{\prime})$ over the curve with $\theta =2.84\times {10}^{-8}$ (region I), which exhibit the same energy but different temperatures. The during discontinuous jump from the profile c towards the profile ${c}^{\prime}$, the system temperature grows and there exist a redistribution of the mass that leads to the formation of a dense degenerate core. After [25].

**Table 1.**Observed values of the DM halos parameters [${r}_{h}$ (radius), ${\rho}_{0}$ (central density) and ${M}_{h}$ (mass)] employed by Destric and co-workers covering from ultracompact galaxies to large spiral galaxies (obtained from [53,54,55,56,57,58,59,60,61,62]). Assuming that the dwarf galaxy Willman 1 belongs to the region III of fermionic King model, ${\theta}_{2}\le \theta \le {\theta}_{m}$, the mass m of WDM particles should belong to the range ${m}_{min}=1.2$ keV $\le m\le {m}_{max}=2.6$ keV. For this range of values, it was calculated the mass ratio $\theta ={M}_{F}/M$ for the rest of galaxies. Here, ${\theta}_{f}$ is the pre-factor of Equation (36), which corresponds to the mass ratio for a WDM particle with mass $m=2$ keV, while ${\theta}_{max}$ and ${\theta}_{min}$ are the mass ratios corresponding to the minimum and maximum masses ${m}_{min}$ and ${m}_{max}$, respectively. The black values of the mass ratio $\theta $ belong to region I ($0<\theta <{\theta}_{1}$), the red values belong to the region II (${\theta}_{1}<\theta <{\theta}_{2}$), while the green values to the region I (${\theta}_{2}<\theta \le {\theta}_{m}$).

Galaxy | ${\mathit{r}}_{\mathit{h}}\phantom{\rule{3.33333pt}{0ex}}\left[\mathbf{pc}\right]$ | ${\mathit{\rho}}_{0}$$\left[{\mathit{M}}_{\odot}{\mathbf{pc}}^{-3}\right]$ | ${\mathit{M}}_{\mathit{h}}$$\left[{10}^{6}{\mathit{M}}_{\odot}\right]$ | ${\mathit{\theta}}_{min}\phantom{\rule{3.33333pt}{0ex}}\left[{\mathit{m}}_{max}\right]$ | ${\mathit{\theta}}_{\mathit{h}}\phantom{\rule{3.33333pt}{0ex}}$ | ${\mathit{\theta}}_{max}\phantom{\rule{3.33333pt}{0ex}}\left[{\mathit{m}}_{min}\right]$ |
---|---|---|---|---|---|---|

Willman 1 | 19 | $6.3$ | $0.029$ | $1.1\times {10}^{-2}$ | $0.085$ | $4.0$ |

Segue 1 | 48 | $2.5$ | $1.93$ | $1.0\times {10}^{-5}$ | $8.0\times {10}^{-5}$ | $3.7\times {10}^{-3}$ |

Coma-Berenices | 123 | $2.09$ | $0.14$ | $8.4\times {10}^{-6}$ | $6.5\times {10}^{-5}$ | $3.0\times {10}^{-3}$ |

Leo T | 170 | $0.79$ | $12.9$ | $3.5\times {10}^{-8}$ | $2.7\times {10}^{-7}$ | $1.2\times {10}^{-5}$ |

Canis Venatici II | 245 | $0.49$ | $4.8$ | $3.1\times {10}^{-8}$ | $2.4\times {10}^{-7}$ | $1.1\times {10}^{-5}$ |

Draco | 305 | $0.5$ | $26.5$ | $2.9\times {10}^{-9}$ | $2.3\times {10}^{-8}$ | $1.0\times {10}^{-6}$ |

Leo II | 320 | $0.34$ | $36.6$ | $1.8\times {10}^{-9}$ | $1.4\times {10}^{-8}$ | $6.6\times {10}^{-7}$ |

Hercules | 387 | $0.1$ | $25.1$ | $1.5\times {10}^{-9}$ | $1.2\times {10}^{-8}$ | $5.5\times {10}^{-7}$ |

Boötes I | 362 | $0.38$ | $43.2$ | $1.1\times {10}^{-9}$ | $8.3\times {10}^{-9}$ | $3.9\times {10}^{-7}$ |

Carina | 428 | $0.15$ | $32.2$ | $8.7\times {10}^{-10}$ | $6.7\times {10}^{-9}$ | $3.2\times {10}^{-7}$ |

Ursa Major I | 504 | $0.25$ | $33.2$ | $5.1\times {10}^{-10}$ | $4.0\times {10}^{-9}$ | $1.9\times {10}^{-7}$ |

Sculptor | 480 | $0.25$ | $78.8$ | $2.5\times {10}^{-10}$ | $2.0\times {10}^{-9}$ | $9.1\times {10}^{-8}$ |

Leo IV | 400 | $0.19$ | 200 | $1.7\times {10}^{-10}$ | $1.3\times {10}^{-9}$ | $6.2\times {10}^{-8}$ |

Leo I | 518 | $0.22$ | 96 | $1.6\times {10}^{-10}$ | $1.3\times {10}^{-9}$ | $6.0\times {10}^{-8}$ |

Ursa Minor | 750 | $0.16$ | 193 | $2.7\times {10}^{-11}$ | $2.1\times {10}^{-10}$ | $9.8\times {10}^{-9}$ |

NGC 185 | 450 | $4.09$ | 975 | $2.5\times {10}^{-11}$ | $1.9\times {10}^{-10}$ | $9.0\times {10}^{-9}$ |

Sextans | 1290 | $0.02$ | 116 | $8.8\times {10}^{-12}$ | $6.8\times {10}^{-11}$ | $3.2\times {10}^{-9}$ |

Canis Venatici I | 1220 | $0.08$ | 344 | $3.5\times {10}^{-12}$ | $2.7\times {10}^{-11}$ | $1.3\times {10}^{-9}$ |

Fornax | 1730 | $0.053$ | 1750 | $2.4\times {10}^{-13}$ | $1.9\times {10}^{-12}$ | $8.8\times {10}^{-11}$ |

NGC 855 | 1063 | $2.64$ | 8340 | $2.2\times {10}^{-13}$ | $1.7\times {10}^{-12}$ | $7.9\times {10}^{-11}$ |

NGC 4478 | 1890 | $3.7$ | $6.55\times {10}^{4}$ | $4.9\times {10}^{-15}$ | $3.8\times {10}^{-14}$ | $1.8\times {10}^{-12}$ |

Small Spiral | 5100 | $0.029$ | 6900 | $2.4\times {10}^{-15}$ | $1.8\times {10}^{-14}$ | $8.7\times {10}^{-13}$ |

NGC 3853 | 5220 | $0.77$ | $2.87\times {10}^{5}$ | $5.4\times {10}^{-17}$ | $4.2\times {10}^{-16}$ | $1.9\times {10}^{-14}$ |

NGC 731 | 6160 | $0.47$ | $2.87\times {10}^{5}$ | $3.3\times {10}^{-17}$ | $2.5\times {10}^{-16}$ | $1.2\times {10}^{-14}$ |

NGC 499 | 7700 | $0.91$ | $1.09\times {10}^{6}$ | $4.4\times {10}^{-18}$ | $3.4\times {10}^{-17}$ | $1.6\times {10}^{-15}$ |

Medium Spiral | $1.9\times {10}^{4}$ | $0.0076$ | $1.01\times {10}^{5}$ | $3.2\times {10}^{-18}$ | $2.4\times {10}^{-17}$ | $1.1\times {10}^{-15}$ |

Large Spiral | $5.9\times {10}^{4}$ | $2.3\times {10}^{-3}$ | $1.0\times {10}^{6}$ | $1.1\times {10}^{-20}$ | $8.3\times {10}^{-20}$ | $3.9\times {10}^{-18}$ |

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Velazquez, L.
New Bounds for the Mass of Warm Dark Matter Particles Using Results from Fermionic King Model. *Universe* **2021**, *7*, 308.
https://doi.org/10.3390/universe7080308

**AMA Style**

Velazquez L.
New Bounds for the Mass of Warm Dark Matter Particles Using Results from Fermionic King Model. *Universe*. 2021; 7(8):308.
https://doi.org/10.3390/universe7080308

**Chicago/Turabian Style**

Velazquez, Luisberis.
2021. "New Bounds for the Mass of Warm Dark Matter Particles Using Results from Fermionic King Model" *Universe* 7, no. 8: 308.
https://doi.org/10.3390/universe7080308