# Imprint of Pressure on Characteristic Dark Matter Profiles: The Case of ESO0140040

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## Abstract

**:**

## 1. Introduction

## 2. DM Distribution in Spirals

- 1.
- The ISO profile [14]:$${\rho}_{Iso}\left(r\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{{\rho}_{0}}{1+{(r/{r}_{0})}^{2}}\phantom{\rule{0.166667em}{0ex}},$$
- 2.
- Exponential sphere [15]:$${\rho}_{Exp}\left(r\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}{e}^{-r/{r}_{0}}.\phantom{\rule{4pt}{0ex}}$$The model depends on the same constants as in the ISO profile.
- 3.
- Burkert profile [16]:$${\rho}_{Bur}\left(r\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{{\rho}_{0}}{(1+r/{r}_{0})(1+{(r/{r}_{0})}^{2})}.\phantom{\rule{4pt}{0ex}}$$The model depends upon the same constants as above and overcomes the issue related to the cusp in galaxies. Despite its experimental triumph, the Burkert profile is not theoretically motivated and remains a phenomenological approach to face the DM problem.
- 4.
- The NFW profile [17]$${\rho}_{NFW}\left(r\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\frac{{\rho}_{0}}{(r/{r}_{0}){(1+r/{r}_{0})}^{2}}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{4pt}{0ex}}$$
- 5.
- Moore profile [18]:$${\rho}_{Moo}\left(r\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}{(r/{r}_{0})}^{-1.16}{(1+r/{r}_{0})}^{-1.84}\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{4pt}{0ex}}$$
- 6.
- Einasto profile [19]:$${\rho}_{Ein}\left(r\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\rho}_{0}exp\left[2\alpha (1-{(r/{r}_{0})}^{1/\alpha})\right]\phantom{\rule{0.166667em}{0ex}},\phantom{\rule{4pt}{0ex}}$$

#### 2.1. Galaxy Eso140040 within Newtonian Gravity

#### 2.2. Perturbations and Optical Properties

#### 2.3. The Role of the Dm Equation of State

## 3. Methods and Analyses

#### 3.1. Numerical Results

**$\alpha =3.0\pm 0.5$**. The ${\chi}^{2}$ values are also shown in the last column of Table 1.

## 4. Final Outlooks and Perspectives

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Color online. Left panel: Different phenomenological DM density profiles and we choose $\alpha =1.5$ for the Einasto profile. Right panel: RCs of galaxy ESO0140040 and fitted profiles.

**Figure 2.**Color online. The refractive index for DM in galaxy ESO0140040. Left panel: The refractive index of the Einasto, NFW and Moore profiles. Right panel: The refractive index of the ISO, exponential sphere and Burkert profiles.

**Figure 3.**Color online. Left panel: Dimensionless state parameter $\omega $ as a function of radial coordinate r. Right panel: Dimensionless state parameter $\omega $ as a function of density $\rho $.

**Figure 4.**Color online. Left panel: Logarithmic density profiles of DM in the halo. Right panel: Logarithmic pressure profiles of DM in the halo.

**Figure 5.**Color online. Left panel: The equation of state of DM halo. Right panel: The speed of sound for the DM fluid for different DM profiles.

**Table 1.**Model parameters for the analyzed galaxy ESO0140040. We reported for each profile the density ${\rho}_{0}$, ${r}_{0}$ and the masses expressed in terms of ${M}_{\odot}$. For every column we report the error bars. The last two columns report the BIC statistical outputs and the corresponding chi squares used for computing the BIC values.

Profiles | ${\mathit{\rho}}_{0}\pm {\mathit{\sigma}}_{{\mathit{\rho}}_{0}}$ [${10}^{-3}{\mathit{M}}_{\odot}/{\mathit{pc}}^{3}$] | ${\mathit{r}}_{0}\pm {\mathit{\sigma}}_{{\mathit{r}}_{0}}$ [kpc] | $\mathit{M}\pm {\mathit{\sigma}}_{\mathit{M}}$ [${\mathit{M}}_{\odot}$] ${}^{\mathit{a}}$ | $\mathit{M}\pm {\mathit{\sigma}}_{\mathit{M}}$ [${\mathit{M}}_{\odot}$] ${}^{\mathit{b}}$ | $\Delta $BIC ${}^{\mathit{c}}$ | ${\mathit{\chi}}^{2}$ |
---|---|---|---|---|---|---|

Burkert | $175\pm 18$ | $6\pm 0.4$ | $4.1\pm 0.7$ | $6\pm 1.7$ | 14 | 3.3 |

NFW | $25\pm 3$ | $16\pm 1$ | $4.4\pm 0.8$ | $26\pm 8.4$ | 3 | 1 |

ISO | $250\pm 27$ | $3\pm 0.2$ | $4.3\pm 0.7$ | $1\pm 0.3$ | 5 | 1 |

Moore | $12\pm 2$ | $23\pm 3$ | $4.4\pm 1.2$ | $4\pm 2$ | 7 | 1.4 |

Einasto | $10\pm 2$ | $13\pm 1.5$ | $4.3\pm 1.4$ | $20\pm 10.6$ | - | 0.4 |

Exp. Sphere | $158\pm 15$ | $5\pm 0.3$ | $4.1\pm 0.7$ | $4\pm 1$ | 18 | 5.5 |

1. | One is forced to exclude dark baryons since the numerical simulations provide radically different large-scale structure of the Universe. |

2. | As morphological and clustering properties of galaxies, abundances of rich clusters, halo masses degenerate if we consider cold, warm or collisional DM. We therefore seek tests of DM’s nature that could be sensitive to its presumed microphysics and/or to its interaction properties and we can assume to consider density profiles only without taking care at this stage of other velocity corrections. |

3. | In general, all model parameters can be estimated by analyzing galaxies, albeit with a strong unvoidable degeneracy [12]. |

4. | With increasing interests, models that aim to unify DM with dark energy are always more studied. A final goal could be to unify the dark Universe under the same standards. |

5. | For a review of galaxy morphology and properties of DM see [38]. |

6. | The Levenberg-Marquardt algorithm is an iterative technique that locates the minimum of a function that is expressed as the sum of squares of nonlinear functions. It consists of a combination of the Gauss–Newton algorithm and the method of the steepest descent gradient. |

7. | BIC is a selection criterion among a finite set of models, conceived to solve the overfitting issue when increasing the number of parameters in the fitting function. For the sake of completeness, please notice that other selection criteria could be used, e.g., the Akaike information and/or DIC criteria [42,43]. |

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**MDPI and ACS Style**

Boshkayev, K.; Konysbayev, T.; Kurmanov, E.; Luongo, O.; Muccino, M.
Imprint of Pressure on Characteristic Dark Matter Profiles: The Case of ESO0140040. *Galaxies* **2020**, *8*, 74.
https://doi.org/10.3390/galaxies8040074

**AMA Style**

Boshkayev K, Konysbayev T, Kurmanov E, Luongo O, Muccino M.
Imprint of Pressure on Characteristic Dark Matter Profiles: The Case of ESO0140040. *Galaxies*. 2020; 8(4):74.
https://doi.org/10.3390/galaxies8040074

**Chicago/Turabian Style**

Boshkayev, Kuantay, Talgar Konysbayev, Ergali Kurmanov, Orlando Luongo, and Marco Muccino.
2020. "Imprint of Pressure on Characteristic Dark Matter Profiles: The Case of ESO0140040" *Galaxies* 8, no. 4: 74.
https://doi.org/10.3390/galaxies8040074