# Assessment of Dark Matter Models Using Dark Matter Correlations across Dwarf Spheroidal Galaxies

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Layout

#### 2.1. Density Correlations from Observations

#### 2.2. Density Correlations from DM models

## 3. Showcase I: Eagle Simulation

_{⊙}. We retrieve the DM particle’s positions in the halo using the Python code snippets provided in [25]. In the following, we use the positions of DM particles to estimate the DM mass density $\rho $ and subsequently estimate the correlations. The whole process as well as statistical error estimation is implemented in a Python code that is publicly available at [26]. The code is intended to work with Gaia dataset, but the following procedures can be achieved with a minimal change.

## 4. Showcase II: Gaia Data

#### 4.1. Selecting Stars

#### 4.2. Coordinate System

#### 4.3. Estimation of the Phase-Space Density

#### 4.4. Results

- A Classic Weakly Collisional Gas of DM

- DM Models at the Critical Temperatures

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

MDPI | Multidisciplinary Digital Publishing Institute |

DOAJ | Directory of open access journals |

TLA | Three letter acronym |

LD | Linear dichroism |

## Appendix A. Classic Gas Model of DM

## Appendix B. Free Energy Functional of an Ideal Gas

## Appendix C. Additional Figures

**Figure A1.**The ratio of the dot products of the unit vectors in the spherical coordinate system for the stars that belong to the Fornax dSph as reported by Gaia. The figure indicates that all of the ratios are negligible for any of the stars.

**Figure A2.**The position and velocity of the stars in the comoving coordinate system of the Fornax dSph. Each point in the two top plots represents a start. The rest of the plots are the histograms of the four dimensions of the phase-space. The histograms indicate that the system is fairly spherical. The height and the width of the histograms of x and y are approximately equal supporting the spherical symmetry in the position-space. The same is approximately valid for ${v}_{x}$ and ${v}_{y}$ histograms supporting the spherical symmetry in the velocity-space. The reason we have used a cylindrical symmetry to carry out the Jeans equations in Section 4 is the slight discrepancy between the width, but not the height, of ${v}_{x}$ and ${v}_{y}$ histograms.

**Figure A3.**The probability distribution of stars’ velocities at the center, $x=y=0$ in the co-moving coordinate system, of the Fornax dSph estimated using the kernel density estimator as described in Section 4. The distribution is close to the Maxwell–Boltzmann distribution function although a slight deviation can be seen. The deviations are slightly different at other $(x,y)$ positions leading to the mass density fluctuations of Figure 2.

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**Figure 1.**We use the hydrodynamic simulations of the Eagle project and select a halo of mass ${10}^{9}$ M

_{⊙}at zero redshift. We estimate the DM mass density across the halo and use that to find the two-point correlations $C\left(\mathsf{\Delta}\right)$ between DM mass density fluctuations as a function of the distance between the position of mass densities $\mathsf{\Delta}$. We observe no statistically significant correlation. The statistical errors are estimated and shown in the figure.

**Figure 2.**The mass density distribution of DM at the center of the Fornax dSph galaxy on the x–y plane of the comoving coordinate system. We have randomly picked 10,000 points in this $x-y$ region and have calculated the DM mass density at those points. The plot indicates a core distribution of DM mass at the center of the Fornax dSph, with mean DM mass density of 0.5 $({\mathrm{M}}_{\odot}/{\mathrm{pc}}^{3})$. The mass density fluctuations can be readily seen. There are ingenuine correlations between the points closer than 100 (pc), i.e., similar colors grouped together, which are induced by the kernel density estimator.

**Figure 3.**The two-point correlations $C\left(\mathsf{\Delta}\right)$ between DM mass density fluctuations as a function of the distance between the position of mass densities $\mathsf{\Delta}$ in the Fornax dSph. Correlations at $\mathsf{\Delta}<100$ (pc) are induced by the kernel density estimator and are not genuine. We observe no statistically significant correlation at larger distances in the galaxy.

**Figure 4.**Bounds on the coefficients of the free energy expansion at 5% significance level using $\mathsf{\Delta}>100\phantom{\rule{3.33333pt}{0ex}}$(pc) of Figure 3. The vertical line is at $\beta \gamma =10\phantom{\rule{3.33333pt}{0ex}}\left({\mathrm{pc}}^{-1}\right)$ and the diagonal red line shows $\beta \gamma =\beta {\nu}^{2}$. The solid gray region is excluded for DM models that live far from their critical temperature. An example is a dark classic gas. The solid grey region and the hatched blue region, i.e., the entire region left to the vertical line, are excluded for DM models that live close to their critical temperature. Superfluid models of DM for example. In any model of DM, there is a map between $\gamma $ and ${\nu}^{2}$ parameters and the underlying principles of the model. Therefore, this plot can be used to explore the allowed regions of the parameter space of DM models.

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**MDPI and ACS Style**

Borzou, A.
Assessment of Dark Matter Models Using Dark Matter Correlations across Dwarf Spheroidal Galaxies. *Universe* **2022**, *8*, 386.
https://doi.org/10.3390/universe8070386

**AMA Style**

Borzou A.
Assessment of Dark Matter Models Using Dark Matter Correlations across Dwarf Spheroidal Galaxies. *Universe*. 2022; 8(7):386.
https://doi.org/10.3390/universe8070386

**Chicago/Turabian Style**

Borzou, Ahmad.
2022. "Assessment of Dark Matter Models Using Dark Matter Correlations across Dwarf Spheroidal Galaxies" *Universe* 8, no. 7: 386.
https://doi.org/10.3390/universe8070386