# Dark Matter as a Non-Relativistic Bose–Einstein Condensate with Massive Gravitons

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## Abstract

**:**

## 1. Introduction

## 2. Self-Gravitating, Spherically Symmetric Bec Distribution in the Thomas-Fermi Approximation

#### 2.1. Mass Density and the Gravitational Potential inside the Condensate

#### 2.2. Gravitational Potential Outside the Condensate

## 3. Rotation Curves in Case of Massive Gravitons

## 4. Best-Fit Rotational Curves

#### 4.1. Contribution of the Baryonic Matter in Newtonian and in Yukawa Gravitation

#### 4.2. Testing Pure Baryonic and Baryonic + Dark Matter Models

## 5. Discussion and Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Theoretical rotational curves of the dwarf galaxy sample. The dots with error-bars denote archive rotational velocity curves. The model rotation curves are denoted as follows: pure baryonic in Newtonian gravitation with dotted line, baryonic + BEC with massless gravitons in Newtonian gravitation with dashed line, and baryonic + BEC with the upper limit on ${m}_{g}$ in Yukawa gravitation with continuous line.

**Table 1.**Parameters describing the theoretical rotational curve models of the 12 dwarf galaxies. Best-fit parameters of the pure baryonic model in the first group of columns: central surface brightness ${S}_{0}$, scale parameter b, $M/L$ ratio $\mathsf{{\rm Y}}$, along with the ${\chi}^{2}$ of the fit. This model results in best-fit model-rotation curves above $5\sigma $ significance level for all galaxies. Best-fit parameters of the baryonic matter + BEC with massless gravitons appear in the second group of columns: $M/L$ ratio $\mathsf{{\rm Y}}$, characteristic density ${\rho}^{(c)}$, distance parameter ${R}_{*}$, along with the ${\chi}^{2}$ of the fit and the respective significance levels. Constraints on the parameter ${m}^{2}/\lambda $ are also derived. In five cases, the fits ${\chi}^{2}$ are within $1\sigma $ and marked as boldface. The fits are between $1\sigma $ and $2\sigma $ in three cases, between $2\sigma $ and $3\sigma $ in one case, between $3\sigma $ and $4\sigma $ in one case and above $5\sigma $ in two cases. Best-fit parameters of the baryonic matter + BEC with massive gravitons are given in the third group of columns only for the well-fitting galaxies: the range for ${R}_{BEC}$ and the upper limit on ${m}_{g}$ are those for which the fit remains within $1\sigma $. Corresponding constraints on the parameter $m/\mu $ are also derived.

Pure Baryonic | Baryonic + BEC with ${\mathit{m}}_{\mathit{g}}=0$ | Baryonic + BEC with ${\mathit{m}}_{\mathit{g}}>0$ | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

ID | ${S}_{0}$ | b | $\mathsf{{\rm Y}}$ | ${\chi}^{2}$ | $\mathsf{{\rm Y}}$ | ${\rho}^{(c)}$ | ${R}_{*}$ | $\frac{{m}^{2}}{\lambda}$ | ${\chi}^{2}$ | sign. lev. | ${R}_{BEC}$ | ${m}_{g}$ | $\frac{m}{\mu}$ | sign. lev. |

${10}^{8}\frac{{L}_{\odot}}{kp{c}^{2}}$ | $kpc$ | ${10}^{7}\frac{{M}_{\odot}}{kp{c}^{3}}$ | $kpc$ | ${10}^{-31}\frac{kg{s}^{2}}{{m}^{5}}$ | $kpc$ | ${10}^{-26}\frac{eV}{{c}^{2}}$ | ${10}^{-10}\frac{{s}^{2}}{{m}^{2}}$ | |||||||

UGC12060 | 0.7 | 0.90 | 11.23 | 155 | 5.50 ± 0.33 | 1.07 ± 0.11 | 2.650 ± 0.118 | 1.78 ± 0.16 | 1.69 | $1\sigma =5.89$ | [7.3 ÷ 10.6] | $<0.95$ | $<7.02$ | $1\sigma =7.08$ |

UGC7278 | 6.1 | 0.49 | 2.59 | 499 | 0.81 ± 0.06 | 3.53 ± 0.23 | 1.702 ± 0.048 | 4.32 ± 0.24 | 7.91 | $1\sigma =21.36$ | [4.6 ÷ 6.8] | $<1.40$ | $<5.46$ | $1\sigma =22.44$ |

UGC6446 | 1.9 | 1.00 | 3.89 | 809 | 1.37 ± 0.11 | 1.02 ± 0.09 | 3.040 ± 0.128 | 1.36 ± 0.11 | 7.91 | $1\sigma =8.18$ | [9.2 ÷ 10] | $<0.42$ | $<4.27$ | $1\sigma =9.86$ |

UGC3851 | 0.5 | 1.80 | 2.74 | 86 | 0.74 ± 0.18 | 1.91 ± 0.22 | 1.509 ± 0.038 | 5.50 ± 0.28 | 11.30 | $1\sigma =20.28$ | [4.3 ÷ 5.5] | $<1.26$ | $<11.4$ | $1\sigma =21.36$ |

UGC7125 | 1.2 | 2.20 | 4.50 | 285 | 1.78 ± 0.18 | 2.26 ± 0.21 | 2.670 ± 0.071 | 1.76 ± 0.93 | 11.82 | $1\sigma =12.64$ | [8.2 ÷ 8.6] | $<0.31$ | $<2.44$ | $1\sigma =13.74$ |

UGC3711 | 5.2 | 0.46 | 4.40 | 232 | 2.00 | 8.06 | 1.212 | - | 5.11 | $2\sigma =6.18$ | - | - | - | - |

UGC4499 | 1.4 | 0.75 | 6.30 | 603 | 1.00 | 1.34 | 2.590 | - | 8.51 | $2\sigma =11.31$ | - | - | - | - |

UGC7603 | 2.1 | 1.00 | 1.88 | 462 | 0.40 | 1.07 | 2.470 | - | 13.46 | $2\sigma =15.78$ | - | - | - | - |

UGC8490 | 2.8 | 0.40 | 9.52 | 1350 | 4.06 | 3.35 | 1.715 | - | 40.27 | $3\sigma =50.55$ | - | - | - | - |

UGC5986 | 4.4 | 1.20 | 3.95 | 1682 | 0.48 | 3.17 | 2.620 | - | 32.12 | $4\sigma =38.54$ | - | - | - | - |

UGC1281 | 1.0 | 1.60 | 1.33 | 231 | 0.53 | 0.75 | 3.70 | - | 48.74 | $5\sigma =43.98$ | - | - | - | - |

UGC5721 | 4.9 | 0.40 | 5.79 | 1388 | 1.75 | 2.84 | 1.982 | - | 88.56 | $5\sigma =50.21$ | - | - | - | - |

**Table 2.**Constraints for both the upper limit for the mass of the graviton (first from the existence of $\mathsf{\Lambda}$, second from the rotation curves) and for the velocity-type and density-type BEC parameters (related to the mass of the BEC particle, scattering length and chemical potential) in the case of the five well-fitting galaxies.

ID | ${\mathit{m}}_{\mathit{g}}(\mathsf{\Lambda}\in \mathbf{I}\mathbf{R})$ | ${\mathit{m}}_{\mathit{g}}$ | ${\overline{\mathit{v}}}_{\mathit{BEC}}$ | ${\overline{\mathit{\rho}}}_{\mathit{BEC}}$ |
---|---|---|---|---|

${10}^{-26}\frac{\mathit{eV}}{{\mathit{c}}^{2}}$ | ${10}^{-26}\frac{\mathit{eV}}{{\mathit{c}}^{2}}$ | $\frac{\mathit{m}}{\mathit{s}}$ | ${10}^{6}\frac{{\mathit{M}}_{\odot}}{{\mathit{kpc}}^{3}}$ | |

UGC12060 | $<1.51$ | $<0.95$ | $\mathrm{37,724}$ | $3.75$ |

UGC7278 | $<2.35$ | $<1.40$ | $\mathrm{42,800}$ | $11.69$ |

UGC6446 | $<1.32$ | $<0.42$ | $\mathrm{48,383}$ | $4.68$ |

UGC3851 | $<2.65$ | $<1.26$ | $\mathrm{29,571}$ | $7.1$ |

UGC7125 | $<1.5$ | $<0.31$ | $\mathrm{63,964}$ | $10.61$ |

**Table 3.**Constraints on $\mu $ and $\lambda $ assuming $m={m}_{g}$ in case of the five well fitting galaxies.

ID | $\mathit{\mu}(\mathit{m}={\mathit{m}}_{\mathit{g}})$ | $\mathit{\lambda}(\mathit{m}={\mathit{m}}_{\mathit{g}})$ |
---|---|---|

${10}^{-53}\frac{{\mathit{m}}^{2}}{{\mathit{s}}^{2}}$ kg | ${10}^{-94}\frac{{\mathit{m}}^{5}}{{\mathit{s}}^{2}}$ kg | |

UGC12060 | $<2.41$ | $<16.08$ |

UGC7278 | $<4.57$ | $<14.40$ |

UGC6446 | $<1.75$ | $<4.14$ |

UGC3851 | $<1.96$ | $<9.17$ |

UGC7125 | $<2.26$ | $<1.74$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Kun, E.; Keresztes, Z.; Das, S.; Gergely, L.Á.
Dark Matter as a Non-Relativistic Bose–Einstein Condensate with Massive Gravitons. *Symmetry* **2018**, *10*, 520.
https://doi.org/10.3390/sym10100520

**AMA Style**

Kun E, Keresztes Z, Das S, Gergely LÁ.
Dark Matter as a Non-Relativistic Bose–Einstein Condensate with Massive Gravitons. *Symmetry*. 2018; 10(10):520.
https://doi.org/10.3390/sym10100520

**Chicago/Turabian Style**

Kun, Emma, Zoltán Keresztes, Saurya Das, and László Á. Gergely.
2018. "Dark Matter as a Non-Relativistic Bose–Einstein Condensate with Massive Gravitons" *Symmetry* 10, no. 10: 520.
https://doi.org/10.3390/sym10100520