# Dark Matter in Fractional Gravity II: Tests in Galaxy Clusters

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background and Data Analysis

#### 2.1. DM in Fractional Gravity

#### 2.2. Forward Modeling of the ICM Thermodynamics

#### 2.3. Bayesian Data Analysis

`emcee`[45], running it with ${10}^{4}$ steps and 200 walkers for every individual cluster; each walker was initialized with a random position uniformly sampled from the (flat) priors. After checking the auto-correlation time, we removed the first $20\%$ of the flattened chain to ensure burn-in; the typical acceptance fractions of the various runs were in the range 30–40%.

## 3. Results

## 4. Summary and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Notes

1 | Hereafter, ${R}_{\Delta}$ indicates the radius where the average DM density is $\Delta $ times the critical density ${\rho}_{\mathrm{c}}\left(z\right)$ at the redshift z of the cluster. |

2 | See https://dominiqueeckert.wixsite.com/xcop/about-x-cop (accessed on 30 May 2023). |

3 | Note that in computing the overall gravitational potential $\Phi $, we have neglected the stellar contribution ${\Phi}_{\u2605}$, mainly originated by the brightest central galaxy; this would add a term $\mathrm{d}{\Phi}_{\u2605}/\mathrm{d}r=G\phantom{\rule{0.166667em}{0ex}}{M}_{\u2605}(<r)/{r}^{2}$ to the integrand on the right-hand side of Equation (11). For the X-COP cluster sample exploited in this work (stellar profiles were available for 5 out of 12 clusters), the related contribution has been shown by [38] to become relevant only for $r\lesssim 0.02\phantom{\rule{0.166667em}{0ex}}{R}_{500}\sim 20$ kpc and as such can barely influence the innermost available data point of the pressure profile; as a consequence, our results were negligibly affected, as we also checked numerically. |

4 | Actually, in principle, fractional gravity can also alter somewhat the total depth of the gravitational potential, thus biasing the overall WL mass estimates; however, given that the fractional gravity masses estimated without WL prior and the Newtonian ones are consistent with each other within $2\sigma $ (see fifth and last column in Table 1), we ignored such a small bias and used the Newtonian WL masses as prior, with their uncertainties, in the fractional gravity analysis. |

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**Figure 1.**Effective DM mass profile (appropriately normalized in units of ${M}_{s}\phantom{\rule{0.166667em}{0ex}}{(\ell /{r}_{s})}^{2-2s}$) in fractional gravity as a function of the radial coordinate (normalized to the NFW scale radius ${r}_{s}$). The profile is shown for different values of the fractional index s (color-coded) from 1 (solid black line, corresponding to Newtonian gravity) to the limiting value $s=1.5$ (dotted black line).

**Figure 2.**Fits to the individual pressure profiles of the X-COP clusters in fractional gravity, according to the Bayesian analysis described in Section 2. Circles refer to X-ray and squares to SZ data. Solid lines illustrate the median, and the shaded areas show the $2\sigma $ credible interval from sampling the posterior distribution of the fits. The vertical dotted lines mark the reference radius ${R}_{500}$. The reduced ${\chi}_{r}^{2}$ value of the joint fit to the pressure and density profiles is reported in each panel.

**Figure 3.**Same as Figure 2 for the density profiles.

**Figure 4.**MCMC posterior distributions of the fractional parameter s, the fractional length scale ℓ, the DM concentration ${c}_{500}$, and the DM mass ${M}_{500}$, for two representative clusters of the X-COP sample: A2255 (

**top**) and ZW1215 (

**bottom**). Magenta contours/lines refer to the analysis with no mass prior, and cyan contours/lines to that with a weak lensing mass prior (only available for ZW1215 in the bottom panel). The contours show 1, 2, and 3$\sigma $ confidence intervals, white crosses mark the maximum likelihood estimates, and the marginalized distributions are in arbitrary units (normalized to unity at their maximum value).

**Figure 5.**Fits to the stacked sample profiles of the X-COP clusters in fractional gravity, according to the Bayesian analysis described in Section 2. Crosses show the stacked sample profiles, while gray lines show the individual ones. Colored solid lines illustrate the median, and the shaded areas show the $2\sigma $ credible interval from sampling the posterior distribution of the fits. The vertical dotted lines marks the average reference radius ${R}_{500}$ of the sample. The ${\chi}_{r}^{2}$ value of the joint fit to the density and pressure profile is reported in the right panel.

**Figure 6.**Posterior distributions (normalized to unity at their maximum values) of fractional index s (

**top left**), fractional length-scale ℓ (

**top right**), concentration ${c}_{500}$ (

**bottom left**) and DM mass ${M}_{500}$ (

**bottom right**), from the fits to the X-COP density and pressure profiles in fractional gravity. Colored areas refer to individual clusters (as detailed in the legend), and the black dashed line to the stacked sample.

**Figure 7.**Concentration ${c}_{200}$ vs. DM mass ${M}_{200}$ relation for the X-COP sample in fractional gravity. Gray shaded area is the relation predicted from N-body simulations in the $\Lambda $CDM cosmology [47]. Filled magenta circles refer to individual clusters and the magenta cross to the stacked sample, while open magenta circles show (with no error bars for clarity) the corresponding ${c}_{500}$ and ${M}_{500}$ values that are the actual fitting parameters.

**Figure 8.**Scaling relations in fractional gravity from galaxies to clusters: fractional index s (

**left**) and length-scale ℓ (

**right**) vs. DM mass ${M}_{200}$. Magenta circles refer to individual clusters and magenta crosses to the stacked cluster sample from this work, while cyan stars refer to the stacked rotation curve of galaxies from [34]. Dashed lines display an ODR linear fit to the overall data, while a solid line in the left panel shows a nonlinear ODR fit with limiting values 1 and $1.5$ from large to small masses, and a dotted line in the right panel illustrates the scale radius ${r}_{s}$ of the NFW profile (see Section 3 for further details).

**Table 1.**Marginalized posterior estimates (mean and $68\%$ confidence limits are reported) for the parameters from the MCMC analysis of the X-COP sample in fractional gravity. Columns report the values of the fractional parameter s, the fractional length scale ℓ, the DM mass ${M}_{500}$, the halo concentration ${c}_{500}$, and the reduced ${\chi}_{r}^{2}$ of the joint fit to the density and pressure profiles. The different portions of the table refer to the fit to individual clusters with no mass prior, to individual clusters with weak lensing mass priors (marked WL, and only available for five clusters from [46]), and to the stacked sample. For reference, the last column reports the best-fit DM mass ${M}_{500}$ from [38].

Cluster | s | $log\ell $ (Mpc) | $log{\mathit{c}}_{500}$ | $log{\mathit{M}}_{500}$ $\left({\mathit{M}}_{\odot}\right)$ | ${\mathit{\chi}}_{\mathit{r}}^{2}$ | $log{\mathit{M}}_{500}^{\mathbf{Eck}22}$ $\left({\mathit{M}}_{\odot}\right)$ |
---|---|---|---|---|---|---|

A85 | ${1.11}_{-0.08}^{+0.04}$ | $-{1.2}_{-0.8}^{+0.5}$ | ${0.49}_{-0.1}^{+0.07}$ | ${14.27}_{-0.23}^{+0.37}$ | $2.02$ | $14.70$ |

A644 | ${1.02}_{-0.01}^{+0.01}$ | $-{0.8}_{-1.6}^{+1.6}$ | ${0.66}_{-0.03}^{+0.03}$ | ${14.56}_{-0.04}^{+0.09}$ | $1.94$ | $14.89$ |

A1644 | ${1.19}_{-0.09}^{+0.05}$ | $-{0.4}_{-0.9}^{+0.5}$ | ${0.49}_{-0.09}^{+0.09}$ | ${14.19}_{-0.35}^{+0.46}$ | $3.01$ | $14.49$ |

A1795 | ${1.05}_{-0.05}^{+0.01}$ | $-{1.9}_{-1.1}^{+0.2}$ | ${0.54}_{-0.04}^{+0.02}$ | ${14.28}_{-0.12}^{+0.29}$ | $1.46$ | $14.65$ |

A2029 | ${1.03}_{-0.05}^{+0.05}$ | $-{0.4}_{-1.6}^{+1.6}$ | ${0.55}_{-0.06}^{+0.02}$ | ${14.70}_{-0.07}^{+0.18}$ | $1.36$ | $14.84$ |

A2142 | ${1.03}_{-0.02}^{+0.02}$ | $-{0.2}_{-2.1}^{+1.4}$ | ${0.42}_{-0.07}^{+0.04}$ | ${14.76}_{-0.11}^{+0.15}$ | $2.91$ | $14.95$ |

A2255 | ${1.15}_{-0.06}^{+0.04}$ | $+{0.5}_{-0.9}^{+0.7}$ | ${0.42}_{-0.10}^{0.09}$ | ${14.81}_{-0.31}^{+0.31}$ | $0.89$ | $14.69$ |

A2319 | ${1.01}_{-0.02}^{+0.02}$ | $-{0.2}_{-1.6}^{+1.6}$ | ${0.59}_{-0.03}^{+0.02}$ | ${14.69}_{-0.02}^{+0.04}$ | $6.46$ | $14.90$ |

A3158 | ${1.11}_{-0.09}^{+0.03}$ | $-{0.9}_{-1.2}^{+0.6}$ | ${0.48}_{-0.12}^{+0.07}$ | ${14.18}_{-0.25}^{+0.43}$ | $1.72$ | $14.63$ |

A3266 | ${1.09}_{-0.09}^{+0.02}$ | $+{0.8}_{-0.9}^{+0.4}$ | ${0.36}_{-0.21}^{+0.09}$ | ${14.93}_{-0.07}^{+0.07}$ | $6.86$ | $14.87$ |

RXC1825 | ${1.03}_{-0.02}^{+0.01}$ | $+{1.4}_{-1.1}^{+0.6}$ | ${0.56}_{-0.06}^{+0.04}$ | ${14.59}_{-0.04}^{+0.04}$ | $2.92$ | $14.59$ |

ZW1215 | ${1.15}_{-0.08}^{+0.05}$ | $-{0.3}_{-1.0}^{+0.6}$ | ${0.41}_{-0.11}^{+0.11}$ | ${14.61}_{-0.31}^{+0.40}$ | $0.68$ | $14.85$ |

A85 +WL | ${1.04}_{-0.02}^{+0.01}$ | $+{1.3}_{-0.9}^{+0.8}$ | ${0.49}_{-0.06}^{+0.06}$ | ${14.79}_{-0.09}^{+0.09}$ | $2.02$ | $14.70$ |

A1795 +WL | ${1.04}_{-0.01}^{+0.01}$ | $+{2.3}_{-1.5}^{+1.5}$ | ${0.68}_{-0.05}^{+0.05}$ | ${14.81}_{-0.08}^{+0.08}$ | $1.47$ | $14.65$ |

A2029 +WL | ${1.03}_{-0.01}^{+0.01}$ | $+{2.0}_{-0.3}^{+0.9}$ | ${0.61}_{-0.05}^{+0.05}$ | ${14.95}_{-0.06}^{+0.06}$ | $1.36$ | $14.84$ |

A2142 +WL | ${1.04}_{-0.02}^{+0.01}$ | $+{1.9}_{-0.5}^{+0.8}$ | ${0.51}_{-0.06}^{+0.05}$ | ${15.01}_{-0.06}^{+0.06}$ | $2.91$ | $14.95$ |

ZW1215 +WL | ${1.10}_{-0.05}^{+0.04}$ | $+{0.3}_{-0.5}^{+0.3}$ | ${0.38}_{-0.09}^{+0.09}$ | ${14.86}_{-0.08}^{+0.08}$ | $0.68$ | $14.85$ |

Stacked | ${1.03}_{-0.02}^{+0.02}$ | $+{0.3}_{-1.3}^{+1.3}$ | ${0.43}_{-0.07}^{+0.04}$ | ${14.76}_{-0.08}^{+0.08}$ | $0.84$ | $14.75$ |

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**MDPI and ACS Style**

Benetti, F.; Lapi, A.; Gandolfi, G.; Haridasu, B.S.; Danese, L.
Dark Matter in Fractional Gravity II: Tests in Galaxy Clusters. *Universe* **2023**, *9*, 329.
https://doi.org/10.3390/universe9070329

**AMA Style**

Benetti F, Lapi A, Gandolfi G, Haridasu BS, Danese L.
Dark Matter in Fractional Gravity II: Tests in Galaxy Clusters. *Universe*. 2023; 9(7):329.
https://doi.org/10.3390/universe9070329

**Chicago/Turabian Style**

Benetti, Francesco, Andrea Lapi, Giovanni Gandolfi, Balakrishna Sandeep Haridasu, and Luigi Danese.
2023. "Dark Matter in Fractional Gravity II: Tests in Galaxy Clusters" *Universe* 9, no. 7: 329.
https://doi.org/10.3390/universe9070329