# Refracted Gravity Solutions from Small to Large Scales

## Abstract

**:**

## 1. Introduction

## 2. Refracted Gravity

## 3. Dynamics of Disk Galaxies

#### 3.1. Mass Model

#### 3.2. Dynamical Model

#### 3.3. Results

## 4. Dynamics of Elliptical Galaxies

#### 4.1. Mass Model

#### 4.2. Dynamical Model

#### 4.3. Results

## 5. Dynamics of Galaxy Clusters

## 6. Covariant Refracted Gravity

## 7. Discussion and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BTFR | baryonic Tully–Fisher relation |

CFHTLS | Canada-France-Hawaii Telescope Legacy Survey |

CMB | cosmic microwave background |

CRG | covariant refracted gravity |

DE | dark energy |

DES | Dark Energy Survey |

DESI | Dark Energy Spectroscopic Instrument |

DM | dark matter |

DMS | DiskMass Survey |

dSph | dwarf spheroidal |

FLRW | Friedmann–Lema$\widehat{i}$tre–Robertson–Walker |

GCs | globular clusters |

GR | general relativity |

HSB | high surface brightness |

ICM | intracluster medium |

KIDS | Kilo-Degree Survey |

$\mathsf{\Lambda}$CDM | $\mathsf{\Lambda}$ cold dark matter |

LSB | low surface brightness |

MDAR | mass discrepancy–acceleration relation |

MGE | multi-Gaussian expansion |

MOND | modified Newtonian dynamics |

RAR | radial acceleration relation |

RG | refracted gravity |

RMS | root-mean-square |

SKA | Square Kilometer Array |

SMBH | supermassive black hole |

SNe Ia | Ia supernovae |

SPARC | Spitzer Photometry and Accurate Rotation Curves |

SPS | stellar population synthesis |

TeVeS | tensor vector scalar gravity |

WFL | weak field limit |

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**Figure 1.**Comparison between the behaviour of Newtonian (

**top panels**) and RG (

**bottom panels**) gravitational field lines for flat (

**left panels**) and spherical (

**right panels**) systems. The formulae reported for flattened systems, in the left part, are equivalent formulae for spherical systems to approximately illustrate the dependence of the field on the distance from the galaxy centre. The figure is reproduced from Figure 16 in [45].

**Figure 2.**Gravitational permittivity $\u03f5\left(\rho \right)$ adopted in RG (Equation (10)) for three different values of the RG free parameter Q and for ${\u03f5}_{0}=0.25$. The larger the value of Q, the steeper the transition between the two asymptotic limits of Equation (2). The black dashed line represents ${\u03f5}_{0}=0.25$. The figure is reproduced from Figure 1 in [48]. Credit: Cesare V., Diaferio A., Matsakos T., and Angus G., A&A, 637, A70, 2020, reproduced with permission © ESO.

**Figure 3.**RG models (blue solid lines) of the rotation curves and the vertical velocity dispersion profiles against the DMS data (red dots with error bars) for three DMS galaxies (UGC 3091, UGC 3701, and UGC 4256). Panels (

**a**): Rotation curves computed with the parameters found by modelling the rotation curves alone of each galaxy. Panels (

**b**): Rotation curves computed with the parameters found by modelling the rotation curves and vertical velocity dispersion profiles of each galaxy at the same time. Panels (

**c**): Vertical velocity dispersion profiles computed with the parameters found by modelling the rotation curves and vertical velocity dispersion profiles of each galaxy at the same time. The cyan solid lines are the RG vertical velocity dispersion profiles calculated with the same parameters as the blue curves but with disk-scale heights ${h}_{z}={h}_{z,\mathrm{SR}}$ (Equation (21)), and the dashed magenta and green vertical lines show the bulge effective radius (${R}_{\mathrm{e}}$ in Equation (15)) and the bulge radius adopted in the disk-bulge decomposition for the surface brightness modelling, respectively. The figure is readapted from Figures D.1 and D.2 in [48]. Credit: Cesare V., Diaferio A., Matsakos T., and Angus G., A&A, 637, A70, 2020, reproduced with permission © ESO.

**Figure 4.**Free parameters of the RG gravitational permittivity $\u03f5\left(\rho \right)$ estimated from the DMS and the three elliptical E0 galaxies. The purple squares with error bars represent the means of the permittivity parameters estimated at the same time from the rotation curves and the vertical velocity dispersion profiles of each DMS galaxy. The light blue shaded areas indicate the posterior distributions of the unique combination of RG parameters estimated from the entire DMS sample at the same time, where their medians and 1$\sigma $, 2$\sigma $, and 3$\sigma $ levels are shown as green dots and yellow, red, and black contours, respectively. The orange squares with error bars are the permittivity parameters derived from each elliptical E0 galaxy. The figure is reproduced from Figure 9 in [47]. Credit: Cesare V., Diaferio A., and Matsakos T., A&A, 657, A133, 2022, reproduced with permission © ESO.

**Figure 5.**RAR models calculated with RG (blue solid lines) with the parameters obtained from the MCMC analysis of the rotation curves and vertical velocity dispersion profiles of each galaxy at the same time, superimposed onto the RAR of DMS data (red points with error bars). The black solid line is the RAR fitting relation (Equation (22)) obtained by McGaugh et al. [41] from SPARC galaxies and the black dashed line is the ${g}_{\mathrm{obs}}={g}_{\mathrm{bar}}$ relation, plotted as a reference. The figure is readapted from Figure 11 in [48]. Credit: Cesare V., Diaferio A., Matsakos T., and Angus G., A&A, 637, A70, 2020, reproduced with permission © ESO.

**Figure 6.**Root-mean-square velocity dispersion profiles computed with RG with the parameters resulting from the MCMC analysis of [47], considering the stars, blue GCs, and red GCs’ kinematic profiles at the same time for each elliptical E0 galaxy. (

**Top panels**) the green, blue, and red solid lines represent the RG models for the stars, blue GCs, and red GCs, respectively. The dashed lines with the same colours are computed with the same $\mathsf{{\rm Y}}$ and $\beta $ parameters as the solid lines but with Newtonian gravity without DM. (

**Bottom panels**) zoom-in of the upper panels for the stars alone. Green solid and dashed lines have the same meaning as in the top panels. Both in the top and in the bottom panels, the black, blue, and red dots with error bars show the kinematic data for the stars, blue GCs, and red GCs, respectively. The figure is reproduced from Figure 8 in [47]. Credit: Cesare V., Diaferio A., and Matsakos T., A&A, 657, A133, 2022, reproduced with permission © ESO.

**Figure 7.**Emission-weighted projected temperature radial profiles of the hot, X-ray-emitting gas in the low-redshift and relaxed galaxy clusters A1991 (

**top panel**) and A1795 (

**bottom panel**). The solid and dashed lines are the RG and Newtonian models, respectively, and the dots with error bars are the measurements taken with the Chandra satellite from [92]. The Newtonian expectations were calculated with the same mass model as in RG without considering the presence of DM. A Hubble constant of ${H}_{0}=71$ ${\mathrm{km}\mathrm{s}}^{-1}$ ${\mathrm{Mpc}}^{-1}$ was adopted. The figure is reproduced from Figure 14 in [45].

**Figure 8.**The BTFR up to mass and velocity scales to include galaxy groups and clusters, besides galaxies. The black solid line is the BTFR given by Equation (8), and the black dashed line is the BTFR given by Equation (42) for $f/{\u03f5}_{0}=1.3$. The open circles and squares represent simulated galaxies [45]. The black dots that concentrate in the bottom-left part of the plot are observational data from [93] and refer to galaxies. The black dots that concentrate in the top-right part of the plot are observational data from [94,95] and refer to galaxy groups and clusters. The two set of measurement points concentrate around the corresponding BTFR model. The figure is reproduced from Figure 17 in [45].

**Figure 9.**Hubble diagram of SNe Ia modelled with CRG. The dashed and solid lines are the CRG− and CRG+ models, respectively. For the CRG− and CRG+ curves, the ${H}_{0}=67.7$ ${\mathrm{km}\mathrm{s}}^{-1}$ ${\mathrm{Mpc}}^{-1}$, ${\mathsf{\Omega}}_{0}=0.31$, and ${\mathsf{\Omega}}_{\mathsf{\Xi},0}=0.65$ parameters were adopted. The open circles with error bars are the data from the Supernova Cosmology Project Union $2.1$ Compilation [114]. The figure is reproduced from Figure E.2 in [46].

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**MDPI and ACS Style**

Cesare, V.
Refracted Gravity Solutions from Small to Large Scales. *Astronomy* **2024**, *3*, 68-99.
https://doi.org/10.3390/astronomy3020006

**AMA Style**

Cesare V.
Refracted Gravity Solutions from Small to Large Scales. *Astronomy*. 2024; 3(2):68-99.
https://doi.org/10.3390/astronomy3020006

**Chicago/Turabian Style**

Cesare, Valentina.
2024. "Refracted Gravity Solutions from Small to Large Scales" *Astronomy* 3, no. 2: 68-99.
https://doi.org/10.3390/astronomy3020006