Distinct radial acceleration relations of galaxies and galaxy clusters supports hyperconical modified gravity

TL;DR

The paper tests a relativistic MOND-like gravity, the hyperconical modified gravity (HMG), against the radial acceleration relation in 10 galaxy clusters and 60 SPARC galaxies. By modeling the RAR with a two-parameter, density-linked interpolation governed by a projective angle, the authors achieve good fits (global $R^2 \approx 0.83$) and derive physically meaningful ranges for the neighborhood density parameter $\varepsilon_H$ and the central projection $\gamma_{cen}$. The approach yields consistent interpolating behavior with MOND-like dynamics while naturally distinguishing galaxy- and cluster-scale anomalies, and it makes falsifiable predictions for small systems (e.g., Solar System, Oort cloud, wide binaries) that are near or below current detection limits. If validated, HMG provides a coherent relativistic framework for modifying gravity on cosmic scales without invoking cold dark matter, with implications for late-time cosmology and structure growth.

Abstract

General relativity (GR) is the most successful theory of gravity, with great observational support on local scales. However, to keep GR valid over cosmic scales, some phenomena (such as flat galaxy rotation curves and the cosmic expansion history) require the assumption of exotic dark matter. The radial acceleration relation (RAR) indicates a tight correlation between dynamical mass and baryonic mass in galaxies. This suggests that the observations could be better explained by modified gravity theories without exotic matter. Modified Newtonian Dynamics (MOND) is an alternative theory that was originally designed to explain flat galaxy rotation curves by using a new fundamental constant acceleration $a_0$, the so-called Milgromian parameter. However, this non-relativistic model is too rigid (with insufficient parameters) to fit the large diversity of observational phenomena. In contrast, a relativistic MOND-like gravity naturally emerges from the hyperconical model, which derives a fictitious acceleration compatible with observations. This study analyses the compatibility of the hyperconical model with respect to distinct RAR observations of 10 galaxy clusters obtained from HIFLUGCS and 60 high-quality SPARC galaxy rotation curves. The results show that a general relation can be fitted to most cases with only one or two parameters, with an acceptable $χ^2$ and $p$-value. These findings suggest a possible way to complete the proposed modification of GR on cosmic scales.

Figures (5)

• Figure 1: RAR modelling with HMG for the total acceleration ($a_{tot}$) compared to the Newtonian acceleration ($a_{N}$). Left: Individual fitting (Equations \ref{['eq:RAR_aN']} and \ref{['eq:model_clust2']}) for the galaxy clusters considered Li2023, using two parameters ($\varepsilon_H$ and $\gamma_{cen}$; Table \ref{['table:table2']}). Right: Global fitting for all data according to three models: MOND-like constant (blue band), general model (grey band, Equations \ref{['eq:RAR']} and \ref{['eq:model_general2']}), and the cluster model (red band, Equations \ref{['eq:RAR']} and \ref{['eq:model_clust2']}). The MOND-like model with constant $\gamma_0^{-1} = \gamma_{sys}^{-1}\cos\gamma_{sys}$ was considered with $\gamma_{sys} = 0.466^{+0.011}_{-0.01}\mathrm{\pi}$, which corresponds to Milgrom's constant $a_0 = 2\gamma_0^{-1}c/t = 1.01_{-0.32}^{+0.33} \times 10^{-10} \mathrm{m\,s}^{-2}$Monjo2023. In the right panel, both the general model and the specific approach for clusters use only one free parameter ($\varepsilon_H$), done by setting the projective angle of galaxies to $\gamma_{cen} = \mathrm{\pi}/2$ and $\gamma_U = \mathrm{\pi}/3$. The general model (grey band) is represented by $\varepsilon_H = 56_{-12}^{+22}$, while the cluster approach (red band) considers an average of $\varepsilon_H = 40_{-6}^{+8}$. The shaded areas represent the 90% confidence intervals.
• Figure 2: Galaxy rotation curves McGaugh2007 compared with the cluster RAR Li2023. Top left: Fitting of galaxy rotation curves according to the HMG model Monjo2023, where $\gamma_0$ is modeled by Equation \ref{['eq:model_general2']} with one free parameter ($\epsilon_H$, while $\gamma_{cen} = 0.48\mathrm{\pi}$; dashed lines) or two parameters ($\epsilon_H$ and $\gamma_{cen}$; solid lines). Top right: Performance of the 1-parameter (dashed grey lines) and 2-parameter (solid grey lines) model for the ratio of predicted ($a_{pred}$) and observed ($a_{obs}$) centripetal acceleration for the 60 galaxies (light grey lines) and comparison with MOND interpolation functions (MLS, simple, standard, and sharp functions, fitted to 153 suitable galaxies). $1\sigma$ confidence intervals for all functions (upper and lower lines of each color) were found as in figure 23 of Banik2023. Bottom left: Global fitting of the dataset according to Equations \ref{['eq:RAR']} and \ref{['eq:model_general2']}. Bottom right: Zoom in on the theoretical prediction made for galaxies fitted using Equation \ref{['eq:model_general2']} with $\varepsilon_H = 21_{-11}^{+32}$. As in Figure \ref{['fig:Fig01']}, the general model prediction for galaxies (grey band) corresponds to the parameter $\varepsilon_H = 56_{-12}^{+22}$ as an average value of the cluster fitting, with fixed $\gamma_{cen} = \mathrm{\pi}/2$ and $\gamma_{U} = \mathrm{\pi}/3$. To compare, the MOND-like model and the observational constraint of the general HMG model are also shown, with the shaded area representing the 90% confidence interval.
• Figure C.1: Conceptual model of the projective angle $\gamma_{sys}$ as a function of the relative geometry (orbital speed $v_N$ over the Hubble flux $\varepsilon_Hv_H$), for each gravitational system (black curves), with respect to the maximum background spacetime (red curves). To simplify the scheme, the projection factor $\gamma_0^{-1} \equiv \gamma_{sys}^{-1}\cos \gamma_{sys}$ is graphically represented by an auxiliary angle $\phi$, defined as the arc between the hyperplane of the gravitational system (dashed black line) and the maximum background hyperplane (dashed red line) curved by the dominant system (purple), such that $\gamma_{sys} = \mathrm{\pi}/2 - \phi$. The distance scale $r$ is represented by $t_{sys}$ (Equation \ref{['eq:t_sys']}). In addition to the cosmic scale, six gravitational systems are considered: i) a small system (e.g. Solar System); ii) an early-type galaxy with a dominant nuclear bulge; iii) a typical spiral galaxy; iv) a late-type spiral galaxy; v) an irregular cluster; and vi) a homogeneous cluster. The dominant objects (purple text) are the star for a small system, the galactic center for galaxies, and the brightest galaxy for a given cluster.
• Figure C.2: Observational constraints on the proposed models fitted to RAR data of the ten galaxy clusters considered Li2023. Left: Best values (red points) and uncertainty area (green shaded regions showing the $1\sigma$ confidence level) of the two parameters ($\varepsilon_H$ and $\gamma_{cen}$; see Table \ref{['table:table2']}) used in fits to individual clusters according to the specific approach (Equation \ref{['eq:model_clust2b']} is used in Equation \ref{['eq:RAR_aNb']}). Right: Best fit of the single free parameter ($\varepsilon_H$) used in the general model (Equation \ref{['eq:model_general2b']} used in Equation \ref{['eq:RAR_aNb']}), with fixed $\gamma_{cen} = 0.48\mathrm{\pi}$. In both cases (1 or 2 free parameters), the projective angle of the neighborhood ($\gamma_U$) was fixed to $\gamma_U = \mathrm{\pi}/3$. Notice that cluster A2029 did not pass the $\chi^2$ test even with the 2-parameter model. However, it passed the test for $\gamma_U = 0.235\mathrm{\pi}$, corresponding to $\gamma_U^{-1}\cos{\gamma_U} \approx 1$.
• Figure C.3: Model parameters fitted to the 60 high-quality galaxy rotation curves (squares), empirical relationships (black lines), and comparison with the fitting to the RAR data of 10 clusters (circles). Left: Constraint on the 1-parameter HMG model for galaxies, with fixed $\gamma_{cen} = 0.48\mathrm{\pi}$ and free $\varepsilon_H$. This shows an empirical identity between the single parameter $\epsilon_H$ and the square root of the relative density $\rho_{typ}/\rho_{vac}$. In other words, in units of vacuum density $\rho_{vac} \equiv 3/(8\mathrm{\pi}Gt^2)$, an observed density $\rho_{typ} \equiv \rho(r_{typ})$ is defined at a typical equilibrium distance of $r_{typ} \sim 50-200\,$kpc, whose fit has been found ($R = 0.85$, $p$-value $<0.0001$) as four times the maximum radius ($r_{typ} \approx 4\,r_{max}$) for each galaxy rotation curve, and equal to the minimum radius ($r_{typ} \approx r_{min}$) for the data of each cluster. Right: Constraint on the 2-parameter HMG model for galaxies, which shows a relationship between $\epsilon_H$ and $\gamma_{cen}$ such that $\cos(\gamma_{cen}) = \cos(0.460\mathrm{\pi}) - 0.020\ln\varepsilon_H$ for $1 \le \varepsilon_H < 400$, with a Pearson coefficient of $R = 0.80$ ($p$-value $<0.0001$; black regression line).

Theorems & Definitions (1)

• Definition B.1: Mass of perturbation