How Well Does Refracted Gravity Fit the Kinematics of Stacked Galaxy Clusters?

Refracted Gravity replaces dark matter with a density-dependent permittivity

Refracted Gravity (RG) is a phenomenological modified gravity model. Instead of adding a dark matter particle, it modifies the Poisson equation by introducing an effective gravitational permittivity ε(ρbar) that depends on the baryonic matter density ρbar. Below a critical density, this permittivity enhances the gravitational field, mimicking the behaviour of dark matter in cosmic structures.

The model has three parameters that matter here. The vacuum permittivity ε0 sets the asymptotic enhancement of gravity in the low-density regime. A critical density ρc marks where the transition occurs, and an exponent Q controls how steep that transition is. A covariant scalar-tensor formulation exists, but the paper tests the phenomenological version, and it notes that a recent analysis found that specific covariant model cannot form cosmic structure early enough — so RG remains a test-bed rather than an established alternative.

This matters for how the results should be read. The paper is not asking whether RG is correct in general. It is asking how tightly cluster kinematics can pin down ε0, and how much that answer moves when the baryonic mass model changes.

Stacking 74 clusters in three velocity-dispersion bins suppresses shape systematics

The sample is 74 galaxy clusters from the Cluster Infall Regions in the Sloan Digital Sky Survey (CIRS). The authors stack them into three bins of increasing velocity dispersion σ, which serves as a proxy for total dynamical mass. Stacking is a deliberate trade: it averages over deviations from spherical symmetry and from dynamical equilibrium, the two assumptions most likely to bias a single-cluster kinematical fit.

The analysis uses the MG-MAMPOSSt code to reconstruct the gravitational potential and fit the velocity anisotropy profile of each stacked system. Only member galaxies with projected radius R ≥ 0.05 Mpc and R ≤ 1.2 × mean r200 enter the fit. The inner cut removes the region where dynamics are dominated by the brightest cluster galaxy; the outer cut removes galaxies that may no longer be in equilibrium or may belong to surrounding large-scale structure. The authors report that varying the upper limit by up to 10% does not significantly change results.

A second, smaller set of stacks is built from only the less-disturbed clusters, selected with A2 < 0.7. This selection is the paper's internal control: it tests whether the answer depends on how relaxed the systems are.

The hot-gas slope moves ε0 by nearly an order of magnitude

The intracluster medium dominates the baryonic mass budget, and its radial profile is parametrised with a beta-model. The authors run the fit twice, with β = 2/3 and β = 1, and marginalise over the gas and stellar mass-profile parameters so that baryonic uncertainty propagates into ε0.

For the three main stacks, the central estimates of ε0 range from 0.04 to 0.08 for β = 2/3, and from 0.19 to 0.37 for β = 1. That is the headline spread: the same clusters, the same kinematics, and a factor of roughly five in the inferred vacuum permittivity, driven by the assumed gas-density slope. The β = 1 case prefers larger ε0, most visibly in Bin 3.

The mechanism is explicit in the paper. At radii much larger than the gas scale radius, the beta-model density falls as ρgas ∝ r^(−3β). For β = 2/3 the density falls as r^(−2) and the cumulative gas mass grows roughly linearly with radius; for β = 1 it falls as r^(−3) and the cumulative mass grows only logarithmically. Because RG's gravitational field is sourced by baryons alone, the shallower profile produces a larger gravitational contribution in the outskirts, and the fit compensates by shifting the inferred gas normalisation, scale radius, and ε0.

Selecting relaxed clusters shrinks the gas-model shift in ε0

The two stacks built from clusters with A2 < 0.7 give a much more consistent picture across gas models. Their central estimates are ε0 = 0.04 and 0.06 for β = 2/3, and ε0 = 0.04 and 0.05 for β = 1. The shift induced by changing the gas-density slope is far smaller than in the three main stacks.

This is an internal consistency result within the adopted modelling framework, and the authors present it as such: a dynamical-state-based selection may help reduce systematic uncertainty. It is not a claim that relaxed clusters are intrinsically better probes of gravity. The comparison also changes binning and the reconstructed baryonic profiles, so part of the difference could come from those changes rather than from relaxation alone.

The practical reading is that the sensitivity of cluster-scale RG constraints to baryonic modelling is itself sample-dependent. Two analyses of the same theory on the same survey can differ in how much they depend on the gas prescription, depending on which clusters are stacked.

Newtonian fits barely notice the gas slope that RG is sensitive to

The paper first fits each stack in Newtonian gravity with a Navarro-Frenk-White dark matter profile plus separate gas and galaxy components, as a baseline. There, the β = 2/3 and β = 1 constraints agree well within uncertainties and produce a negligible difference in χ². The authors conclude that in Newtonian gravity the systematics from the intracluster medium parametrisation are subdominant to statistical uncertainties.

The contrast with the RG runs is the point. Adding a dark matter halo absorbs much of what the gas profile would otherwise control, so the Newtonian fit is largely indifferent to β. In RG, where the baryonic distribution alone sources the field, the same choice of β propagates directly into ε0.

The paper also flags a subtlety about radii. The r200 values fitted by MG-MAMPOSSt do not coincide exactly with the tabulated mean r200 values, because the two are not equivalent estimators: the tabulated values average individual caustic-based measurements, while the fit determines an effective dynamical scale for a single model fitted to the full stacked galaxy distribution. The fitted r200 should be read as the characteristic dynamical radius of the ensemble cluster, not as a reconstruction of the mean.

ρc is only bounded from below and Q is essentially unconstrained

Only ε0 is well localised. The transition exponent Q is totally unbound, and the critical density only gets a lower limit, log10[ρc/(M⊙/Mpc³)] ≳ 12.3, consistent across all bins. The RG parameters show no relevant correlations among themselves or with the anisotropy and number-density parameters, though a slight positive correlation appears between the vacuum permittivity and the baryonic profile parameters.

The paper explains why. Most of the radial range probed by the kinematics lies in the low-density regime ρbar ≪ ρc, where the permittivity approaches ε(ρbar) ≈ ε0. In that regime the dynamics constrain the asymptotic enhancement of the gravitational field but carry little information about where the transition happens or how steep it is. Excluding the innermost 50 kpc and not modelling the brightest cluster galaxy density profile separately further reduce sensitivity to the transition region.

So this analysis can test whether ε0 is consistent across stacks. It cannot test whether ρc and Q are universal, and the authors say so directly.

The BIC prefers Newtonian gravity, but the likelihoods are comparable

Using BIC = −2 lnLmax + k ln(N), the Newtonian model with an NFW dark matter component is significantly preferred over the baryon-only RG model, with ΔBIC > 6 in all bins. The RG analysis has 10 free parameters including baryonic ones, the Newtonian analysis has 9, and N is the total number of galaxies.

The authors are careful about what this means. The maximum likelihoods of the two models are never significantly different, so the BIC preference is mostly driven by the penalty for the extra parameter rather than by a better fit. And because ρc and Q are only weakly identified, the magnitude of the BIC difference should be treated with caution: repeating the analysis with Q fixed to 3, for example, produces no statistical preference for either framework.

That is the honest summary of the model comparison. The data at this precision do not separate the two frameworks on fit quality; they separate them on parameter economy, and only when the weakly constrained parameters are left free.

ε0 overlaps earlier cluster and elliptical-galaxy estimates, not disk galaxies

The inferred ε0 values are comparable to existing RG estimates. They agree with the bounds from the two massive clusters analysed previously, overlapping with both MACS 1206 and Abell S1063 within quoted uncertainties, and they also agree with the best-fit vacuum permittivity for elliptical galaxies. Against the estimate from disk galaxies, the results show a tension of roughly 2σ or more, with a significance that depends on the bin and the gas prescription.

The paper cautions against over-reading the cluster-to-cluster comparison. The earlier cluster constraints came from two individual, massive, relatively high-redshift clusters with resolved gas and stellar mass profiles plus the BCG stellar velocity-dispersion profile. This analysis uses lower-redshift stacked systems and effective baryonic profiles reconstructed from scaling relations and average bin properties. Any comparison therefore mixes possible intrinsic variation in RG parameters with differences in cluster mass, radial coverage, and baryonic modelling.

The consistency that does hold concerns ε0 specifically. The strongest tensions reported previously involved ρc and Q, and those are not constrained well enough here to reassess.

Baryonic assumptions shared by all stacks cannot be averaged away

The main limitations sit with the baryonic profiles and with treating each stack as a single spherical system in dynamical equilibrium. The gas distribution is inferred indirectly from an LX–Mgas scaling relation calibrated on resolved profiles for only a small subset of the sample, and the stellar component is tied to the gas through an assumed universal galaxy-to-gas mass ratio. These assumptions are shared by every stack, so stacking cannot average them down.

Neither the brightest cluster galaxy nor the diffuse intracluster light is modelled explicitly. That omission matters more in RG than in Newtonian fits, because the predicted acceleration depends on both the enclosed baryonic mass Mbar(<r) and the local density ρbar(r) entering the permittivity. The authors also note that simulations will be needed to quantify how much stacking actually reduces cluster-specific systematics, and whether residual tensions come from baryonic and dynamical modelling or from the stacking procedure itself.

At the present level of precision, the paper argues that a simple single-beta gas parametrisation is adequate, and that a more flexible gas model with additional poorly constrained parameters is not yet justified. That conclusion is specific to this data set and radial coverage.

The priority is controlling baryons, not adding clusters

The most decision-relevant result in this paper is not the value of ε0. It is that ε0's central estimate moves from about 0.04–0.08 to about 0.19–0.37 when the gas-density slope changes from β = 2/3 to β = 1, while the same shift nearly vanishes for the A2 < 0.7 stacks. For anyone comparing RG constraints across samples, that means the gas parametrisation and the dynamical-state selection are not nuisance details — they set the size of the error bar on the theory's main parameter.

The paper's own forward look follows the same logic. It points to resolved cluster-by-cluster gas-density and stellar-mass profiles, explicit treatment of the BCG and intracluster light, and independent validation of the spectroscopic selection function, membership, dynamical state, and stacking procedure. It notes that surveys such as 4MOST eROSITA and 4MOST CHANCES will supply far more cluster members, but states plainly that increased statistics alone will not control the dominant systematics.

So the threshold to watch is not how many clusters enter a stack. It is whether the baryonic distribution is measured rather than assumed, and whether the stacked sample is selected on dynamical state. Until both change, cluster-scale RG constraints will keep reporting an ε0 whose uncertainty is dominated by modelling choices rather than by the data.

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