STVG-MOG Cluster Dynamics and the Cosmological $1/r^2$ Force Law from Pairwise kSZ Data

Abstract: We investigate whether Scalar-Tensor-Vector Gravity in its weak-field modified gravity form can account for the cluster-scale inverse-square force law inferred from recent kinematic Sunyaev-Zeldovich measurements of cluster pairwise motions. The starting point is the X-COP cluster fit of STVG-MOG, for which a representative baryonic cluster mass $M\sim 10^{15}M_\odot$ together with parameters $α\sim 9.11$ and $μ\sim 0.196~{\rm Mpc}^{-1}$ provides a successful description of cluster dynamics without particle dark matter. We extrapolate this fit to the separation range $30$ to $230~{\rm Mpc}$, relevant for the pairwise kSZ analysis. Since the Yukawa transition length $μ^{-1}\simeq 5.1~{\rm Mpc}$ is much smaller than these separations, the STVG-MOG acceleration law reduces to an effective inverse-square form. This explains why the theory can satisfy the observed Newtonian behavior while remaining distinct from MOND-like long-distance modifications. We derive the corresponding pairwise velocity curve and show that, after fitting a single overall kSZ amplitude, the extrapolated STVG-MOG prediction reproduces the measured trend of the pairwise kSZ data. The analysis shows that the X-COP cluster fit and the cosmological-scale kSZ force-law result are mutually consistent within STVG-MOG.

• The paper’s main contribution is demonstrating that STVG-MOG, calibrated with X-COP cluster data, yields a 1/r² force law at cosmological distances with deviations below 1%.
• It employs a Yukawa-corrected gravitational acceleration model, highlighting how the amplitude and transition scale reconcile internal cluster dynamics with pairwise kSZ observations.
• The findings imply that modified gravity can mimic Newtonian behavior on large scales without invoking particle dark matter, emphasizing the critical role of the transition scale.

STVG-MOG, Cluster Dynamics, and the Cosmological $1/r^2$ Force Law Inferred from Pairwise kSZ Data

Introduction

The paper "STVG-MOG Cluster Dynamics and the Cosmological $1/r^2$ Force Law from Pairwise kSZ Data" (2604.23440) addresses the consistency between cluster-scale gravitational phenomenology under the Scalar-Tensor-Vector Gravity (STVG, or MOG) framework and cosmological-scale constraints derived from recent pairwise kinematic Sunyaev-Zeldovich (kSZ) measurements. Central to the analysis is whether the scale-dependent modifications to gravity introduced by STVG-MOG remain compatible with the empirically favored inverse-square ($1/r^2$) force law on scales spanning several tens to hundreds of megaparsecs, as inferred from kSZ data, while still providing a successful description of internal cluster dynamics without particle dark matter.

Theoretical Framework and Parameter Regime

STVG-MOG modifies the weak-field gravitational potential via a Yukawa-like correction, parameterized by an amplitude $\alpha$ and a scale $\mu^{-1}$. The acceleration law takes the form

$$g_{\rm MOG}(r) = \frac{G_N M}{r^2}\left[1+\alpha-\alpha e^{-\mu r}(1+\mu r)\right],$$

where $M$ is the baryonic mass, $\alpha$ controls the long-distance renormalization of the gravitational coupling, and $\mu^{-1}$ is the characteristic transition length between intermediate-range modification and asymptotic behavior. For the representative fit to X-COP cluster data ($M\sim 10^{15} M_\odot$, $1/r^2$0, $1/r^2$1 Mpc$1/r^2$2), $1/r^2$3 is approximately 5.1 Mpc. On scales $1/r^2$4, the Yukawa term is exponentially suppressed, and the force law reverts to a Newtonian form with an enhanced amplitude: $1/r^2$5.

Comparison with Cosmological-scale kSZ Data

Recent results from Gallardo et al. have provided direct observational constraints on the radial dependence of the gravitational force law at cosmological distances using the pairwise kSZ momentum of galaxy clusters. Their analysis disfavors modified gravity models that produce a long-range $1/r^2$6 (MOND-like) field, instead supporting $1/r^2$7 (inverse-square) behavior across $1/r^2$8 to $1/r^2$9 Mpc.

This paper demonstrates that the parameter regime preferred by galaxy cluster data for STVG-MOG ensures that the transition to the inverse-square law occurs at scales well below those tested by the kSZ analysis. Specifically, over $1/r^2$0 to $1/r^2$1 Mpc, the STVG-MOG force law—after normalization—traces the $1/r^2$2 behavior nearly identically, with deviations below the percent level. The only residual freedom is the amplitude, absorbed in fitting the pairwise kSZ normalization parameter, which is degenerate with other uncertain astrophysical and observational quantities (e.g., optical depth).

Consequences for Modified Gravity and Dark Matter Paradigms

By tracing the connection between the cluster-scale (X-COP) fit and the kSZ force law constraint, the paper establishes that STVG-MOG remains viable under both data sets: the strong claim is that contrary to the interpretation that cosmological inverse-square constraints necessarily exclude modified gravity, STVG-MOG, with a finite Yukawa scale, is not ruled out. In contrast, models like MOND, whose modifications persist to large $1/r^2$3, are excluded by the kSZ force-law data. Thus, the Gallardo et al. measurement discriminates among modified gravity models based on the scale of force-law transition, rather than simply between modified gravity and dark matter.

The implications are clear: achieving consistency with both cluster dynamics (without dark matter) and cosmological force-law constraints does not require introducing particle dark matter, provided the modified gravity behaves Newtonian on sufficiently large scales due to an appropriately localized transition in force law. This result holds provided that future, more precise kSZ and other momentum correlation data do not detect subtle deviations at the percent level or below.

Numerical Results and Robustness

The extrapolated STVG-MOG acceleration law at cosmological separations is shown to be indistinguishable, at the current precision, from a pure $1/r^2$4 form with an adjusted amplitude. The derived pairwise velocity and kSZ momentum curves, after normalization, closely follow the measured trends from Gallardo et al. The dominant uncertainty affecting direct comparability is related to the amplitude normalization, which is influenced by optical depth modeling and survey selection function, rather than radial dependence.

While the present analysis uses a simplified quasi-static approach for estimating pairwise velocities, the result is robust insofar as the force law's radial shape is the primary discriminant. A full likelihood or covariance-based treatment—embedding the STVG-MOG kernel directly into correlation-function pipelines—would refine the quantitative comparison but is not expected to change the substantive qualitative conclusion.

Implications for Future Theoretical and Observational Work

The findings have significant implications for the ongoing search for a fundamental description of cosmic-scale gravitational phenomena:

• Model Selection: The scale at which a modified gravity theory transitions to Newtonian behavior is a key parameter in determining its compatibility with large-scale structure and CMB-probe constraints.
• Broader Discriminatory Power: Upcoming high-precision kSZ, weak lensing, and other dynamical probes will further constrain the allowed transition scales in non-GR models.
• Theoretical Viability: STVG-MOG's compatibility with both cluster internal dynamics and cosmological force-law tests provides motivation to pursue further analytic and simulation-based work within its framework, especially at the interface between intermediate and large scales.
• Degeneracies Remain: The amplitude degeneracy in the large-scale force law means empirical determination of the gravitational constant in cosmological contexts remains entangled with baryonic physics and selection effects.

Conclusion

This study substantiates that STVG-MOG, installed with X-COP cluster parameters, naturally extrapolates to a cosmological-scale $1/r^2$5 force law, matching the radial dependence required by the latest pairwise kSZ data over tens to hundreds of megaparsecs. In this regime, the Yukawa corrections are negligible and the normalization is enhanced relative to GR, but the shape is observationally indistinguishable from Newtonian gravity. These findings clarify that cosmological force-law constraints, as currently measured, do not preclude all modified gravity alternatives to particle dark matter, provided the transition scale is sufficiently small. Future work refining both empirical and theoretical aspects will further elucidate the viability of such models in cosmological and astrophysical contexts.