Isotropic Equivalence of STVG--MOG and $Λ$CDM and Its Breakdown in Large--Scale Anisotropic Cosmological Observables

Abstract: We show that Scalar-Tensor-Vector Gravity (STVG-MOG) is observationally equivalent to the standard model $Λ$CDM cosmological model for all probes that depend on isotropic and linear gravitational dynamics, including galaxy rotation curves, cluster lensing, the linear matter power spectrum P(k), $σ_8$, baryon acoustic oscillations, and the cosmic microwave background (CMB). This degeneracy arises from the scale-dependent effective gravitational coupling $G_{\mathrm{eff}}$, which ensures identical background evolution, transfer functions, and linear growth. Consequently, all early-universe, low and intermediate scale cosmological observables are equally well described by STVG-MOG without invoking non-baryonic dark matter. We argue that the equivalence implies that isotropic cosmological data alone cannot establish the physical existence of dark matter. The degeneracy is broken only by observables sensitive to large-scale, anisotropic gravitational response. In particular, recent measurements of enhanced radio-galaxy and quasar number-count dipoles at gigaparsec scales probe a regime where $G_{\mathrm{eff}}$ departs from its $Λ$CDM limit, allowing STVG-MOG to generate anisotropic bulk flows, while preserving consistency with all isotropic constraints. These observations provide a concrete pathway for empirically distinguishing modified gravity from particle dark matter.

• The paper argues for the observational equivalence of Scalar–Tensor–Vector Gravity (STVG) and ΛCDM for isotropic cosmological probes. They assert that anisotropic observables are needed to discriminate between the two.
• The theory reduces to Newton's constant G_N at early times and small scales, recovering standard General Relativity (GR) and matching specific isotropic cosmological observables.
• The theoretical framework of STVG-MOG backbone is augmented by scalar fields inducing scale-selective enhancements that mimic cold dark matter effects.

Overview and central claim

This paper by J. W. Moffat argues that Scalar–Tensor–Vector Gravity (STVG, also known as MOG) is observationally degenerate with $\Lambda$CDM for the entire class of cosmological probes that depend on isotropic, linear gravitational dynamics: galaxy rotation curves, cluster dynamics, weak and strong lensing, the linear matter power spectrum $P(k)$, $\sigma_8$, redshift-space distortions (RSD), baryon acoustic oscillations (BAO), and the CMB temperature and polarization spectra. The claimed origin of this degeneracy is the scale- and time-dependent effective gravitational coupling $G_{\rm eff}(k,a)$, which reduces to Newton's constant $G_N$ at early times and on small physical scales while reproducing the effective sourcing of cold dark matter on intermediate scales. The paper's strongest—and most contestable—claim is that isotropic cosmological data alone cannot establish the physical existence of dark matter; discrimination requires observables sensitive to large-scale anisotropic response, specifically Gpc-scale number-count dipoles and bulk flows.

The STVG framework

STVG augments the metric with a massive vector field $\phi_\mu$ (the "phion" field) and two scalar fields $G(x)$ and $\mu(x)$ governing the coupling strength and vector mass. The action comprises an Einstein–Hilbert term with variable $1/G$, a Proca-type vector action, scalar kinetic/self-interaction terms, and the matter action. In phenomenological applications the scalars are treated as slowly varying or constant, reducing the theory to the parameters $(\alpha, \mu)$.

The weak-field limit yields a Newtonian-plus-Yukawa potential,

$P(k)$0

with $P(k)$1 setting the long-range enhancement and $P(k)$2 the Yukawa range. The theory is constructed so that $P(k)$3 for $P(k)$4 and at early times, recovering standard GR during recombination and passing Solar-System tests.

Derivation of $P(k)$5

Fourier transforming the convolution of the Yukawa-modified kernel with a density field gives a Poisson-like equation with a scale-dependent prefactor:

$P(k)$6

This is a purely algebraic consequence of the Fourier transforms of $P(k)$7 and $P(k)$8 kernels, valid locally in FRW neglecting curvature corrections. The limits are explicit: $P(k)$9 for $\sigma_8$0 and $\sigma_8$1 for $\sigma_8$2. The modification is therefore scale-selective: negligible on small scales, a constant enhancement $\sigma_8$3 on ultra-large scales where coherent bulk flows and projected number-count dipoles are sourced. The paper stresses that this form is not inserted phenomenologically but follows from the weak-field limit of the covariant action under stated late-time approximations (constant $\sigma_8$4 and $\sigma_8$5; negligible vector stress-energy at the background level).

The isotropic degeneracy argument

The core analytical device is the correspondence

$\sigma_8$6

where $\sigma_8$7. Under this mapping:

• Galaxy and cluster dynamics: the circular velocity satisfies $\sigma_8$8, exactly degenerate with Newtonian gravity acting on baryons plus CDM.
• Lensing: since lensing probes the sum of metric potentials through the same effective coupling, deflection angles are indistinguishable between the frameworks.
• Linear growth: both theories obey structurally identical growth equations once the correspondence is imposed, yielding identical growth factors $\sigma_8$9.
• $G_{\rm eff}(k,a)$0 and RSD: because $G_{\rm eff}(k,a)$1 depends only on $G_{\rm eff}(k,a)$2 and the geometrical window function, and $G_{\rm eff}(k,a)$3 only on $G_{\rm eff}(k,a)$4, both observables are necessarily degenerate.
• CMB: with $G_{\rm eff}(k,a)$5 at recombination, acoustic peak structure, damping tail, and polarization spectra match $G_{\rm eff}(k,a)$6CDM exactly.

The paper's conclusion here is epistemological rather than merely technical: isotropic data constrain only the effective gravitational response, not its microscopic origin. The empirical success of $G_{\rm eff}(k,a)$7CDM is thus presented as consistent with—but not uniquely diagnostic of—particle dark matter.

BAO suppression via window functions

A notable quantitative argument addresses why baryonic features remain visible in a baryon-only model yet are suppressed observationally. Decomposing the baryonic transfer function into a no-wiggle part times $G_{\rm eff}(k,a)$8, the observed spectrum is a convolution with a finite survey window $G_{\rm eff}(k,a)$9. Because $G_N$0 induces smooth scale dependence in $G_N$1, the no-wiggle power varies across the window support, and the leading oscillatory contribution averages to zero; the residual scales as

$G_N$2

Smooth differential growth across the binning window thereby suppresses BAO contrast, partially mimicking the smoothing attributed to dark matter. This is a specific mechanism, though its magnitude relative to actual survey systematics is not quantified against data in this paper.

Breaking the degeneracy: anisotropic large-scale observables

The degeneracy fails for $G_N$3, where enhanced coherent accelerations sourced purely by baryons generate bulk flows and number-count dipoles that $G_N$4CDM strongly suppresses through the decay of long-wavelength potentials. The paper points to reported excess radio-galaxy and quasar dipole amplitudes at Gpc scales as probing precisely this regime (Moffat, 12 Jan 2026), offering what it characterizes as a concrete discriminator between modified gravity and particle dark matter.

The paper also connects to the DESI low-$G_N$5 trend (Karim et al., 2024): by allowing modest suppression of gravitational response on quasi-linear scales ($G_N$6–$G_N$7) at late times while preserving early-universe physics, STVG-MOG can accommodate reduced clustering amplitude without dark matter. The author concedes plainly that the current statistical significance of this tension is limited by imaging systematics, dust corrections, and bias modeling, so it does not yet constitute decisive evidence.

Limitations and open questions

Several caveats bear directly on the strength of the claims. First, the equivalence is established at the level of linear, quasi-Newtonian perturbation theory under controlled approximations (constant scalars, negligible vector background stress-energy); whether the degeneracy survives full relativistic perturbation evolution—including vector-field contributions to the CMB—is asserted but not demonstrated here. Second, the identification $G_N$8 is exact only if the mapping holds mode-by-mode across all epochs; any deviation would propagate into $G_N$9 or the CMB. Third, the BAO-suppression mechanism is derived analytically but not validated against measured survey window functions. Fourth, the decisive test rests on the interpretation of the radio-galaxy/quasar dipole excesses, whose systematics (source catalog hemispheric asymmetries, selection effects) are acknowledged implicitly but not analyzed in this work. Whether the same $\phi_\mu$0 can simultaneously fit the low-$\phi_\mu$1 trend and the dipole amplitude within a single parameter set $\phi_\mu$2 remains an open quantitative question.

Conclusion

The paper establishes, within its stated approximations, a formal observational equivalence between STVG-MOG and $\phi_\mu$3CDM across all isotropic, linear-probe cosmology, rooted in the scale-dependent coupling $\phi_\mu$4 and the correspondence $\phi_\mu$5. Its methodological thesis—that agreement with isotropic data is necessary but insufficient for establishing the physical existence of dark matter—is clearly articulated, and the proposed breakdown channel via Gpc-scale anisotropic dipoles and bulk flows provides a falsifiable pathway. The argument's force ultimately depends on the robustness of those anisotropic measurements and on extending the analysis beyond the quasi-Newtonian, effectively-static regime adopted here.