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

TL;DR

Isotropic cosmological data can be explained equally well by ΛCDM and STVG--MOG due to a scale- and time-dependent Geff(k,a) that reproduces background evolution, transfer functions, and linear growth with baryons alone. The key mechanism is a Yukawa-type modification encoded in Geff(k,a) = GN[1 + αeff(k,a)], which recovers GR on small scales and enhances gravity on large scales, leading to identical P(k), σ8, and CMB signatures under the mapping Geffρb ≡ GNρm. However, this degeneracy is not universal; STVG--MOG predicts anisotropic, ultra-large-scale effects such as enhanced bulk flows and number-count dipoles that ΛCDM cannot easily mimic. Therefore, large-scale anisotropic observables offer a concrete empirical route to distinguish modified gravity from particle dark matter while remaining consistent with isotropic cosmological constraints.

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.