Bimetric MOND (BIMOND) is used as a platform for variable-$G$ theories that have MOND-specific idiosyncrasies. E.g., MOND premises dictate return to standard dynamics in the high-acceleration limit, predicting the standard value of $G$ for high-acceleration systems. This automatically ensures compliance of such theories with all the constraints on inconstancy of $G$ that emerge from the study of high-acceleration systems: geophysics, solar system, pulsars, supernovae, stellar evolution, emission of gravitational waves, etc. In MOND, constraints deduced from such phenomena have no bearing on possible $G$ variability in cosmology. My guiding motivation is to see if such theories may account for some roles of dark matter in cosmology; e.g., in accounting for the expansion history of the Universe in the matter-dominated era, by having a $G_e\approx 2\pi G$ govern the later stages of the expansion, instead of invoking matter density $\approx 2\pi \times$ baryon density. BIMOND is a relativistic MOND theory whose gravitational degrees of freedom are two metrics: $g_{\mu \nu }$, to which matter couples minimally, and ${\widehat{g}}_{\mu \nu }$, defining the geometry of a twin sector, presumably with its own “twin-matter.” The BIMOND gravitational action for the metric-symmetric subclass is $I^B\propto G^{-1}\int d^{4}x[|g{|}^{1/2}R+|\widehat{g}{|}^{1/2}\widehat{R}+v(g,\widehat{g}){\ell }_M^{-2}\tilde{\mathcal{M}}]$, where the metric-interaction term, $\tilde{\mathcal{M}}$, is a function of dimensionless scalars constructed from ${\ell }_M{C}_{\beta \gamma }^{\alpha }$, where the “relative-acceleration” tensor, $C_{\beta \gamma }^{\alpha }\equiv {\mathrm{\Gamma }}_{\beta \gamma }^{\alpha }-{\widehat{\mathrm{\Gamma }}}_{\beta \gamma }^{\alpha }$, is the difference of the Levi-Civita connections, and ${\ell }_M=c^2/a_0$ is the MOND length. Locally, where the random fluctuations in the two metrics are different, $\tilde{\mathcal{M}}$ encapsulates MOND effects in galactic systems, and the average of $\tilde{\mathcal{M}}$ on cosmological scales acts as a (dynamical) dark energy. Without adding degrees of freedom, or new dimensionful constants, BIMOND can be extended to a class of theories that entail what is best described as phenomenon-dependence of Newton’s constant, $G$: The Lagrangian density in $I^B$ is multiplied by $1+{\mathcal{M}}_G$, where ${\mathcal{M}}_G$ too is a function of the above scalars. $G_e\equiv G/(1+{\mathcal{M}}_G)$ plays the role of the effective gravitational constant. I consider the standard and the Einstein-Palatini formulations for the above action. My aim here is to present the general framework, while specific predictions would depend crucially on details of the structure of the theory—e.g., on the choice of variables, and the dependence of $\tilde{\mathcal{M}}$ and ${\mathcal{M}}_G$ on them—and, in cosmology, on unknown inputs: cosmological initial conditions, material content of the twin sector, etc. I cannot yet present a consistent model that complies with all the observations in cosmology, including the expansion history, with all its details. Instead, I describe some examples of theories in the class that predict different values of $G_e$ in different circumstances, including one where $G$ takes its standard value for all subcosmological systems—even if they are deep in the MOND regime. I also discuss scenarios in which $G_e\approx G$ in the early Universe, as required by constraints from big bang nucleosynthesis, but with $G_e>G$ setting in at later times, where it can affect the expansion history during the matter-dominated era.

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• Alternative gravity theories
• Cosmological parameters
• Dark energy
• Gravitation
• Large scale structure of the Universe
• Relativistic aspects of cosmology