We show that a large class of modified gravity theories (MOG) with the
    Jordan-frame Lagrangian $f(R)$ translate into scalar-field (scalaron) models
    with hilltop potentials in the Einstein frame. (A rare exception to this rule
    is provided by the Starobinsky model for which the corresponding scalaron
    potential is plateau-like for $\phi > 0$.) We find that MOG models featuring
    two distinct mass scales lead to scalaron potentials that have a flattened
    hilltop, or tabletop. Inflationary evolution in tabletop models agrees very
    well with CMB observations. Tabletop potentials therefore provide a new and
    compelling class of MOG-based inflationary models. By contrast, MOG models with
    a single mass scale generally correspond to steep hilltop potentials and fail
    to reproduce the CMB power spectrum. Inflationary evolution in hilltop/tabletop
    models can proceed in two alternative directions: towards the stable point at
    small $R$ describing the observable universe, or towards the asymptotic region
    at large $R$. The MOG models which we examine have several new properties
    including the fact that gravity can become asymptotically vanishing, with
    $G_{\rm eff} \to 0$, at infinite or large finite values of the scalar curvature
    $R$. A universe evolving towards the asymptotically vanishing gravity region at
    large $R$ will either run into a ‘Big-Rip’ singularity, or inflate eternally.

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