Using the Mach–Sciama theory of gravitation to study dark-sector interactions and the cosmic coincidence problem

We propose a minimal interacting dark-sector model in which the coupling between cold dark matter and scalar dark energy is fixed by a Mach–Sciama selection rule. The model is formulated in frame-invariant scalar-tensor variables, where the invariant matter coupling induces a scalar-dependent dark-matter mass, and hence, an energy transfer controlled by the scalar velocity. The Mach–Sciama prescription selects this velocity as the scalar response sourced by the cosmological matter distribution, with the independent source-free mode removed. Thus, the interaction is not introduced as a phenomenological ansatz, but follows from the selected scalar dynamics. We study the autonomous dynamical system relevant to the present cosmological epoch and show that it admits stable accelerated attractors for which cold dark matter and scalar dark energy remain of the same order. We also study the sub-horizon linear growth of cold dark matter perturbations in Newtonian gauge, including the coupled scalar fluctuation. The resulting growth equation contains a scale-dependent effective gravitational coupling and an interaction-dependent friction term, while the scalar perturbation has a positive effective mass for the exponential potential. For the representative attractor considered here, the linear coefficients do not lead to tachyonic scalar growth or anti-friction enhancement, providing a first consistency check of the Mach–Sciama selected interacting model.