Time-dependent $G$ in Einstein's equations as an alternative to the
  cosmological constant
  In this work, we investigate cosmologies where the gravitational constant
varies in time, with the aim of explaining the accelerated expansion without a
cosmological constant. We achieve this by considering a phenomenological
extension to general relativity, modifying Einstein's field equations such that
$G$ is a function of time, $G(t)$, and we preserve the geometrical consistency
(Bianchi identity) together with the usual conservation of energy by
introducing a new tensor field to the equations. In order to have concrete
expressions to compare with cosmological data, we posit additional properties
to this tensor field, in a way that it can be interpreted as a response of
spacetime to a variation of $G$. Namely, we require that the energy this tensor
represents is nonzero only when there is a time variation of $G$, and its
energy depends on the scale factor only because of its coupling to $G$ and the
matter and radiation energy densities. Focusing on the accelerated expansion
period, we use type Ia supernovae and baryon acoustic oscillation data to
determine the best fit of the cosmological parameters as well as the required
variation in the gravitational constant. As a result, we find that it is
possible to explain the accelerated expansion of the Universe with a variation
of $G$ and no cosmological constant. The obtained variation of $G$ stays under
10 \% of its current value in the investigated redshift range and it is
consistent with the local observations of $\dot{G}/G$.
