  We investigate the cosmological applications of a bi-scalar modified gravity
that exhibits partial conformal invariance, which could become full conformal
invariance in the absence of the usual Einstein-Hilbert term and introducing
additionally either the Weyl derivative or properly rescaled fields. Such a
theory is constructed by considering the action of a non-minimally
conformally-coupled scalar field, and adding a second scalar allowing for a
nonminimal derivative coupling with the Einstein tensor and the energy-momentum
tensor of the first field. At a cosmological framework we obtain an effective
dark-energy sector constituted from both scalars. In the absence of an explicit
matter sector we extract analytical solutions, which for some parameter regions
correspond to an effective matter era and/or to an effective radiation era,
thus the two scalars give rise to "mimetic dark matter" or to "dark radiation"
respectively. In the case where an explicit matter sector is included we obtain
a cosmological evolution in agreement with observations, that is a transition
from matter to dark energy era, with the onset of cosmic acceleration.
Furthermore, for particular parameter regions, the effective dark-energy
equation of state can transit to the phantom regime at late times. These
behaviours reveal the capabilities of the theory, since they arise purely from
the novel, bi-scalar construction and the involved couplings between the two
fields.
