  Cosmological solutions with a scalar field behaving as radiation are
obtained, in the context of gravitational theory with dynamical time. The
solution requires the spacial curvature of the universe k, to be zero, unlike
the standard radiation solutions, which do not impose any constraint on the
spacial curvature of the universe. This is because only such $ k=0 $ radiation
solutions poses a homothetic Killimg vector. This kind of theory can be used to
generalize electromagnetism and other gauge theories, in curved space time, and
there are no deviations from standard gauge filed equation (like Maxwell
equations) in the case there exist a conformal Killing vector. But there could
be departures from Maxwell and Yang Mills equations, for more general space
times.
