  The warm dark matter (WDM) can be described by simple and useful model called
reduced relativistic gas (RRG). In this work, it is analytically constructed
the scalar field actions minimally and non-minimally coupled to gravity, which
are equivalent to RRG in the sense they produce the same cosmological solutions
for the conformal factor of the metric. In particular, we construct the scalar
theory which corresponds to the model of ultra-relativistic ideal gas of
spinless particles possessing conformal symmetry. Finally, the possibility of
supplementing our scalar field model with a dynamical dark energy in the form
of a running cosmological constant (RCC) is also considered.
