  Recently the Chiral Cosmological Model (CCM) coupled to cold dark matter
(CDM) has been investigated as $\sigma$CDM model to study the observed
accelerated expansion of the Universe. Dark sector fields (as Dark Energy
content) coupled to cosmic dust were considered as the source of Einstein
gravity in Friedmann-Robertson-Walker (FRW) cosmology. Such model had a
beginning at the matter-dominated era. The purposes of our present
investigation are two folds: to extend <<life>> of the $\sigma$CDM for earlier
times to radiation-dominated era and to take into account variation of the
exponential potential via variation of the interaction parameter $\lambda $.
  We use Markov Chain Monte Carlo (MCMC) procedure to investigate possible
values of initial conditions constrained by the measured amount of the dark
matter, dark energy and radiation component today.
  Our analysis includes dark energy contribution to critical density, the ratio
of the kinetic and potential energies, deceleration parameter, effective
equation of state and evolution of DE equation of state with variation of
coupling constant $\lambda $. A comparison with the $\Lambda$CDM model was
performed. A new feature of the model is the existence of some values of
potential coupling constant, leading to a $\sigma$CDM solution without transit
into accelerated expansion epoch.
