  We investigate an evolutional scenario of the FRW universe with the Casimir
energy scaling like $(-)(1+z)^4$. The Casimir effect is used to explain the
vacuum energy differences (its value measured from astrophysics is so small
compared to value obtained from quantum field theory calculations). The
dynamics of the FRW model is represented in terms of a two-dimensional
dynamical system to show all evolutional paths of this model in the phase space
for all admissible initial conditions. We find also an exact solution for non
flat evolutional paths of Universe driven by the Casimir effect. The main
difference between the FRW model with the Casimir force and the $\Lambda$CDM
model is that their generic solutions are a set of evolutional paths with a
bounce solution and an initial singularity, respectively. The evolutional
scenario are tested by using the SNIa data, FRIIb radiogalaxies, baryon
oscillation peak and CMB observation. We compare the power of explanation of
the model considered and the $\Lambda$CDM model using the Bayesian information
criterion and Bayesian factor. Our investigation of the information criteria of
model selection showed the preference of the $\Lambda$CDM model over the model
considered. However the presence of negative like the radiation term can remove
a tension between the theoretical and observed primordial ${}^4$He and D
abundance.
