Scale invariant cosmology III: dynamical models and comparisons with
  observations
  We examine the properties of the scale invariant cosmological models, also
making the specific hypothesis of the scale invariance of the empty space at
large scales. Numerical integrations of the cosmological equations for
different values of the curvature parameter k and of the density parameter
Omega_m are performed. We compare the dynamical properties of the models to the
observations at different epochs. The main numerical data and graphical
representations are given for models computed with different curvatures and
density parameters. The models with non-zero density start explosively with
first a braking phase followed by a continuously accelerating expansion. The
comparison of the models with the recent observations from supernovae SN Ia,
BAO and CMB data from Planck 2015 shows that the scale invariant model with k=0
and Omega_m=0.30 very well fits the observations in the usual Omega_m vs.
Omega_Lambda plane and consistently accounts for the accelerating expansion or
dark energy. The expansion history is compared to observations in the plot H(z)
vs. redshift z, the parameters q_0 is also examined, as well the recent data
about the redshift z_trans of the transition between braking and acceleration.
These dynamical tests are fully satisfied by the scale invariant models. The
past evolution of matter and radiation density is studied, it shows small
differences with respect to the standard case. These first comparisons are
encouraging further investigations on scale invariant cosmology with the
assumption of scale invariance of the empty space at large scales.
