A matter-dominated cosmological model with variable $G$ and $\Lambda$.
  Confrontation of theoretical predictions with observational data
  In the framework of renormalization-group improved cosmologies, we analyze
both theoretically and observationally the exact and general solution of the
matter-dominated cosmological equations, using the expression of \Lambda =
\Lambda(G) already determined by the integration method employed in a previous
paper. A rough comparison between such a model and the concordance \LambdaCDM
model as to the magnitude-redshift relationship has been already done, without
showing any appreciable differences. We here perform a more refined study of
how astrophysical data (Union2 set) on type-I supernovae, gamma ray bursts (in
a sample calibrated in a model independent way with the SneIa dataset), and gas
fraction in galaxy clusters (using a sample of Chandra measurements of the
X-ray gas mass fraction) affect the model and constrain its parameters. We also
apply a cosmographic approach to our cosmological model and estimate the
cosmographic parameters by fitting both the supernovae and the gamma ray bursts
datasets. We show that this matter-dominated cosmological model with variable
Newton parameter and variable cosmological term is indeed compatible with the
observations above (on type Ia supernovae, the gamma ray bursts Hubble diagram,
and the gas mass fraction in X-ray luminous galaxy clusters). The cosmographic
approach adopted confirms such conclusions. Finally, it seems possible to
include radiation into the model, since numerical integration of the equations
derived by the presence of both radiation and matter shows that, after
inflation, the total density parameter is initially dominated by the radiation
contribution and later by the matter one.
