Accelerated expansion in a stochastic self-similar fractal Universe

In a recent paper, a cosmological model based on El Naschie {\it E} infinity cantorian spacetime was presented [Iovane, G.; Chaos, Solitons and Fractals, 20: 657-667, (2004)]. In that work it was claimed that the present accelerated expansion of the Universe can be obtained as the effect of a scaling law on newtonian cosmology with a certain time dependent gravitational constant ($G$). In the present work we show that it may be problematic to explain the present accelerated expansion of the Universe using the approach presented in \cite{iovane}. As a better alternative we apply the same scaling law and a time-dependent gravitational constant, that follows from the observational constraints, to relativistic cosmology, i.e. the Friedmann's model. We are able to show that for a matter-dominated flat Universe, with the scaling law and a varying $G$, an accelerated expansion emerges in such a way that the function luminosity distance vs redshift can be made close to the corresponding function that comes from the usual Friedmann's model supplemented with a cosmological constant of value $\Omega_{\Lambda}\simeq0.7$. Then the measurements of high redshift supernovae, could be interpreted as a consequence of the fractal-self similarity of the $G$ varying relativistic universe.