Porosity Criterion for Hyperbolic Voids and CMB

At the study of non-Gaussian signatures in CMB we recently have shown that the low density regions, the voids, can induce hyperbolicity properties to null geodesics in globally flat and slightly positively curved Universe with matter inhomogeneities. In terms of an introduced porosity parameter, we now obtain the criterion for hyperbolicity: high underdensity regions, i.e. almost empty voids and not too dense walls. That criterion seems to be supported by the parameters of 30 Mpc scale voids revealed by the galactic redshift surveys. The temperature independent ellipticity of the excursion sets measured at the CMB maps, if due to the discussed geometrical effect of void hyperbolicity, would require a porosity parameter p > 0.7. Then CMB maps can probe the effective distance scale of such a filamentary Universe. The role of possible larger scale inhomogeneities, from superclusters to semi-Hubble scale, in this effect remains yet unclear.