used to probe the nature of dark matter [@sh99; @Metcalf1999; @Metcalf2006]). Therefore, it is expected that lensing shall induce a dispersion of the luminosities of the sources, and thus an extra scatter in the Hubble diagram [@scatterHubble]. Indeed, such an effect does also appear at the perturbation level—i.e. with light propagating in a perturbed FL spacetime—and it was investigated in Refs. [@Valageas2000; @Bonvin:2005ps; @Meures:2011gp; @Marozzi2012; @DiDio2012; @Marozzi2013]. The dispersion due to the large-scale structure becomes comparable to the intrinsic dispersion for redshifts $z > 1$ [@hl] but this dispersion can actually be corrected [@Cooray:2005yp; @Cooray:2005yr; @Sarkar:2007sp; @Cooray:2008qn; @Vallinotto:2010qm; @menard03]. Nevertheless, a considerable fraction of the lensing dispersion arises from sub-arc minute scales, which are not probed by shear maps smoothed on arc minute scales [@Dalal2002]. The typical angular size of the light beam associated with a supernova (SN) is typically of order $10^{-7}$ arc sec (e.g. for a source of physical size $\sim1$ AU at redshift $z\sim1$), while the typical observational aperture is of order 1 arc sec. This is smaller than the mean distance between any massive objects.

One can estimate [@Metcalf1999] that a gas composed of particles of mass $M$ can be considered diffuse on the scale of 