  The scatter in the apparent magnitude of type Ia supernovae induced by
stochastic gravitational lensing is highly dependent on the nonlinear growth of
cosmological structure. In this paper, we show that such a dependence can
potentially be employed to gain significant information about the mass
clustering at small scales. While the mass clustering ultimately hinges on
cosmology, here we demonstrate that, upon obtaining more precise observational
measurements through future cosmological surveys, the lensing dispersion can
very effectively be used to gain information on the poorly understood
astrophysical aspects of structure formation, such as the clumpiness of dark
matter halos and the importance of gas physics and star formation into shaping
the large-scale structure. In order to illustrate this point we verify that
even the tentative current measurements of the lensing dispersion performed on
the Supernova Legacy Survey sample favor a scenario where virialized structures
are somewhat less compact than predicted by $n-$body cosmological simulations.
Moreover, we are also able to put lower limits on the slope of the
concentration-mass relation. By artificially reducing the statistical
observational error we argue that with forthcoming data the stochastic lensing
dispersion will allow one to importantly improve constraints on the baryonic
physics at work during the assembly of cosmological structure.
