space information. This may allow them to solve the coarse graining problem and reduce the impact of non-physical two body relaxations between the macroscopical particles. This formalism has lead to promising results in the study of WDM cosmology and the differences between the CDM and WDM halo mass functions [@Angulo2013].

All the potential shortcomings of the [$N$-body ]{}formalism and the difficulty to evaluate their impact on the simulation results make it important to develop another framework not based on a particle approach.

An alternative framework
------------------------

Our framework resembles the attempt by [@Peebles1987] to use only the density field $\rho(x)$ and potential $U(x)$. The main problem of such an approach is that there is no closed system of equations that includes only the density and gravitational potential.

The situation is different when looking at quantum physics. In this realm, all the phase-space information can be encoded in a single function, the wavefunction $\psi(x)$ which does not depend on the velocity $v$. It is thus possible to write a closed Schrödinger-Poisson system that would replace the Vlasov-Poisson one and that would only depend on the spatial variable $x$ (See also [@Short2006] for a similar idea). This would effectively be a 3D 