free parameters: the normalization of the stellar-to-halo mass ratio $(m/M)_0$, a characteristic mass $M_1$, where the SHM ratio is equal to $(m/M)_0$, and two slopes $\beta$ and $\gamma$ which indicate the behavior of $m/M$ at the low and high mass ends respectively. We use the same parameters for the central and satellite populations, since – unlike luminosity – the stellar mass of satellites changes only slightly after they are accreted by the host halo.

Note that though both $\beta$ and $\gamma$ are expected to be positive, they are not restricted to be so. The SHM relation is therefore not necessarily monotonic.

Constraining the free parameters {#constraining}
--------------------------------

Having set up the model we now need to constrain the four free parameters $M_1$, $(m/M)_0$, $\beta$ and $\gamma$. To do this, we populate the halos in the simulation with galaxies. The stellar masses of the galaxies depend on the mass of the halo and are derived according to our prescription (equation \[eqnmmap\]). The positions of the galaxies are given by the halo positions in the $N$-body simulation.

Once the simulation box is filled with galaxies, it is straightforward to compute the SMF $\Phi_{mod}(m)$. As we want to fit this model mass function 