A direct and robust method to observationally constrain the halo mass
  function via the submillimeter magnification bias: Proof of concept
  Aims. The main purpose of this work is to provide a method to derive
tabulated observational constraints on the halo mass function (HMF) by studying
the magnification bias effect on high-redshift submillimeter galaxies. Under
the assumption of universality, we parametrize the HMF according to two
traditional models, namely the Sheth and Tormen (ST) and Tinker fits and assess
their performance in explaining the measured data within the {\Lambda} cold
dark matter ({\Lambda}CDM) model. We also study the potential influence of the
halo occupation distribution (HOD) parameters in this analysis and discuss two
important aspects regarding the HMF parametrization. Methods. We measure the
cross-correlation function between a foreground sample of GAMA galaxies with
redshifts in the range $0.2<z<0.8$ and a background sample of H-ATLAS galaxies
with redshifts in the range $1.2<z<4.0$ and carry out an MCMC algorithm to
check this observable against its mathematical prediction within the halo model
formalism. Results. If all HMF parameters are assumed to be positive, the ST
fit only seems to fully explain the measurements by forcing the mean number of
satellite galaxies in a halo to increase substantially from its prior mean
value. The Tinker fit, on the other hand, provides a robust description of the
data without relevant changes in the HOD parameters, but with some dependence
on the prior range of two of its parameters. When the normalization condition
for the HMF is dropped and we allow negative values of the $p_1$ parameter in
the ST fit, all the involved parameters are better determined, unlike the
previous models, thus deriving the most general HMF constraints. While all
cases are in agreement with the traditional fits within the uncertainties, the
last one hints at a slightly higher number of halos at intermediate and high
masses, raising the important point of the allowed parameter range.
