  Context. As recently demonstrated, high-z submillimetre galaxies (SMGs) are
the perfect background sample for tracing the mass density profiles of galaxies
and clusters (baryonic and dark matter) and their time-evolution through
gravitational lensing. Their magnification bias, a weak gravitational lensing
effect, is a powerful tool for constraining the free parameters of a halo
occupation distribution (HOD) model and potentially also some of the main
cosmological parameters. Aims. The aim of this work is to test the capability
of the magnification bias produced on high-z SMGs as a cosmological probe. We
exploit cross-correlation data to constrain not only astrophysical parameters
($M_{min}$, $M_1$, and $\alpha$), but also some of the cosmological ones
($\Omega_m$, $\sigma_8$, and $H_0$) for this proof of concept. Methods. The
measured cross-correlation function between a foreground sample of GAMA
galaxies with spectroscopic redshifts in the range 0.2 < z < 0.8 and a
background sample of H-ATLAS galaxies with photometric redshifts >1.2 is
modelled using the traditional halo model description that depends on HOD and
cosmological parameters. These parameters are then estimated by performing a
Markov chain Monte Carlo analysis using different sets of priors to test the
robustness of the results and to study the performance of this novel observable
with the current set of data Results. With our current results, $\Omega_m$ and
$H_0$ cannot be well constrained. However, we can set a lower limit of >0.24 at
95\% confidence level (CL) on $\Omega_m$ and we see a slight trend towards
$H_0>70$ values. For our constraints on $\sigma_8$ we obtain only a tentative
peak around 0.75, but an interesting upper limit of $\sigma_8\lesssim 1$ at
95\% CL. We also study the possibility to derive better constraints by imposing
more restrictive priors on the astrophysical parameters.
