Cosmology with the sub-millimetre galaxies magnification bias:
  Tomographic Analysis
  As in Gonzalez-Nuevo et al. 2017 and Bonavera et al. 2019, the high-z
sub-millimetre galaxies can be used as background sample for gravitational
lensing studies thanks to their magnification bias. In particular, as in
Bonavera et al. 2020 the magnification bias can be exploited in order to
constrain the free parameters of a Halo Occupation Distribution (HOD) model and
some of the main cosmological parameters. In this work the magnification bias
has been evaluated as cosmological tool in a tomographic set up. The
cross-correlation function (CCF) data have been used to jointly constrain the
astrophysical parameters $M_{min}$, $M_{1}$ and $\alpha$ in each one of the
selected redshift bins and the $\Omega_{M}$, $\sigma_{8}$, and $H_0$
cosmological ones ($\Lambda$CDM). Moreover, we explore the possible time
evolution of the dark energy density introducing also the $\omega_0, \omega_a$
parameters in the joint analysis ($\omega_0$CDM and $\omega_0\omega_a$CDM). The
CCF has been measured between a foreground spectroscopic sample of GAMA
galaxies that has been divided into four redshift bins (0.1-0.2, 0.2-0.3,
0.3-0.5 and 0.5-0.8) and a sample of H-ATLAS galaxies with photometric
redshifts >1.2. The CCF is modelled using a description that depends on HOD and
cosmological parameters that are estimated with MCMC in different cases. For
the $\Lambda$CDM model, the analysis yields a maximum posterior value at 0.26
with $[0.17,0.41]$ 68\% C.I. for $\Omega_M$ and at 0.87 with $[0.75,1]$ 68\%
C.I. for $\sigma_8$. With our current results $H_0$ is not yet constrained.
With the $\omega_0$CDM model, the constraints on $\Omega_M$ and $\sigma_8$ are
similar, but we found a maximum posterior value for $\omega_0$ at -1 with
$[-1.56, -0.47]$ 68\% C.I. In the $\omega_0\omega_a$CDM model, the results are
-1.09 with $[-1.72, -0.66]$ 68\% C.I. for $\omega_0$ and -0.19 with $[-1.88,
1.48]$ 68\% C.I. for $\omega_a$.
