The XXL survey: XLVI. Forward cosmological analysis of the C1 cluster
  sample
  We present the forward cosmological analysis of an $XMM$ selected sample of
galaxy clusters out to a redshift of unity. Following our previous 2018 study
based on the dn/dz quantity alone, we perform an upgraded cosmological analysis
of the same XXL C1 cluster catalogue (178 objects), with a detailed account of
the systematic errors. We follow the ASpiX methodology: the distribution of the
observed X-ray properties of the cluster population is analysed in a 3D
observable space (count rate, hardness ratio, redshift) and modelled as a
function of cosmology. Compared to more traditional methods, ASpiX allows the
inclusion of clusters down to a few tens of photons. We obtain an improvement
by a factor of 2 compared to the previous analysis by letting the normalisation
of the M-T relation and the evolution of the L-T relation free. Adding
constraints from the XXL cluster 2-point correlation function and the BAO from
various surveys decreases the uncertainties by 23 and 53 % respectively, and
62% when adding both. Switching to the scaling relations from the Subaru
analysis, and letting free more parameters, our final constraints are
$\sigma_8$ = $0.99^{+0.14}_{-0.23}$, $\Omega_m$ = 0.296 $\pm$ 0.034 ($S_8 =
0.98^{+0.11}_{-0.21}$) for the XXL sample alone. Finally, we combine XXL ASpiX,
the XXL cluster 2-point correlation function and the BAO, with 11 free
parameters, allowing for the cosmological dependence of the scaling relations
in the fit. We find $\sigma_8$ = $0.793^{+0.063}_{-0.12}$, $\Omega_m$ = 0.364
$\pm$ 0.015 ($S_8 = 0.872^{+0.068}_{-0.12}$), but still compatible with Planck
CMB at 2.2$\sigma$. The results obtained by the ASpiX method are promising;
further improvement is expected from the final XXL cosmological analysis
involving a cluster sample twice as large. Such a study paves the way for the
analysis of the eROSITA and future Athena surveys.
