The XXL survey. XLVI. Forward cosmological analysis of the C1 cluster sample

Context. We present the forward cosmological analysis of an XMM-selected sample of galaxy clusters out to a redshift of unity. We derive mass-observable relations in a self-consistent manner using the sample alone. Special care is given to the modelling of selection effects.Aims. Following our previous 2018 study based on the dn/dz quantity alone, we perform an upgraded cosmological analysis of the same XXL Cl cluster catalogue (178 objects), with a detailed account of the systematic errors. The results are combined with external constraints from baryon acoustic oscillations (BAO) and the cosmic microwave background (CMB).Methods. This study follows the ASpiX methodology: we analysed the distribution of the observed X-ray properties of the cluster population in a 3D observable space (count rate, hardness ratio, redshift) and modelled as a function of cosmology along with the scaling relations and the selection function. Compared to more traditional methods, ASpiX allows the inclusion of clusters down to a few tens of photons and is much simpler to use. Two M - T relations are considered: that from the Canada-France-Hawaii Telescope (hereafter CFHT) and another from the more recent Subaru lensing analyses.Results. We obtain an improvement by a factor of two compared to the previous analysis, which dealt with the cluster redshift distribution for the XXL sample alone, letting the normalisation of the M - T relation and the evolution of the L-T relation free. Adding constraints from the XXL cluster two-point correlation function and the BAO from various surveys decreases the uncertainties by 23% and 53%, respectively, and 62% when adding both. The central value is in excellent agreement with the Planck CMB constraints. Switching to the scaling relations from the Subaru analysis and leaving more parameters free to vary provides less stringent constraints, but those obtained are still consistent with the Planck CMB at the 1-sigma level. Our final constraints are sigma(8) = 0.99(-0.23)(+0.14), Omega(m) = 0.296 +/- 0.034 (S-8 = 0.98(-0.)(2)(1)(+0.11)) for the XXL sample alone. Combining XXL ASpiX, the XXL cluster two-point correlation function, and the BAO, leaving 11 parameters free to vary, and allowing for the cosmological dependence of the scaling relations in the fit induces a shift of the central values, which is reminiscent of that observed for the Planck S-Z cluster sample. We find sigma(8) = 0.793(-0.12)(+0.063) and Omega(m) = 0.364 +/- 0.015 (S-8 = 0.872(-0.12)(+0.068)), which are still compatible with Planck CMB at 2.2 sigma.Conclusions. The results obtained by the ASpiX method are promising; further improvement is expected from the final XXL cosmological analysis involving a cluster sample that is twice as large. Such a study paves the way for the analysis of the eROSITA and future Athena surveys.