The cosmological analysis of X-ray cluster surveys V. The potential of cluster counts in the $1<z<2$ range

Abstract: Cosmological studies have now entered Stage IV according to the Dark Energy Task Force prescription, thanks to new missions (Euclid, Rubin Observatory, SRG/eROSITA) that are expected to provide the required ultimate accuracy in the dark energy (DE) equation of state (EoS). However, none of these projects have the power to systematically unveil the galaxy cluster population at $z>1$. There therefore remains the need for an ATHENA-like mission to run independent cosmological investigations and scrutinise the consistency between the results from the $0<z\<1$ and $1<z\<2$ epochs. We study the constraints on the DE EoS and on primordial non-Gaussanities for typical X-ray cluster surveys executed by ATHENA. We consider two survey designs: 50 deg$^2$ at 80ks (survey A) and 200 deg$^2$ at 20ks (survey B). We analytically derive cluster counts in a space of observable properties, and predict the cosmological potential of the corresponding samples with a Fisher analysis. The achieved depth allows us to unveil the halo mass function down to the group scale out to $z=2$. We predict the detection of thousands of clusters down to a few 10$^{13} h^{-1} M_{\odot}$, in particular 940 and 1400 clusters for surveys A and B, respectively, at $z\>1$. Such samples will allow a detailed modelling of the evolution of cluster physics along with a standalone cosmological analysis. Our results suggest that survey B has the optimal design as it provides greater statistics. Remarkably, high-$z$ clusters, despite representing 15% or less of the full samples, allow a significant reduction of the uncertainty on the cosmological parameters: $\Delta w_a$ is reduced by a factor of 2.3 and $\Delta f_{NL}^{loc}$ by a factor of 3. Inventorying the high-$z$ X-ray cluster population can play a crucial role in ensuring overall cosmological consistency. This will be the major aim of future new-generation ATHENA-like missions.