Forecast on the generalised dark matter properties from a Euclid-like survey

The Stage~IV \textit{Euclid} mission will deliver spectroscopic galaxy redshifts together with photometric positions and shapes, enabling cosmological analyses through spectroscopic galaxy clustering (GCsp), photometric galaxy clustering (GCph), weak-lensing cosmic shear (WL), and their cross-correlation (XC). In this work we forecast the constraining power of a Euclid-like survey on the Generalised Dark Matter (GDM) parameters $w_{\rm gdm}$ and $c^{2}_{s,{\rm gdm}}$. Our analysis extends previous forecasting pipeline used for standard cold dark matter. For GCsp, we adopt a semi-analytic nonlinear RSD model, with free terms for each bin. For the photometric probes, we compute the nonlinear GDM matter power spectrum using dedicated simulations, and we modify the lensing and clustering window and the intrinsic-alignment prescription. We consider several survey configurations and explore three fiducial values of $\sigma_8$ motivated by current CMB and low-redshift measurements. In an optimistic setting, for fiducial values $\sigma_8 \simeq 0.81$ and $\sigma_8 \simeq 0.77$, we find relative errors of $4.01\%$ (GCsp), $5.01\%$ (GCph+WL+XC), and $1.96\%$ (all probes) on $c^{2}_{s,{\rm gdm}}$, and $3.26\%$ (GCph+WL+XC) and $1.85\%$ (all probes) on $w_{\rm gdm}$. For a lower fiducial value $\sigma_8 \simeq 0.67$, that could strongly disfavor $\Lambda$GDM, we find constraints of $5\%$ (GCsp), $5\%$ (GCph+WL+XC), and $2.45\%$ (all probes) on $c^{2}_{s,{\rm gdm}}$, and $3.43\%$ (GCph+WL+XC) and $2.04\%$ (all probes) on $w_{\rm gdm}$. We also found that, combining all probes, whether in the pessimistic or optimistic settings, a Euclid-like survey will be able to disentangle between the three scenarios. These results show that the survey will be able to constrain the GDM parameters and distinguish between normalisations of the matter fluctuations.(Abridged)