Euclid preparation: LXVI. Impact of line-of-sight projections on the covariance between galaxy cluster multi-wavelength observable properties: Insights from hydrodynamic simulations

Context. Cluster cosmology can benefit from combining multi-wavelength studies. In turn, these studies benefit from a characterisation of the correlation coefficients among different mass-observable relations. Aims. In this work, we aim to provide information on the scatter, skewness, and covariance of various mass-observable relations in galaxy clusters in cosmological hydrodynamic simulations. This information will help future analyses improve the general approach to accretion histories and projection effects, as well as to model mass-observable relations for cosmology studies. Methods. We identified galaxy clusters in Magneticum Box2b simulations with masses of M_200c > 10¹⁴ M_⊙ at redshifts of z = 0.24 and z = 0.90. Our analysis included Euclid-derived properties such as richness, stellar mass, lensing mass, and concentration. Additionally, we investigated complementary multi-wavelength data, including X-ray luminosity, integrated Compton-y parameter, gas mass, and temperature. We then examined the impact of projection effects on mass-observable residuals and correlations. Results. We find that at intermediate redshift (z = 0.24), projection effects have the greatest impact of lensing concentration, richness, and gas mass in terms of the scatter and skewness of the log-residuals of scaling relations. The contribution of projection effects can be significant enough to boost a spurious hot-versus cold-baryon correlations and consequently hide underlying correlations due to halo accretion histories. At high redshift (z = 0.9), the richness has a much lower scatter (of log-residuals), while the quantity that is most impacted by projection effects is the lensing mass. The lensing concentration reconstruction, in particular, is affected by deviations of the reduced-shear profile shape from that derived using a Navarro-Frenk-White (NFW) profile; the amount of interlopers in the line of sight, on the other hand, is not as important.