Detecting galaxy groups populating the local Universe in the eROSITA era

eROSITA will deliver an unprecedented volume of X-ray survey observations, $20-30$ times more sensitive than ROSAT in the soft band ($0.5-2.0$ keV) and for the first time imaging in the hard band ($2-10$ keV). The final observed catalogue of sources will include galaxy clusters and groups along with obscured and unobscured ( active galactic nuclei ) AGNs. This calls for a powerful theoretical effort to mitigate potential systematics and biases that may influence the data analysis. We investigate the detection technique and selection biases in the galaxy group and AGN populations within a simulated X-ray observation conducted at the depth equivalent to a four-year eROSITA survey (eRASS:4). We generate a mock observation spanning $30 30$ deg2 based on the cosmological hydrodynamical simulation Magneticum Pathfinder from $z=0$ up to redshift $z=0.2$, mirroring the depth of eRASS:4 (with an average exposure of$ 600$ s). We combined a physical background from the real eFEDS background analysis with realistic simulations of X-ray emission for the hot gas, AGNs, and XRB. Using a detection method similar to that utilised for eRASS data, we assessed completeness and contamination levels to reconstruct the luminosity functions for both extended and point sources within the catalogue. We define the completeness of extended detections as a function of the input X-ray flux $S_ $ and halo mass $M_ $ at the depth of eRASS:4. Notably, we fully recovered the brightest (most massive) galaxy clusters and AGNs. However, a significant fraction of galaxy groups ($M_ M_ odot $) remain undetected. Examining gas properties between the detected and undetected galaxy groups at a fixed halo mass, we observe that the detected population typically displays higher X-ray brightness compared to the undetected counterpart. Furthermore, we establish that X-ray luminosity primarily correlates with the hot gas fraction, rather than temperature or metallicity. Our simulation suggests a systematic selection bias in current surveys, leading to X-ray catalogues predominantly composed of the lowest-entropy, gas-richest, and highest surface brightness halos on galaxy group scales.