Brightest group and cluster galaxies as indicators of relaxation
Context. Galaxy groups and clusters are widely used to probe the evolution of the cosmic web and cosmology, while assuming that they are relaxed. Aims. We identify the properties of the brightest halo galaxies (BHGs) that can be used to predict the most likely sample of dynamically relaxed host halos. Our work combines thoroughly studied galaxy clusters with less frequently analysed groups. Methods. Our analysis was based on data from the IllustrisTNG simulations. We considered several observationally motivated parameters, including the offset of the BHG from the potential well of the host system ($d_\text{off}$) and from the r-band luminosity centre ($d_\text{lum}$), the distance between the brightest and second-brightest galaxies ($d_{12}$), and the r-band magnitude gap between them ($\Delta m_{12}$). The primary analysis was performed at redshift $z=0$, with an additional investigation of the redshift evolution of halo relaxation up to $z=1$. The observable proxies were applied to construct a halo mass function (HMF), which was then compared to the HMF of the relaxed sample defined from 3D information commonly used in theoretical approaches. Results. We find that $d_\text{off}$ and $\Delta m_{12}$ are effective indicators of group and cluster relaxation, particularly when used in combination. The selection criteria of $d_\text{off}<0.05~R_{200}$ and $\Delta m_{12}>1.6$ mag allowed us to reproduce an HMF that closely matches that of the relaxed halo population. These criteria can be applied to observations up to $z\sim0.2$ within a mass range $\text M_{200}\geq10^{12.5}\text M_\odot$ ($\text M_\text{*, BHG}\gtrsim10^{10.9}\text M_\odot$), including groups and clusters in the selection. In this mass range, $15-23\%$ of the systems are considered fully relaxed at $z=0$. The fraction of relaxed haloes decreases with redshift up to $z\sim0.4$, after which the decrease is far slower.