A WISE view of the composite luminosity function of cluster members.

Galaxy clusters play a crucial role in constraining cosmological parameters. Cluster detection and characterization have therefore become an important field of modern cosmology. In this work, we aim at constructing reasonably representative composite luminosity function of cluster members in the WISE survey that can be used in the future as an input in matching algorithms for cluster detection. We use a sample of massive clusters from the redMaPPer catalog to match the positions of the WISE sources. We build the composite luminosity function of the WISE members in different redshift bins. We find that galaxy cluster members have a characteristic composite luminosity function, with a clear change in the slope at a given magnitude apparent $M^{*}$, which becomes fainter with increasing redshift. The best-fit bright-end slope $\beta$ is compatible with a constant value of $3.54$ with no clear trend with redshift, while the faint-end slope $\alpha$ is remarkably different, ranging from $2$ to $3$. We present the first characterization of the composite luminosity function of the WISE counterparts of redMaPPer cluster members and in this way we provide an element for building cluster detection techniques based on matching algorithms.