  Atomically thin heterostructures formed by twisted transition metal
dichalcogenides can be used to create periodic moir\'e patterns. The emerging
moir\'e potential can trap interlayer excitons into arrays of strongly
interacting bosons, which form a unique platform to study strongly correlated
many-body states. In order to create and manipulate these exotic phases of
matter, a microscopic understanding of exciton-exciton interactions and their
manifestation in these systems becomes indispensable. Recent density-dependent
photoluminescence (PL) measurements have revealed novel spectral features
indicating the formation of trapped multi-exciton states providing important
information about the interaction strength. In this work, we develop a
microscopic theory to model the PL spectrum of trapped multi-exciton complexes
focusing on the emission from moir\'e trapped single- and biexcitons. Based on
an excitonic Hamiltonian we determine the properties of trapped biexcitons as
function of twist angle and use these insights to predict the luminescence
spectrum of moir\'e excitons for different densities. We demonstrate how side
peaks resulting from transitions to excited states and a life time analysis can
be utilized as indicators for moir\'e trapped biexcitons and provide crucial
information about the excitonic interaction strength.
