Characterising the Structure of Halo Merger Trees Using a Single Parameter: The Tree Entropy

Linking the properties of galaxies to the assembly history of their dark matter haloes is a central aim of galaxy evolution theory. This paper introduces a dimensionless parameter \(s\in[0,1]\), the "tree entropy", to parametrise the geometry of a halo's entire mass assembly hierarchy, building on a generalisation of Shannon's information entropy. By construction, the minimum entropy (\(s=0\)) corresponds to smoothly assembled haloes without any mergers. In contrast, the highest entropy (\(s=1\)) represents haloes grown purely by equal-mass binary mergers. Using simulated merger trees extracted from the cosmological \(N\)-body simulation SURFS, we compute the natural distribution of \(s\), a skewed bell curve peaking near \(s=0.4\). This distribution exhibits weak dependences on halo mass \(M\) and redshift \(z\), which can be reduced to a single dependence on the relative peak height \(\delta_{\rm c}/\sigma(M,z)\) in the matter perturbation field. By exploring the correlations between \(s\) and global galaxy properties generated by the SHARK semi-analytic model, we find that \(s\) contains a significant amount of information on the morphology of galaxies \(-\) in fact more information than the spin, concentration and assembly time of the halo. Therefore, the tree entropy provides an information-rich link between galaxies and their dark matter haloes.