Characterising the Structure of Halo Merger Trees Using a Single Parameter: The Tree Entropy.- [PDF] - [Article]

Danail Obreschkow, Pascal J. Elahi, Claudia del P. Lagos, Rhys J. J. Poulton, Aaron D. Ludlow

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 $s=0.3$ for first generation haloes. This distribution exhibits weak dependences on halo mass $M$ and redshift $z$, which can be reduced to a single dependence of $\langle s\rangle$ 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.