# Origins of scaling relations of globular cluster systems

## TL;DR

This paper presents a detailed analysis of the physical origins and evolution of the scaling relations of globular cluster systems, using an analytic model that matches observed relations across a wide range of galaxy masses.

## Contribution

It provides new insights into the evolution of the GC-to-halo mass ratio, the role of accreted satellites, and the formation history of metal-poor and metal-rich clusters.

## Key findings

- The GC mass to halo mass ratio is nearly constant over redshift but evolves in normalization.
- The shape of the stellar mass-halo mass relation influences the GC specific frequency.
- Accreted satellites contribute increasingly to GC systems in more massive galaxies.

## Abstract

Globular cluster (GC) systems demonstrate tight scaling relations with the properties of their host galaxies. In previous work, we developed an analytic model for GC formation in a cosmological context and showed that it matches nearly all of the observed scaling relations across 4 orders of magnitude in host galaxy mass. Motivated by the success of this model, we investigate in detail the physical origins and evolution of these scaling relations. The ratio of the combined mass in GCs $M_{\rm GC}$ to the host dark matter halo mass $M_h$ is nearly constant at all redshifts, but its normalization evolves by a factor of $\sim$10 from birth to $z=0$. The relation is steeper than linear at halo masses $M_h \lesssim 10^{11.5} M_{\odot}$, primarily due to non-linearity in the stellar mass-halo mass relation. The near constancy of the ratio $M_{\rm GC}/M_h$, combined with the shape of the stellar mass-halo mass relation, sets the characteristic $U-$shape of the GC specific frequency as a function of host galaxy mass. The contribution of accreted satellite galaxies to the buildup of GC systems is a strong function of the host galaxy mass, ranging from $\approx$0% at $M_h \approx 10^{11} M_{\odot}$ to 80% at $M_h \approx 10^{15} M_{\odot}$. The metal-poor clusters are significantly more likely to form ex-situ relative to the metal-rich clusters, but a substantial fraction of metal-poor clusters still form in-situ in lower mass galaxies. Similarly, the fraction of red clusters increases from $\approx 10$% at $M_h = 10^{11} M_{\odot}$ to $\approx 60$% at $M_h \approx 10^{13} M_{\odot}$, and flattens at higher $M_h$. Clusters formation occurs essentially continuously at high redshift, while at low redshift galactic mergers become increasingly important for cluster formation.