Scaling relations 

Despite the complexity of galaxy formation and the number of processes that are thought to play important roles, galaxies obey remarkably tight scaling relations. These are correlations between different physical properties that can span, in some cases, several orders of magnitude. The study of these relations and, in particular, their evolution with cosmic time is of great interest and subject of frenetic current investigations. Scaling relations constitute benchmarks for any theoretical model of galaxy formation and evolution.

One of my favoured scaling relation is the mass versus (specific) angular momentum relation, also called the "Fall relation" (Fall 1983). This relation links two fundamental quantities in galaxy formation and stems directly from one of the key theories of galaxy formation that explains the acquisition of spin by dark matter halo and eventually galaxies: the tidal torque theory. In observations, the Fall relation can be investigated for star and baryonic matter and for galaxies of different types. Here below a version of the Fall relation for disc galaxies and dwarf irregulars in the local Universe (from Mancera Piña, Fraternali et al. 2020) in the stellar component and in the baryonic (stars + gas) component.

Left: stellar j-M (specific angular momentum vs mass) relation for a sample of nearby galaxies. The circles represent the observed galaxies, while the black dashed line and grey region show, respectively, the best-fitting relations and their perpendicular intrinsic scatter. Right: same as the left panel but for the baryonic (stars+gas j and M) relation.

Sometime considered a scaling relation but also a fundamental law of galaxies and their interstellar medium, the Schmidt-Kennicutt law of star formation links the density of gas and the density of star formation on galactic scales. 

The density involved are typically column (or surface) density (integrated along the line of sight) but more fundamentally one should find a relation between volume  densities, which likely have more physical meaning. Recently, we have done just that and determined the volumetric star formation law. Here below a comparison plot between the surface-based star formation law (left) and the volumetric star formation law (from Bacchini, Fraternali et al. 2020).

Star formation laws based on the surface densities (left) and the volume densities (right) of the total gas and the SFR for a sample of dwarf and disc galaxies. Each color indicates the azimuthally averaged radial profile of a dwarf galaxy according to the color-bar. The light blue points and the green crosses respectively indicate the star-forming regions within and beyond the optical radius for a sample of 12 disc galaxies, while the yellow stars are for the Milky Way. In the left panel, the grey dashed line is the Schmidt-Kennicutt law. In the right panel, the red solid line is the best-fit of the volumetric star formation law.