# Halo Mass Definitions¶

This document describes the Colossus mechanisms for dealing with halo mass and radius definitions.

## Spherical overdensity basics¶

Halo masses and radii are most commonly defined using spherical overdensity mass definitions. A spherical overdensity radius is the radius within which the halo has an overdensity $$\Delta$$ with respect to some reference density, usually the mean or critical density of the universe:

$M_{\rm \Delta} \equiv \frac{4 \pi}{3} \rho_{\rm mdef} R_{\rm \Delta}^3$

where the mdef parameter determines the density threshold $$\rho_{\rm mdef}$$. Another common mass definition is via the friends-of-friends algorithm (Davis et al. 1985). The mdef parameter can take on the following values in Colossus:

Type

mdef

Examples

Explanation

Matter

<int>m

178m, 200m

Integer times the mean matter density of the universe

Critical

<int>c

200c, 500c, 2500c

Integer times the critical density of the universe

Virial

vir

vir

An overdensity that varies with redshift (Bryan & Norman 1998)

Any SO

*

*

Any spherical overdensity mass definition

Friends-of-friends

fof

fof

Some functions furthermore use an rmdef parameter which denotes spherical overdensity radii by R<mdef> (e.g., R200m) and masses by M<mdef>, e.g. M500c. Not all mass definitions are allowed in all functions. For example, functions dealing with SO masses will typically not accept * or fof.

## Modules¶

The functions in the halo.mass module are split into various sub-modules:

• The mass_so module contains functions that are purely related to spherical overdensity, i.e., that do not depend on the particular form of the density profile. This includes density thresholds, converting mass to radius and vice versa, and the dynamical time.

• The mass_defs module contains functions related to converting SO mass definitions into one another, as well as computing pseudo-evolution.

• The mass_adv module contains advanced functions that rely on other modules such as concentration, as well as some non-SO mass definitions.

• The splashback module contains functions related to the splashback radius and mass.