# SBCS 1.2 SO Masses

## 1.2 Spherical Overdensity (SO) Masses

This method of mass definition is frequently used in observational analyses of clusters, theoretical analyses of halos in cosmological simulations, and analytic models such as the Halo Occupation Distribution (HOD). Spherical overdensity algorithms measure mass within a sphere encompassing a fixed interior density, and that density value (specified as a contrast, or "overdensity", relative to a reference density) is a key control parameter. However, SO masses are not uniquely defined by choice of overdensity; an algorithm must also define a method for determining the centers of the spheres and for assigning masses to objects in cases where two or more spheres overlap one another. See Section 1.3 for discussions of such methods.

## 1.2.0 Generic SO mass

Notation:
$M_{\Delta}$
(This notation has often been used with an appropriate qualifier specifying which method below; e.g., critical
$M_{200}$
for a density contrast of 200 relative to the critical density. This usage should be demoted in favor of the specific notation given below.)

Introduction:
The spherical overdensity mass of a galaxy cluster is defined as the mass within the radius
$R_\Delta$
enclosing a given density contrast
$\Delta$
with respect to some reference density
$\rho \; ,$
so that the cluster's mass is
$M_{\Delta} = (4 \pi / 3) R^3_{\Delta} \cdot \Delta \cdot \rho \; .$
Different values of the density contrast
$\Delta$
and different choices for the reference density are used in the literature, and the three most common choices are given below.

The classic Press and Schechter (1974) paper employed an SO approach to identify collapsed structures in 1000-particle N-body experiments. Using a threshold of 10 times the mean density, they found a multiplicity function (now called the mass function) in the simulations that modestly agreed with their analytic model expectations. Turner and Gott (1976) applied an SO method to identify galaxy groups as projected density enhancements in the Zwicky catalog, and Turner et al (1979) applied SO methods to 1000-particle N-body simulations in both 3D and on projected 2D sky maps of galaxies' assigned to simulation particles. The latter paper is among the first to employ simulations to assess the quality of an optical cluster finder, particularly the effects of projected blending along the line-of-sight.

Algorithmic Description: See specific cases below.

Links to Software: See specific cases below.

Strengths:
• Simplicity. SO mass can be derived for observed clusters using mass profiles derived from hydrostatic equilibrium analyses of galaxy velocity dispersions or X-ray derived gas and temperature profiles or from modeling of the weak lensing shear. The mass measured in simulations can therefore be directly compared to identically defined mass derived from observations (which is not the case, e.g., for the FoF mass).
• Clear relationship to observables. The SO mass in simulated halos tightly correlates with cluster observables such as gas mass, temperature, and Ysz or Yx (e.g., Bialek et al. 2001; da Silva et al. 2004; Motl et al. 2005; Nagai 2006, Kravtsov et al. 2006; Nagai et al. 2007; Stanek et al. 2010).

Weaknesses:
• Forced geometry. Simulated halos are aspherical and their boundaries can be complex, especially during mergers.
• Non-uniqueness. Multiple choices of density method and overdensity value are employed throughout the literature. This may also be seen as a strength, in that one can adjust the mass measure to accommodate different observables probing different cluster regions.
• Dynamically agnostic. The mass definition does not take into account whether mass that falls within the spherical boundary is gravitationally bound to the object. This is also a weakness of the basic FOF method.

Discussion:
The existence of a particular density contrast delineating a halo boundary is predicted only in the limited context of spherical collapse of a density fluctuation with the top-hat profile (i.e., uniform density, sharp boundary), as originally argued by Gunn and Gott (1972). Collapse in such cases proceeds on the same time scale at all radii and the collapse time and "virial radius" of collapsed object are well defined. However, the peaks in an initial Gaussian density field are not uniform in density, are not spherical, and do not have a sharp boundary. Existence of a radial density profile results in different times of collapse for different radial shells, but power-law profiles can admit self-similar collapse solutions that retain a fixed interior density contrast (e.g., Bertschinger 1985). Triaxiality of the density peak makes tidal effects of the surrounding mass distribution important. The continuous initial density field, along with effects of non-uniform density, triaxiality and nonlinear effects during collapse of smaller scale fluctuations within each peak, produce an outer density structure without a well-defined radial boundary.

While SO mass is sometimes referred to as a virial mass', it is more correct to consider it a `hydrostatic mass', meaning the mass interior to a radius within which the mean radial velocity is very close to zero. Early gas dynamic simulations (Evrard et al 1996, Eke et al. 1998) supported critical overdensity values near 500 as marking the hydrostatic-to-infall transition boundary. Subsequent N-body studies have shown that this boundary depends on mass and epoch, with dynamically older systems possessing hydrostatic regions that extend to lower density contrasts (Cuesta et al. 2008). In the future of a concordance LCDM cosmology, all halos become dynamically isolated and extend to a critical density contrast of ~6, essentially the classical turnaround scale of the spherical collapse model (Busha et al. 2005).

A further possible complication arises if one takes a literal interpretation of the spherical model that postulates a well-defined formation epoch for a halo. The virial density contrast formally applies only at the time of collapse; after a given density peak collapses its internal density will stay constant while reference density will change simply due to cosmological expansion. The actual overdensity of collapsed tophat initial fluctuations will therefore grow larger than initial virial overdensity at times later than the time of collapse.

The weaknesses listed above are perceived weakness of the SO mass definition that are sometimes cited. However, these are not real weakness in practice because 1) mass defined within a spherical boundary correlates with some cluster observables with scatter of <~10%, 2) in observations all mass that falls within
$R_\Delta$
is assigned to
$M_{\Delta}$
; likewise, doing the same in simulations, even though ability to differentiate between bound and unbound material exists, is appropriate if mass is to be compared to observations. Finally, although there is certain arbitrariness in the choice of boundary corresponding to a given density contrast Delta, this is not a problem as long as one compares to the mass measured within the same boundary in observations. In observations, the choice of the boundary is often dictated by data limitations (e.g., extent of measured X-ray emission and spectra) rather by physical considerations. Theoretical mass determinations often have to conform to the observational definitions of mass. Thus, for example, although it is possible to define the entire mass that will ever collapse onto a halo in simulations (e.g., Cuesta et al. 2008, Anderhalden & Diemand 2011), it is impossible to measure such mass in observations, which makes such mass of interest only from the standpoint of theoretical models of halo collapse. (Andrey Kravtsov)

## 1.2.1 SO mass relative to the mean matter density

Notation:
$M_{\Delta m}$
(The numerical value of the chosen density contract factor
$\Delta$
is given in the subscript, e.g.
$M_{180m}$
for a density contrast factor of 180.)

Introduction: One useful way to define a cluster's mass is as the mass within a sphere of radius
$R_{\Delta m}$
chosen so that it encloses a mean density
$\Delta$
times the mean matter density of the universe
$\bar{\rho} \; .$
The cluster's mass is then
$M_{\Delta m} = (4 \pi / 3 ) R^3_{\Delta m} \Delta \bar{\rho} \; ,$
or equivalently,
$M_{\Delta m} = ( \Omega_{m0} H_0^2 \Delta / 2 G ) R^3_{\Delta m} (1+z)^3 \; .$

Algorithmic Description: One must first choose a density contrast factor
$\Delta \; .$
Some common choices for this mass definition are
$\Delta$
$R_{\Delta m}$
is then defined by solving the implicit equation
$\Delta = 3 M(R) / 4 \pi R^3 \bar{\rho} \; .$
The cluster mass profile
$M(R)$
needed for specifying
$M_{\Delta c}$
can be measured in simulations but must be determined in real clusters from modeling of observational data. If
$M(R)$
is binned, the exact value of
$R$
needs to be obtained by interpolation between bins or via modeling of the binned profile by spline.

In practice, an algorithm for determining the centers of the spheres and assigning masses to overlapping spherical halos must be specified in order for the halo masses to be uniquely defined. See Section 1.3 for discussions of such algorithms.

Strengths: (Note generic SO strengths in section 1.2.0)
• Cosmology independence of reference density evolution. The (1+z)^3 evolution of the mean matter density is independent of cosmological parameters, making it easy to apply to observations and simulations.

Weaknesses: (Note generic SO weaknesses in section 1.2.0)
• A fixed threshold does not match well the behavior of the hydrostatic boundary across a wide range of halo masses and epochs (Cuesta et al. 2008).
• A mean matter density threshold breaks down in the far future of a concordance cosmology, when the mean matter density becomes negligible relative to the critical density.

Discussion:
Using the Mean Mass Density as a Reference Density: Please discuss the specific pros and cons of
$M_{\Delta m}$
here.

## 1.2.2. SO mass relative to critical density

Notation:
$M_{\Delta c}$
(The numerical value of the chosen density contract factor
$\Delta$
is given in the subscript, e.g.
$M_{200c}$
for a density contrast factor of 200.)

Introduction. Another useful way to define a cluster's mass is as the mass within a sphere of radius
$R_{\Delta c}$
chosen so that it encloses a mean density
$\Delta$
times the critical density. The cluster's mass is then
$M_{\Delta c} = \frac {4\pi} {3} R^3_{\Delta c} \Delta \rho_{\rm cr} \; .$

Algorithmic Description: One must first choose a density contrast factor
$\Delta \; .$
Some common choices are
$\Delta$
= 180, 200, 500, 2500. The radius
$R_{\Delta c}$
is then defined by solving the implicit equation
$\Delta = 3 M(R) / 4 \pi R^3 \rho_{\rm cr}$
where
$\rho_{\rm cr}(z) \equiv 3 H^2(z) / 8 \pi G$
and
$H(z)$
is the Hubble constant at redshift
$z \; .$

In the LCDM model with negligible contributions from relativistic components, the Hubble constant at a given redshift depends on cosmological parameters as
$H^2(z) = H_0^2 [\Omega_{m0}(1+z)^3 + (1-\Omega_{m0}-\Omega_{\Lambda})(1+z)^2 + \Omega_{\Lambda}] \; ,$
where
$\Omega_{m0}$
is the mean matter density in units of the critical density at the present time and
$\Omega_\Lambda$
is constant.

In more general homogeneous dark energy models with equation of state
$p = w(z) \rho \; ,$
the Hubble constant is given by (e.g., Percival 2005)
$H^2(z) = H_0^2 [\Omega_{m0}(1+z)^3 + (1-\Omega_{m0}-\Omega_{\rm X})(1+z)^2 + \Omega_{\rm X}(1+z)^{f(z)}] \; ,$
where
$f(a) = \frac {3} {\ln(a)} \int^{\ln(a)}_0 [1+w(a^{\prime})] d\ln a^{\prime}$
and
$a(z) \equiv 1/ (1+z) \; .$
If
$w$
is constant, then
$f(z)=3(1+w) \; .$

The cluster mass profile
$M(R)$
needed for specifying
$M_{\Delta c}$
can be measured in simulations but must be determined in real clusters from modeling of observational data. If
$M(R)$
is binned, the exact value of
$R$
needs to be obtained by interpolation between bins or via modeling of the binned profile by spline.

In practice, an algorithm for determining the centers of the spheres and assigning masses to overlapping spherical halos must be specified in order for the halo masses to be uniquely defined. See Section 1.3 for discussions of such algorithms.

Strengths: (Note generic SO strengths in section 1.2.0)
• Physically motivated. The Hubble parameter that defines the critical density sets the drag term in single-particle and linear-theory mode dynamics.
• Broadly applicable. A critical density threshold matches the behavior of halos in the far future of a concordance cosmology (Busha et al. 2005).

Weaknesses: (Note generic SO weaknesses in section 1.2.0)
• Cosmology dependence. The reference density depends on cosmological parameters, which complicates observational analysis.

Discussion:
Using the Critical Density as a Reference Density: Different thresholds, such as
$\Delta$
= 200, 500, or 2500, are useful in different contexts. Please discuss their pros and cons here.

## 1.2.3. SO mass relative to the virial density contrast

Notation:
$M_{\Delta_{\rm v}}$

Introduction. Yet another way to define a cluster's mass is as the mass within a sphere of radius
$R_{\Delta_{\rm v}}$
chosen so that it encloses a mean density
$\Delta_{\rm v}$
times the critical density of the universe at the cluster's redshift. This density contrast factor is not freely chosen but is derived from a spherical-collapse model of cluster formation.

Algorithmic Description: Spherical collapse is an idealized model of cluster formation in which the cluster is represented as a series of concentric spherical shells that expand to a turnaround radius
$R_{\rm ta}$
and then collapse to the origin at a time
$t_{\rm coll} \; .$
The cluster is assumed to virialize at this moment, and its radius after virialization
$R_{\rm v}$
is assumed to be half the turnaround radius, for reasons inspired by the virial theorem. According to this idealized model, the virial density contrast
$\Delta_{\rm v}$
is equal to the cluster's mass divided by
$4 \pi R_{\rm v}^3 / 3 \; .$

For the CDM and LCDM cosmologies the virial density contrast can be approximated with an accuracy of better then 1% (for
$\Omega_m(z)=0.1-1$
) by the following formulae (cf. Bryan & Norman 1998) for the density contrast factor. In a flat LCDM model,
$\Delta_{\rm v}=18\pi^2+82x-39x^2$
with
$x \equiv \Omega_m(z)-1 \; .$
In a CDM model with no cosmological constant or dark energy,
$\Delta_{\rm v} = 18\pi^2+60x-32x^2 \; .$

$R_{\Delta_{\rm v}}$
for a given value of
$\Delta_{\rm v}$
is defined by solving the implicit equation
$\Delta_{\rm v} = 3 M(R) / 4 \pi R^3 \rho_{\rm cr}(z) \; .$
Section 1.2.2 gives expressions for determining
$\rho_{\rm cr}(z) \; .$

The cluster mass profile
$M(R)$
needed for specifying
$M_{\Delta_{\rm v}}$
can be measured in simulations but must be determined in real clusters from modeling of observational data. If
$M(R)$
is binned, the exact value of
$R$
needs to be obtained by interpolation between bins or via modeling of the binned profile by spline.

In practice, an algorithm for determining the centers of the spheres and assigning masses to overlapping spherical halos must be specified in order for the halo masses to be uniquely defined. See Section 1.3 for discussions of such algorithms.