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:

(This notation has often been used with an appropriate qualifier specifying which method below; e.g., critical

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

enclosing a given density contrast

with respect to some reference density

so that the cluster's mass is

Different values of the density contrast

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).

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

is assigned to

; 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:

(The numerical value of the chosen density contract factor

is given in the subscript, e.g.

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

chosen so that it encloses a mean density

times the mean matter density of the universe

The cluster's mass is then

or equivalently,

Algorithmic Description: One must first choose a density contrast factor

Some common choices for this mass definition are

= 180, 200. The radius

is then defined by solving the implicit equation

The cluster mass profile

needed for specifying

can be measured in simulations but must be determined in real clusters from modeling of observational data. If

is binned, the exact value of

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.

Links to Software:

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

here.

1.2.2. SO mass relative to critical density

Notation:

(The numerical value of the chosen density contract factor

is given in the subscript, e.g.

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

chosen so that it encloses a mean density

times the critical density. The cluster's mass is then

Algorithmic Description: One must first choose a density contrast factor

Some common choices are

= 180, 200, 500, 2500. The radius

is then defined by solving the implicit equation

where

and

is the Hubble constant at redshift

In the LCDM model with negligible contributions from relativistic components, the Hubble constant at a given redshift depends on cosmological parameters as

where

is the mean matter density in units of the critical density at the present time and

is constant.

In more general homogeneous dark energy models with equation of state

the Hubble constant is given by (e.g., Percival 2005)

where

and

If

is constant, then

The cluster mass profile

needed for specifying

can be measured in simulations but must be determined in real clusters from modeling of observational data. If

is binned, the exact value of

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.

Links to Software:

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

= 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:

Introduction. Yet another way to define a cluster's mass is as the mass within a sphere of radius

chosen so that it encloses a mean density

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

and then collapse to the origin at a time

The cluster is assumed to virialize at this moment, and its radius after virialization

is assumed to be half the turnaround radius, for reasons inspired by the virial theorem. According to this idealized model, the virial density contrast

is equal to the cluster's mass divided by

For the CDM and LCDM cosmologies the virial density contrast can be approximated with an accuracy of better then 1% (for

) by the following formulae (cf. Bryan & Norman 1998) for the density contrast factor. In a flat LCDM model,

with

In a CDM model with no cosmological constant or dark energy,

The radius

for a given value of

is defined by solving the implicit equation

Section 1.2.2 gives expressions for determining

The cluster mass profile

needed for specifying

can be measured in simulations but must be determined in real clusters from modeling of observational data. If

is binned, the exact value of

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.

Links to Software:

Strengths: (Note generic SO strengths in section 1.2.0)

Weaknesses: (Note generic SO weaknesses in section 1.2.0)

Relative complexity. Applying a variable threshold instead of a fixed value requires extra effort.

Discussion: For a general discussion of spherical overdensity mass definitions, see section 1.2.1.

Using the Virial Density Contrast as a Reference Density: Please discuss here the specific pros and cons of mass defined with respect to the virial density contrast.

## 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:(This notation has often been used with an appropriate qualifier specifying which method below; e.g., critical

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

enclosing a given density contrast

with respect to some reference density

so that the cluster's mass is

Different values of the density contrast

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.See specific cases below.Algorithmic Description:See specific cases below.Links to Software:Strengths:Weaknesses: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

is assigned to

; 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 densityNotation:(The numerical value of the chosen density contract factor

is given in the subscript, e.g.

for a density contrast factor of 180.)

One useful way to define a cluster's mass is as the mass within a sphere of radiusIntroduction:chosen so that it encloses a mean density

times the mean matter density of the universe

The cluster's mass is then

or equivalently,

One must first choose a density contrast factorAlgorithmic Description:Some common choices for this mass definition are

= 180, 200. The radius

is then defined by solving the implicit equation

The cluster mass profile

needed for specifying

can be measured in simulations but must be determined in real clusters from modeling of observational data. If

is binned, the exact value of

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.

Links to Software:(Note generic SO strengths in section 1.2.0)Strengths:(Note generic SO weaknesses in section 1.2.0)Weaknesses:Discussion:Using the Mean Mass Density as a Reference Density:Please discuss the specific pros and cons ofhere.

1.2.2. SO mass relative to critical densityNotation:(The numerical value of the chosen density contract factor

is given in the subscript, e.g.

for a density contrast factor of 200.)

Another useful way to define a cluster's mass is as the mass within a sphere of radiusIntroduction.chosen so that it encloses a mean density

times the critical density. The cluster's mass is then

One must first choose a density contrast factorAlgorithmic Description:Some common choices are

= 180, 200, 500, 2500. The radius

is then defined by solving the implicit equation

where

and

is the Hubble constant at redshift

In the LCDM model with negligible contributions from relativistic components, the Hubble constant at a given redshift depends on cosmological parameters as

where

is the mean matter density in units of the critical density at the present time and

is constant.

In more general homogeneous dark energy models with equation of state

the Hubble constant is given by (e.g., Percival 2005)

where

and

If

is constant, then

The cluster mass profile

needed for specifying

can be measured in simulations but must be determined in real clusters from modeling of observational data. If

is binned, the exact value of

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.

Links to Software:(Note generic SO strengths in section 1.2.0)Strengths:(Note generic SO weaknesses in section 1.2.0)Weaknesses:Discussion:Using the Critical Density as a Reference Density:Different thresholds, such as= 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:Yet another way to define a cluster's mass is as the mass within a sphere of radiusIntroduction.chosen so that it encloses a mean density

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.

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 radiusAlgorithmic Description:and then collapse to the origin at a time

The cluster is assumed to virialize at this moment, and its radius after virialization

is assumed to be half the turnaround radius, for reasons inspired by the virial theorem. According to this idealized model, the virial density contrast

is equal to the cluster's mass divided by

For the CDM and LCDM cosmologies the virial density contrast can be approximated with an accuracy of better then 1% (for

) by the following formulae (cf. Bryan & Norman 1998) for the density contrast factor. In a flat LCDM model,

with

In a CDM model with no cosmological constant or dark energy,

The radius

for a given value of

is defined by solving the implicit equation

Section 1.2.2 gives expressions for determining

The cluster mass profile

needed for specifying

can be measured in simulations but must be determined in real clusters from modeling of observational data. If

is binned, the exact value of

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.

Links to Software:(Note generic SO strengths in section 1.2.0)Strengths:(Note generic SO weaknesses in section 1.2.0)Weaknesses:For a general discussion of spherical overdensity mass definitions, see section 1.2.1.Discussion:Using the Virial Density Contrast as a Reference Density:Please discuss here the specific pros and cons of mass defined with respect to the virial density contrast.