1.3 Algorithms for Centering and Mass Assignment of Spherical Overdensity Halos

1.3.1. The Tinker et al. 2008 implementation of the SO halo finder (contributed by Andrey Kravtsov and Jeremy Tinker)

Introduction: The standard spherical overdensity algorithm is described in detail in Lacey & Cole (1994). However, in the approach of Tinker et al. several significant modifications have been made. First, in Lacey & Cole (1994) the center of a halo is located on the center of mass of the particles within the sphere. The center of mass generally cannot be unambiguously identified in observations, which typically associate cluster center with the peak of X-ray emission, position of the brightest galaxy, centroid of weak lensing shear, etc. Due to substructure, the center of mass may be displaced significantly from the main peak in the density field. The Tinker et al. implementation of the SO algorithm therefore locates the centers of halos at their density peaks. The algorithm also allows mass in regions of overlap between the spherical volumes of neighboring halos to be assigned to both halos - i.e., a certain amount of mass double-counting is allowed. As long as the halo center does not reside within the radius of another neighboring halo, the algorithm identifies these halos as distinct objects. Rather than attempt to determine which halo each particle belongs to, or to divide each particle between the halos, the mass is double counted. No solution to this problem is ideal (see discussion section below), but experiments show that at low redshifts the total amount of double-counted mass is only ~0.75% of all the mass located within halos, with no dependence on halo mass.

Algorithmic Description: The Tinker et al. implementation of the SO halo finder starts by estimating the local density around each particle within a fixed top-hat aperture with radius approximately 3 times the force softening of the simulation (the radius is one of the input parameters of the algorithm). Beginning with the highest density particle, a sphere is grown around the particle until the mean interior density is equal to the input value of overdensity

where

is the density contrast within a sphere of radius

with respect to the mean density of the universe at the epoch of analysis

Since local densities smoothed with a top-hat kernel are somewhat noisy, the location of the peak of the halo density is refined with an iterative procedure. Starting with a radius of

the center of mass of the halo is calculated iteratively until convergence. The value of r is reduced iteratively by 1%, and the new center of mass found until a final smoothing radius of

or until only 20 particles are found within the smoothing radius. At this small aperture, the center of mass corresponds well to the highest density peak of the halo. This process is computationally efficient and both eliminates noise and accounts for the possibility that the chosen initial halo location resides at the center of a large substructure; in the latter case, the halo center will wander toward the larger mass and eventually settle on its center. Once the new halo center is determined, the sphere is regrown and the mass is determined. All particles within

are marked as members of a halo and skipped when encountered in the loop over all particle densities. Particles located just outside of a halo can be chosen as candidate centers for other halos, but the iterative halo-centering procedure will wander into the parent halo. Whenever two halos have centers that are within the larger halo's

the halo with the largest maximum circular velocity, defined as the maximum of the circular velocity profile

, is taken to be the parent halo and the other halo is discarded.

Links to Software: tar file with the Tinker et al. SO implementation

Forces a spherical boundary to a generally non-spherical distribution of matter

No accounting for whether particles assigned to halo are bound to it or not

A small fraction of mass can be assigned to more than a single halo (mass double counting)

Discussion: The two ambiguous and arguable features of the algorithm are the choice in determining location of the halo center and mass assignment (in particular, the double counting of mass). Different choices for centers are possible. However, if the ultimate goal is to compare properties of halos identified by the algorithm to observed objects, the main guiding principle should be to identify centers as closely matching center identification in observations as possible. In this sense, identification of centers with the global density maximum is preferrable to the center of mass, as the former should be closer to observational assignment of center as an X-ray peak or location of the brightest cluster galaxy. The global minimum of the potential may be the center choice most closely related to observations. However, such center definition requires values of potential at particle locations that are not always available in simulation outputs.

Identification of distinct halos by the Tinker et al. algorithm aims to mimic an observational procedure that would identify each peak of X-ray emission separated by more than a virial radius as a distinct object. Thus two neighboring halos separated by a distance larger than the largest of the virial radii of the pair are counted as distinct objects. Their volumes thus can overlap significantly. The overlap volume may contain particles. Rather than attempt to determine which halo each particle belongs to, or to divide each particle between the halos, the mass is double counted. The fraction of mass in such overlapping regions is small, however (~1%) and thus in practice would not change global properties, such as the halo mass function or observable-mass correlations substantially. {Andrey Kravtsov)

1.3.2. Name of Algorithm

Introduction: A brief explanation of the rationale for the algorithm.

Algorithmic Description: An unambiguous description of the algorithm to identify halos. Make sure the description is so clear that anyone following it will get the same result.

Links to Software: Add links to any publicly available software. This will be a great way to get others to use it.

Strengths:

Make these brief bullet-like points.

Weaknesses:

Make these brief bullet-like points.

Discussion: Expand upon the strengths and weaknesses here, and sign your comments in parentheses. (Your Name Here)

## 1.3 Algorithms for Centering and Mass Assignment of Spherical Overdensity Halos

## 1.3.1.

The Tinker et al. 2008 implementation of the SO halo finder(contributed by Andrey Kravtsov and Jeremy Tinker)The standard spherical overdensity algorithm is described in detail in Lacey & Cole (1994). However, in the approach of Tinker et al. several significant modifications have been made. First, in Lacey & Cole (1994) the center of a halo is located on the center of mass of the particles within the sphere. The center of mass generally cannot be unambiguously identified in observations, which typically associate cluster center with the peak of X-ray emission, position of the brightest galaxy, centroid of weak lensing shear, etc. Due to substructure, the center of mass may be displaced significantly from the main peak in the density field. The Tinker et al. implementation of the SO algorithm therefore locates the centers of halos at their density peaks. The algorithm also allows mass in regions of overlap between the spherical volumes of neighboring halos to be assigned to both halos - i.e., a certain amount of mass double-counting is allowed. As long as the halo center does not reside within the radius of another neighboring halo, the algorithm identifies these halos as distinct objects. Rather than attempt to determine which halo each particle belongs to, or to divide each particle between the halos, the mass is double counted. No solution to this problem is ideal (see discussion section below), but experiments show that at low redshifts the total amount of double-counted mass is only ~0.75% of all the mass located within halos, with no dependence on halo mass.Introduction:The Tinker et al. implementation of the SO halo finder starts by estimating the local density around each particle within a fixed top-hat aperture with radius approximately 3 times the force softening of the simulation (the radius is one of the input parameters of the algorithm). Beginning with the highest density particle, a sphere is grown around the particle until the mean interior density is equal to the input value of overdensityAlgorithmic Description:where

is the density contrast within a sphere of radius

with respect to the mean density of the universe at the epoch of analysis

Since local densities smoothed with a top-hat kernel are somewhat noisy, the location of the peak of the halo density is refined with an iterative procedure. Starting with a radius of

the center of mass of the halo is calculated iteratively until convergence. The value of

ris reduced iteratively by 1%, and the new center of mass found until a final smoothing radius ofor until only 20 particles are found within the smoothing radius. At this small aperture, the center of mass corresponds well to the highest density peak of the halo. This process is computationally efficient and both eliminates noise and accounts for the possibility that the chosen initial halo location resides at the center of a large substructure; in the latter case, the halo center will wander toward the larger mass and eventually settle on its center. Once the new halo center is determined, the sphere is regrown and the mass is determined. All particles within

are marked as members of a halo and skipped when encountered in the loop over all particle densities. Particles located just outside of a halo can be chosen as candidate centers for other halos, but the iterative halo-centering procedure will wander into the parent halo. Whenever two halos have centers that are within the larger halo's

the halo with the largest maximum circular velocity, defined as the maximum of the circular velocity profile

, is taken to be the parent halo and the other halo is discarded.

tar file with the Tinker et al. SO implementatioLinks to Software:nStrengths:Weaknesses:The two ambiguous and arguable features of the algorithm are the choice in determining location of the halo center and mass assignment (in particular, the double counting of mass). Different choices for centers are possible. However, if the ultimate goal is to compare properties of halos identified by the algorithm to observed objects, the main guiding principle should be to identify centers as closely matching center identification in observations as possible. In this sense, identification of centers with the global density maximum is preferrable to the center of mass, as the former should be closer to observational assignment of center as an X-ray peak or location of the brightest cluster galaxy. The global minimum of the potential may be the center choice most closely related to observations. However, such center definition requires values of potential at particle locations that are not always available in simulation outputs.Discussion:Identification of distinct halos by the Tinker et al. algorithm aims to mimic an observational procedure that would identify each peak of X-ray emission separated by more than a virial radius as a distinct object. Thus two neighboring halos separated by a distance larger than the largest of the virial radii of the pair are counted as distinct objects. Their volumes thus can overlap significantly. The overlap volume may contain particles. Rather than attempt to determine which halo each particle belongs to, or to divide each particle between the halos, the mass is double counted. The fraction of mass in such overlapping regions is small, however (~1%) and thus in practice would not change global properties, such as the halo mass function or observable-mass correlations substantially. {Andrey Kravtsov)

## 1.3.2. Name of Algorithm

A brief explanation of the rationale for the algorithm.Introduction:An unambiguous description of the algorithm to identify halos. Make sure the description is so clear that anyone following it will get the same result.Algorithmic Description:Add links to any publicly available software. This will be a great way to get others to use it.Links to Software:Strengths:Weaknesses:Expand upon the strengths and weaknesses here, and sign your comments in parentheses. (Your Name Here)Discussion: