VORONOI TESSELLATION AND NON-PARAMETRIC HALO CONCENTRATION

The concentration parameter is traditionally defined in an NFW halo as

$$\begin{eqnarray}&&{c}_{{\rm{nfw}}}=\displaystyle \frac{{R}_{200}}{{R}_{{\rm{s}}}}\end{eqnarray} \tag{ 1 }$$

where R₂₀₀ is the radius enclosing a mean density that is 200 times the critical density of the universe and R_s is the scale radius. Since R₂₀₀ can be easily found from the particle data, concentration is typically obtained by fitting Equation (2) to the radially enclosed mass profile to find R_s.

$$\begin{eqnarray}&&{M}_{{\rm{enc}}}(r)=4\pi {\rho }_{0}{R}_{{\rm{s}}}^{3}\left[\mathrm{ln}\left(\displaystyle \frac{{R}_{{\rm{s}}}+r}{{R}_{{\rm{s}}}}\right)-\displaystyle \frac{r}{{R}_{{\rm{s}}}+r}\right].\end{eqnarray} \tag{ 2 }$$

Although this is simple in theory, fitting even spherical halos without substructure can be difficult. Fits can be highly sensitive to choice of center, resolution at the center, deviation from the expected NFW power laws, and choice of binning (see Prada et al. 2012). These effects can be alleviated in practice by avoiding fitting altogether. Instead, the unknown profile parameters are related to other halo properties that can be robustly measured. For example, if R_half (the radius enclosing half the mass) and R₂₀₀ of a halo is known, it is possible to numerically solve

$$\begin{eqnarray}&&\displaystyle \frac{1}{2}=\displaystyle \frac{\mathrm{ln}\left[({R}_{{\rm{s}}}+{R}_{{\rm{half}}})/{R}_{{\rm{s}}}\right]-{R}_{{\rm{half}}}/({R}_{{\rm{s}}}+{R}_{{\rm{half}}})}{\mathrm{ln}\left[({R}_{{\rm{s}}}+{R}_{{\rm{200}}})/{R}_{{\rm{s}}}\right]-{R}_{{\rm{200}}}/({R}_{{\rm{s}}}+{R}_{{\rm{200}}})}\end{eqnarray} \tag{ 3 }$$

for R_s.

This can be done for any two independent halo properties, typically characteristic radii or velocities (Avila-Reese et al. 1999; Thomas et al. 2001; Alam et al. 2002; Gao et al. 2004; Klypin et al. 2011; Prada et al. 2012). While such techniques are more robust against deviations from NFW and yield more accurate results than fitting for dense halos with under-sampled central regions, even these techniques still assume that halos are spherical, centered appropriately, and do not contain substructure.

In principle, more accurate concentrations for non-spherical halos could be obtained by fitting to triaxial or ellipsoidal bins. Such techniques have been found to provide more accurate mass estimates in both simulations and observations, but generally assume that the axis ratio and alignment remains constant throughout the halo (Warren et al. 1992; Jing & Suto 2002; Allgood et al. 2006; Despali et al. 2013, 2014). In reality, simulations show that halo shape is highly dependent on radius, becoming less and less spherical as you look deeper in the halo (Allgood et al. 2006; Vera-Ciro et al. 2011). This makes measurements assuming a constant shape dependent upon the location within the halo at which shape is measured. In addition, despite allowing for more freedom in halo shape, non-spherical fitting is still victim to the same caveats as spherical fitting and relies on the additional measurement of halo shape.

While both non-spheroidal binning and non-parametric spherical techniques have advantages, neither is completely free of assumptions or can handle substructure. However, using Voronoi tessellation, we can construct a technique that does not rely on fitting, does not required the halo to be centered, does not make any assumptions of spherical symmetry, and allows for substructure.

Given a set of seed points $\{{p}_{1},\ldots ,{p}_{n}\}$ in some space, Voronoi tessellation divides the space between the seeds such that each seed is the closest seed to its Voronoi region. In this way, each seed (p_i) has a corresponding Voronoi region of volume V_i encompassing all points in space that are closest to that seed. In astrophysics, Voronoi tessellation and its dual, Delaunay triangulation, have been used to reconstruct density fields from discrete points (Schaap & van de Weygaert 2000), non-parametrically identify galaxy clusters in galaxy surveys (Soares-Santos et al. 2011), identify dark matter halos (Neyrinck et al. 2004) and voids (Neyrinck 2008) in cosmological simulations, and improve the treatment of hydrodynamics in simulations (Springel 2010; Mocz et al. 2013; Hopkins 2015). We take this one step further. Once halos are identified in cosmological simulations (either by friends-of-friends, spherical over-density, or tessellation), the additional information provided by the particles' associated volumes can be used to derive halo properties (like concentration) without imposing any additional functional form.

To determine concentration from the particle volumes, a profile is constructed that describes how mass scales with volume rather than radius. For a particle p_i with mass m_i and a Voronoi volume V_i, the volume V_enc,i "enclosed" by that particle is taken to be the sum of all particle volumes which are smaller than V_i or

$$\begin{eqnarray}&&{V}_{{\rm{enc}},i}=\displaystyle \sum _{j=0}^{n}{V}_{j}[{V}_{j}\leqslant {V}_{i}].\end{eqnarray} \tag{ 4 }$$

Generally speaking, a particle's volume V_i will increase with its distance from the center of the halo as density drops off. This means the above summation is likely to include particles that are interior to particle p_i and will not include particles at the edges of the halo which can have infinite volumes. However, since particle volumes are formally independent of where the center of the halo is, this summation will be the same regardless of how the halo is centered. Similarly, the mass M_enc,i "enclosed" by a particle p_i is

$$\begin{eqnarray}&&{M}_{{\rm{enc}},i}=\displaystyle \sum _{j=0}^{n}{m}_{j}[{V}_{j}\leqslant {V}_{i}].\end{eqnarray} \tag{ 5 }$$

Each particle can then be assigned a "pseudo radius" ${R}_{i}^{\prime }={(3{V}_{{\rm{enc}},i}/4\pi )}^{1/3}$ which is defined as the radius the particle would have if V_enc,i were spherical. The result is a volume-based mass profile ${M}_{{\rm{enc}}}(R^{\prime} ).$ Naively, the volume-based concentration could then be defined as

$$\begin{eqnarray}&&{c}_{{\rm{vol}}}={\left(\displaystyle \frac{{V}_{200}}{{V}_{{\rm{s}}}}\right)}^{1/3}\end{eqnarray} \tag{ 6 }$$

where V₂₀₀ is the smallest volume containing an average density that is 200 times the critical density of the universe and V_s is some scale volume. If the pseudo radii (${R}_{200}^{\prime }$ and ${R}_{{\rm{s}}}^{\prime }$) associated with these volumes converged to the corresponding physical radii (R₂₀₀ and R_s) in the case of a spherical halo, c_vol would equal c_nfw and this would be sufficient. However, this is not strictly true.

For even a spherical halo, the relationship between a particle's physical radius R_i and pseudo radius ${R}_{i}^{\prime }$ is not 1:1. Due to the intrinsic scatter in the inter-particle spacing at a given physical radius, particles with slightly larger/smaller volumes will be scattered to larger/smaller pseudo radii. Since the density of particles is generally greater toward smaller physical radii, it is more likely for particles inside R_i to be scattered to larger pseudo radii. As a result, there will then be systematically fewer particles considered "enclosed" by particle p_i and ${R}_{i}^{\prime }$ will be systematically lower.

In order to correct for this and preserve the same numerical values for c_vol and c_nfw in the case of spherical halos, the volume based concentration is defined as

$$\begin{eqnarray}&&{c}_{{\rm{vol}}}=\beta {\left(\displaystyle \frac{{V}_{{\rm{200}}}}{{V}_{{\rm{s}}}}\right)}^{\alpha /3}.\end{eqnarray} \tag{ 7 }$$

β = 0.8062 and α = 1.0417 were obtained by fitting to measurements of V₂₀₀ and V_s for 10 spherical halos with known concentrations between 5 and 70 containing 1 × 10⁶ particles. Fits for α and β were independent of particle number for N ≥ 1 × 10⁵, indicating that the quoted values are free from contamination by noise from under-sampling the profile. While this treatment is simplistic, tests performed in Section 3.1 indicate that it should be sufficient for most studies.

Equation (7) can then be simplified in terms of the radii a halo would have if it were spherical.

$$\begin{eqnarray}&&{c}_{{\rm{vol}}}=\beta {\left(\displaystyle \frac{{R}_{{\rm{200}}}^{\prime }}{{R}_{{\rm{s}}}^{\prime }}\right)}^{\alpha }.\end{eqnarray} \tag{ 8 }$$

Note that since V₂₀₀ is defined as the smallest volume containing an average density that is 200 times the critical density of the universe, ${R}_{{\rm{200}}}^{\prime }$ can be found directly from the Voronoi volumes. However, V_s and ${R}_{{\rm{s}}}^{\prime }$ are somewhat arbitrary in the absence of fitting. Instead, we choose to define the scale radius and volume in terms of a quantity we can measure. We adopt ${V}_{{\rm{half}}}=4\pi {R}_{{\rm{half}}}^{\prime 3}/3,$ the smallest volume enclosing M₂₀₀/2 (half the mass within V₂₀₀).

Once ${R}_{{\rm{half}}}^{\prime }$ is known, the corresponding ${R}_{{\rm{s}}}^{\prime }$ for a spherical halo can be found by numerically solving Equation (3). This relationship yields unique concentrations for each R_half when 1 < c < 100. Above c = 100, R_half/R₂₀₀ becomes somewhat degenerate about 0.1.

The TesseRACt procedure for finding c_vol is then:

• 1.  
  Run Voronoi tessellation to determine the volumes and densities at each particle.
• 2.  
  Rank particles in order of decreasing density (increasing volume).
• 3.  
  Calculate enclosed mass (M_enc) and volume (V_enc) at each particle by summing the masses and volumes of "denser" particles.
• 4.  
  Calculate mean enclosed density at each particle $\left(\left\langle{\rho }_{{\rm{enc}}}\right\rangle={M}_{{\rm{enc}}}/{V}_{{\rm{enc}}}\right)$.
• 5.  
  Find V₂₀₀ and M₂₀₀ (the volume and mass enclosed by the particle at which the mean enclosed density reaches 200 times the critical density of the universe).
• 6.  
  Find V_half (the volume enclosed by the particle at which the enclosed mass reaches M₂₀₀/2).
• 7.  
  Calculate ${R}_{{\rm{200}}}^{\prime }$ and ${R}_{{\rm{half}}}^{\prime }$ from V₂₀₀ and V_half.
• 8.  
  Numerically solve Equation (3) for ${R}_{{\rm{s}}}^{\prime }.$
• 9.  
  Calculate c_vol from Equation (8).

Here V_half is defined in terms of M₂₀₀, which is calculated from the particle data using the 200ρ_crit density threshold. While other definitions of halo mass can also be used after making the appropriate substitutions for V₂₀₀ and ${R}_{200}^{\prime }$ above, this is equivalent to redefining concentration and can significantly change the measured value.

We first tested the performance of each method on halos with concentrations of 5, 10, 25, and 50. Results are shown in Figure 1.

Zoom In Zoom Out Reset image size

As these are idealized halos, all of the tested techniques return accurate measures of concentration as expected. For the idealized halos, TesseRACt is the most accurate for all but the c = 5 halo for which the measured concentration is still within 2% of the correct value. Of the techniques which assume spherical symmetry, all three perform similarly with the maximum circular velocity and fitting techniques returning marginally more accurate concentrations at the low and high ends respectively. All of the techniques tested were consistent across the 10 halo realizations with standard deviations of <0.6%.

The accuracy of the techniques that assume spherical symmetry does not appear to be overly concentration dependent. However, as the modified concentration definition used for TesseRACt in Equation (8) was fit to measurements across a range of concentrations, TesseRACt is slightly more accurate for halos with intermediate concentrations. If Equation (8) were fit to a wider range of concentrations that extended to c < 5, we would expect performance to improve for c = 5 halos. However, since the returned concentration is still with 2% of the correct value, this effect would contribute a negligible amount of error for real halos.

3.2.1. Oblateness

To test how each method performed on oblate halos, the c = 10 halo from above was squeezed along the z axis using the volume preserving transformation

$$\begin{eqnarray}&&\left[\begin{array}{c}x^{\prime} \\ y^{\prime} \\ z^{\prime} \end{array}\right]={\left(\displaystyle \frac{a}{b}\right)}^{1/3}\left[\begin{array}{ccc}a & 0 & 0\\ 0 & a & 0\\ 0 & 0 & b\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right],\end{eqnarray} \tag{ 10 }$$

for 7 different values of ellipticity where e_oblate = b/a and a = 1. Results are shown in Figure 2.

Zoom In Zoom Out Reset image size

For all but the least spherical halos, TesseRACt consistently recovers concentrations within 0.5% of the correct value, with a minor dependence on halo shape (overestimated by 0.4% for e_oblate = 0.3 versus underestimated by 0.02% for spherical halos). However, the performance of the techniques that assume spherical symmetry is highly dependent on halo shape. Fitting performs the worst overall, with increasingly underestimated values for the least spherical halos (10% for e_oblate = 0.3). The performance of the two non-parametric spherical techniques is dependent on halo shape in a more complicated manner.

For decreasing values of e_oblate (decreasing spherical symmetry), both non-parametric spherical techniques have moderately increasing positive residuals until around e_oblate = 0.6–0.7 where the residuals begin to rapidly decrease resulting in underestimates of 6% and 8% for the half-mass and peak-velocity techniques respectively. These dependencies arise because the deformation of the halo flattens the radial mass profile. As fitting uses the entire profile, the concentration is underestimated for all non-spherical halos. However, non-parametric techniques use only two points in the profile (R₂₀₀ and some inner radius) and their performance will depend on how these parts of the profile are affected by the transformation. R₂₀₀ continually increases as compression of the halo edge decreases the density at the edge of the profile. The inner radius will increase as well, but at a slower rate resulting in an overestimation of the concentration until the edge of the halo is compressed to a size comparable to the inner radius. Once this occurs, the inner radius increases more rapidly than R₂₀₀ resulting in underestimates. The precise value of e_oblate at which this occurs will depend on the radius that a particular non-parametric technique utilizes. Since the velocity peaks at a smaller radius than the half-mass radius for an NFW profile, the technique that uses the peak velocity is less sensitive to this effect.

The test halos here have the same ellipticity at all radii by design. However, real halos are found to be decreasingly spherical at their centers (Allgood et al. 2006; Vera-Ciro et al. 2011). While TesseRACt would not be affected, it is likely that the spherical techniques would produce even less accurate concentrations. This would be particularly pronounced for fitting due to sensitivity to the inner profile slope.

3.2.2. Prolateness

To test how each method performed on prolate halos, the c = 10 halo from above was squeezed along its z and y axes using the volume preserving transformation

$$\begin{eqnarray}&&\left[\begin{array}{c}x^{\prime} \\ y^{\prime} \\ z^{\prime} \end{array}\right]={\left(\displaystyle \frac{{a}^{2}}{{b}^{2}}\right)}^{1/3}\left[\begin{array}{ccc}a & 0 & 0\\ 0 & b & 0\\ 0 & 0 & b\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right],\end{eqnarray} \tag{ 11 }$$

for 7 different values of ellipticity where e_prolate = b/a = c/a and a = 1. Results are shown in Figure 3.

Zoom In Zoom Out Reset image size

The performance across all techniques is similar to the above tests for oblate halos. However, there does appear to be a stronger dependence on ellipticity at intermediate values for prolate halos than for oblate halos for techniques which assume spherical symmetry. This is because the prolate transformation flattens the radial mass profile to a greater degree than the oblate transformation.

3.2.3. Triaxiality

We then tested the performance of each method on triaxial halos (See Figure 4). For 9 different values of triaxiallity ($T=\frac{{c}^{2}-{b}^{2}}{{c}^{2}-{a}^{2}},$ for $c\leqslant b\leqslant a,$ Franx et al. 1991), the c = 10 halo from above was squeezed along the z and y axes using the volume preserving transformation

$$\begin{eqnarray}&&\left[\begin{array}{c}x^{\prime} \\ y^{\prime} \\ z^{\prime} \end{array}\right]={\left(\displaystyle \frac{{a}^{2}}{{bc}}\right)}^{1/3}\left[\begin{array}{ccc}a & 0 & 0\\ 0 & b & 0\\ 0 & 0 & c\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right],\end{eqnarray} \tag{ 12 }$$

where a = 1, c/a = 0.3, and $b=\sqrt{{c}^{2}-T({c}^{2}-{a}^{2})}.$

Zoom In Zoom Out Reset image size

TesseRACt outperforms all spherical techniques at <0.2% for all triaxialities. The spherical techniques return concentrations that are only slightly less accurate overall (<4%) than in the case where T = 1. While TesseRACt's performance does not depend on triaxiality, the accuracy of the techniques which assume spherical symmetry does to a small degree. The two non-parametric techniques become less accurate for lower values of T, while fitting only exhibits this behavior until T ∼ 0.3, where this trend is reversed. However, since the residuals never exceed 4% of the fiducial value, which is itself off by up to 10% from the true concentration, it is reasonable to conclude that only the ratio between the largest and smallest halo axes appears to have a significant impact on the accuracy of the concentration estimate.

To assess how substructure influences concentration measurement using TesseRACt and the other techniques tested here, idealized subhalos of varying mass and concentration were added to the c = 10 halo at varying radii. In each case the mass of the subhalo was set by downsampling the test halo from the previous section with the desired concentration. The size of the subhalo was then scaled such that the mean density within 0.2 R₂₀₀ was half that of the parent halo.

It should be noted that, when substructure is present, the traditional definition of concentration loses its physical meaning. While it can be used after removing the substructure, this is problematic in several ways. First, what remains within the parent halo will depend upon the technique used to identify the substructure. Second, from a physical standpoint, the substructure is a part of the parent halo that should contribute to its properties. Instead, concentration can be redefined as a measure of how compact the halo is overall, independent of the number of density peaks the mass is distributed between. While TesseRACt does this naturally, techniques assuming spherical symmetry cannot without first identifying substructure.

3.3.1. Substructure Mass

Figure 5 shows the results from varying the mass of the subhalo across 1%–20% of the parent halo's M₂₀₀. The subhalos had a concentration of 50 and were placed 0.5 R₂₀₀ from the center of the parent halo. Since subhalos were scaled to have a central density within 0.2 R₂₀₀ equal to half that of the parent halo, increasing the mass of the subhalo also increases its size.

Zoom In Zoom Out Reset image size

As expected, all techniques return concentrations that depend on the mass (and size) of the subhalo, but TesseRACt is less dependent overall. For the most massive subhalo, 20% the mass of the parent, TesseRACt estimates the concentration to be 8% larger than that of the parent halo alone, while techniques assuming spherical symmetry estimate the concentration to be up to 30% smaller. TesseRACt returns larger concentrations because particles in the subhalo have small enough volumes that they are assumed to be within the inner parts of the parent halo and thus result in a smaller half-mass volume. The techniques assuming spherical symmetry return smaller concentrations in the presence of substructure because the subhalo contributes mass to the outer profile. However, if concentration is meant to measure how compact the matter is within the halo, even if its not all centrally located, halos with more substructure should have higher concentrations.

3.3.2. Substructure Radius

Figure 6 shows the results from varying the distance of the subhalo from the center of the parent halo from 0.05 to 0.5 R₂₀₀. The subhalos each had a concentration of 50% and 1% the mass of the parent halo.

Zoom In Zoom Out Reset image size

At small radii, the subhalo almost coincides with the center of the parent halo, causing all techniques to return higher concentrations than they would had there been no substructure present. As the subhalo is placed at larger radii, the concentrations returned by all techniques become lower. However, while techniques assuming spherical symmetry return concentrations that are lower than that of the parent halo alone by up to 20% as additional mass is added to the outer profile, TesseRACt estimates the concentration to be only 0.7% lower at 0.75 R₂₀₀. This occurs because, when the subhalo is placed at larger radii where the density of the parent is lower, the densities assigned to the subhalo particles using tessellation become lower and contribute less to the half-mass volume. The radius at which the spherical techniques change from returning concentrations that are higher versus lower than the parent halo depends on which part of the mass profile is used to calculate concentration.

3.3.3. Substructure Concentration

Figure 7 shows the results from varying the concentration of the subhalo from 5 to 50. The subhalos were placed 0.5 R₂₀₀ from the center of the parent halo and had 1% the mass of the parent.

Zoom In Zoom Out Reset image size

For subhalos of equal or greater concentration than the parent (c_sub ≥ 10), TesseRACt returns concentrations closer to that of the parent halo than the spherical techniques (<3% versus <20%). While the spherical techniques show little dependence on concentration in this regime, TesseRACt does. Since TesseRACt places no constraints on where the center of the halo is, the measured concentration is essentially an average of the concentrations of the two halos present, the parent and the subhalo. As a result, the same dependence on halo concentration that was seen in Section 3.1 is also present for subhalo concentration.

However, the case of a subhalo that is less concentrated than the parent halo (c_sub = 5) is less physical. In order to maintain a central density half that of the c = 10 parent halo with 1% of the mass, the c = 5 halo must be scaled to less than half of its original size. This means that the substructure contributes to a narrower region of the radial mass profile, resulting in lower estimates for R₂₀₀ and M₂₀₀ that are closer to those for the parent halo alone. For fitting, which uses the entire profile, the result is a significantly lower concentration (20%). The non-parametric techniques, which are not dependent on the whole profile, return concentrations that are closer to the parent halo in this case. The half-mass technique is slightly closer (2%) than the peak velocity technique (3%) because while R_half changes with R₂₀₀, the peak velocity occurs well within the radius at which the substructure was placed.

Next, we tested the performance of each technique for different resolutions (See Figure 8). Beginning with the c = 10 halo containing 10⁶ particles, transformed to be prolate with an e_prolate = 0.3, random subsets of the particles were selected to create halos of the same size and shape, but lower resolution.

Zoom In Zoom Out Reset image size

Above 2 × 10³ particles, all of the tested techniques are reasonably convergent (<10%) around the fiducial concentration for N = 1 × 10⁶. For fewer particles, the residuals for the peak velocity technique climb quickly to >80% of the fiducial value for 100 particles. TesseRACt and the half-mass non-parametric technique perform similarly on average, returning concentrations within 20% of the fiducial value at the lowest resolution. However, TesseRACt was more precise than the half-mass technique. Across the 10 halo realizations, the standard deviation in the concentrations returned by TesseRACt and the half-mass technique were 40% and 50% respectively at N = 100. Fitting was surprisingly accurate at low resolutions with residuals <10% and standard deviations similar to TesseRACt.

Finally, we tested each technique on four different halos from the Ganesh cosmological simulation described in M. Sinha et al. (2015, in preparation). Figure 9 shows the projected mass density for each of the selected halos. These halos cover a range of masses (6 × 10¹¹ to 1 × 10¹⁴ M_⊙), have a range of shapes, and offer varying amounts of substructure.

Zoom In Zoom Out Reset image size

Different techniques can measure very different concentrations for the same halo. For each halo, TesseRACt returns a concentration that is systematically larger than that returned by the techniques that assume spherical symmetry, particularly for halos that contain significant substructure (the top two panels in Figure 9). This is consistent with the previous tests on idealized halos and highlights the differences between TesseRACt and traditional techniques for measuring concentration. While TesseRACt measures an increase in the concentration of the halo when the halo contains additional density peaks (subhalos), techniques that assume spherical symmetry do not. Rather, the presence of substructure in the outer halo can cause such techniques to measure a lower concentration. If concentration is interpreted as measuring how compact the halo is, regardless of how many density peaks the mass is distributed between, then TesseRACt is a better tool.

In addition, while the traditional techniques require a choice of center, TesseRACt does not. For the halos in Figure 9, the choice of center had a significant influence on the concentrations measured using the techniques that assume spherical symmetry, particularly for halos with substructure. The concentrations measured using fitting for halos (a) and (b) in Figure 9 each changed by 4% when the center was changed by 5% of the virial radius as measured by TesseRACt. Given that there are a wide variety of methods for centering currently in use (e.g., center of mass, density peak, potential minimum), this could make direct comparison of concentrations from different studies difficult. However, TesseRACt measures the same concentration regardless of how the halo is centered as long as it contains the same particles.

It is important to note that, regardless of technique, concentrations measured for halos in cosmological simulations will be dependent on the technique used to identify the halos and the selected definition of concentration. For this study, concentrations were defined in terms of R₂₀₀, the radius (or pseudo radius) where $\frac{3{M}_{{\rm{enc}}}({\lt R}_{200})}{4\pi {R}_{200}^{3}}=200{\rho }_{{\rm{crit}}}.$ R₂₀₀ can be measured directly from the particle data as long as the halo extends past this point and only includes halo particles. However, different halo finders use different criteria for including/excluding particles from a halo and the halo may be truncated interior to R₂₀₀. An alternate radius may be defined in terms of the mass determined by the halo finder or a larger density threshold, but direct comparison between concentrations defined in terms of different radii should be avoided. Additional work is needed to assess how TesseRACt and other techniques for measuring concentration depend upon choice of halo finder and will be the subject of a future paper.