Biases in the estimation of velocity dispersions and dynamical masses for galaxy clusters

Using a set of 73 numerically simulated galaxy clusters, we have characterised the statistical and physical biases for three velocity dispersion and mass estimators, namely biweight, gapper and standard deviation, in the small number of galaxies regime ($N_{gal} \leq 75$), both for the determination of the velocity dispersion and the dynamical mass of the clusters via the $\sigma-M$ relation. These results are used to define a new set of unbiased estimators, that are able to correct for those statistical biases. By applying these new estimators to a subset of simulated observations, we show that they can retrieve bias-corrected values for both the mean velocity dispersion and the mean mass.


Introduction
Several authors have used the velocity dispersion mass proxy to study and characterise scaling relations between SZ and dynamical mass [1][2][3]. In order to have sample with enough statistical power, it is necessary to estimate the velocity dispersion for hundreds of galaxy clusters (GCs). Although this goal can be achieved through spectroscopic follow-up [e.g. 4, 5], these studies are extremely expensive in terms of observational time and data reduction. For these reasons, it is extremely difficult to estimate radial velocities for more than few members (typically ∼ 20) for each cluster target.
In this article we present our study of statistical and physical biases introduced in the estimation of velocity dispersion and dynamical mass.

Statistical biases in Velocity Dispersion estimation
For our analysis we use a sample of 73 simulated massive clusters selected from the simulations described in [6]. Our selected sample contains clusters with masses M 200 > 2 × 10 14 M and located at five redshifts between 0.12 ≤ z ≤ 0.82.
There are several method to estimate mean and scale of a distribution. We focus our attention on the estimators that, in the last decades, became standard tools in GC analyses,  namely biweight and gapper, compared with the standard deviation. A detailed description of these estimators can be found in [7]. In order to investigate bias and variance of the three scale estimators S X (N gal ), being X= "std", "bwt" or "gap", for each one of the three cases, we have explored the low number of galaxies regime between N gal = 8 and N gal = 75. We have generated 2500 configurations of randomly selecting galaxies within the projected circle of radius R 200 and then we averaged the 73 × 2250 velocity dispersions (750 for each mayor axis). Left panel of Fig. 1 shows the average S X (N gal ) normalised with respect to S std (< R 200 ), which represents the velocity dispersion of all the galaxies in the simulation within a circle of projected radius R 200 , and calculated using the standard deviation estimator. In the low-N gal regime, all estimators are biased showing very different behaviours. The standard deviation estimator (green line) shows a dependence with the number of elements used for the estimation. However, this dependence can be theoretically predicted to be 1 − 1/(4(N gal − 1)). The biweight (blue line) shows a stronger drop for N gal ≤ 30 underestimating the reference dispersion by up to 4 % at N gal = 7. Finally, the gapper (red line) shows an estimate of the velocity dispersion almost constantly biased at any N gal .
In order to construct an unbiased velocity dispersion estimators, S X (N gal ), we use a parametrisation of the curves in left panel of Fig. 1: (1) Table 1 shows the best-fit values for the parameters D, β and B, for each one of the three estimators (biweight, gapper and standard deviation). Right panel of Fig. 1 shows that the corrected estimators S X (N gal ) are actually unbiased by construction. Figure 2. Effect of interlopers on S X /S std (< R 200 ), as a function of the number of galaxies. The velocity dispersion, S X (N gal ), is first computed using a pure galaxie sample (solid lines). We also evaluated the response of biweight (blue lines), standard deviation (green lines), and gapper (red lines) using samples contaminated by 0% (solid line), 5% (dashed lines), 10% (dot dashed lines), 15% (three-dot-dashed lines), 20% (two-dot-long-dashed lines) and 30% (dotted lines) of interlopers at any N gal .

Biases by interlopers contamination
Any spectroscopic sample of cluster members is contaminated by galaxies belonging to the large scale structure that surrounds the cluster itself. This population of 'pseudo cluster members', called interlopers, modifies the velocity distribution and therefore affects the estimation of velocity dispersion. According to the definition of interlopers given in [8], one must distinguish between two very different types of contaminants: i. galaxies gravitationally bound to the clusters that are outside the virial sphere (according to the definition given in [9]), but that, due to projection effects appear within a smaller radius; ii. background/foreground galaxies with similar redshifts to the cluster, but belonging to the large scale structure that surrounds the cluster itself. Detailed study of these interlopers is beyond the scope of this work. Here, we illustrate the robustness of the three estimators in exam, by fixing the fraction of contaminants at any N gal . Fig. 2 shows the results obtained for five different fractions of interlopers: 5% (dashed lines), 10% (dot-dashed lines), 15%(three-dot-dashed lines), 20% (two-dot-longdashed lines) and 30% (dotted lines). We note that the three estimators are similarly affected by interloper contamination. The effect of the interlopers contamination consists of a velocity dispersion overestimation which is as high as their relative fraction (up to 30%) representing the most damaging source of biases in the velocity estimation estimation.

Physical biases in Velocity Dispersion estimation
Observational strategies and technical limitations generally force us to observe only massive clusters members sampled in a fraction of R 200 . We studied how these limitations might produce biased velocity dispersion estimates.

Effects due to the selected fraction of massive galaxies
In the ideal case we can observe any cluster member regardless of its brightness. However, the telescope aperture limits the detection magnitude and prevents us from detecting faint objects, for a fixed integration time. Therefore, cluster samples contain only a fraction of the brightest galaxy members, which are also the most massive.
In order to simulate this effect, we mimicked observational conditions by selecting three percentages of all visible galaxies in the simulation, i.e. 50 %, 33 %, and 25 %, by sorting the cluster members by mass and dividing the sample in 2, 3, and 4 mass bins, starting from the most massive object. For each case, we reproduce the procedure explained in Sect. 2. Fig. 3 (left panel) shows S std (N gal )/S std (< R 200 ) as function of N gal calculated with the corrected standard deviation estimator, and using galaxies picked up from 100 % (black line), 1/2 (blue line), 1/3 (red line) and 1/4 (green line) of the complete cluster member samples. We see that the velocity dispersion is sensitive to the fraction of massive galaxies used to estimate it, reaching a bias of almost 2% using only the most massive fraction. However, it is almost insensitive to N gal . For this reason, we can use the curves in the left panel of Fig. 3 to correct this physical effect.

Effect of aperture sub-sampling
There are evidence in the literature that the velocity dispersion has a radial dependence [e.g., 2,9]. This implies that sampling galaxies in different fractions of the cluster's virial radius should lead to a biased velocity dispersion. In order to investigate and quantify this effect, we averaged the 73 velocity dispersions calculated using all galaxies inside a cylinder of variable radius 0.2 ≤ r/R 200 ≤ 1.2. Fig. 3 (right panel) shows that the velocity dispersion is (on average) overestimated in region smaller than R 200 and slightly underestimated for r > R 200 , in analogy with that recover by [2].

Statistical bias in the estimation of M 200
When we use velocity dispersion as a mass proxy we apply a scaling relation σ 1D − M, it is a power law that has been previously calibrated either with simulations [6,10,11]. The left panel of Fig. 4 shows how applying the scaling relation σ 1D − M to the unbiased velocity dispersion estimator S X the resulting masses M(S X )/M(S std (< R 200 )) are biased.  We can calculate analytically the functional form of the mass estimator as a function of the number of galaxies 1−2α 4α 2 (N gal −1) , with A = 1177.0 km s −1 and α = 0.364 parameters of the [6] scaling relation.
As in Sect. 2, we fitted a parametric form description of the bias as a function of N gal by using three parameters (E ,F and γ , listed in Table 2): ( The right panel of Fig. 4 shows the bias of M (S X ). The new mass estimator is actually unbiased by construction.
Finally, in order to test the corrections described in previous sections, we have applied them to a set of mock cluster catalogues. To do this we have simulated a realistic observational strategy based on the observations described in [4,5]. We generated 100 mock samples out of the 73 GCs object simulated in this study. For each of these samples we calculated the mean ratio between the estimated and the reference velocity dispersion of each cluster, as well as the velocity dispersion calculated using S X defined in eq. 1 with the parameters in Table 1. Averaging over all the mock samples we obtained a biased mean velocity dispersion S std (N gal , r)/S bwt (< R 200 ) = 0.96 ± 0.02, S std (N gal , r)/S bwt (< R 200 ) = 1.00 ± 0.02.
The normal S std estimator shows to be biased, whereas the S X lead to an unbiased estimation of the velocity dispersion.