4.3 Results
4.3.1 Cross Calibration
distribution
P(n|NˆRM(j))∝e−nnNRM(j), (4.22) which, if interpreted as a distribution in ˆNRM(j) would be the aforementioned Poissonian, but for given ˆNRM(j) is a distribution in NRM(j). This provides as sample of NRM(j).
We then use our posterior on the scaling relation parameters to create a sample of N(j).
We also use ΩM = 0.276±0.047 and σ8 = 0.781±0.037 from Bocquet et al. (2019a) to properly account for uncertainties in cosmology when sampling N(j). For pairs of entries from the NRM(j) sample and the N(j) sample we estimate πpur,NC(j), thus creating a sample thereof which account for systematic uncertainties on the richness–mass relation and on cosmology as well as the statistical uncertainties of the number counts. Note that under these assumptions, the inferred purity can also be larger than 1, indicating excess incompleteness instead.
Abbildung 4.6: Posteriors on the parameters of the richness–mass scaling relation, the scatter around that relation and the correlation coefficient between the intrinsic scatter in SZe signal and in richness. Furthermore, where applicable, also the purities and the outlier fractions posterior are shown. We show the results for different combinations of likelihoods and optical error models: blue and yellow the C19 model, and the S15 model, respectively, when sampling just the cross calibration likelihood, while in green and red the C19 model, and the S15 model, respectively, when also considering the detection probability likelihood.
In black the richness mass relation obtained by S15 from a SPT-DES matched sample of 25 clusters.
Tabelle 4.1: Mean and standard deviation estimated from the one dimensional marginal posterior plots for the parameters the richness scaling relation, the purity and the outlier fractions.
Aλ Bλ Cλ σλ ρ πpur πlow πhigh
SPT calibr. (C19) 75.7±8.8 0.91±0.07 0.34±0.34 0.18±0.05 – – – – + det. prob. 69.9±9.2 1.04±0.07 -0.09±0.31 0.22±0.06 – ¿0.80 ¡0.20 0.022±0.016 SPT calibr. (S15) 83.7±11.5 1.03±0.10 0.71±0.41 0.22±0.06 – – – – + det. prob. 78.6±10.1 0.94±0.07 0.28±0.28 0.22±0.06 – ¿0.76 ¡0.22 0.033±0.018
10
24 × 10
16 × 10
12 × 10
23 × 10
210
110
0f(S PT |, z)
z (0.20,0.65)
S15 C19
Abbildung 4.7: SPT confirmation fraction as a function of measured richness, predicted from the posteriors on the scaling relation parameters for the S15 error model (blue) and when considering projection effects (red), overlaid with the measurement as black points.
This comparison reveals no indication for significant contamination or low purity above measured richness ˆλ >40, consistent with our statistical analysis.
Scaling relation
As outlined in section 4.2.3 the measured richnesses and measured SZe signals for clusters in the cross matched RM-SPT sample can be used to transfer the calibration of the SZe signal–mass relation given by the priors discussed in section 4.2.6 onto the richness–mass relation. To this end, the sum of the likelihoods given in equation 4.8 are sampled with the aforementioned priors. The scatter plot of the matched sample in redshift bins is shown in figure 4.5 with black points.
The resulting posteriors on the parameters of the richness–mass scaling relation are summarized in terms of their means and standard deviations in Table 4.1 and shown with their marginal distributions in Fig. 4.6. In blue the constraints from adopting the C19 optical error model and sampling the cross calibration likelihood, while in yellow the posteriors when using the S15 optical error model. The constraints are in very good agreement, highlighting that at the level of statistical constraining power of the cross matched sample the two error models are not distinguishable. The minor changes induced by the projection effects can be seen on the mean relation over plotting the data scatter plot in figure 4.5. We also plot in black previous results by S15 on DES-SV.
Purity and Outlier Fraction
As discussed at length in section 4.2.4 the probability of detecting a RM cluster is SPT is very sensitive on the respective scaling relation parameters of the two selection obser-vables. Consequently, one needs to marginalize over a reasonable range of scaling relation parameters when inferring the purity of the RM sample and the outlier fractions with the likelihood given in equation 4.13. This is accomplished by sampling the likelihood of detection probabilities simultaneously with the cross calibration likelihood (equation 4.8).
This ensures proper accounting for the systematic uncertainties on the scaling relation when inferring the purity and outlier fraction. The resulting posteriors, depending on the assumed optical error model, are shown as green and red contours in Fig. 4.6 for the C19 and the S15 model, respectively, and summarized in Table 4.1.
The posteriors on the scaling relation parameters are generally in good agreement with the results without the detection probability likelihood. We find a upper limit πlow <0.20 and πlow < 0.22 (at 97.5%) on the low outlier fraction for the C19 model and the S15 model, respectively. We also find an lower limit for the purity of πpur > 0.80 and πpur >
0.76 (at 2.5%) for the C19 model and the S15 model, respectively. For the high outlier fraction, we find πhigh= 0.022±0.016 and πhigh = 0.033±0.018 for the two error models.
Neither of the two measurements in statistically significant. Yet the projection effect model shows consistently larger purity and smaller outlier fraction, hinting at a more adequate description of the scatter.
Interestingly, the detection probability likelihood slightly alters the posteriors on the scaling relation parameters, with the largest impact on the slope and the redshift evolution.
This is not surprising when considering that not detecting a RM object in SPT is equivalent to the measurementξ <4, which given priors on the SZe signal–mass relation carries some
mass information, at least in the form of an upper limit. This information is however quite weak, as can be seen by directly comparing the improvement in measurement uncertainties.
Furthermore, it is consistent with the information recovered from the matched sample alone, as the shifts is mean values also do not exceed 1σ.
The measurement of purity consistent with unity is confirmed when comparing the SPT confirmation fraction of RM objects as a function of richness to the measured fraction, as done in Fig. 4.7. Following equation 4.16, we predict the SPT confirmation fraction of the RM (ˆλ > 40) sample as a function of measured richness. The shaded region reflects the 1 σ systematic uncertainty as propagated from the posterior samples. Note also here that the difference between the predictions for the two error models is small. As discussed by C19 the values of the scaling relation and scatter around it absorb a majority of the differences between the two error models. We over plot the predicted fraction with the mea-sured fraction in richness bins and associated statistical uncertainties. Within the statistic and systematic uncertainties the prediction and the measurement are in good agreement, confirming the high purity of the RM (ˆλ >40) sample.
4.3.2 Comparison to λ > ˆ 20
In the previous section we determined the systematic uncertainty on the richness–mass relation in the regime of measured richness ¿40, that is at the high mass end. In this section we extrapolate the relation determined at high masses to lower masses. We first compare our prediction for the mean mass in richness bins to a measurement thereof from stacked weak lensing by McC19. Second we predict the number counts of RM clusters in redshift and richness bins to our prediction. Finally, both these comparison are turned into estimates of purity as a function of redshift and richness, extending to ˆλ >20.
Stacked Weak Lensing
The mean mass in redshift and richness bin for our scaling relation parameters constraints are derived following equation 4.18 for the richness bins with edges (20,30,45,60,300) and redshift bins with edges (0.20,0.35,0.50,0.60) as in McC19. The resulting masses are shown in Fig. 4.8 with the systematic uncertainties as blue and red boxes for the S15 and the C19 error model, respectively. Over plotted are also the measurements by McC19 with statistical uncertainties. The resulting tensionT(j) in the binj between the measurements MˆWL(j)±ˆσ and the prediction ¯M(j)±σsys is estimated as
T =
MˆWL(j)−M¯(j)
qσˆ2+σsys2 . (4.23)
We report the resulting tension as numbers in the plot. While at high mass the agreement is good, it worsens when moving to lower richness and higher redshift. In the lowest richness, highest redshift bin it clearly exceeds 2 σ for both optical models.
10
15M
200m(20, 30) (30, 45) (45, 60) (60, 300)
1.86 0.77 1.28
0.22
2.06 0.62 0.91
-0.33
z (0.20, 0.35)
10
15M
200m1.89 0.84 0.47
0.00
2.17 0.70 0.07
-0.64
z (0.35, 0.50)
10
1410
15M
200m2.34 0.33 0.08
-0.09
2.46 0.02 -0.49
-0.75
z (0.50, 0.65)
McC 19 C19 S15
Abbildung 4.8: Mean mass in redshift and richness bins predicted from our posteriors for the S15 error model (blue), and the C19 error model (red), where the filled region represents 1σ and the empty one 2σ. Over plotted the mean masses reported by McC19 from stacked weak lensing as black points with errorbars. We also report the levels of tension between our prediction and the measurement as numbers next to the respective predictions (red for C19, blue for S15). In the lowest richness bins the tension exceeds 2 σ for the C19 model, while for the S15 model it does so in the highest redshift, lowest richness bin.
10
210
110
010
110
210
3N
RMz (0.20, 0.35)
10
2z (0.35, 0.50)
10
2z (0.50, 0.65)
S15 C19
Abbildung 4.9: Number of RM objects in bins of richness and redshift. In color the predic-tion from our richness–mass relapredic-tion constraints and SPT cluster number counts for the different optical error models (blue S15, red C19). The shaded region represents the 1σ sy-stematic uncertainty. Over plotted in black the number of RM objects with the associated Poissonian error bar. The prediction matched the data within 1 σ systematic uncertainty.