tokens: before the analyses pipelines were finalized. Our decision to focus on the NGC exclusively did come after a comparison with S16 had occurred, though we believe that we are well justified in our choice. S16 provide the current best weak lensing mass calibration of redMaPPer clusters, and they discuss in detail how their mass calibration compares to others in the literature. While the constraints on the slope of the mass–richness relation are in good agreement between the two works, there is mild tension on the recovered amplitude assuming our fiducial cosmology, with the clustering-derived constraints preferring a somewhat higher amplitude. At the richness pivot point used in this work, λ 0 = 35, S16 predict log 10 ( M /M ⊙ ) = 14.43 ± 0.04, while our constraints yield log 10 ( M /M ⊙ ) = 14.62 ± 0.08; this discrepancy corresponds to roughly 2.2σ tension. Because the clustering analysis prefers a slightly lower value of α than the weak lensing analysis, this tension is reduced for λ > 35. For λ 35, the tension remains roughly the same since the error bars increase as one moves away from the λ pivot point. We postpone a discussion of the origin of this tension, and a more detailed comparison of the two results, to future work. That is, our current results reflect our best understanding of both the weak lensing and clustering analysis prior to the comparison of the two works. We have also considered a power-law bias–richness relation as per equation ( 10 ). This model has five parameters: A λ , α λ , β λ and two bias parameters for 5 λ 20 clusters in the two redshift bins. Fitting this model to the data, we find A λ = 3.54 ± 0.13, α λ = 0.57 ± 0.08 and β λ = 2.0 ± 1.1 with χ 2 /d.o.f. = 57.7/75. 7 CHARACTERIZING THE UNCERTAINTY IN THE MASS–RICHNESS RELATION The uncertainties on the mass–richness parameters presented in Fig. 6 and Table 1 receive contributions from several sources: statistical noise in the measurement of the angular correlation functions, uncertainty on the parameters of the line-of-sight projection model, uncertainty on the parameters of the assembly bias model and scatter around the b ( M ) relation from Tinker et al. ( 2010 ). In addition to these sources of scatter that are explicitly included in our model, it is also possible that our measurement is affected by systematic biases that we have not modelled. Below, we determine the approximate contributions of both modelled sources of scatter and unmodelled systematic errors to the uncertainty budget of our parameter constraints. 7.1 Statistical noise in the correlation function measurement We first investigate how statistical noise in the measurement of w αβ (θ) contributes to our parameter uncertainties. The contribution of statistical noise to our parameter uncertainties can be determined by setting the parameters of the line-of-sight projection model, the parameters of the assembly bias model and b scaling to constant values rather than allowing these parameters to vary.
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