Secondly, we focus on galaxies to discuss some remaining solutions. The miscentring (e.g. A1986, A1961) decreases the X-ray luminosity/SZ signal, as some flux moves outside of the aperture. Sehgal et al. (2013) demonstrated the effect of miscentring on decreasing SZ signal. They also proposed that the miscentring effect causes their lower measured SZ signal compared to Planck due to the finer resolution of ACT. However, as they point out, the miscentring distribution from their sample alone can only explain part of the discrepancy between optical and SZ, unless an unrealistic larger offset exists. Moreover, most miscentred clusters are merging or disturbed clusters with lower X-ray or SZ surface brightness, which makes them more difficult to be detected. The discrepancy of SZ signal is at a level of 10 per cent in our richness range (Biesiadzinski et al. 2012). The projection (e.g. A750, A1319) will increase the optical richness. As the cluster mass-richness relation is close to a linear relation, projection causes the projected clusters to simply slide up and down the mass-richness relation, without deviating from it (Simet et al. 2017). However, the LX−M or YSZ−M relation is a power-law relation with an index greater than one (∼1.6; Rozo et al. 2014a). For example, if we have two clusters with N200 = 80 ($M_{500|N=80}=5\times 10^{14} \, \mathrm{M}_{\odot}$, bolometric X-ray luminosity LX|N = 80 = 11 × 1044 erg s−1) projected together, they will be detected as a N = 160 cluster, the corresponding expected mass and X-ray luminosity are $M_{500|N=160}=10\times 10^{14} \, \mathrm{M}_{\odot}$ and LX|N = 160 = 43 × 1044 erg s−1, respectively. Though the mass is equal to the sum of two subclusters, the total X-ray luminosity is overestimated by a factor of ∼2 than the linear combination of 2LX|N = 80 = 22 × 1044 erg s−1. Thus, projection causes the expected X-ray luminosity or the SZ signal from the summed optical richness to be higher than the actual summed values. The projection fraction of samples extending to much lower richness is around 10 per cent (Simet et al. 2017), even higher for these richest maxBCG clusters (Fig. 2). The above two effects can act together, especially in super clusters and large-scale filaments (Fig. D1). The contamination of low-mass haloes, whose true halo mass is far below the value suggested by the optical richness, would also dilute and reduce the mean mass of the sample. We note that there is contamination of such low-mass systems based on Fig. A2. These low-mass haloes are mostly blended systems with boosted richness affected by nearby large-scale structure. Thus there is a mixture of halo masses at very high N200, the clean and the blended, and the PDF of M given N200 (or λ) will be asymmetric with a low-mass tail. A skew-normal or Hermite polynomial expansion (Shaw, Holder & Dudley 2010) are good alternatives to mixture modelling. Next, we roughly estimate the contamination fraction. There are four low LX systems beyond the 2σ line of LX−N relation from Rozo et al. (2014b) in Fig. A2 (and seven beyond the 1.5σ line towards lower LX versus 1 beyond the 1.5σ line towards higher LX). Taking these numbers at face value, the contamination is 10–15 per cent.
Citation: Biesiadzinski et al. 2012
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