In this work we present an application of general extreme value statistics ( GEV ) to very massive single clusters at high and low redshifts .
After introducing the formalism , we apply this statistics to four very massive high redshift clusters .
Those clusters comprise ACT-CL J0102-4915 with a mass of M _ { 200 m } = ( 2.16 \pm 0.32 ) \times 10 ^ { 15 } M _ { \odot } at a redshift of z = 0.87 , SPT-CL J2106-5844 with a mass of M _ { 200 m } = ( 1.27 \pm 0.21 ) \times 10 ^ { 15 } M _ { \odot } at z = 1.132 and two clusters found by the XMM-Newton Distant Cluster Project survey : XMMU J2235.32557 with a mass of M _ { 200 c } = ( 7.3 \pm 1.3 ) \times 10 ^ { 14 } M _ { \odot } located at a redshift z = 1.4 and XMMU J0044.0-2033 having a mass in the range of M _ { 200 c } = ( 3.5 - 5.0 ) \times 10 ^ { 14 } M _ { \odot } at z = 1.579 .
By relating those systems to their corresponding distribution functions of being the most massive system in a given survey area and redshift interval , we find that none of the systems alone is in tension with \Lambda cold dark matter ( \Lambda CDM ) .
We confront these results with a GEV analysis of four very massive low redshift clusters : A2163 , A370 , RXJ1347-1145 and 1E0657-558 , finding no tendency of the high- z systems to be more extreme than the low- z ones . In addition , we study the extreme quantiles of single clusters at high- z and present contour plots for fixed quantiles in the mass vs. survey area plane for four redshift intervals , finding that , in order to be significantly in conflict with \Lambda CDM , cluster masses would have to be substantially higher than the currently observed ones .