The cosmological principle is not in the sky

The 7th KIAS Workshop on Cosmology and Structure Formation Oct. 30 Nov. 4 2016 The cosmological principle is not in the sky Chan-Gyung Park (Division of Science Education, Chonbuk National University) in collaboration with Hyerim Noh (KASI) Hwasu Hyun, Jai-chan Hwang (KNU) 1

Homogeneity on large scales - Einstein s Cosmological Principle The Universe is homogeneous and isotropic on large scales. Spatial homogeneity No variation in matter density at sufficiently large scales. - Is the present Universe homogeneous on large scales? Homogeneity Mean density Correlation function & Power spectrum 2

Mainstream results on homogeneity test - Hogg et al. (2005) SDSS DR7 LRG (0.2 < z < 0.4) Scrimgeour et al. (2012) Ntelis (2016) Counting galaxies within a sphere, N < r Averaged N < r or D 2 r Homogeneity at r > 70 h 1 Mpc. WiggleZ Dark Energy Survey (0.1 < z < 0.9) BOSS DR12 CMASS (0.4 < z < 0.7) Homogeneity N < r r 3 D 2 r = dln N(< r ) = 3 dln r Scrimgeour+(2012) 1% deviation Transition scale of homogeneity Ntelis (2016) 3

Misconception about spatial homogeneity 1 L 1 L The homogeneity is achieved if the averaged amount within a volume is very close to the expectation of homogeneous distribution. fluctuations in the individual estimates 1 L 1 L averaging 1 L 4

Correct concept of spatial homogeneity 1 L 1 L 1 L Scatter expected in the homogeneous distribution 1 L 1 L The amount contained in a volume is the same everywhere within the scatter expected in the homogeneous distribution (due to finite data points). 5

SDSS DR7 LRG (Luminous Red Galaxies) sample - A good tracer of massive halos - LRG catalog provided by Kazin et al. (2010, ApJ 710, 1444) DR7-Full (0.16 < z < 0.47, N=105,831, V eff = 1.6 h 3 Gpc 3 ) z c = 0. 40 z c = 0. 35 z c = 0. 25 ΔR = 50h 1 Mpc Z 6

Generating random and mock catalogs LRG survey information 1,000 random catalogs Criterion for spatial homogeneity Lightcone halo catalogs from Horizon Run 3 N-body simulation (Kim et al. 2011) 1,296 mock catalogs Consistency of LRG statistics with current paradigm of cosmology 7

LRG Observed, Random, Mock data sets LRG (0.338 < z < 0.362; N=11711) Survey incompleteness map of DR7 LRG data Random 0001 (0.338 < z < 0.362; N=11854) Mock 0001 (0.338 < z < 0.362; N=11727) 8

Average and Standard deviation Counting galaxies within a sphere R.......... LRG...... R...... Random points Scaled N R = Measured N R from LRG Expected N R from random distribution 1 (homogeneity) LRG Mock Random Distribution of scaled N(R) s 9

Counting galaxies within a truncated cone - A self-consistent test of homogeneity not relying on the radial selection function and the random data points. - Count galaxies within a truncated cone that is circumscribed about the sphere of radius R located around a galaxy within a thin slice (at z c =0.35). Observer ξ R number density within a truncated cone = number density within the whole slice of 2R thickness 2R 1 (homogeneity) 10

Average and Standard deviation R=100 Mpc/h z c =0.35 LRG Mock Random Distribution of ξ s R=200 Mpc/h R=300 Mpc/h Mock (min σ) Mock (max σ) 11

Angular distributions of ξ (z c =0.35 & R=300 Mpc/h) LRG data Random Mock [max σ] Mock [min σ] ξ 12

LRG versus random & mock catalogs (ξ mean & std) R=70 Mpc/h R=100 Mpc/h z c =0.35 Mock x 1296 Random x 1000 LRG R=200 Mpc/h R=300 Mpc/h 17σ 11σ 13

Redshift dependence of ξ mean & std The LRG deviations on large scales (300 Mpc/h) from random and mock distributions still persist at lower and higher redshifts. z c =0.25 z c =0.35 z c =0.40 Mock LRG Mock LRG Mock LRG Random Random Random 14

Conclusions Homogeneity not reached even at 300 h -1 Mpc. - Consistent with current paradigm of cosmology 15

Conclusions Homogeneity not reached even at 300 h -1 Mpc. - Consistent with current paradigm of cosmology z c =0.35 slice of Δr=600 Mpc/h 16

Conclusions Homogeneity not reached even at 300 h -1 Mpc. - Consistent with current paradigm of cosmology Where is the Cosmological Principle? - Not in the observed sky (in the present epoch) - CP may stay in the theoretical cosmology (in the early era) - Needs a careful study of light propagation in nonlinear clustering stage of the Universe. 17

Supplementary Slides 18

The Astrophysical Journal, 624:54 58, 2005 May 1 Transition to homogeneity at 70 Mpc/h 19

Scrimgeour et al. (2012, MNRAS, 425, 116) Average of N(< r) and D 2 (r) WiggleZ Dark Energy Survey (N=179,599. 0.1 < z < 0.9. V eff 1h 3 Gpc 3 ) Homogeneity scale of the Universe, R H =70~80 Mpc/h http://wigglez.swin.edu.au/site/design.html 20

P. Ntelis (2016) BOSS DR12 CMASS (N=786,950. 0.43 < z < 0.70. V eff = 5.1 + 2.3 h 3 Gpc 3 ) Homogeneity scale : R H =60~70 Mpc/h bias correction 1% deviation Transition scale of homogeneity 21

Average and Standard deviation LRG Mock Random Distribution of scaled N(R) s 22

LRG versus random & mock catalogs (ξ mean & std) z c =0.35, R=70 & 100 Mpc/h Random x 1000 LRG Random x 1000 LRG Mock x 1296 Mock x 1296 23

LRG versus random & mock catalogs (ξ mean & std) z c =0.35, R=200 & 300 Mpc/h Random x 1000 LRG Mock x 1296 Random x 1000 Mock x 1296 LRG 24

Redshift dependence of ξ mean & std The LRG deviations (at 100 Mpc/h scale) from random and mock distributions still persist at lower and higher redshifts. z c =0.25 z c =0.35 z c =0.40 Mock LRG Mock LRG Mock LRG Random Random Random 25

Angular distributions of LRG s Shown are galaxies within the redshift slice with z c =0.35 & Δr=140 Mpc/h. 26

Random data points within zc=0.35 slice of Δr=600 Mpc/h 27

Angular distributions of LRG s LRG data Random 0001 R = 300 h 1 Mpc Mock 1216 [max σ(ξ)] Mock 1227 [min σ(ξ)] Shown are galaxies within the redshift slice with z c =0.35 & Δr=600 Mpc/h. 28

Counts-in-sphere using SDSS DR7 LRG data Radius (Mpc/h) Scaled N(R) mean Scaled N(R) mean (UNI) Scaled N(R) Std Scaled N(R) Std (UNI) <all galaxies used, weight average> Scaled N(R) mean Scaled N(R) mean (UNI) Scaled N(R) Std Scaled N(R) Std (UNI) <galaxies with P>0.95 used> 1 10 70272 7.13159 7.13186 5.30532 5.30547 51391 7.15964 7.15973 5.34672 5.34687 2 20 70272 2.33819 2.33819 1.52089 1.52075 51391 2.35555 2.35542 1.54078 1.54062 3 30 70272 1.55914 1.55916 0.81308 0.81306 51391 1.56972 1.56964 0.81561 0.81562 4 40 70272 1.29610 1.29610 0.53730 0.53732 51391 1.30545 1.30536 0.53655 0.53663 5 50 70272 1.17812 1.17811 0.39678 0.39679 51391 1.18700 1.18692 0.39479 0.39482 6 60 70272 1.11775 1.11775 0.31307 0.31308 51391 1.12668 1.12661 0.31053 0.31054 7 70 70272 1.08255 1.08255 0.25818 0.25817 51391 1.09122 1.09115 0.25565 0.25564 8 80 70272 1.06035 1.06034 0.21981 0.21979 51391 1.06914 1.06905 0.21715 0.21713 9 90 70272 1.04575 1.04573 0.19212 0.19209 51391 1.05477 1.05468 0.18980 0.18976 10 100 70272 1.03708 1.03706 0.17075 0.17071 51391 1.04618 1.04609 0.16844 0.16839 11 110 70272 1.03118 1.03115 0.15346 0.15341 51391 1.04037 1.04028 0.15148 0.15142 12 120 70272 1.02679 1.02676 0.13894 0.13889 51391 1.03601 1.03592 0.13712 0.13705 13 130 70272 1.02338 1.02335 0.12702 0.12695 51391 1.03282 1.03273 0.12523 0.12515 14 140 70272 1.02101 1.02097 0.11719 0.11711 51391 1.03058 1.03049 0.11543 0.11533 15 150 70272 1.01916 1.01912 0.10873 0.10865 51391 1.02863 1.02855 0.10696 0.10685 16 160 70272 1.01753 1.01750 0.10161 0.10151 51391 1.02689 1.02682 0.09995 0.09984 17 170 70272 1.01632 1.01629 0.09525 0.09515 51391 1.02551 1.02545 0.09369 0.09357 18 180 70272 1.01527 1.01523 0.08948 0.08938 51391 1.02419 1.02413 0.08790 0.08777 19 190 70272 1.01415 1.01412 0.08434 0.08423 51391 1.02271 1.02266 0.08265 0.08253 20 200 70272 1.01317 1.01314 0.07981 0.07969 51391 1.02136 1.02131 0.07796 0.07783 21 210 70272 1.01237 1.01233 0.07580 0.07568 51391 1.02024 1.02019 0.07377 0.07364 22 220 70272 1.01169 1.01166 0.07208 0.07196 51391 1.01918 1.01914 0.06988 0.06975 23 230 70272 1.01117 1.01114 0.06877 0.06865 51391 1.01833 1.01828 0.06633 0.06619 24 240 70272 1.01082 1.01079 0.06600 0.06588 51391 1.01765 1.01761 0.06329 0.06315 25 250 70272 1.01050 1.01047 0.06348 0.06337 51391 1.01704 1.01699 0.06050 0.06037 26 260 70272 1.01031 1.01028 0.06117 0.06106 51391 1.01652 1.01648 0.05793 0.05780 27 270 70272 1.01015 1.01013 0.05901 0.05889 51391 1.01609 1.01605 0.05551 0.05538 28 280 70272 1.01004 1.01001 0.05683 0.05672 51391 1.01571 1.01567 0.05311 0.05299 29 290 70272 1.00992 1.00990 0.05467 0.05457 51391 1.01537 1.01533 0.05075 0.05063 30 300 70272 1.00982 1.00980 0.05251 0.05241 51391 1.01504 1.01501 0.04843 0.04831 Number of galaxies at z=0.20~0.40 Number galaxies with completeness P>0.95 29

SDSS DR7 LRG data (for galaxies on a slice at z c = 0.35 and with Δr=50 Mpc/h) Radius (Mpc/h) Number galaxies within slice ξ mean ξ std ξ mean (UNI) 1 3 30 0.35000 50.00 9731 8691 1.31192 0.60974 1.31181 0.60961 1.32412 0.61085 1.32387 0.61067 1 4 40 0.35000 50.00 9731 8485 1.17188 0.43370 1.17182 0.43402 1.18383 0.43772 1.18369 0.43803 1 5 50 0.35000 50.00 9731 8328 1.10531 0.32506 1.10522 0.32498 1.11918 0.32727 1.11900 0.32719 1 6 60 0.35000 50.00 9731 8096 1.07385 0.26749 1.07378 0.26748 1.08746 0.26869 1.08731 0.26871 1 7 70 0.35000 50.00 9731 7902 1.05416 0.22670 1.05414 0.22668 1.06718 0.22825 1.06707 0.22827 1 8 80 0.35000 50.00 9731 7701 1.04233 0.19781 1.04232 0.19780 1.05381 0.19800 1.05374 0.19804 1 9 90 0.35000 50.00 9731 7546 1.03316 0.17157 1.03316 0.17158 1.04826 0.17166 1.04820 0.17170 1 10 100 0.35000 50.00 9731 7396 1.02742 0.15168 1.02744 0.15170 1.04324 0.14954 1.04318 0.14959 1 11 110 0.35000 50.00 9731 7304 1.02254 0.13810 1.02257 0.13805 1.03763 0.13614 1.03759 0.13612 1 12 120 0.35000 50.00 9731 7190 1.01969 0.12558 1.01970 0.12544 1.03445 0.12301 1.03437 0.12289 1 13 130 0.35000 50.00 9731 7001 1.01769 0.11476 1.01771 0.11461 1.03277 0.11141 1.03270 0.11127 1 14 140 0.35000 50.00 9731 6810 1.01522 0.10600 1.01524 0.10585 1.03083 0.10194 1.03077 0.10180 1 15 150 0.35000 50.00 9731 6636 1.01393 0.09477 1.01395 0.09464 1.02947 0.09002 1.02941 0.08991 1 16 160 0.35000 50.00 9731 6407 1.01291 0.08687 1.01292 0.08671 1.02906 0.08185 1.02897 0.08171 1 17 170 0.35000 50.00 9731 6216 1.01220 0.08032 1.01219 0.08014 1.02846 0.07477 1.02836 0.07461 1 18 180 0.35000 50.00 9731 6064 1.01187 0.07407 1.01185 0.07388 1.02836 0.06793 1.02824 0.06774 1 19 190 0.35000 50.00 9731 5868 1.01080 0.06796 1.01077 0.06778 1.02799 0.06186 1.02785 0.06169 1 20 200 0.35000 50.00 9731 5680 1.00982 0.06318 1.00979 0.06302 1.02741 0.05671 1.02727 0.05656 1 21 210 0.35000 50.00 9731 5531 1.00907 0.05903 1.00904 0.05887 1.02630 0.05219 1.02617 0.05204 1 22 220 0.35000 50.00 9731 5347 1.00895 0.05597 1.00892 0.05581 1.02642 0.04828 1.02628 0.04813 1 23 230 0.35000 50.00 9731 5184 1.00842 0.05274 1.00839 0.05259 1.02601 0.04513 1.02587 0.04498 1 24 240 0.35000 50.00 9731 5017 1.00821 0.05133 1.00818 0.05116 1.02723 0.04317 1.02707 0.04300 1 25 250 0.35000 50.00 9731 4856 1.00843 0.04893 1.00840 0.04877 1.02842 0.03995 1.02827 0.03979 1 26 260 0.35000 50.00 9731 4713 1.00848 0.04635 1.00845 0.04621 1.02864 0.03656 1.02849 0.03641 1 27 270 0.35000 50.00 9731 4567 1.00874 0.04459 1.00870 0.04446 1.03027 0.03349 1.03010 0.03335 1 28 280 0.35000 50.00 9731 4443 1.00881 0.04187 1.00876 0.04176 1.03042 0.03065 1.03027 0.03053 1 29 290 0.35000 50.00 9731 4305 1.00851 0.03876 1.00846 0.03866 1.02987 0.02762 1.02971 0.02750 1 30 300 0.35000 50.00 9731 4147 1.00846 0.03732 1.00841 0.03721 1.02984 0.02582 1.02968 0.02570 ξ std (UNI) <all galaxies used, weight average> ξ mean ξ std ξ mean (UNI) <galaxies with P>0.95 used> ξ std (UNI) Number galaxies with completeness P>0.95 30

Random catalog 0001 (for galaxies on a slice at z c = 0.35 and with Δr=50 Mpc/h) Radius (Mpc/h) Number galaxies within slice ξ mean ξ std ξ mean (UNI) 1 3 30 0.35000 50.00 9832 8656 1.03047 0.24834 1.03051 0.24838 1.02974 0.24563 1.02978 0.24567 1 4 40 0.35000 50.00 9832 8401 1.01196 0.15645 1.01197 0.15640 1.01295 0.15374 1.01295 0.15368 1 5 50 0.35000 50.00 9832 8187 1.00486 0.11336 1.00486 0.11331 1.00422 0.11197 1.00424 0.11191 1 6 60 0.35000 50.00 9832 7981 1.00196 0.08628 1.00196 0.08619 1.00101 0.08433 1.00103 0.08426 1 7 70 0.35000 50.00 9832 7767 1.00109 0.06844 1.00108 0.06836 1.00059 0.06725 1.00058 0.06721 1 8 80 0.35000 50.00 9832 7573 1.00091 0.05439 1.00091 0.05432 1.00002 0.05372 1.00002 0.05366 1 9 90 0.35000 50.00 9832 7420 1.00067 0.04593 1.00067 0.04585 1.00017 0.04509 1.00015 0.04502 1 10 100 0.35000 50.00 9832 7251 1.00079 0.03803 1.00078 0.03795 1.00008 0.03760 1.00006 0.03752 1 11 110 0.35000 50.00 9832 7158 1.00104 0.03375 1.00104 0.03370 1.00107 0.03304 1.00105 0.03300 1 12 120 0.35000 50.00 9832 7052 1.00092 0.03087 1.00093 0.03084 1.00127 0.03001 1.00127 0.02999 1 13 130 0.35000 50.00 9832 6883 1.00089 0.02770 1.00088 0.02767 1.00178 0.02718 1.00176 0.02717 1 14 140 0.35000 50.00 9832 6692 1.00058 0.02615 1.00057 0.02611 1.00091 0.02484 1.00091 0.02482 1 15 150 0.35000 50.00 9832 6533 1.00055 0.02421 1.00055 0.02418 1.00131 0.02269 1.00132 0.02269 1 16 160 0.35000 50.00 9832 6371 1.00046 0.02254 1.00046 0.02248 1.00126 0.02083 1.00127 0.02080 1 17 170 0.35000 50.00 9832 6200 1.00041 0.02053 1.00041 0.02049 1.00159 0.01920 1.00160 0.01920 1 18 180 0.35000 50.00 9832 6035 1.00025 0.01932 1.00026 0.01930 1.00115 0.01869 1.00117 0.01871 1 19 190 0.35000 50.00 9832 5877 0.99990 0.01899 0.99991 0.01897 0.99992 0.01825 0.99995 0.01826 1 20 200 0.35000 50.00 9832 5718 1.00021 0.01788 1.00022 0.01785 1.00029 0.01709 1.00031 0.01711 1 21 210 0.35000 50.00 9832 5534 1.00014 0.01573 1.00014 0.01568 1.00010 0.01482 1.00012 0.01484 1 22 220 0.35000 50.00 9832 5356 1.00003 0.01498 1.00003 0.01494 1.00010 0.01411 1.00012 0.01412 1 23 230 0.35000 50.00 9832 5186 1.00007 0.01383 1.00008 0.01380 0.99991 0.01301 0.99993 0.01303 1 24 240 0.35000 50.00 9832 5021 1.00015 0.01393 1.00015 0.01390 1.00006 0.01331 1.00008 0.01332 1 25 250 0.35000 50.00 9832 4843 0.99993 0.01339 0.99995 0.01337 0.99937 0.01286 0.99940 0.01289 1 26 260 0.35000 50.00 9832 4667 1.00006 0.01267 1.00007 0.01265 0.99957 0.01233 0.99959 0.01234 1 27 270 0.35000 50.00 9832 4497 1.00018 0.01202 1.00019 0.01201 0.99952 0.01185 0.99954 0.01187 1 28 280 0.35000 50.00 9832 4327 1.00036 0.01104 1.00036 0.01103 1.00008 0.01064 1.00009 0.01066 1 29 290 0.35000 50.00 9832 4157 1.00022 0.01053 1.00022 0.01054 0.99990 0.01027 0.99991 0.01030 1 30 300 0.35000 50.00 9832 4014 1.00024 0.01054 1.00024 0.01055 1.00007 0.01014 1.00006 0.01017 ξ std (UNI) <all galaxies used, weight average> ξ mean ξ std ξ mean (UNI) <galaxies with P>0.95 used> ξ std (UNI) Number galaxies with completeness P>0.95 31

Mock catalog 0001 (for galaxies on a slice at z c = 0.35 and with Δr=50 Mpc/h) Radius (Mpc/h) Number galaxies within slice ξ mean ξ std ξ mean (UNI) 1 3 30 0.35000 50.00 9736 8516 1.37249 0.61299 1.37241 0.61299 1.36716 0.61411 1.36704 0.61402 1 4 40 0.35000 50.00 9736 8266 1.20306 0.41709 1.20300 0.41694 1.19478 0.41452 1.19470 0.41438 1 5 50 0.35000 50.00 9736 8069 1.12614 0.31353 1.12612 0.31346 1.11769 0.30707 1.11763 0.30701 1 6 60 0.35000 50.00 9736 7881 1.08267 0.25274 1.08261 0.25265 1.07874 0.24827 1.07860 0.24817 1 7 70 0.35000 50.00 9736 7688 1.05821 0.21438 1.05817 0.21430 1.05644 0.21052 1.05634 0.21047 1 8 80 0.35000 50.00 9736 7495 1.04173 0.18125 1.04170 0.18115 1.04162 0.17819 1.04153 0.17812 1 9 90 0.35000 50.00 9736 7306 1.03083 0.15470 1.03080 0.15460 1.03391 0.15218 1.03385 0.15211 1 10 100 0.35000 50.00 9736 7158 1.02443 0.14016 1.02441 0.14010 1.03013 0.13953 1.03009 0.13945 1 11 110 0.35000 50.00 9736 7130 1.01919 0.12536 1.01919 0.12534 1.02532 0.12343 1.02531 0.12340 1 12 120 0.35000 50.00 9736 7086 1.01562 0.11491 1.01563 0.11493 1.01968 0.11172 1.01969 0.11172 1 13 130 0.35000 50.00 9736 6942 1.01273 0.10497 1.01274 0.10499 1.01634 0.10134 1.01635 0.10133 1 14 140 0.35000 50.00 9736 6796 1.01009 0.09658 1.01011 0.09660 1.01247 0.09195 1.01249 0.09195 1 15 150 0.35000 50.00 9736 6660 1.00769 0.09087 1.00771 0.09088 1.00927 0.08542 1.00930 0.08540 1 16 160 0.35000 50.00 9736 6490 1.00606 0.08493 1.00608 0.08496 1.00695 0.07982 1.00699 0.07980 1 17 170 0.35000 50.00 9736 6275 1.00481 0.07683 1.00485 0.07687 1.00421 0.07194 1.00428 0.07194 1 18 180 0.35000 50.00 9736 6084 1.00376 0.07116 1.00379 0.07122 1.00199 0.06598 1.00205 0.06601 1 19 190 0.35000 50.00 9736 5912 1.00294 0.06624 1.00297 0.06632 0.99986 0.05946 0.99991 0.05950 1 20 200 0.35000 50.00 9736 5744 1.00284 0.06100 1.00287 0.06107 0.99895 0.05216 0.99900 0.05219 1 21 210 0.35000 50.00 9736 5581 1.00258 0.05610 1.00261 0.05617 0.99865 0.04735 0.99870 0.04739 1 22 220 0.35000 50.00 9736 5408 1.00299 0.05242 1.00303 0.05249 0.99803 0.04360 0.99809 0.04364 1 23 230 0.35000 50.00 9736 5192 1.00304 0.05002 1.00309 0.05007 0.99682 0.03988 0.99687 0.03990 1 24 240 0.35000 50.00 9736 4993 1.00324 0.04697 1.00327 0.04703 0.99647 0.03667 0.99651 0.03670 1 25 250 0.35000 50.00 9736 4796 1.00257 0.04402 1.00260 0.04406 0.99465 0.03388 0.99469 0.03389 1 26 260 0.35000 50.00 9736 4644 1.00188 0.04330 1.00192 0.04335 0.99200 0.03337 0.99203 0.03339 1 27 270 0.35000 50.00 9736 4498 1.00135 0.04114 1.00138 0.04116 0.99118 0.03049 0.99120 0.03048 1 28 280 0.35000 50.00 9736 4321 1.00121 0.03799 1.00123 0.03801 0.99177 0.02678 0.99178 0.02677 1 29 290 0.35000 50.00 9736 4147 1.00093 0.03580 1.00095 0.03581 0.99177 0.02427 0.99177 0.02425 1 30 300 0.35000 50.00 9736 3959 1.00050 0.03344 1.00051 0.03345 0.99106 0.02260 0.99106 0.02258 ξ std (UNI) <all galaxies used, weight average> ξ mean ξ std ξ mean (UNI) <galaxies with P>0.95 used> ξ std (UNI) Number galaxies with completeness P>0.95 32

SDSS DR7 LRG data (for galaxies on a slice at z c = 0.35 and with Δr=50 Mpc/h) R(Mpc/h) Number of all galaxies within slice Mean of ξ STD of ξ Skewness of ξ Kurtosis of ξ Number of galaxies with P>0.95 within slice Mean of ξ STD of ξ Skewness of ξ Kurtosis of ξ 3 30 1 9731 1.31192 0.60974 0.98144 1.92384 8691 1.32412 0.61085 0.99429 2.01207 4 40 1 9731 1.17188 0.43370 1.14820 3.03514 8485 1.18383 0.43772 1.18275 3.15204 5 50 1 9731 1.10531 0.32506 1.04308 2.67258 8328 1.11918 0.32727 1.10777 2.79467 6 60 1 9731 1.07385 0.26749 1.05036 2.47187 8096 1.08746 0.26869 1.14070 2.607 77 7 70 1 9731 1.05416 0.22670 0.94110 1.49527 7902 1.06718 0.22825 0.98392 1.49129 8 80 1 9731 1.04233 0.19781 0.83489 0.91226 7701 1.05381 0.19800 0.84763 0.84305 9 90 1 9731 1.03316 0.17157 0.69804 0.49745 7546 1.04826 0.17166 0.69908 0.443 24 10 100 1 9731 1.02742 0.15168 0.55119 0.35253 7396 1.04324 0.14954 0.54207 0.38406 11 110 1 9731 1.02254 0.13810 0.37271-0.00301 7304 1.03763 0.13614 0.33687-0.00265 12 120 1 9731 1.01969 0.12558 0.26455-0.15459 7190 1.03445 0.12301 0.24830-0.20129 13 130 1 9731 1.01769 0.11476 0.21790-0.20308 7001 1.03277 0.11141 0.21655-0.26828 14 140 1 9731 1.01522 0.10600 0.18101-0.27608 6810 1.03083 0.10194 0.19807-0.43411 15 150 1 9731 1.01393 0.09477 0.08829-0.31413 6636 1.02947 0.09002 0.12670-0.56634 16 160 1 9731 1.01291 0.08687 0.04911-0.39940 6407 1.02906 0.08185 0.05434-0.67158 17 170 1 9731 1.01220 0.08032 0.02741-0.48519 6216 1.02846 0.07477-0.00407-0.70223 18 180 1 9731 1.01187 0.07407-0.01337-0.64337 6064 1.02836 0.06793-0.09348-0.69197 19 190 1 9731 1.01080 0.06796-0.04492-0.72654 5868 1.02799 0.06186-0.19508-0.58098 20 200 1 9731 1.00982 0.06318-0.10540-0.70903 5680 1.02741 0.05671-0.31097-0.36094 21 210 1 9731 1.00907 0.05903-0.14613-0.75148 5532 1.02630 0.05218-0.35864-0.31722 22 220 1 9731 1.00895 0.05597-0.21185-0.74858 5347 1.02642 0.04828-0.41482-0.20435 23 230 1 9731 1.00842 0.05274-0.20501-0.73906 5184 1.02601 0.04513-0.33218-0.28444 24 240 1 9731 1.00821 0.05133-0.21535-0.75317 5019 1.02720 0.04318-0.30677-0.26050 25 250 1 9731 1.00843 0.04893-0.19940-0.72645 4856 1.02842 0.03995-0.20874-0.31988 26 260 1 9731 1.00848 0.04635-0.26049-0.72799 4713 1.02864 0.03656-0.21538-0.42088 27 270 1 9731 1.00874 0.04459-0.32435-0.70109 4567 1.03027 0.03349-0.23229-0.48609 28 280 1 9731 1.00881 0.04187-0.34088-0.66239 4443 1.03042 0.03065-0.19951-0.53924 29 290 1 9731 1.00851 0.03876-0.34773-0.62939 4305 1.02987 0.02762-0.17236-0.58881 30 300 1 9731 1.00846 0.03732-0.37512-0.55638 4147 1.02984 0.02582-0.11020-0.65208 33

SDSS DR7 LRG data (for galaxies on a slice at z c = 0.25 and with Δr=50 Mpc/h) R(Mpc/h) Number of all galaxies within slice Mean of ξ STD of ξ Skewness of ξ Kurtosis of ξ Number of galaxies with P>0.95 within slice Mean of ξ STD of ξ Skewness of ξ Kurtosis of ξ 1 50 1 4993 1.12243 0.31105 0.17644-0.25604 4101 1.13245 0.30999 0.20167-0.25966 2 100 1 4993 1.02335 0.13570-0.08058-0.09055 3565 1.03055 0.13591-0.18524-0.17861 3 150 1 4993 1.00823 0.06958-0.69991 0.39602 3052 1.02274 0.06742-0.96353 1.11023 4 200 1 4993 1.00892 0.05437-0.59895-0.07582 2509 1.02816 0.04483-0.66990 0.49947 5 250 1 4993 1.00946 0.04499-0.82104 0.25276 1964 1.02851 0.03061-0.70381-0.36169 6 300 1 4993 1.00871 0.03297-0.65849 0.21436 1450 1.02472 0.01937-0.44837-0.54607 SDSS DR7 LRG data (for galaxies on a slice at z c = 0.40 and with Δr=50 Mpc/h) R(Mpc/h) Number of all galaxies within slice Mean of ξ STD of ξ Skewness of ξ Kurtosis of ξ Number of galaxies with P>0.95 within slice Mean of ξ STD of ξ Skewness of ξ Kurtosis of ξ 1 50 1 6918 1.13512 0.35224 0.39120-0.13289 5812 1.13259 0.34615 0.36241-0.12048 2 100 1 6918 1.02584 0.14475-0.00815-0.47354 5177 1.03400 0.14354-0.03870-0.44587 3 150 1 6918 1.01362 0.09350 0.08219-0.26963 4823 1.02705 0.09134 0.10715-0.41563 4 200 1 6918 1.00968 0.06850 0.26320-0.36570 4385 1.02661 0.06459 0.24429-0.59090 5 250 1 6918 1.00938 0.05804 0.07774-0.72071 3858 1.02893 0.04931 0.04851-0.80426 6 300 1 6918 1.00816 0.04686-0.11717-0.94335 3338 1.02834 0.03703-0.18347-0.70313 34

SDSS DR7 LRG data (for Healpix pixels on a slice at z c = 0.35 and with Δr=50 Mpc/h) - Count the number of galaxies within the specified truncated cone with R=300 Mpc/h centered on each Healpix pixel with Nside=128 (55563/196608 [28%]) Catalog Number LRG Radius (Mpc/h) redshift Number galaxies within slice <all galaxies used, weight average> ξ mean ξ std <galaxies with P>0.95 used> ξ mean ξ std Random Mock 1 1 300 0.35000 50.00 9731 55563 1.00689 0.03706 1.02895 0.02589 1 1 300 0.35000 50.00 9832 55563 1.00010 0.01063 1.00004 0.01017 653 1 300 0.35000 50.00 9754 55563 0.99329 0.02931 0.98101 0.02365 79 1 300 0.35000 50.00 9729 55563 1.00767 0.03163 1.02200 0.02579 1216 1 300 0.35000 50.00 9733 55563 0.99942 0.01539 0.99732 0.01426 1227 1 300 0.35000 50.00 9750 55563 0.99979 0.04690 0.99971 0.04478 35