AR_final file_2018-19

10 − 7 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0 FF overdense 10 − 8 10 − 7 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0 ff largest ff other LCS 1 − LCS (a) HI overdense segment 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0 FF underdense 10 − 7 10 − 6 10 − 5 10 − 4 10 − 3 10 − 2 10 − 1 10 0 ff largest ff other LCS 1 − LCS (b) HI underdense segment Figure 5: The cluster statistics at z = 13: The four fractions, ff largest , ff other , LCS and (1 − LCS) for HI overdense and underdense segments are plotted against corresponding filling factors in left and right panel respectively. In both panels the percolation transition for overdense region is shown by black dotted vertical line while the cyan dotted vertical line represents the percolation in the underdense segment. In both the cases, during percolation, the fractions ff largest and LCS rise steeply while the other two fractions ff other and (1 − LCS) decay sharply. These sharp changes in these ‘ordered parameters’ can also independently define the percolation transition in either segment. 10 2 10 3 10 4 10 5 Volume in (0 . 56 × Mpc ) 3 1 . 0 1 . 2 1 . 4 1 . 6 1 . 8 2 . 0 2 . 2 2 . 4 2 . 6 T and B in (0 . 56 × Mpc ) 10 0 10 1 10 2 10 3 10 4 L in (0 . 56 × Mpc ) Thickness T Breadth B Length L (a) Thickness, breadth and length 10 2 10 3 10 4 10 5 Volume in (0 . 56 × Mpc ) 3 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 Planarity and Filamentarity 10 − 1 10 0 10 1 10 2 10 3 10 4 Genus Planarity P Filamentarity: F Genus (b) Planarity and filamentarity Figure 6: Shapefinders, planarity, filamentarity and genus of HI overdense clusters at z = 13 are binned in volume for a threshold just before percolation. Note that the values of Shapefinders are quoted in comoving scale. In the left panel T , B and L are fitted to straight lines shown by the dotted lines with respective colours. The slopes of the best fit straight lines to T , B are negligible. On the other hand, the best fit straight line to log L vs log V has slope 0 . 945 ± 0 . 01 which is O (1). This shows T and B of sufficiently large clusters increase very slowly with the cluster volume V but L almost increases linearly V . From the right panel it is evident that the filamentarity increases with cluster volume and almost saturates to unity for large clusters. Filamentarity in volume bins are joined by dashed line for visual guidance. But the planarity does not increase much, hence the best fit straight line again has a negligible slope. Genus of clusters also increases almost linearly with the volume. ( 63 )