AR_final file_2018-19

given density threshold, for both the HI overdense and underdense excursion sets separately: The filling factor ( FF ) FF = total vol . of all neutral / ionized regions volume of the simulation box . (5) The “largest cluster statistic” (LCS) LCS = vol . of the largest cluster total vol . of all the clusters , (6) ff largest = vol . of the largest cluster vol . of the simulation box (7) = FF ionized × LCS (8) ff other = vol of all clusters except largest cluster vol . of the simulation box (9) ≡ FF ionized − ff largest = FF ionized × (1 − LCS) . The fractions ff largest and ff other are essentially the filling factors of the largest cluster and the rest of the clusters (all the clusters excluding the largest cluster) respectively. Satadru Bag , et al. have studied the shapes of the ionized regions at different redshifts (various stages of reionization) using Shapefinders, which are derived from Minkowski functionals. The mor- phology of a closed two dimensional surface em- bedded in three dimensions is well described by the four Minkowski functionals: (i) Volume: V , (ii) Surface area: S , (iii) Integrated mean cur- vature (IMC): C = 1 2 H ( κ 1 + κ 2 ) dS , (iv) Inte- grated Gaussian curvature or Euler characteristic: χ = 1 2 π H ( κ 1 κ 2 ) dS Here κ 1 and κ 2 are the two principle curvatures at any point on the surface. The fourth Minkowski functional (Euler characteristic) is a measure of the topology of the surface. It can be written in terms of the genus (G) as G = 1 − χ/ 2 . The ‘Shapefinders’ are ratios of these Minkowski functionals, namely (i) Thickness: T = 3 V/S , (ii) Breadth: B = S/C , (iii) Length: L = C/ (4 π ). The Shapefinders T, B, L , have dimension of length, and can be interpreted as providing a mea- sure of the three physical dimensions of an object. Using the Shapefinders they have determined the morphology of an ionized region by means of the following dimensionless quantities which character- ize its planarity and filamentarity Planarity : P = B − T B + T , Filamentarity : F = L − B L + B . (10) For a planar object (such as a sheet) P ≫ F , while the reverse is true for a filament which has F ≫ P . A ribbon will have P ∼ F ≫ 0 whereas P ≃ F ≃ 0 for a sphere. In all cases 0 ≤ P, F ≤ 1. The results in Figures 5 and 6 can be summarized as follows: 1. The overdense and underdense segments per- colate at different critical density thresholds corresponding to different values of the respec- tive filling factors. To explore this asymme- try in percolation more quantitatively, one can study the so called “percolation curves” which are essentially plots of the filling factors of the largest cluster against the corresponding FF ionized for overdense and underdense seg- ments. The area under the percolation curve is a robust geometric measure of non-gaussianity. They find that the area under the hysteresis in the percolation curves at redshifts 13,11 and 10 increases as reionization proceeds which in turn indicates that the non-gaussianity in the HI density field increases. 2. Since most of the large clusters appear just before percolation, they have studied the be- haviour of the Minkowski functionals for val- ues of density thresholds corresponding to the onset of percolation in both overdense and un- derdense segments. The clusters have different Minkowski functionals but their ratios, defined as Shapefinders, show interesting properties. The thickness ( T ) and breadth ( B ) of large clusters in both overdense and underdense seg- ments are of similar values but the length ( L ) increases almost linearly with the volume of ( 62 )