AR_final file_2018-19

the cluster. Hence the filamentarity ( F ) of large clusters in both overdense and under- dense segments increases with their volume and saturates to unity for extremely large clus- ters. The similar values of T and B imply that the filament-like large clusters have simi- lar cross sections (estimated by the product of the first two Shapefinders T × B ) and their shapes only differ in terms of their lengths. The reason is that the larger clusters form due to mergers of relatively smaller clusters which are themselves large enough to possess similar values of T and B . They find that the genus of the cluster too increases with the volume. A high value of genus makes the clusters porous with many tunnels passing through. One can thus imagine the large clusters have multiply connected structures made of many filamen- tary branches and sub-branches. Spherical collapse of fuzzy dark mat- ter Fuzzy Dark Matter (FDM) has been proposed as an alternative to Cold Dark Matter (CDM). FDM is comprised of ultra light Bosons, of masses in the range of 10 − 24 − 10 − 22 eV, which exists as a Bose- Einstein condensate. Due to the very low mass of FDM, the de Broglie wavelength of these Bosons is of the order of kpc and the quantum effects man- ifest at those scales. Hence, unlike CDM, FDM experiences quantum pressure along with gravita- tional attraction. In this work, V. Sreenath in- vestigated the gravitational collapse of a spherically symmetric FDM overdensity. The equation of motion of a spherically symmet- ric shell containing an overdense region of FDM contains contributions from both gravity as well as quantum pressure. Assuming a power law density profile, he could obtain a parametric solution to the evolution of radius L , of the spherical shell for the case of non-interacting Bosons, as L = A (1 − e cos ϑ ) and t = p mA 3 /k ( ϑ − e sin ϑ ), where A and k are quantities that depend on the mass of the Bosons m , the mass of the overdense region M , and the first integral of motion E . In the above expression, e is given by e = q 1 + E ~ 2 G 2 M 2 m 3 (2 γ − γ 2 ) 2 , where γ is the index of the power law profile. Since, for a bound system E < 0, e < 1, from the solutions for L , one can see that the shell will initially expand along with the Hubble flow, then turn around and start to collapse. Since e < 1, unlike CDM, the shell will not collapse to zero radius, but it will start expanding again once it reaches the minimum radius L = A (1 − e ). Thus, this toy model captures partly the competing effects of gravity and pressure. From the expression for e , one can also see that in the limit ~ /m → 0, CDM. behaviour is retrieved. Finally, as a shell is contracting, it will interact with inner shells which are expanding again after their initial contraction. When they interact, dif- ferent shells will repel each other due to the quan- tum pressure and repel or attract each other ac- cording to their force of interaction. This would cause the density profile to depart from its initial power law shape, which in turn would imply that the solutions derived above will not be valid much beyond the turn around radius. Using virial theo- rem, he derived expressions for averaged overden- sity in collapsed halo in the linear theory (taking the small ϑ limit) and the full theory. The results have been summarized in the Table 1. It shows that in this model, when the linear averaged over- density reaches a critical value, ¯ δ ≃ 1 . 69 /e 2 / 3 , the overdense region would have virialized to form a halo. t Linear theory Full theory t ta 1 . 06 e 2 / 3 9 2 π 2 (1+ e ) 3 − 1 t vir 1 . 69 e 2 / 3 18 π 2 8 (1 + e ) 3 − 1 Table 1: The averaged overdensity ¯ δ , in the linear and the full theory at turn around and virialization. It can be verified that the averaged overdensity matches with the CDM result in the limit e → 1. ( 64 )