Jeans instability in GR and f(R) gravity

In this thesis, gravitational instability of self-gravitating systems was studied using Jeans analysis, by coupling the collisionless Boltzmann equation with the weak-field, non-relativistic limit of GR and f(R). By studying how a system in equilibrium responds to perturbations in its density, Jeans analysis allows us to determine the conditions under which these disturbances grow exponentially, rendering the system gravitationally unstable and subject to collapse. First, the standard results of the response of self-gravitating and non-self gravitating fluids to perturbations in their densities is reviewed. For non-self gravitating fluids density perturbations oscillate throughout the system, whereas, for self-gravitating fluids in the Newtonian limit of GR, the arising dispersion relation exhibits a critical limit, defined by the Jeans wavenumber kJ, below which the perturbations exponentially grow, and the system becomes unstable. Then, the dispersion relation that arises for stellar systems described by a Maxwell Boltzmann distribution is analysed in the Newtonian limit of GR, in response to small perturbations in their densities. For one-component stellar systems, composed entirely of baryonic matter, the instability limit is defined by the critical Jeans mass MJ, which if exceeded the system is unstable and subject to collapse upon its own gravitation. The work of [42] in two-component stellar systems, composed of dark and baryonic matter is revisited. Particularly, for the mass density and velocity dispersion ratios between dark and baryonic matter given by Ïd = 5.5 σ Ïb and d = 1.83, first the result of [42] is reproduced, in which the two-component σb system seems to lower the limit of instability in comparison with the a dark matter dominated system such that the new Jeans mass M(db)J ≈ 0.58MJd. Then this re- sult is extended by comparing the two-component system to a system dominated by baryonic matter, in which case, the instability limit appears to be raised such that M(db)J ≈ 1.5MJ . Lastly, the work of [5, 18, 68] in analysing the dispersion relations for one-component stellar systems in f(R) is revisited, and then it is extend to two- components. It is shown that f(R) appears to lower the limit of instability in both cases, lowering the critical mass required to initiate the exponential growth in the perturbations