| 74 | | [[latex($\rho_1 =$)]] |
| | 74 | [[latex($\rho_1 \propto e^{-i i [4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]t}$)]] |
| | 75 | |
| | 76 | which gives |
| | 77 | |
| | 78 | [[latex($\rho_1 = C \cos(kx) e^{[4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})]t}$)]] |
| | 79 | |
| | 80 | |
| | 81 | Thus we have found the condition on [[latex($\omega$)]] which gives unstable perturbation modes. Now, the growth rate of these modes is given by |
| | 82 | |
| | 83 | [[latex($\Gamma = 4 \pi G \rho_0 (1-\frac{\lambda_J^2}{\lambda^2})$)]] |
| | 84 | |
| | 85 | and so the characteristic timescale for exponential growth, that is the time in which the perturbation increases by a factor of e, is given by |
| | 86 | |
| | 87 | [[latex($\tau = \frac{1}{\Gamma}$)]] |
