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($\tau = \frac{1}{\Gamma}$)]] |