25 When T < T ∗ c , the system is unstable for the modes such that relating σ$_{k}$ to the wavenumber k . The amplitudes A$_{k}$ and B$_{k}$ are determined by the initial condition. We see that σ$_{k}$ is real so that the perturbation either grows or decays exponentially rapidly. The homogeneous phase will be spectrally stable if σ$_{k}$ < 0 for all k and it will be spectrally unstable if there exists one or several modes for which σ$_{k}$ > 0. We note that the dispersion relation (108) is the same as in an infinite domain [57]. However, in a finite domain, the allowed wavenumbers k are quantized while they are continuous in an infinite domain. According to Eq. (108), the system will be unstable if there exists k = 0 such that Therefore, a necessary condition of instability is that The growth rate can be written where It achieves its maximum value for k = k$_{∗}$ ( T ) where The corresponding value of the growth rate is k < k$_{0}$ ( T$_{c}$ T − 1 ) 1 / 2 ≡ k$_{m}$ ( T ) . (117) σ$_{k}$ = k$_{B}$T m k $^{2}$( k$_{m}$ ( T ) 2 − k $^{2}$) k 2 0 + k 2 , (118) k$_{m}$ ( T ) = k$_{0}$ ( T$_{c}$ T − 1 ) 1 / 2 . (119) k$_{∗}$ ( T ) = k$_{0}$ [ ( T$_{c}$ T ) 1 / 2 − 1 ] 1 / 2 . (120) S$_{d}$Gρ k 2 + k 2 0 > k$_{B}$T m . (109) k$_{B}$T m < S$_{d}$Gρ k 2 f + k 2 0 ≡ k$_{B}$T ∗ c m , (110) σ$_{∗}$ ( T ) = k$_{B}$T$_{c}$ m k 2 0 [ 1 − ( T T$_{c}$ ) 1 / 2 ] 2 . (121) k 2 < S$_{d}$Gmρ k$_{B}$T − k 2 0 ≡ k 2 $_{m}$. (111) k$_{B}$T ∗ c m = S$_{d}$Gρ k 2 f , (112) k 2 < S$_{d}$Gmρ k$_{B}$T ≡ k 2 $_{J}$. (113) The number of clusters that is expected to form in the linear regime is N ( T ) = R/ (2 π/k$_{∗}$ ( T )). For a fixed value of k$_{0}$ , this number increases as the temperature decreases. The behaviour of the different quantities defined above is represented in Figs. 49 and 50. where k$_{f}$ is the smallest non-zero wavenumber. For T > T ∗ c , the homogeneous distribution is stable for per- turbations with arbitrary wavenumbers. For T < T ∗ c , the homogeneous distribution is unstable for perturba- tions with wavenumbers For k$_{0}$ = 0, the critical temperature is and we recover the Jeans criterion In the general case, the instability criterion can be writ- ten We see that, for the screened Newtonian potential ( k$_{0}$ = 0), the instability occurs for larger wavelengths as com- pared to the Newtonian model ( k$_{0}$ = 0). Let us introduce the notation 0 0.5 1 1.5 2 k 2 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 k$_{m}$(T) k$_{*}$(T) FIG. 49: Growth ( σ > 0) or decay ( σ < 0) rate as a function of the wavenumber k . The system is unstable for k < k m ( T ) and the maximum growth rate is reached for k = k$_{∗}$ ( T ). The parameters have been scaled such that k 0 = 1, T c = 1, T = 1 / 2. which corresponds to the critical temperature in an infi- nite domain ( k$_{f}$ = 0). Since the dispersion relation (108) does not explicitly depend on k$_{f}$ , it is convenient to in- troduce the notation (115). We have Let us consider some particular cases: • For T = T$_{c}$ , we have k$_{m}$ = 0, k$_{∗}$ = 0, σ$_{∗}$ = 0 and k 2 < k 2 J − k 2 0 ≡ k 2 $_{m}$. (114) k$_{B}$T$_{c}$ = S$_{d}$Gmρ k 2 0 , (115) T ∗ c = T$_{c}$ 1 + ( k$_{f}$/k$_{0}$ ) $_{2}$. (116) σ$_{k}$ = − k$_{B}$T$_{c}$ m k 4 k 2 0 + k 2 < 0 . (122)