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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 

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 

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 

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When T < T ∗ c , the system is unstable for the modes such 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 

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. 

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Let us consider some particular cases: 

• For T = T$_{c}$ , we have k$_{m}$ = 0, k$_{∗}$ = 0, σ$_{∗}$ = 0 and 

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