26 unstable wavenumbers correspond to k < k$_{J}$ where the Jeans wavenumber is given by Eq. (113). The growth rate is maximum for k$_{∗}$ = k$_{f}$ i.e. for the maximum wavelength λ$_{f}$ = 2 π/k$_{f}$ . In that case, we have only one cluster. The corresponding value of the growth rate is σ$_{∗}$ = S$_{d}$Gρ − k$_{B}$T k 2 $_{f}$/m . The two limit cases discussed above are illustrated in Fig. 51. We now analyze the thermodynamical stability of the homogeneous phase by using variational principles. Ba- sically, we have to solve the maximization problem (3) in MCE and the minimization problem (17) in CE. How- ever, for spatially homogeneous systems, it is shown in Appendix A that they are both equivalent to the mini- mization problem (24). Therefore, the system is stable iff the second order variations of free energy (27) are posi- tive definite for any perturbations δρ that conserve mass, i.e. ∫ δρ d r = 0. We are led therefore to considering the eigenvalue problem FIG. 50: Evolution of k m , k$_{∗}$ and σ$_{∗}$ as a function of the temperature. The parameters have been scaled such that k 0 = 1 and T c = 1. 0 0.2 0.4 0.6 0.8 1 T 1 2 3 4 k, k$_{m}$(T) k$_{*}$(T) $_{*}$(T) T$_{c}$ T$_{c}$ * B. Thermodynamical stability 0 2 4 6 8 10 k 2 0.2 0.4 0.6 0.8 1 k$_{0}$ = 0 T  T = 0 k$_{J}$ k$_{0}$ Many clusters One cluster δ Φ$_{λ}$ + k$_{B}$T ρm δρ$_{λ}$ = λδρ$_{λ}$, (125) Δ δ Φ$_{λ}$ − k 2 $_{0}$δ Φ$_{λ}$ = S$_{d}$Gδρ$_{λ}$. (126) δρ ( r ) = A$_{k}$ψ$_{k}$ ( r ) , δ Φ( r ) = − S$_{d}$GA$_{k}$ k 2 + k 2 0 ψ$_{k}$ ( r ) , (127) λ ( k ) = − S$_{d}$G k 2 + k 2 0 + k$_{B}$T ρm , (128) S$_{d}$G k 2 + k 2 0 − k$_{B}$T ρm < 0 , (129) σ$_{k}$ = k$_{B}$T$_{c}$ m k 2 $_{0}$k 2 k 2 0 + k $_{2}$. (123) FIG. 51: Growth ( σ > 0) rate as a function of the wavenumber k in two limits: (i) the Newtonian limit k 0 = 0 (and T = 0) for which the maximum growth rate corresponds to k$_{∗}$ = k$_{f}$ << 1 (large scales), and (ii) the cold limit T = 0 (and k 0 = 0) for which the maximum growth rate corresponds to k$_{∗}$ → $^{+}$∞ (small scales). If all the eigenvalues λ are positive, then the system is stable since δ $^{2}$F = 1 2 ∑ λ λa 2 λ > 0 where the perturbation has been decomposed under the form δρ = ∑ λ a$_{λ}$δρ$_{λ}$ . If at least one eigenvalue is negative, the system is unstable since δ $^{2}$F = 1 $_{2}$λ ∫ ( δρ$_{λ}$ ) 2 d r < 0 for that perturbation. It is easy to see that the eigenfunctions are and that the corresponding eigenvalues are for all quantized k (see Sec. V A). We note that ∫ ψ$_{k}$ d r = − 1 k 2 ∫ Δ ψ$_{k}$ d r = − 1 k 2 ∮ ∇ ψ$_{k}$ · d S = 0, so that δρ d r = 0 as required. Regrouping all these results, we conclude that the system is stable iff Therefore, the system is stable. More generally, for T ≥ T$_{c}$ , the system is stable. For T → + ∞ , we have σ$_{k}$ = − k$_{B}$T m k $^{2}$. • For T = 0, we have k$_{m}$ → + ∞ , k$_{∗}$ → + ∞ , σ$_{∗}$ → k$_{B}$T$_{c}$k 2 $_{0}$/m and for all (quantized) k . This returns the stability condi- tion obtained in Sec. V A. Therefore, the system is stable iff T > T ∗ c . If T < T ∗ c , the homogeneous phase is an unstable saddle point of free energy at fixed mass. This method proves the thermodynamical stability of the homogeneous phase in the canonical and microcanonical ensembles. This implies the stability with respect to the The growth rate is maximum for k$_{∗}$ → + ∞ , i.e. for very small wavelengths λ$_{∗}$ → 0. In that case, we expect a very large number of clusters in the linear regime. • For k$_{0}$ = 0 (modified Newtonian model), we have The system is unstable for T < T ∗ c where the critical temperature is given by Eq. (112). Furthermore, the σ$_{k}$ = S$_{d}$Gρ − k$_{B}$T m k $^{2}$. (124)