Dynamical instability of polytropic spheres in spacetimes with a cosmological constant

The dynamical instability of relativistic polytropic spheres, embedded in a spacetime with a repulsive cosmological constant, is studied in the framework of general relativity. We apply the methods used in our preceding paper to study the trapping polytropic spheres with $\Lambda = 0$, namely, the critical point method and the infinitesimal and adiabatic radial perturbations method developed by Chandrasekhar. We compute numerically the critical adiabatic index, as a function of the parameter $\sigma = p_{\mathrm{c}}/(\rho_{\mathrm{c}} c^2)$, for several values of the cosmological parameter $\lambda$ giving the ratio of the vacuum energy density to the central energy density of the polytrope. We also determine the critical values for the parameter $\sigma_{\mathrm{cr}}$, for the onset of instability, by using both approaches. We found that for large values of the parameter $\lambda$, the differences between the values of $\sigma_{\mathrm{cr}}$ calculated by the critical point method differ from those obtained via the radial perturbations method. Our results, given by both applied methods, indicate that large values of the cosmological parameter $\lambda$ have relevant effects on the dynamical stability of the polytropic configurations.