Dynamical instability of polytropic spheres in spacetimes with a cosmological constant

TL;DR

This paper investigates how a positive cosmological constant affects the dynamical stability of relativistic polytropic spheres, revealing that larger values of the cosmological parameter significantly influence their stability thresholds.

Contribution

It extends previous stability analyses by incorporating a non-zero cosmological constant and compares two methods to determine the critical stability parameters.

Findings

Large cosmological constant values alter the critical adiabatic index.

Differences between methods increase with higher cosmological parameters.

Cosmological constant significantly impacts the stability of polytropic spheres.

Abstract

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.