Restrictions on the polytrope parameters. Unified (n–σ) plot for possible finite polytropic configurations distinguished by the cosmological parameter λ. Considered polytropes have the parameter pairs n–σ limited by the σ_causal boundary (black line) and the boundaries determined by the static radius given by a chosen λ (thick colored line); monotonically decreasing pressure is expected in the whole interior of the configuration. Thin colored borderlines depict the transition between polytropes stable and unstable against the radial oscillations. For given n the parameter σ needs to be lower than that depicted by this border. For λ < 1/10⁴ these borderlines are all hardly distinguishable from the case λ = 0. They are determined using both trial functions, and the plotted ones represent the more restrictive for the chosen value of λ. Generally, no stable configurations having σ < σ_casual and n > 3 exist. A detailed view of the boundaries for lower λ are on the next figure.