General relativistic polytropes with a repulsive cosmological constant

Abstract

Spherically symmetric equilibrium configurations of perfect fluid obeying a polytropic equation of state are studied in spacetimes with a repulsive cosmological constant. The configurations are specified in terms of three parameters---the polytropic index $n$, the ratio of central pressure and central energy density of matter $\ensuremath{\sigma}$, and the ratio of energy density of vacuum and central density of matter $\ensuremath{\lambda}$. The static equilibrium configurations are determined by two coupled first-order nonlinear differential equations that are solved by numerical methods with the exception of polytropes with $n=0$ corresponding to the configurations with a uniform distribution of energy density, when the solution is given in terms of elementary functions. The geometry of the polytropes is conveniently represented by embedding diagrams of both the ordinary space geometry and the optical reference geometry reflecting some dynamical properties of the geodesic motion. The polytropes are represented by radial profiles of energy density, pressure, mass, and metric coefficients. For all tested values of $n>0$, the static equilibrium configurations with fixed parameters $n$, $\ensuremath{\sigma}$, are allowed only up to a critical value of the cosmological parameter ${\ensuremath{\lambda}}_{\mathrm{c}}={\ensuremath{\lambda}}_{\mathrm{c}}(n,\ensuremath{\sigma})$. In the case of $n>3$, the critical value ${\ensuremath{\lambda}}_{\mathrm{c}}$ tends to zero for special values of $\ensuremath{\sigma}$. The gravitational potential energy and the binding energy of the polytropes are determined and studied by numerical methods. We discuss in detail the polytropes with an extension comparable to those of the dark matter halos related to galaxies, i.e., with extension $\ensuremath{\ell}>100\text{ }\text{ }\mathrm{kpc}$ and mass $M>1{0}^{12}\text{ }{\mathrm{M}}_{\ensuremath{\bigodot}}$. For such largely extended polytropes, the cosmological parameter relating the vacuum energy to the central density has to be larger than $\ensuremath{\lambda}={\ensuremath{\rho}}_{\text{vac}}/{\ensuremath{\rho}}_{\mathrm{c}}\ensuremath{\sim}{10}^{\ensuremath{-}9}$. We demonstrate that the extension of the static general relativistic polytropic configurations cannot exceed the so-called static radius related to their external spacetime, supporting the idea that the static radius represents a natural limit on the extension of gravitationally bound configurations in an expanding universe dominated by the vacuum energy.