Publication date: Nov 2016
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 σ , and the ratio of
energy density of vacuum and central density of matter λ . 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 , σ , are allowed
only up to a critical value of the cosmological parameter
λc=λc(n ,σ ). In the case of n >3 , the
critical value λc tends to zero for special values of σ . 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 ℓ>100 kpc and mass M
>1 012 M☉ . For such largely extended
polytropes, the cosmological parameter relating the vacuum energy to the
central density has to be larger than λ
=ρvac/ρc̃10-9. 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.
Authors:
Stuchlík, Zdeněk; Hledík, Stanislav; Novotný, Jan;