**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

λ_{c}=λ_{c}(n ,σ ). In the case of n >3 , the

critical value λ_{c} 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 0^{12} M_{☉} . For such largely extended

polytropes, the cosmological parameter relating the vacuum energy to the

central density has to be larger than λ

=ρ_{vac}/ρ_{c}̃10^{-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;