**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 $sigma$, and the

ratio of energy density of vacuum and central density of matter

$lambda$. 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

$n$, $sigma$, are allowed only up to a critical value of the

cosmological parameter

$lambda_{mathrm{c}}=lambda_{mathrm{c}}(n,sigma)$. In the case of

$n>3$, the critical value $lambda_{mathrm{c}}$ tends to zero for

special values of $sigma$. 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 $ell > 100,mathrm{kpc}$ and mass $M >

10^{12},mathrm{M}_{odot}$. …

**Authors:**

Stuchlík, Zdeněk; Hledík, Stanislav; Novotný, Jan;