Research Note
A brief analysis of selfgravitating polytropic models with a nonzero cosmological constant
^{1}
University of Rome “La Sapienza”Department of Physics,
Piazzale Aldo Moro 2,
00185
Rome,
Italy
email: marco.merafina@roma1.infn.it
^{2}
Space Research Institute (IKI) Profsoyuznaya 84/32,
117997
Moscow,
Russia
email: gkogan@iki.rssi.ru
^{3}
National Research Nuclear University MEPHI,
Kashirskoe Shosse
31, 115409
Moscow,
Russia
email: trsvsrg@mail.ru
Received:
21
September
2011
Accepted:
24
February
2012
Context. We investigate the equilibrium and stability of polytropic spheres in the presence of a nonzero cosmological constant.
Aims. We solve the Newtonian gravitational equilibrium equation for a system with a polytropic equation of state of the matter P = Kρ^{γ} introducing a nonzero cosmological constant Λ.
Methods. We consider the cases of n = 1, 1.5, 3 and construct series of solutions with a fixed value of Λ. For each value of n, the nondimensional equilibrium equation has a family of solutions, instead of the unique solution of the LaneEmden equation at Λ = 0.
Results. The equilibrium state exists only for central densities ρ_{0} higher than the critical value ρ_{c}. There are no static solutions at ρ_{0} < ρ_{c}. We investigate the stability of equilibrium solutions in the presence of a nonzero Λ and show that dark energy reduces the dynamic stability of the configuration. We apply our results to the analysis of the properties of the equilibrium states of clusters of galaxies in the present universe with nonzero Λ.
Key words: dark matter / dark energy / galaxies: clusters: general
© ESO, 2012
1. Introduction
Detailed analysis of the observations of distant SN Ia (Riess et al. 1998; Perlmutter et al. 1999) and the spectrum of fluctuations in the cosmic microwave background radiation (CMB) (see e.g. Spergel et al. 2003) have lead to the conclusion that the term representing “dark energy” (DE) contains about 70% of the average energy density in the present universe and its properties are very close (identical) to the properties of the Einstein cosmological Λ term. In the papers of Chernin (see review 2008), the question was raised about the possible influence of any cosmological constant on the properties of the Hubble flow in the local galaxy cluster (LC) and whether the LC can exist in the equilibrium state, at present values of the DE density, where the LC densities of matter consist of the baryonic and dark matter.
Here, we construct Newtonian selfgravitating models with a polytropic equation of state in the presence of DE. In this case, we have a family instead of the single model for each polytropic index n. The additional parameter β represents the ratio of the density of DE to the matter central density of the configuration. For values of n = 1, 1.5, 3, corresponding to the polytropic powers γ = 2, 5/3, 4/3, we find the limiting values of β_{c}, such that at β > β_{c} there are no equilibrium configurations but only an expanding cluster, possibly affected by the Hubble flow.
We derive a virial theorem and analyze the influence of DE on the dynamic stability of the equilibrium models, by using an approximate energetic method. It is shown that DE produces an effect that counteracts the stabilizing influence of the cold dark matter (McLaughlin & Fuller 1996; BisnovatyiKogan 1998).
2. Main equations
We consider a spherically symmetric equilibrium configuration in Newtonian gravity, in the presence of DE, represented by the cosmological constant Λ. In this case, the gravitational force F_{g} that a unit mass undergoes in a spherically symmetric body is written as , where m = m(r) is the mass inside the radius r. Its connections with the matter density ρ and the equilibrium equation are written respectively as (1)and the DE density ρ_{v} is connected with Λ as . We consider a polytropic equation of state P = Kρ^{γ}, with . By introducing the nondimensional variables ξ and θ_{n} such that (2)we obtain the LaneEmden equation for polytropic models with DE (see also BalagueraAntolínez et al. 2007) (3)where ρ_{0} is the matter central density, α is the characteristic radius, and β = Λ/4πGρ_{0} = 2ρ_{v}/ρ_{0} is twice the ratio of the DE density to the central density of the configuration.
3. The virial theorem
We first calculate the Newtonian gravitational energy of the configuration in the presence of the cosmological constant. The spherically symmetric Poisson equation for the gravitational potential ϕ_{∗} in the presence of DE is given by (4)The gravitational energy of a spherical body ε_{g} is given by (5)where R is the total radius. For ϕ_{Λ} with uniform density ρ_{v} the normalization ϕ = 0 at r = ∞ is impossible. We can then choose ϕ_{Λ} = 0 at r = 0 as the most convenient normalization. This choice, using Eq. (4), leads to the potential ϕ_{Λ} = −4πGρ_{v}r^{2}/3. Consequently, the energy ε_{Λ}, representing the interaction of the matter with DE, is given by (6)We find the relations between the gravitational ε_{g} and thermal ε_{th} energies, and the energy ε_{Λ}. For the gravitational energy, we have (7)where M = m(R), and m is written using Eq. (1). For adiabatic systems with a polytropic equation of state, we have ρE = nP and , where E and I are thermal energy and enthalpy per mass unit. After some transformations, we obtain (8)where ε_{tot} = ε_{th} + ε_{g} + ε_{Λ}, , while is defined by Eq. (6), and the additive constant in the energy definition of ε_{Λ} is chosen so that ε_{Λ} = 0 at Λ = 0 or M = 0. The gravitational energy may also be written as (9)We can transform the last integral for polytropic matter by using Eq. (7) and making partial integrations. We have (10)Then, by using Eqs. (8) and (10), we obtain from Eq. (9) the relations Finally, by inserting Eq. (12) into (8), we get (13)We can calculate ε_{tot} for some particular cases. For n = 3, 1, and 0, we have, respectively, , and . The LaneEmden model with n = 5 has an analytical solution with finite mass M, finite values of the energies, and an infinite radius R, so that must be (5 − n)R → constant (const.) at n → 5. In the presence of DE, the finiteness of values of all kinds of energies requires that The LaneEmden solution (without DE) at n = 3 has zero total energy at any given radius and corresponds to a neutral equilibrium. Hence, the knowledge of the total energy of the configuration permits us to identify the boundary between dynamically stable (n < 3, ε_{tot} < 0) and unstable (n > 3, ε_{tot} > 0) configurations. In our case, the virial theorem does not permit us to do this, because the value of ε_{Λ} is not properly defined, while the presence of DE in the whole space does not permit us to choose, in a simple way, a universal additive constant of the energy. Therefore, in spite of ε_{tot} = 3ε_{Λ} < 0 at n = 3 and in accordance with the stability analysis made in Sect. 4, the polytropic solution at n = 3 in the presence of DE becomes unstable. Some aspects of the virial theorem in the presence of Λ were investigated by BalagueraAntolínez et al. (2007).
4. Equilibrium solutions
The equilibrium mass M_{n} for a generic polytropic configuration that is a solution of the LaneEmden equation is written as (14)Using Eq. (3), the integral on the right side may be calculated by partial integration, giving the relation for the mass of the configuration (15)where θ_{n}(ξ) is not a unique function, but depends on the parameter β, according to Eq. (3). For the limiting configuration, with β = β_{c}, we have on the outer boundary , and the mass M_{n,lim} of the limiting configuration is written as (16)such that the limiting value β_{c} is exactly equal to the ratio of the average matter density of the limiting configuration to its central density ρ_{0c}: . For the LaneEmden solution (with β = 0), we have for n = 1, 1.5, 3, respectively. We consider the curve M(ρ_{0}) for a constant DE density ρ_{v} = Λ/8πG. In order to plot this curve in the nondimensional form, we introduce an arbitrary scaling constant ρ_{ch} and write the expression for the mass in the form with (17)where is the nondimensional central density. We also introduced the nondimensional mass .
At n = 1, Eq. (3) is linear and has an analytic solution (Chandrasekhar 1939). The solution satisfying the boundary conditions at the center, θ_{1}(0) = 1, θ_{1}′(0) = 0, is written as . The radius of the configuration is determined by the transcendental equation . This equation only has real solutions at β < β_{c}, such that at the outer boundary not only does θ_{1} = 0, but also θ_{1}′ = 0 for β = β_{c}. We have . Therefore, the parameters β_{c} and ξ_{out,c} of the limiting equilibrium solution in the presence of DE are determined by the algebraic equations and tanξ_{out,c} = ξ_{out,c}, where π < ξ_{out,c} < 3π/2. At large ξ, the solutions asymptotically approach the horizontal line θ_{1} = β. Our numerical analysis indicates that ξ_{out} = π, 3.490, 4.493, for β = 0, β = 0.5β_{c} = 0.089, and β = β_{c} = 0.178, respectively. We plot the nondimensional curve , at constant ρ_{v} = βρ_{0}/2. We construct the curve starting from the model with at different β, and then following the sequence by varying the central density assuming that , at β ≤ β_{c}. For n = 1, we have (18)where . The behavior of is given in Fig. 1 for β_{in} = 0, β_{in} = 0.5β_{c}, and β_{in} = β_{c}, for which at . We note that for β_{in} = β_{c} there are equilibrium models only for .
Fig. 1 Nondimensional mass of the equilibrium polytropic configurations at n = 1 as a function of the nondimensional central density , for different values of β_{in}. The cosmological constant Λ is the same along each curve. The curves at β_{in} ≠ 0 are limited by the configuration with β = β_{c}. 

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At n = 3, the mass of the configuration is given by , where is derived by Eq. (17). The LaneEmden model (β = 0) has a unique value of the mass that is independent of the density (equilibrium configuration with neutral dynamical stability). At β ≠ 0, the model’s dependence on the density appears because the function θ_{3} is different for different values of β and, along the curve , the value of β is inversely proportional to .
The density distribution for equilibrium configurations with β = 0, β = 0.5β_{c}, and β = β_{c} is shown in Fig. 2. At large ξ, these solutions asymptotically approach the horizontal line θ_{3} = β^{1/3}, with damping oscillations around this value. The numerical solution of the equilibrium equation gives ξ_{out} = 6.897, 7.489, 9.889, for β = 0, β = 0.5β_{c} = 0.003, and β = β_{c} = 0.006, respectively. In Fig. 3, we show the behavior of , for different values of β_{in} = 0, β_{in} = 0.5β_{c}, and β_{in} = β_{c}, for which , at , respectively.
Fig. 2 The density distribution for configurations at n = 3 with β = 0, β = 0.5β_{c}, and β = β_{c}. The curves are marked with the values of β. The nonphysical solution at β = 1.5β_{c}, which does not have an outer boundary, is given by the dashdot line. The nonphysical parts of the solutions at β ≤ β_{c}, behind the outer boundary, are given by the dash lines. The solutions asymptotically approach, at large ξ, the horizontal line θ_{3} = β^{1/3}. 

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Fig. 3 Same as in Fig. 1, for n = 3. 

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The behavior of in Fig. 3, showing a decreasing mass with increasing central density, corresponds, for an adiabatic index equal to the polytropic one, to dynamically unstable configurations, according to the static criterion of stability (Zel’dovich 1963). When the vacuum influence is small, it is possible to investigate the stability of the adiabatic configuration by the approximate energetic method (Zeldovich & Novikov 1966; BisnovatyiKogan 2001). For n = 3, at ρ_{0} ≫ ρ_{v}, the density in the configuration is distributed according to the LaneEmden solution . In this case, we may investigate the stability to homologous perturbations by changing only the central density at fixed density distribution, given by the function θ_{3}.
We consider configurations close to the polytropic (adiabatic) equilibrium solution at n = 3 (and β = 0), where the turning point of stability is expected. In this case, the presence of DE does not affect significantly the gravitational equilibrium, thus the unperturbed polytropic solution at n = 3 can be used to calculate the gravitational energy ε_{g}. The energy ε_{∗} will then be given by (19)thus , where, taking into account the nondimensional variables in Eq. (2), the energy ε_{Λ} can be written as (20)where (see BisnovatyiKogan 2001). In the analysis of the dynamical stability, we consider the total energy ε of the configuration, taking into account a small correction ε_{GR} due to general relativistic effects. We have (21)where we used the relations for the polytropic configuration with n = 3, ξ_{out} = 6.897, and . The equilibrium configuration is determined by the zero of the first derivative of ε over ρ_{0}, at constant entropy S and mass M, while the stability of the configuration is analyzed in terms of the sign of the second derivative: if positive, the configuration is dynamically stable, if negative, the configuration is unstable. It is more convenient to take derivatives over than over ρ_{0}. Thus (22)for the equilibrium, and the sign of the second derivative (23)for the analysis of the dynamical stability, where and are the adiabatic index γ at constant entropy S and the nondimensional function φ, which both remain constant during homologous perturbations, respectively. It follows from Eq. (23) that DE input in the stability of the configuration is negative, as in the general relativistic correction (Chandrasekhar 1964; Merafina & Ruffini 1989). Therefore, an adiabatic star with a polytropic index of 4/3 becomes unstable in the presence of DE. The dynamic stability of pure polytropic models was also investigated by BalagueraAntolínez et al. (2006, 2007), by using a static criterion of stability. Our criterion is valid for any equation of state P(ρ,T).
At n = 1.5, the mass of the configuration is written as , where is derived by Eq. (17). At large ξ, these solutions asymptotically approach the horizontal line θ_{3/2} = β^{2/3}. The numerical solution gives ξ_{out} = 3.654, 3.984, 5.086, for β = 0, β = 0.5β_{c} = 0.041, β = β_{c} = 0.082, respectively. In Fig. 4, we show the behavior of , for different values of β_{in} = 0, β_{in} = 0.5β_{c}, and β_{in} = β_{c}, for which , at , respectively.
Fig. 4 Same as in Fig. 1, for n = 3/2. 

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5. Discussion
The question about the importance of DE to the dynamics of the Local Cluster (LC) was raised by Chernin (2008). For presently accepted values of the DE density ρ_{v} = (0.72 ± 0.03) × 10^{29} g/cm^{3}, the mass of the Local Group, including its dark matter input, is between M_{LC} ~ 3.5 × 10^{12} M_{⊙}, according to Chernin et al. (2009), and M_{LC} ~ 1.3 × 10^{12} M_{⊙}, according to Karachentsev et al. (2006). The radius R_{LC} of the LC is even more poorly known. It can be estimated by measuring the velocity dispersion v_{t} of galaxies in the LC and by the application of the virial theorem, such that . The estimated velocity dispersion of galaxies in the LC, which has been found to equal v_{t} = 63 km s^{1}, is very close to the value of the local Hubble constant H = 68 km s^{1} Mpc^{1} (Karachentsev et al. 2006). The similarity between these values indicates the great difficulties in dividing the measured velocities between regular and chaotic components. The radius of the LC may be estimated to be , and to have values between 1.5 Mpc and 4 Mpc and a very large error box that we cannot estimate properly. Chernin et al. (2009) identifies the radius R_{LC} with the radius R_{v} of the zerogravity force, which is identical to the one corresponding to our critical model with β = β_{c}, in which the average matter density is equal to 2ρ_{v}: 1.2 < M_{LC} < 3.7 × 10^{12} M_{⊙} and 1.1 < R_{v} < 1.6 Mpc. All these estimations show the importance of the presently accepted value of DE density on the structure and dynamics of the outer parts of LC, and its vicinity. Polytropic solutions with DE are inappropriate for describing the LC, but may be more applicable to rich galactic clusters.
Acknowledgments
The work of GSBK and SOT was partially supported by RFBR grant 110200602, the Presidium RAN program P20 and RF President Grant NSh3458.2010.2.
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All Figures
Fig. 1 Nondimensional mass of the equilibrium polytropic configurations at n = 1 as a function of the nondimensional central density , for different values of β_{in}. The cosmological constant Λ is the same along each curve. The curves at β_{in} ≠ 0 are limited by the configuration with β = β_{c}. 

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In the text 
Fig. 2 The density distribution for configurations at n = 3 with β = 0, β = 0.5β_{c}, and β = β_{c}. The curves are marked with the values of β. The nonphysical solution at β = 1.5β_{c}, which does not have an outer boundary, is given by the dashdot line. The nonphysical parts of the solutions at β ≤ β_{c}, behind the outer boundary, are given by the dash lines. The solutions asymptotically approach, at large ξ, the horizontal line θ_{3} = β^{1/3}. 

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In the text 
Fig. 3 Same as in Fig. 1, for n = 3. 

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In the text 
Fig. 4 Same as in Fig. 1, for n = 3/2. 

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In the text 