Astrophysical Configurations with Background Cosmology:
Probing Dark Energy at Astrophysical Scales
We explore the effects of a positive cosmological constant on astrophysical and cosmological configurations described by a polytropic equation of state. We derive the conditions for equilibrium and stability of such configurations and consider some astrophysical examples where our analysis may be relevant. We show that in the presence of the cosmological constant the isothermal sphere is not a viable astrophysical model since the density in this model does not go asymptotically to zero. The cosmological constant implies that, for polytropic index smaller than five, the central density has to exceed a certain minimal value in terms of the vacuum density in order to guarantee the existence of a finite size object. We examine such configurations together with effects of in other exotic possibilities, such as neutrino and boson stars, and we compare our results to N-body simulations. The astrophysical properties and configurations found in this article are specific features resulting from the existence of a dark energy component. Hence, if found in nature would be an independent probe of a cosmological constant, complementary to other observations.
keywords:Cosmology – Theory – Dark Energy – Structure Formation
Some of the most relevant properties of the universe have been established through astronomical data associated to light curves of distant Supernova Ia riess (), the temperature anisotropies in the cosmic microwave background radiation wmap3 () and the matter power spectrum of large scale structures tegmark (). Such observations give a strong evidence that the geometry of the universe is flat and that our Universe is undertaking an accelerated expansion at the present epoch. This acceleration is attributed to a dominant dark energy component, whose most popular candidate is the cosmological constant, .
The present days dominance of dark energy make us wonder if this component may affect the formation and stability of large astrophysical structures, whose physics is basically Newtonian. This is in fact an old question put forward already by Einstein einstein () and pursued by many other authors noer (); chernin2 (); manera (); nunes (); baryshev () and kagramanova (); jetzer1 (). In general, the problem is rooted in the question whether the expansion of the universe, which in the Newtonian sense could be understood as a repulsive force, affects local astrophysical properties of large objects. The answer is certainly affirmative if part of the terms responsible for the Universe expansion survives the Newtonian limit of the Einstein equations. This is indeed the case of which is part of the Einstein tensor. In fact, as explained in the main text below, all the effects of the universe expansion can be taken into account, regardless of the model, by generalizing the Newtonian limit. This approach allow us to calculate the impact of a given cosmological model on astrophysical structures.
Although, there are several candidates to dark energy which have their own cosmological signature, e.g. Koivisto1 (); daly (); wang2 (); koivisto (); koivisto2 (); shaw2 () and seo (); brook (); daly2 (); Koivisto3 (); shaw3 (), in this paper we will investigate the CDM model only. Such consideration is in fact not restrictive and our results will be common to most dark energy models. At astrophysical scales and within the Newtonian limit one does not expect to find important differences among the different dark energy models. This, however should not be interpreted as if has no effect at smaller astrophysical scales. In fact, the effects of a cosmological constant on the equilibrium and stability of astrophysical structures is not negligible, and can be of relevance to describe features of astrophysical systems such as globular clusters, galaxy clusters or even galaxies cher (); iorio (); bala1 (); nowakowski1 (); cardoso (). Motivated by this, we investigate the effects of a dark energy component on the Newtonian limit of Einstein gravity and its consequences at astrophysical scales.
In this article we investigate how the cosmological constant changes certain aspects of astrophysical hydrostatic equilibrium. In particular we search for specific imprints which are unique to the existence of a dark energy fluid. For instance, the instability of previously viable astrophysical models when is included. We explore such possibility using spherical configurations described by a polytropic equation of state (e.o.s) . The polytropic equation of state derives its importance from its success and consistency, and it is widely used in determining the properties of gravitational structures ranging from stars chan () to galaxies binney (). It leads to an acceptable description of the behavior of astrophysical objects in accordance with observations and numerical simulations kennedy1 (); gruzinov (); kaniadakis (); sadeth (); pinzon1 (); ruffet (). The description of such configurations can be verified in the general relativistic framework herrera () and applications of these models to the dark energy problem have in fact been explored mukhop ().
The effect of a positive cosmological constant can be best visualized as a repulsive non local force acting on the matter distribution. It is clear that this extra force will result into a minimum density (either central or average) which is possible for the distribution to be in equilibrium. This minimum density is a crucial crossing point: below this value no matter can be in equilibrium, above this value low density objects exist lahav91 (). Both effects are novel features due to . We will demonstrate such inequalities, which are generalizations of corresponding inequalities found in nowakowski1 () and bala2 (), for every polytropic index . However, the most drastic effect can be found in the limiting case of the polytropic equation of state, i.e, the isothermal sphere where the polytropic index goes to infinity. This case captures, as far as the effects of are concerned, many features also for higher, but finite . The model of the isothermal sphere is often used to model galaxies and galactic clusters natarajan () and used in describing effects of gravitational lensing kawano (); sereno (); maccio (). Herein lies the importance of the model. Regarding the isothermal sphere we will show that renders the model unacceptable on general grounds. This essentially means that the model does not even have an appealing asymptotic behavior for large radii and any attempt to definite a physically acceptable radius has its severe drawbacks.
The positive cosmological constant offers, however, yet another unique opportunity, namely the possible existence of young low density virialized objects, understood as configurations that have reached virial equilibrium just at the vacuum dominated epoch (in contrast to the structures forming during the matter dominated era, where the criteria for virialization is roughly ). This low density hydrostatic/virialized objects can be explained again due to which now partly plays the role of the outward pressure.
The applicability of fluid models, virial theorem and hydrostatic equilibrium to large astrophysical bodies has been discussed many times in the literature. For a small survey on this topic we refer the reader to (jackson, ; bala5, ) where one can also find the relevant references.
It is interesting to notice that dark matter halos represent a constant density background which, in the Newtonian limit, objects embedded in them feel the analog to a negative cosmological constant. The equilibrium analysis for such configurations has been performed in Umemura (); Horedt (). A negative will just enhance the attractive gravity effect, whereas a positive one opposes this attraction. As a result the case reveals different physical concepts as discussed in this paper.
The article is organized as follows. In the next section we introduce the equations relevant for astrophysical systems as a result of the weak field limit and the non-relativistic limit of Einstein field equations taking into account a cosmological constant. There we derive such limit taking into account the background expansion independently of the dark energy model. In section 3 we derive the equations governing polytropic configurations, the equilibrium conditions and stability criteria. In section 4 we describe the isothermal sphere and investigate its applicability in the presence of . In section 5 we explore some examples of astrophysical configurations where the cosmological constant may play a relevant role. In particular, we probe into low density objects, fermion (neutrino) stars and boson stars. Finally we perform an important comparison between polytropic configurations with and parameterized density of Dark Matter Halos. We end with conclusions. We use units except in section 5.4 where we restore and use natural units .
2 Local dynamics in the cosmology background
The dynamics of the isotropic and homogeneous cosmological background is determined by the evolution of the (dimensionless) scale factor given through the Friedman-Robertson-Walker line element as solution of Einstein field equations,
corresponding to the Raychaudhury equation and Friedman equation, respectively. The total energy density is a contribution from a matter component - baryonic plus dark matter - (), radiation () and a dark energy component ( with ). The function is given as
where the term represents the equation of state for the dark energy component. The case corresponds to the cosmological constant . The effects of the background on virialized structures can be explored through the Newtonian limit of field equations from which one can derive a modified Poisson’s equation (see e.g. noer (); nowakowski2 ()). Recalling that pressure is also a source for gravity, the gravitational potential produced by an overdensity is given by
where and . Where is the local overdensity with respect to the background density . Notice that equation (3) reduces to the usual Poisson equation, , when non relativistic matter dominates the Universe, and . However, at present times, when dark energy dominates, the pressure is non-negligible and might even be non zero, such as in the case of quintessence models ml (); wang (); mota1 (). In this work, however, we will focus in the case of an homogeneous dark energy component where . With this in mind, one can then write the modified Poisson equation as
Note that this equation allows one to probe local effects of different Dark Energy models through the term given in Eq.(1).
Since we will be investigating the configuration and stability of astrophysical objects nowadays, when dark energy dominates, it is more instructive to write the above equations in terms of an effective vacuum density i.e.
where by using (1) has been defined as
which reduces to for and negligible contribution from the cold dark matter component with respect to the over-density . With being the solution associated to the pure gravitational interaction, the full solution for the potential can be simply written as
which defines the Newton-Hooke space-time for a scale factor close to the present time (vacuum dominated epoch), and gibbons (); aldro (). For a CDM universe with and we get : that is, the positive density of matter which has an attractive effect opposing the repulsive one of reduces effectively the strength of the ’external force’ in (5). Note that, although in the text we will use the notation which would be vaild in the case of a Newton-Hook space time, it must be understood that we can replace by for a CDM background.
Given the potential , we can write the Euler’s equation for a self-gravitating configuration as
, where is the (statistical) mean velocity and is the total energy density in the system. We can go beyond Euler’s equation and write down the tensor virial equation which reduces to its scalar version for spherical configurations. The (scalar) virial equation with the background contribution reads as bala3 (); caimmi ()
where is the gravitational potential energy defined by
and is the contribution of ordered motions to the kinetic energy. Also, is the moment of inertia about the center of the configuration and is the trace of the dispersion tensor. The full description of a self gravitating configuration is completed with an equation for mass conservation, energy conservation and an equation of state . If we assume equilibrium via , we obtain the known virial theorem jackson (); bala3 ()
where we have neglected the surface term in (8), which is valid in the case when we define the boundary of the configuration where .
With given in (6), one must be aware that an equilibrium configuration is at the most a dynamical one. This is to say that the ’external repulsive force’ in (6) is time dependent through the inclusion of the background expansion and so are the terms in (8). This leads to a violation of energy conservation, which also occurs in the virialization process wang3 (); ml (); caimmi (); mota1 (); shaw ()). Traced over cosmological times, this implies that if we insist on the second derivative of the inertial tensor to be zero, then the internal properties of the object like angular velocity or the internal mean velocity of the components will change with time. Even if in the simplest case, one can assume that the objects shape and its density remain constant. Hence, equilibrium here can be thought of as represented by long time averages in which case the second derivative also vanishes, not because of constant volume and density, but because of stability bala4 (); bala5 ().
The expressions derived in the last section, especially (5) and (10) can be used for testing dark energy models on configurations in a dynamical state of equilibrium. However, in these cases, one should point out that, in this approach there is no energy conservation within the overdensity: dark energy flows in and out of the overdensity. Such feature is a consequence of the assumption that dark energy does not cluster at small scales (homogeneity of dark energy). This is in fact the most common assumption in the literaturewang (); chen (); hore (), with a few exceptions investigated in wang3 (); ml (); caimmi (); mota1 ().
In this paper, we will concentrate on the possible effects of a background dominated by a dark energy component represented by the cosmological constant at late times (1). It implies that the total density involved in the definitions of the integral quantities appearing in the virial equation can be approximated to . In that case the Poisson equation reduces to the form . As mentioned before, the symbol has no multiplicative factors in the case of a Newton-Hooke space time, while for a CDM model it must be understood as . As the reader will see, the most relevant quantities derived here come in forms of ratio of a characterizing density and , and hence the extra factor appearing in the CDM can be re-introduced in the characterizing density. For general consideration of equilibrium in the spherical case see boehmer1 () and boehmer (), while the quasi spherical collapse with cosmological constant has been discussed in debnath ().
3 Polytropic configurations and the -Lane-Emden equation
We can determine the relevant features of astrophysical systems by solving the dynamical equations describing a self gravitating configuration (Euler’s equation, Poisson’s equation, continuity equations). In order to achieve this goal we must first know the potential to be able to calculate the gravitational potential energy. To obtain , one must supply the density profile and solve Poisson’s equation. In certain cases, the potential is given and we therefore can solve for the density profile in a simple way. Here we face the situation where no information on the potential (aside from its boundary conditions) is available and we also do not have an apriori information about the density profile (see for instance binney () for related examples). In order to determine both, the potential and the density profile, a complete description of astrophysical systems required. This means we need to know an equation of state (here we change notation and we call the proper density of the system). The equation of state can take several forms and the most widely used one is the so-called polytropic equation of state, expressed as
where is the polytropic index and is a parameter that depends on the polytropic index, central density, the mass and the radius of the system. The exponent is defined as and is associated with processes with constant (non-zero) specific heat . It reduces to the adiabatic exponent if . The polytropic equation of state was introduced to model fully convective configurations. From a statistical point of view, Eq. (11) represents a collisionless system whose distribution function can be written in the form , with being the relative energy and being the relative potential (where is a constant chosen such that )binney (). In astrophysical contexts, the polytropic equation of state is widely used to describe astrophysical systems such as the sun, compact objects, galaxies and galaxy clusters chan (); binney (); kennedy1 (). We now derive the well known Lane-Emden equation. We start from Poisson equation and Euler equation for spherically symmetric configurations, written as
This set of equations together with Eq. (11) can be integrated in order to solve for the density in terms of the potential as
where is the central density. In view of eq. (13) in conjunction with eq. (7) it is clear that will have the effect to increase the value of . Therefore the boundary of the configuration will be located in a greater as compared to the case . In order to determine the behavior of the density profile, we again combine Eq(12) and Eq.(11) in order to eliminate the potential . We obtain
where we defined the function
We can rewrite Eq (14) by introducing the variable defined by , where is the central density. We also introduce the variable where
This is the -Lane-Emden equation (). Note that for constant density, we recover as the first non trivial solution of equation. This is consistent with the results from virial theorem for constant density spherical objects which tell us that nowakowski1 (). Note that using Eq. (13) we can write the solution with the explicit contribution of as
so that for a given smaller than the radius we will obtain
as already pointed out before. Then the differential equation (17) must be solved with the initial conditions , satisfied by (18). Numerical solutions were obtained for the first time in bala3 (). The solutions presented in Fig. 1 are given in terms of the ratio for and . This choice of variables are useful also since sets a fundamental scale of density (the choice will be explored for the isothermal sphere, where figure 1 will be helpful for discussions). The radius of a polytropic configuration is determined as the value of when the density of matter with the e.o.s (11) vanishes. This happens at a radius located at
Note that equation (17) yields a transcendental equation to determine . Also one notes from Fig. 1 that not all values of yield allowed configurations in the sense that we cannot find a value of such that . This might not be surprising since for and we find the situation where the asymptotic behavior is as (we consider this still as an acceptable behavior). There is, however, one crucial difference when we switch on a non-zero . For not only we cannot reach a definite radius but the derivative of the density changes sign and hence becomes non-physical. The situation for the cases is somewhat similar to the extreme case of (isothermal sphere). Clearly, these features are responsible of the last term in Eq.(18), which for high values of may become dominant over the remaining (gravitational) terms. We will discuss this case in section four where we will attempt another definition of a finite radius with the constraint . For now it is sufficient to mention that, as expected, the radius of the allowed configurations are larger than the corresponding radius when .
3.1 Equilibrium and stability for polytropes
In this section we will derive the equilibrium conditions for polytropic configurations in the presence of a positive cosmological constant. We will use the results of last section in order to write down the virial theorem. The total mass of the configuration can be determined as usual with together with Eq.(17). One then has a relation between the mass, the radius and the central density:
Note that we have introduced the cosmological constant in the equation for the radius, leading to the appearance of the astrophysical length scale (with Mpc Mpc for the concordance values and ). This scale has been already found in the context of Schwarzschild - de Sitter metric where it is the maximum allowed radius for bound orbits. At the same time it is the scale of the maximum radius for a self gravitating spherical and homogeneous configuration in the presence of a positive bala3 (). This also let us relate the mean density of the configuration with its central density and/or cosmological parameters as .
Similarly we can determine the other relevant quantities appearing in the equations for the energy and the scalar virial theorem (10). For the traces of the moment of inertia tensor and the dispersion tensor we can write
where the functions have been defined as
using (21). These functions are numerically determined in the sequence , such that for a given mass we obtain the radius as .
Let us consider the virial theorem (10) for a polytropic configuration. The gravitational potential energy can be obtained following the same arguments shown in chan (). The method consist in integrating Euler’s equation and solve for , then using Eq.(9) one obtains . The final result is written as
This expression shows the typical behavior of a polytrope (even if ): the configuration has an infinite potential energy, due to the fact that the matter is distributed in a infinite volume. The energy of the configuration in terms of the polytropic index can be easily obtained by using (10), (25) and (26)
One is tempted to use as the condition to be fulfilled for a gravitationally bounded system. For we recover the condition () for gravitationally bounded configurations in equilibrium. On the other hand, for this condition might not be completely true due to the following reasoning: The two-body effective potential in the presence of a positive cosmological constant does not go asymptotically to zero for large distances, which is to say that is not stringent enough to guarantee a bound system. Therefore, we rather rely on the numerical solutions from which, for every , we infer the value of such that
This gives us the lowest possible central density in terms of . The behavior of the , the functions , the solution and the values of are shown in fig 2. Note that this inequality can be understood as a generalization of the equilibrium condition , which, when applied for a spherical homogeneous configurations yields with constant bala3 (); bala2 ().
Note that, at the radius of the configuration becomes undefined, as well as the energy. A polytrope is highly concentrated at the center chan (). No criteria can be written since even for there is not a finite radius. But it is this high concentration at the center and a smooth asymptotic behavior which makes this case still a viable phenomenological model if is zero. On the contrary for non-zero, positive the solutions start oscillating around which makes the definition of the radius more problematic. For the polytropic e.o.s describes an ideal gas (isothermal sphere). Since in this limit the expressions derived before are not well definite, this case will be explored in more detail in the next section. In spite of the mathematical differences, the isothermal sphere bears many similarities to the cases and our conclusions regarding the definition of a radius in the case equally apply to finite bigger than .
3.2 Effects with generalized dark energy equation of state
In the last section we have explored the effects of a dark energy-dominated background with the equation of state . Other dark energy models are often used with and or even , in the so-called phantom regime, or even a time dependent dark energy model (quintessence). A simple generalization to such models can be easily done by making the following replacement in our equations
where , and where the function is defined in Eq.(2). Note that for this generalization to be coherent with the derivation of LE in (17), one needs to consider that the equation of state is close to , so that there is almost no time-dependence, and the energy density for dark energy is almost constant. This is indeed the case for most popular candidates of dark energy specially at low redshifts . Also, notice once again, that we are still assuming an homogeneous dark energy component which flows freely to and from the overdensity. Hence, violating energy conservation inside it. Clearly, other models of dark energy will posses dynamical properties that the cosmological constant does not have. For instance, we could allow some fraction of dark energy to take part in the collapse and virialization ml (); caimmi (); mota1 (), which would lead to the presence of self and cross interaction terms for dark energy and the (polytropic like) matter in Euler equation, which at the end modifies the Lane Emden equation. With this simplistic approach, we see that the effects with a general equation of state are smaller than those associated to the cosmological constant. In particular, the equation of state displays a null effects since it implies (note that this equation of state can also resemble the curvature term in evolution equation for the background). On the other hand, phantom models of dark energy, which are associated to equations of state cald (); nojiri (), have quite a strong effect. In Fig. 3 we show numerical solutions of Lane-Emden equations for a background dominated with dark energy with and a phantom dark energy with , with . These curves are to be compared with those at fig.1. Clearly equations of state with will generate larger radius than the case described in the main text. Furthermore, the asymptotic behavior of the ratio between the density and is .
3.3 Stability criteria with cosmological constant
Stability criteria for polytropic configurations can be derived from the virial equation. Using equations (23), (24) and (25) we can write the virial theorem (10) in terms of the radius and the mass :
where we have assumed that the only contribution to the kinetic energy comes from the pressure in the form of . Note that for and finite mass, one obtains while for we would obtain a cubic equation for the radius. Instead of solving for the virial radius, we solve for the mass as a function of central density with the help of Eq (21). We have
The explicit dependence of the mass with respect to the central density splits into two parts: on one hand it has the same form as the usual case with , that is, ; on the other hand the function has a complicated dependence on the central density because of the term . With the help of (21) and (31) we can write a mass-radius relation and the radius-central density relation
Following the stability theorem (see for instance in weinberg ()), the stability criteria can be determined from the variations of the mass in equilibrium with respect to the central density. We derive from Eq. (32):
Stability (instability) stands for (). This yields a critical value of the polytropic exponent when given by
in the sense that polytropic configurations are stable under small radial perturbations if . It is clear that the second term in (34) also depends on the polytropic index and therefore this equation is essentially a transcendental expression for .
It is worth mentioning that by including the corrections due to general relativity, the critical value for is also modified as shapiro () and hence for compact objects the correction to the critical polytropic index is stronger from the effects of general relativity than from the effects of the background. This is as we would expect it. Stability of relativistic configurations with non-zero cosmological constant has been explored in boehmer (); boehmer1 ().
Going back to equation (31), we can write the mass of the configuration as , where is the mass when and is the enhancement factor. Both quantities can be calculated to give
where are numerical factors (tabulated in table 1) that can be determined in a straightforward way. Similarly, by using Eq (21), the radius can be written as , where
In table 1 we show the values of the enhancement factors and for different values of and different polytropic index . We also show the values of the critical ratio which separates the configurations with definite ratio such that a zero exist provided that . We will show some examples where the enhancement factors may be relevant in section 5.
4 The isothermal sphere
The isothermal sphere is a popular model in astrophysics, either to model large astrophysical and cosmological objects (galaxies, galaxy clusters) lynden (); penston (); yabu (); sommer (); chavanis1 (); more (), to examine the so-called gravothermal catastrophe binney (); natarajan (); lombardi () and finally to compare observations with model predictions rines (); more (). In the limit in the polytropic equation of state one obtains the description for an isothermal sphere, (ideal gas configuration) with
where is the velocity dispersion (). The pattern we found in section 3 for gets confirmed here: no finite radius of the configurations is found with , the asymptotic behavior is not as (but rather ) and, as we will show below, other attempts to define a proper finite radius are not satisfactory.
The results for a finite value of the index are defined in the limit only asymptotically in the case . We consider this as an acceptable behavior of the density. Because of the limiting case the analysis for the isothermal sphere must be done in a slightly different way. As was done in Eq. (13), we can integrate the equilibrium equations (Euler and Poisson’s equations) and obtain an explicit dependence of the density with as
The resulting differential equation for the density with cosmological constant can be written as
This differential equation could be treated in the same way as we did for the polytropic equation of state, i.e, by defining a new function , but here we can already use the fact that the cosmological constant introduces scales of density, length and time bala1 (). Let us then define the function and , with the associated length scale. Since we are now scaling the density with , the associated length scale should be also scaled by the length scale imposed by :
For an hydrogen cloud with kms we have kpc which is approximately the radius of an elliptical (E0) galaxy. In terms of the function , the differential equation governing the density profile is then written as 111Compare with Eq. 374 of chan () or Eq.1 of natarajan () where the density is scaled by the central density. The factor on the r.h.s of (40) is due to .
so that according to Eq. (38) we may write
From this we can derive different solutions depending on the initial condition and at . In Fig. 4 we show numerical results for the solutions of equation (40) using different values values of . As it is the case for , the radius cannot be defined by searching the first zero of the density i.e. the value such that (including ). In this case, the behavior of the derivative of the density profile changes as compared with the the case since with increasing the density starts oscillating around the value such that for one has a solution . This can be checked from (40) which corresponds to the first non trivial solution for . This behavior implies that there exist a value of where the derivative changes sign and hence the validity of the physical condition required for any realistic model i.e. should be given up unless we define the size of the configuration as the radius at the value of the first local minimum. We will come back to this option below to show that it is not acceptable. A second option would be to set the radius at the position where the density acquires for the first time its asymptotic value . We could motivate such a definition by demanding that the density at the boundary goes smoothly to the background density. This is for tow reasons, however, not justified. First, we recall that a positive cosmological constant leads to a repulsive ’force’ as it accelerates the expansion. A negative cosmological constant could be modeled in a Newtonian sense by a constant positive density which, however, is strictly speaking still not a background density. Secondly, if we include the background density we would have started with (with a dynamical equation for being constant) in which case the boundary condition would again be to define the extension of the body (or at least, as ). Hence, this second option can be excluded on general grounds. In any case as can be seen from Figure 4 both definitions would yield two different values of radius. Since the first candidate to define a radius is based upon a physical condition of the configuration, we could expect this definition as the more suitable one. However such a definition must be in agreement with the observed values for masses and radius of specific configurations and the validity of this definition can be put to test by the total mass of the configuration, given as
If we define the radius at the first minimum (see Fig. 4), we find . Although this might set the right order of magnitude for the mass of a E0 galaxy if we insist on realistic values for the respective radius, say kpc, the picture changes again as the radius is fixed by (39) which gives . In order to get a radius of the order of kpc with masses of the order of we would require . This differs by almost eight orders of magnitude with the measured values for the velocity dispersion in elliptical galaxies ( kms) or with the Faber-Jackson Law for velocity dispersion padma (). We conclude that defining the radius by the position of the first minimum is not a realistic solution. As the last option to get realistic values for the parameters of the configuration we consider the brute force method to simply fix the value of . It is understood, however, that this method is not acceptable if we insist that the model under consideration has some appealing features (without such features almost any model would be phenomenologically viable). Therefore we discuss this option only for completeness. For a configuration with , say an elliptical galaxy, we fix the radius at kpc in Eq.(39) and using a typical value for the velocity dispersion kms we get which implies . The mass in Eq.(42) is then given as while the density at the boundary is , that is, . In table 2 we perform the same exercise for other radii. The resulting mean density is in accordance with the observed values of the mean density of astrophysical objects ranging from an small elliptical galaxy to a galaxy cluster. However, as mentioned above, the model introduces an arbitrary cut-off and cannot be considered as a consistent model of hydrostatic equilibrium.
In summary, the attempts to define a finite radius for the isothermal sphere fail in the presence of a cosmological constant either because such a model fails to reproduce certain phenomenological values (if the definition of the radius is fixed by the first minimum) or because the definition is technically speaking quite artificial to the extent of introducing arbitrary cut-offs. Note that this conclusion is valid almost for any object as the density of the isothermal sphere with has a minimum whatever the central density we choose.
5 Exotic astrophysical configurations
In this section we will probe into the possibilities of exotic, low density configurations. The global interest in such structures is twofold. First, the cosmological constant will affect the properties of low density objects. Secondly, plays effectively the role of an external, repulsive force. Hence a relevant issue that arises in this context is to see whether the vacuum energy density can partly replace the pressure which essentially is encoded in the parameter (). By the word replace we mean that we want to explore the possibility of a finite radius as long as the pressure effects are small in the presence of .
5.1 Minimal density configurations
As mentioned above the effect of a positive cosmological constant on matter is best understood as an external repulsive force. In previous sections we have probed into one extreme which describes the situation where a relatively low density object is pulled apart by this force (to an extent that we concluded that the isothermal sphere is not a viable model in the presence of ). Limiting conditions when this happens were derived. On the other hand, approaching with our parameters these limiting conditions, but remaining still on the side of equilibrium, means that relatively low density objects can be still in equilibrium thanks to the positive cosmological constant. The best way to investigate low density structures is to to use the lowest possible central density. As explained in section 3, for every there exist a such that which defines the lowest central density. Certainly, a question of interest is to see what such objects would look like. We start with the parameters of the configuration. The radius at the critical value is given by Eq.(21) after taking the limit (see (28)). It is given by
From fig. 5 we can see that the product changes a little round the value as we change the index , so that these polytropic configurations will have roughly the same radius (for a given mass). This implies that such configurations have approximately the same average density . That is, such configurations have a mean density of the order of the of the critical density of the universe. Given such a density we would, at the first glance, suspect that the object described by this density cannot be in equilibrium. However, our result follows strictly from hydrostatic equilibrium and therefore there is no doubt that such object can theoretically exist. Furthermore, satisfies the inequalities derived in nowakowski1 () and bala2 () from virial equations and Buchdahl inequalities which guarantee that the object is in equilibrium (). Interestingly, the central density for such objects has to be much higher than as, e.g., for we have and therefore .
Note that these values have been given from the solution of the Lane-Emden equation, which is a consequence of dynamical equations reduced to describe our system in a steady state. However, we have not tried to solve explicitly quantities from the virial theorem. This makes sense as for Dark Matter Halos (DMH) the parametrized density profiles go often only asymptotically to zero and the radius of DMH is defined as a virial radius where the density is approximately two hundred times over the critical one. Therefore, an analysis using virial equations seems to be adequate here. For constant density Eq.(30) can be expressed as a cubic equation bala1 () for the radius at which the virial theorem is satisfied (let us ignore for these analysis any surface terms coming from the tensor virial equation). However, if the density is not constant, this expression becomes a transcendental equation for the dimensionless radius . This equation is
understanding the functions now as integrals up to the value . Once we fix for a given index , we use as a first guess for the iteration process the value . In fig 5 we show the behavior of the functions and the solutions of Eq.(45). For these values, Eq. (44) gives for a radius
which can be compared with the radius-mass relation derived in the top-hat sphericall collapse padma ()
where is the dimensionless Hubble parameter and is the redshift of virialization. The resulting average density is then of the order of the value predicted by the top-hat sphericall model:
Note with the help of Eq.(16) and (44) that the mass can be written as proportional to the parameter (introduced in the polytropic equation of state Eq.(11)). Therefore is equivalent to choosing a small pressure and, at the same time, a small mass which, in case of a relatively small radius, amounts to a diluted configuration with small density and pressure. Without such configurations would be hardly in equilibrium. Hence, for the configuration which has the extension of pc, Eq.(46), with one solar mass we conclude that the equilibrium is not fully due to the pressure, but partially maintained also by . This is possible, as exerts an outwardly directed non local force on the body. Other mean densities, also independent of are for and , respectively:
The first value is close to and therefore also to . Certainly, if in this example we choose a small mass, equivalent to choosing a negligible pressure, part of the equilibrium is maintained by the repulsive force of . In bala2 () we found a simple solution of the hydrostatic equation which has a constant density of the order of . The above is a non-constant and non-trivial generalization of this solution.
5.2 Cold white dwarfs
The neutrino stars which we will discuss in the subsequent subsection are modeled in close analogy to white dwarfs. Therefore it makes sense to recall some part of the physics of white dwarfs. In addition we can contrast the example of white dwarfs to the low density cases affected by .
In the limit where the thermal energy of a (Newtonian) white dwarf is much smaller than the energy at rest of the electrons (, these configurations can be treated as polytropic configuration with . This is the ultra-relativistic limit where the mass of electrons is much smaller than Fermi’s momentum . In the opposite case we obtain a polytrope or configuration with . weinberg (); shapiro (). In both cases, the parameter from the polytropic equation of state is given as
where is the nucleon mass, is the electron mass and is the number of nucleons per electron. Using the Newtonian limit with cosmological constant, we can derive the mass and radius of these configurations in equilibrium. In the first case, for the mass is written using (31) as which corresponds approximately to the Chandrasekhar’s limit (strictly speaking a configuration would have the critical mass, i.e, the Chandrasekhar’s limit, if its polytropic index is such that ). For this situation one has and . On the other hand, for one obtains and the radius is given by where is the mean density of the sun. Since for these configurations the ratio is much smaller than we see from Fig. 2 that the effects of are almost negligible. The critical value of the ratio gives for the inequality and for the same limit reads . Central densities of white dwarfs are of the order of which corresponds to a deviation of nearly thirty orders of magnitude of .
5.3 Neutrino stars
An interesting possibility is to determine the effects of on configurations formed by light fermions. Such configurations can be used, for instance, to model galactic halos dolgov (); lattanzi (); jetzer (); boerner (). While discussing the phenomenological interest of fermion stars below, we intend to describe such a halo. Clearly, these kind of systems will maintain equilibrium by counterbalancing gravity with the degeneracy pressure as in a white dwarf. For stable configurations, i.e, , one must replace the mass of the electron and nucleon by the mass of the considered fermion and set in (50) and (31). We then get for the mass and the radius: