Astron. Astrophys. 363, 425-439 (2000)

4. Conclusions

The new phenomena in the structure of equilibrium configurations of test perfect fluid, caused by the presence of a repulsive cosmological constant, can be summarized in the following way.

1. There is always an equipotential surface with a cusp for . It is always an open surface.

2. The position of the outer cusp of the equipotential surface with is just at . The value of the potential at the cusp is given by

   

   Because

   

   we find

   

3. The accretion disks around black holes can exist when an inner cusp will appear near the black-hole horizon, in addition to the outer cusp, located nearby the static radius .

4. Closed equipotential surfaces, necessary for the existence of toroidal accretion disks, can exist for . Here () corresponds to the local minimum (maximum) of the function , giving the minimum (maximum) value of stable circular geodesic (Keplerian) orbits. The closed surfaces can exist in the spacetimes with .

5. Accretion onto the central black hole by the Paczynski mechanism is possible, if ; The value corresponds to the marginally bound circular geodesics. Now, they are determined nontrivially: by the condition that for both and the effective potential of geodesic motion have two local maxima with the same value (recall that there). In this case, outflow from the accretion disk is possible through both cusps, if the mechanical equilibrium is destroyed for both the cusps, i.e., if both equipotential surfaces with cusp are filled up: . If , the accretion flow is directed down the black hole only.

6. We stress that for , the equipotential surface with has two cusps. The mass outflow due to mechanical non-equilibrium, i.e., overfilling of the (both-sided) marginally closed equipotential surface, is equally efficient for the inflow down the black hole and the outflow near the static radius. Of course, we could expect significant differences in details of the accretion inflow near the black-hole horizon, and the outflow near the static radius.

7. The outer cusp of the configuration with , and , i.e., the limiting equilibrium configuration which enables accretion into the Schwarzschild-de Sitter black holes, is located at . It is quite interesting that such configurations will approach the static radius, however, they cannot exceed the static radius ( if ). Notice that (Fig. 2), while (Fig. 3).

8. For , the accretion flow down the hole is "switched-off", because an open self-crossing equipotential surface with appears under the inner edge of the toroidal configuration in the equatorial plane. However, the outflow through the cusp near the static radius can still occur due to a possible mechanical non-equilibrium.

9. Toroidal structures of equipotential surfaces, leading to equilibrium configurations of perfect fluid, cannot exist just if . Then, an inner cusp, nearby the black-hole horizon, still exists for equipotential surfaces with . However, these equipotential surfaces are always open, and can exist in spacetimes with .

10. The behavior of the open equipotential surfaces along the axis of rotation gives an important effect-the surfaces become significantly narrower while approaching the static radius and the cosmological horizon. This behavior suggests a strong collimation effect on jets, caused by the influence of a repulsive cosmological constant.

In the case of Schwarzschild-anti-de Sitter spacetimes the situation is different. The presence of an attractive cosmological constant brings no qualitatively new phenomena in comparison with the Schwarzschild case, concerning the character of the equilibrium configurations related to accretion disks. Notice, however, the special shape (resembling a falling wave) of the closed equipotential surfaces which manifests in an illustrative way the interplay of the gravitational, cosmological, and centrifugal forces. Moreover, there exist no open equipotential surfaces around the rotation axis in these spacetimes.

From the astrophysical point of view, the most important phenomena were discovered in spacetimes with a repulsive cosmological constant, if they admit stable circular geodesic orbits. The first is the presence of an outer cusp of toroidal disks nearby the static radius which enables outflow of mass and angular momentum from the accretion disks by the Paczynski mechanism, i.e., due to a violation of the hydrostatic equilibrium. This is the same mechanism that drives the accretion into the black hole through the inner cusp. (Recall that outflow from toroidal disks around a Schwarzschild or Kerr black hole by the Paczynski mechanism is impossible because no outer cusp of toroidal disks exists in the asymptotically flat black-hole spacetimes (Kozlowski et al. 1978; Abramowicz et al. 1978; Jaroszynski et al. 1980).) The second is the possibility of strong collimation effects on jets escaping along the rotation axis of toroidal disks following the open equipotential surfaces that are narrowing strongly when approaching the static radius (and the cosmological horizon). We give an explicit illustration of these two principally new phenomena caused by the repulsive cosmological constant in Fig. 8. Of course, both of those very interesting phenomena deserve further, more detailed studies. Further, the runaway instability of the toroidal disks with respect to the outflow through the outer cusp, and the influence of self-gravitation on their structure, have to be examined. We plan these studies in near future.

Fig. 8. The structure of an accretion disk with a jet is compared in meridian sections. The radial coordinate is expressed in units of M, but the logarithmic scale is not used here, since we are interested in the regions near the static radius where both the outer cusp and the collimation effect are evident.

It is interesting to find astrophysically plausible situations in which these two phenomena could be relevant. We should consider their role in

• (a) quasars and active galactic nuclei during the present period of expansion of the Universe,

• (b) accretion processes onto primordial black holes during the very early stages of expansion of the Universe, when phase transitions connected to symmetry breaking of physical interactions due to Higgs mechanism (e.g., the breaking of electroweak interactions) could take place, and the effective cosmological constant can have values in many orders exceeding its present value (Kolb & Turner 1990).

Recent cosmological observations give strong indications that the present value of the vacuum energy density is (Krauss 1998)

with present values of the critical energy density , and Hubble parameter given by

Taking value of the dimensionless parameter , we arrive at the present value of the "relict" repulsive cosmological constant

Having this value of , we can determine the mass parameter of the spacetime corresponding to any given value of y, and all the relevant parameters of the equilibrium configurations. The results concerning the important radii characterizing the Schwarzschild-de Sitter spacetimes with are summarized in Table 3 and Table 4.

Table 3. Characteristic radii of the Schwarzschild-de Sitter black-hole spacetimes (in units of mass parameter M). The parameter determines the relative extension of the toroidal accretion disks with . The table ends at , corresponding to the marginal spacetime allowing stable circular geodesics. In spacetimes with , stable circular geodesics are not allowed, and both thick and thin accretion disks cannot exist.

Table 4. Mass parameter and the radius determining the outer edge of toroidal disks with in the Schwarzschild-de Sitter black-hole spacetimes, given for (a) the relict repulsive cosmological constant indicated by recent cosmological observations , (b) the primordial effective cosmological constant , and (c) the other possible primordial effective cosmological constant .

We can clearly see that the relict cosmological constant puts a natural limit on the size of equilibrium configurations rotating around black holes. In fact, the outer edge of the accretion disks, where the outflow goes through the outer cusp of the toroidal structure, is located nearby the static radius. It is quite interesting that for black holes of masses -, corresponding to black holes located in the central parts of quasars and active galactic nuclei, the outer edge of the largest accretion disks is located at -100 kpc, and is comparable with maximum extension of large galaxies. Note that extension of quasikeplerian, thin accretion disks is limited by the outer marginally stable circular orbit; if y is small enough (), it can be shown that

and dimensions of these disks are comparable to the static radius, too. Therefore, the relict repulsive cosmological constant can radically influence the behavior of accretion disks in large galaxies with active nuclei, and can even be connected to the limit of extension of these large galaxies.

Moreover, it is clear that the collimation effect of the repulsive cosmological constant could be relevant in these situations, because the largest observed jets extend to distances (Blandford 1990), exceeding dimensions of the "seed" galaxy (comparable to the static radius).

It is well known (Carroll & Ostlie 1996) that dimensions of large galaxies, of both spiral and elliptical type, are in the interval 50-100 kpc, while the extremely large elliptical galaxies of cD type extend up to 1000 kpc. Thus, we can conclude that toroidal disks around a central hole of mass have sizes comparable with the large galaxies and can be related to size-limits on these galaxies. On the other hand, such disks are well inside the cD elliptical galaxies; in order to obtain an accretion disk of dimension kpc, mass parameter of the central black hole have to be .

Of course, if the mass of a protogalactic disk related to a quasar is higher than the mass of the central black hole, the self-gravitational effects of the disk itself have to be taken into consideration. Nevertheless, we can expect that even in the situation like this the repulsive cosmological constant keeps the presence of the outer cusp enabling outflows of matter from the disk. On the other hand, the collimation effect on jets could be efficient even for small toroidal disks, with outer edge located deeply under the static radius. In such disks the self-gravitational effects could usually be neglected.

In the case of accretion onto primordial black holes in the very early universe, with assumed high values of repulsive cosmological constant, we can expect even stronger effects. Considering the electroweak phase transition at  GeV, we obtain an estimate of the primordial effective cosmological constant

Considering the quark confinement at  GeV, we obtain an estimate of the primordial cosmological constant

It follows from the Table 4 that the accretion onto primordial black holes of mass  g, and  g, respectively, is then forbidden in the disk regime because no equilibrium configurations of perfect fluid are allowed in the corresponding Schwarzschild-de Sitter backgrounds. Of course, the accretion can be realized in quasispherical regime in these spacetimes, however, its character represents an open problem.

© European Southern Observatory (ESO) 2000

Online publication: December 11, 2000
helpdesk.link@springer.de