|Annu. Rev. Astron. Astrophys. 1984. 22:
Copyright © 1984 by Annual Reviews. All rights reserved
As the fluid can now move with acceleration and rotation, the Raychaudhuri equation does not directly imply the existence of a singularity in the past. However, the powerful Hawking-Penrose singularity theorems (45, 102) show that as long as suitable energy and causality conditions are satisfied, singularities will indeed occur in a general inhomogeneous universe model [essentially because the matter that thermalized the microwave background radiation has sufficient energy to cause a refocusing of all the null geodesics that generate our past light cone (45)].
In many cases the singularity will be essentially of the same character as in the spatially homogeneous case, often showing the same kind of oscillatory behavior as the Bianchi IX cosmologies (7, 9; but see 5, 8, 57). The major difference is that one can now additionally get timelike singularities.
TIMELIKE INITIAL SINGULARITIES The existence of inhomogeneities at early phases in a universe that is like a FLRW universe at late times implies that "the initial big bang is not necessarily simultaneous. One can formulate initial conditions in such a way that expansion of some parts of the universe are delayed" (116). It has been proposed that such "lagging cores" could be responsible for violent astrophysical phenomena such as QSOs (69, 71a). The situation could be the time-reverse of the collapse to a black hole: this case is called a "white hole" (71a). In this case, the singularity will be surrounded for most of its life by an absolute particle horizon, into which no particles may fall but from which particles may be ejected (76). Difficulties arise concerning the existence of "white holes," particularly in view of the particle creation process that would tend to make them rapidly evaporate away (76, 78).
However, there appears to be nothing preventing the existence of more general timelike singularities that do not occur instantaneously, but continue over a time (22), and that are not associated with such horizons (104); the timelike part of the singularity may have the nature of either a negative- or positive-mass singularity (62, 104). A new dynamics now occurs, for the singularity both emits matter,and radiation to the Universe and receives them from it; thus, it both acts on and can be acted upon by the Universe. This relation of the singularity to the Universe is quite unlike that in the spacelike singularity case where the singularity emits matter and radiation to the Universe, and indeed determines all its initial conditions, but cannot be affected by it. One now has the possibility of investigating the dynamics of interaction of the singularity with the Universe.
These timelike singularities can occur as inhomogeneous segments of a singularity that is spacelike elsewhere (62, 104). Indeed, it is clear that unless one chooses very special conditions in the Universe (by setting to zero the "decaying modes" of density perturbations), inhomogeneities will dominate at early stages (97). Penrose argues on entropy grounds that the early Universe should obey such rather special conditions (77, 78); his argument depends on probability assumptions about the set of all possible universe - assumptions that may be queried in view of the uniqueness of the Universe.
The most intriguing possibility is if one or more isolated timelike singularities occur, with most of the matter passing between them rather than originating at them (45). This situation is closely related to the hope that many people have that the present expanding phase of the Universe should result from a previous collapse phase, with the formation of isolated singularities and subsequent reexpansion. Then the hot big bang phase is like an inhomogeneous SHBB but is preceded by a previous collapsing phase. The overall picture is like "cyclic" FLRW universes collapsing from infinity or from a previous expansion phase and then reexpanding (with most of the matter avoiding the singularities; these must form because the energy conditions are satisfied).
We do not have exact cosmological models where this situation occurs; but two examples indicate it may be possible, namely the Reissner-Nordstrom solution with test matter in it (45), and certain classes of Whimper singularities where two horizons may occur (31) (if these exist; no exact solutions of this kind are known at present). In both these cases the situation is probably unstable, but they show that it is not obviously prohibited by the field equations. If it does occur, the conditions of the present phase of expansion are determined by the two-way interaction between the Universe and the singularity formed from the previous collapse phase; no energy violations or quantum effects are needed to cause such a "bounce."
ESSENTIALLY INHOMOGENEOUS UNIVERSES These are universe models containing no epochs where the space-time is approximately spatially homogeneous; that is, they are never like a FLRW universe model. This assumption, of course, goes against the usual "cosmological principle," which states a priori that the Universe is spatially homogeneous (109); but that is an unverified and, indeed, possibly unverifiable (28) philosophical statement that could be incorrect. (3)
One possibility is asymptotically flat models - an isolated dense universe surrounded by empty space. An example of this kind is the recent Narlikar & Burbidge (68) proposal, in which all detectable matter lies in an expanding super-supercluster. While this kind of situation is a possibility, many features remain to be explained, e.g. the element abundances and the microwave background radiation. The specific Narlikar-Burbidge version appears to be ruled out by QSO absorption-line measurements (81).
A related concept is that of a hierarchical universe, where clustering occurs over all possible scales, so that each observer is within an effective spherical inhomogeneity but the average density of matter goes to zero in larger and larger volumes. In this model the Universe is spatially homogeneous (19). However, this is difficult even to describe mathematically (as the description used depends completely on the scale of averaging); it is not clear that the models proposed so far (10, 110) in fact represent the hierarchical situation fully.
If we abandon statistical homogeneity, the only universe models that give isotropic observations (4) are inhomogeneous, spherically symmetric models, where we are near a center of symmetry (11). In order to explain the microwave background radiation in such models, an intriguing possibility is that there are two centers of symmetry in the Universe, with one in our neighborhood and the other at the location of a timelike singularity. This singularity would be surrounded by a massive fireball emitting blackbody radiation; thus the space sections of the Universe would look like Figure 2b, with the fireball - like a supermassive star - at the antipodes. A space-time diagram can be drawn as in Figure 2a. The timelike singularity at the center of the fireball would continually emit radiation and particles into the Universe as well as receive them from the Universe - it would be dynamically interacting with the Universe (as discussed above). A remarkable duality between such models and the FLRW universes is possible, with each stage of the FLRW universes that occurred in the past as a result of time evolution also occurring in these universes spatially as a result of spatial inhomogeneity (32).
Figure 2. A two-centered universe, with our Galaxy near one of the centers. (a) Space-time diagram of the Universe (two spatial dimensions suppressed). It is a cylinder, with a timelike singularity at the other center. Our past light cone is shown. (b) Space section of the Universe (one spatial dimension suppressed). The singularity lies behind a fireball, which is like a supermassive star "over there."
Perhaps the most radical such proposal is the completely static two-centered universe possibility (17, 32), where the redshifts observed for distant galaxies are purely cosmological gravitational redshifts (cf. Equation 2b). High pressures are needed in these models to cause the acceleration that underlies the gravitational redshift; this is not impossible but is perhaps uncomfortable. Two problems arise: Firstly, because of the spherical symmetry, the (redshift, distance) law in such a universe must be quadratic at the center, so (as in Segal's theory discussed in Section 3.1) we can test this model by seeing if a quadratic relation is indeed observed. The usual data show a linear relation, contrary to expectation in such universes; however, these data are analyzed on the assumption the Universe is a FLRW universe, and selection effects play a crucial role in determining which is the correct law. (Essentially the same analysis will apply here as in the Segal theory.) The data still have not been examined using an analysis that takes surface brightness effects fully into account. Secondly, philosophical problems may arise because of the Earth being near the center in these models. However, these qualms can be offset by noting that physical conditions can favor the occurrence of life at this center in such universes and by considering the a priori improbability of the FLRW universe models (27a). On the positive side, it is intriguing to note that the background radiation, still our best evidence for isotropy of the Universe, would appear isotropic at every point in such universes (because its temperature will be determined by the ratio of gravitational potentials at the points of emission and reception of the radiation; and this is independent of the path the light follows to the observer). Thus, observed isotropy of this radiation does not, in this case, imply that the Universe is isotropic about our position. On the other hand, isotropy of galactic redshifts would carry this implication.
The major interest of this model is the way in which it highlights the quite different role the singularity can play as a continuing influence in the evolution of the Universe. It is not a once-and-for-all event, but is instead a continuing source and sink of matter and information; in this case, it enables a steady state. The singularity has a different meaning than in the FLRW universe.
While it is difficult to make a convincing case for a static universe of this kind, it seems likely that expanding versions maintaining the essential features of Figure 2 could account for all present observations. If the expansion originated in a singularity, the model would then be essentially like the delayed core models mentioned above; but the expansion could, for example, conceivably be preceded by a stationary phase or by a contraction phase to some minimum radius with the singularity always localized in one region of the Universe.
FURTHER EFFECTS As in the previous cases, one can consider the nature of inhomogeneous cosmologies if energy violation, other gravitational theories, or quantum effects are taken into account.
Perhaps the most intriguing new possibility is that the energy conditions may be violated for essentially geometric reasons. The basic point here is that the Einstein field equations are tested, and believed to hold, on a particular scale (say, that of the solar system). In an inhomogeneous situation it is not clear that the geometrically averaged space-time metric representing the situation at other scales (say, that of clusters of galaxies) will obey the same field equations; indeed, one would expect correction terms (similar to the polarization terms in electromagnetic theory) to allow for the spatial averaging that is taking place (29). However, there is no guarantee that these terms will obey the energy conditions and, indeed, in a turbulent situation they might be able to cause effective negative energy densities so large as to avoid a singularity (59). Thus the space-time metric averaged out to describe the Universe at the larger scale could be singularity free (but small-scale singularities could still occur, e.g. local black holes resulting from gravitational collapse of stars).
3 It is false in the currently fashionable inflationary universe scenario, where major inhomogeneities occur at the bubble walls. Back.
4 See, for example (55) for a summary of the evidence that the Universe is isotropic about us. Back.