In a third stage of cosmological modeling, the epoch around and before the Planck time (10-44 s) is briefly dealt with. At such extremely early epochs, quantum mechanics and quantum field theory are applied. At present, a consistent and mathematically rigorous quantum field theory of gravitation, i.e., quantum gravity, is under construction but still not completed. 31 Nevertheless, within general relativity, intriguing schemes like canonical quantization in the geometrodynamics approach [133], [134], [135], its gauge theoretical variant loop quantization [137], [136], [138], [139], covariant quantization, e.g., in the form of Feynman path integral quantization [140], and the (numerically implemented) models of causal dynamical triangulation [141], [142] are pursued with impressive success. 32 Some general hypotheses are made:
- The gravitational field must be quantized around and before the Planck
epoch.
- Unlike in the procedure for other fields, quantization of gravity must
be done in a background independent manner (in canonical quantization).
- All local and global degrees of freedom of the gravitational field
must be taken into account.
- Einstein's field equations hold right up to the big bang singularity.
That gravity ought to be be quantized is the majority vote. Some think that quantization must be performed within a theory in which all fundamental interactions are united, e.g., a claim made by string theory. At present, string theory does not yet noticeably contribute to a solution of the most pressing questions in quantum gravity; it still is in "a rather preliminary stage" ([143], p. 753). Few believe in gravity as a classical field generated, perhaps, as an effective field by the other fundamental interactions. 33 Looked at from usual field quantization, at the root of the difficulties with quantization of gravity is its (perturbative) non-renormalizability. From a more technical point of view, quantization with (Hamiltonian and diffeomorphism) constraints, as in the case of the Hamiltonian formulation of general relativity, is a hurdle. Moreover, it is not entirely clear whether it suffices to quantize the gravitational field on a continuous space-time or, whether the very concept of a manifold ought to be replaced by discrete sets (causal set theory), combinatorily defined discrete structures like graphs, or spin networks (cf. [144], [145]). In loop gravity, while continuous 3-geometries still are investigated, area- and volume operators with a discrete spectrum do appear. Whether they are observables in the usual sense, i.e., commuting with the diffeomorphism constraints, is not entirely clear. 34 Background independence means that quantization should not rely on a metrical structure but, at most, on a differentiable manifold (cf. [146], [147]. Consequently, a lot of advanced mathematics is required. As no empirical input is available at present, "mathematical consistency is the only guiding principle to construct the theory" ([138], p. XX). The recent endeavour to derive rigorous results belongs into mathematical physics. For a critical discussion cf. [148], [149]. Quantum gravity is said to apply to two main systems: the very early universe (quantum cosmology) and to evaporating black holes.
5.2.1. Law of initial conditions?
On the one hand, application of quantum mechanics to the universe is seen as an intermediate step in between the big bang and the inflationary epoch with the aim of providing initial conditions for inflation. But quantum cosmology also has been taken as a program for a cosmogonic theory: an attempt to construct a theory determining uniquely the initial conditions of the universe [150], [151], [152]. Turned around: as a program for a theory avoiding the big bang singularity. 35 Such an endeavor makes sense only if the universe itself carries the rationale for its initial data. If transferred to human life, this would mean that the reason for us coming to life does not lie in our parents but in ourselves. Strange as this thought may be (above the level of protozoans): a human being and the universe are quite different systems. It seems plausible, philosophically, that the cosmos cannot be thought of without the inclusion of a reason for its coming into being. In classical theory, the very idea of prescribing uniquely the initial data of a system by help of its dynamics is violating the spirit of physics. Perhaps, quantum theory could make the difference. For a positive suggestion in this direction within quantum cosmology, cf. [154], [155].
5.2.2. The Wheeler-DeWitt equation
In the Hamiltonian formulation, space-time is foliated into space sections, and the Einstein field equations are decomposed into time-evolution equations and constraint equations on the 3-geometries 3 g. Canonical quantization leads to the Wheeler-DeWitt equation (WDW) for the wave function of the universe ψ, a formal analogue of the stationary Schrödinger equation 36. It is a functional ψ[3 g, ϕ] of the geometry of space sections and the matter fields ϕ and hence defined on an infinite-dimensional space called superspace. The spacetime geometry can be pictured as a trajectory in superspace. The wave function of the universe represents the superposition of all possible space-time geometries correlated with matter functions [156]. It is assumed to be a pure state. Mathematically, the WDW-equation is not well defined (factor ordering and regularization problems). Nevertheless, one of the successes of the canonical approach is that its semiclassical approximation bridges the gap to quantum field theory on a fixed background [153].
In model calculations, isotropy and homogeneity of the space geometry is assumed and leads to a wave function ψ depending on just one geometric variable: the scale factor a of the Friedman models. Usually, only a single scalar matter field ϕ is taken into account such that ψ = ψ[a, ϕ]. In this case, the infinite dimensional superspace is reduced to a finite number of degrees of freedom, i.e. to minisuperspace.
Despite this technical simplification, the main problem cannot be circumnavigated: a unique solution of the Wheeler-DeWitt equation is obtained only if a boundary condition for ψ is chosen. Several suggestions to this end have been made. In the path integral formulation [150], [157] ψ is determined by a summing over all paths describing compact euclidean 4-geometries with regular matter fields. All 4-geometries must have a given 3-geometry as their boundary (no-boundary-condition) 37. An alternative condition is Vilenkin's quantum tunneling from nothing (where "nothing" corresponds to the vanishing of the scale factor a): the universe is nucleating spontaneously as a DeSitter space [159], [160]. This boundary condition has been criticized on the ground that it equally well describe tunneling into nothing. For a detailed discussion cf. ([119], section 8.3, [153], section 4.2). In loop quantum cosmology, the WDW-equation is replaced by a discrete evolution equation.
Because the dynamical equations follow from the constraints on the spatial hypersurfaces, the wave function of the universe cannot depend on an external time parameter as is cosmic time. In minisuperspace, the Wheeler-DeWitt equation is a hyperbolic differential equation the dynamics of which is depending on two variables, a and ϕ, both of which can play the role of an internal time. The ambiguity in the selection of an internal time parameter permits reinterpretation of the WDW-equation as a Klein-Gordon equation. In particular (cosmological) models, the (bounded) volume of the space sections are used as a measure of time. At the big bang, in loop quantum gravity, the (degenerated) eigenvalue of the volume operator is zero.
5.2.3. Puzzles of quantum cosmology
An acceptable quantum cosmology will have to solve three internal problems:
- to give a definition of time,
- to determine the role of "observers",
- to describe the "emergence" of a classical universe from the quantum
one,
plus one external:
- to link quantum cosmology with empirical data.
The striking inequality in the treatment of time and space is an inheritance from non-relativistic quantum mechanics. Presently, at best, time appears as a notion in a semiclassical approximation scheme ([119], section 5.2). For a detailed discussion of the "quantum problem of time" cf. ([138], section 2.4). 38
A straightforward application of the Copenhagen-interpretation of quantum mechanics to the wave function of the universe does not make sense. Who is the classical observer carrying out preparation- and other measurements? A way out is to assume that the (quantum) universe is divided into one part as "the system to be looked at" and the remainder as "the measuring apparatus" [162]. A continuous shift of the borderline between observing and observed parts of the universe would then be necessary. In fact, if quantum gravity is to lead to the existence of a classical limit, i.e., how classical space-time can emerge including Einstein's field equation, another part might have to be defined, the "environment". Its wave function is entangled with the measuring part of the universe ("the apparatus"). The interaction with the environment will lead to "decoherence" and provide classical properties by a continuous measurement process [163], [164]. Possibly, measuring apparatus and environment can be made to coincide in the universe. For the interpretation of the wave function of the universe, it may be unavoidable to employ some version of Everett's interpretation of quantum mechanics; in it the splitting of the wave function by a measurement is equivalent to splitting the universe into many copies. In each of these copies one of the allowed measurement results occurs [165]. 39 Another proposal replaces the "many worlds" of Everett by a "many histories" interpretation in which observers making measurements are within "decohering" histories of the same universe [152].
Originative cosmology is taking place in our minds - as pure mathematics does. By it, awareness of what could be potentially real is produced. Passage from the potentially to the actually real requires a linking to an empirical basis. In the example of Bose condensation, the time span between the suggestion of the idea and its experimental validation was relatively short: it took about 60 years. The agreement among scientists in the case of quantum cosmology may take a very much longer time.
5.3. Make-believe cosmology: the multiverse
The conceptually well founded development of quantum cosmology and quantum gravity is very removed from the multiverse scenario to be briefly sketched now. A multiverse is an ensemble of universes. At best, the elements ("universes") of the set are generated from some underlying theory, e.g., from the "string landscape" (see below). At worst, the ensemble is just assumed to exist. A multiverse can be represented by a higher-dimensional space-time with four or more space dimensions. Often, this is done within the framework of "braneworld", in which a 3-dimensional space resides in a higher dimensional space, called "the bulk" to which time is added. Gravitation can play in the bulk, all other interactions are restricted to the brane. The additional spatial dimensions may be compactified or not. The multiverse can also consist of an infinite number of replica of one and the same universe as the many-worlds interpretation of quantum mechanics would imply. Another case is the multi-domain multiverse with its "universe-bubbles" bifurcating away from another in particular inflationary schemes (eternal inflation). For a discussion of different brands of multiverses cf. [167].
The multiverse-concept is introduced in order to help solving philosophical problems inherent in, or superimposed on cosmology. With the first, avoidance of the singularity at the big bang is meant, with the second an attempt at bringing the biosphere back into the realm of the universe (anthropic principles).
In a special approach in brane cosmology, the ekpyrotic model, the universe is embedded as a 3-(mem)brane in a higher-dimensional space plus time along with other universes ("parallel branes"). All expand independently according to general relativity. The ekpyrotic model hypothesizes that the origin of the observable universe occurred when two parallel branes collided [168]. It is the precurser to cyclic universe models [169]. In them, a periodical big crunch is followed by a big bang with up to trillions of years (~ 1012) in between each bang and crunch. Density and temperature remain finite. The cyclic universes are said to be an alternative to inflation; they produce the right density fluctuation spectrum [170]. A further example for a multiverse scenario is the so-called "string landscape". It is the energy-"manifold" formed by all degenerated string vacuum solutions (their number is given as of the order of ~ 10500). From each vacuum state a universe is assumed to "nucleate" with a certain probability. Relying on an estimate ascribed to R. Penrose ([35], p. 728-730), the nucleation of "our" universe (at energies ~ 1016 GeV) would have had only a probability of 10-10123.
If all this is not solely forming a mental construct, not just philosophers might have difficulties in relating the multiverse with the notion of "all that exists in a physical sense". M. Rees is reducing the problem to a semantical one: what we now call "universe" could be named "metagalaxy"; the "multiverse" would be re-named "universe" ([171], p. 57). This stand hides a change in ontology: the multiverse is taken to exist in the same sense as the solar system does. In a correspondence about whether Everett's "many-worlds" interpretation of quantum mechanics should be taken as describing infinitely many "really existing" universes, or only logical mental possibilities, B. DeWitt sided with the first claim and asked: "Is there any difference" between things "physically real" and "abstractions such as numbers and triangles"? ([172], p. 10). In this spirit, it has been claimed recently that the introduction of the concept multiverse is leading to "an extension of the Copernican Principle": "The universe is not at the center of the world (the multiverse)" [173], p. 13). We cannot but conclude that, in the mind of the author, the multiverse now is "all that exists in a physical sense". A little less daring was, two decades ago, Tipler's definition of the Universe (with a capital U) to consist of all logically possible universes where "Universe" was the totality of everything in existence and "universe" a single Everett-branch [174], [175]. Enthusiasm and playfulness may have seduced some theorists to act on a quip, heard occasionally,: "All that can be thought of and expressed by a mathematical scheme must be realized in nature, somewhere". The "realistic" view of the multiverse leads to the uneasy task of finding a link between this system and empirical data upon which physics as we know it is based. A task which may well be impossible to fulfill (Cf. [73] p. 406). It is not made easier by the fact that in many of the multiverse definitions, their universe-elements are causally disjoint: they cannot be observed from our place. Apparently, on the assumption that quantum mechanics is valid also in the multiverse and that the wavefunctions of the universe-elements can form an entangled state, we are offered imprints of the multiverse on CMB in the form of two underdense regions (voids) one of which is connected with the cold spot ([173], p. 8-9).
A regress ad infinitum is not excluded, with its first step being the introduction of the concept "multi-multiverse" as the set of all multiverses. 40
5.3.3. Multiverse questionaire
The questions asked within the multiverse scenario are quite different from those of "quantum cosmology" (section 4.3), or "physical cosmology" (section 2.2). We list some of them:
- How large is the multiverse (finite, infinite)?
- What is its precise structure?
- Do all members have the same (or similar) properties (dimension, geometry,
physical laws)?
- How can the members be compared (i.e., empirically, not just by a
mathematical classification)?
- Is the multiverse (as an ensemble) a dynamical system (with a history), or
not?
- Why is there a need for a selection principle leading to a particular
universe?
- How can the values for the (dimensionless) physical constants be
derived from the multiverse?
- Can the multiverse provide the initial conditions for a universe like
"ours"?
While, previously, cosmologists were satisfied with trying to find out whether the fundamental physical constants are depending on cosmic time, or not, now the demand is to explain why they have the particular values observed [177]. Cosmological modeling is transformed into a bird's eye view of the universe: scientists working in multiverse theory seemingly put themselves "outside" of "their" universe (mentally, that is). The necessary fine-tuning of some of the parameters required for life to exist seems to be a strong motivation for the concept of multiverse. It appears to me that many of the above questions are meaningless within physics; at this time, they seem to belong into philosophical thinking about the cosmos.
31 This is no surprise, when we think that even quantum field theory in Minkowski space has not yet been made mathematically rigorous in all aspects. Back.
32 It is an open question whether these different approaches will lead to equivalent quantum theories of gravitation. Back.
33 This is not to be mixed up with gravity dealt with as an effective quantum field theory with a high-energy cut-off. Back.
34 For a detailed discussion of the volume operator cf. [138], Secs. 13.1-13.6, pp. 432-457. Back.
35 From quantum cosmology, we may expect more than forming a "toy model for full quantum gravity in which the mathematical difficulties disappear", cf. ([153], p. 894). Back.
36 In reality, WDW comprises an infinite number of equations. Back.
37 Cf. C.J. Isham [158]: "the universe is created ex nihilo since the 4-manifold has only the connected 3-space as its boundary". Back.
38 It has also been argued that time can be eliminated altogether [161]. Back.
40 The plural "multiverses" has already been amply used, albeit only as a logical possibility, not as "reality". Cf. several articles in [8] with ([176], p. 368) as an example. Back.