Multiverses and physical cosmology (original) (raw)

Abstract

The idea of a multiverse — an ensemble of universes — has received increasing attention in cosmology, both as the outcome of the originating process that generated our own Universe, and as an explanation for why our Universe appears to be fine-tuned for life and consciousness. Here we carefully consider how multiverses should be defined, stressing the distinction between the collection of all possible universes, and ensembles of really existing universes that are essential for an anthropic argument. We show that such realized multiverses are by no means unique. A proper measure on the space of all really existing universes or universe domains is needed, so that probabilities can be calculated, and major problems arise in terms of realized infinities. As an illustration we examine these issues in the case of the set of Friedmann–Lemaître–Robertson–Walker universes. Then we briefly summarize scenarios such as chaotic inflation, which suggest how ensembles of universe domains may be generated, and we point out that the regularities which must underlie any systematic description of truly disjoint multiverses must imply some kind of common generating mechanism. Finally, we discuss the issue of testability, which underlies the question of whether multiverse proposals are really scientific propositions.

1 Introduction

The idea of a multiverse has been proposed as the only scientifically based way of avoiding the fine-tuning required to set up the conditions for our seemingly very unlikely Universe to exist. Weinberg (2000a,b), for example, uses it to explain the value of the cosmological constant, which he relates to anthropic issues. Rees (2001a,b,c) employs it to explain the whole set of anthropic coincidences (Barrow & Tipler 1986), that is, to explain why our Universe is a congenial home for life. These and similar proposals have been triggered by the dawning awareness among many researchers that there may be many other existing universes besides ours. This possibility has received strong stimulation from proposals such as Andrei Linde's chaotic inflationary scenario (Linde 1983, 1990), in which the origin of our own observable universe region naturally involves the origin of many other similar expanding universe regions.

There is, however, a vagueness about the proposed nature of multiverses. They might occur in various ways, as discussed by Leslie (1996), Weinberg (2000a,b), Tegmark (2003) and Gardner (2003). They might originate naturally in different times and places (cf. Ellis 1979) through meta-cosmic processes such as chaotic inflation, or in accord with Smolin's (1999) cosmic Darwinian vision. In the latter case, an ensemble of expanding universe regions grows from each other following gravitational collapse and re-expansion, where natural selection of universes through optimization of black hole production leads to bio-friendly universe regions. This is an intriguing idea, but with many uncertain steps — in particular, no proof has been given of the last step, that the physics that maximizes black hole production also favours life. They might be associated with the multi-universe Everett–Wheeler-type interpretation of quantum mechanics (Deutsch 1998), or perhaps multiverses can be truly disjoint collections of universes (see Sciama 1993; Lewis 2000; Rees 2001a,b,c; Tegmark 1998, 2003).

Some refer to the separate expanding universe regions in chaotic inflation as ‘universes’, even though they have a common causal origin and are all part of the same single space–time. In our view (as ‘uni’ means ‘one’) the Universe is by definition the one unique connected1 existing space–time of which our observed expanding cosmological domain is a part. We will refer to situations such as in chaotic inflation as a multidomain universe, as opposed to a completely causally disconnected multiverse. Throughout this paper, when our discussion pertains equally well to disjoint collections of universes (multiverses in the strict sense) and to the different domains of a multidomain universe, we shall for simplicity simply use the word ‘ensemble’. When an ensemble of universes is all subregions of a larger connected space–time — the ‘Universe as a whole’– we have the multidomain situation, which should be described as such. Then we can reserve ‘multiverse’ for the collection of genuinely disconnected ‘universes’– those which are not locally causally related.

So far, none of these ideas has been developed to the point of actually describing such ensembles of universes in detail, nor has it been demonstrated that a generic well-defined ensemble will admit life. Some writers tend to imply that there is only one possible multiverse (characterized by ‘all that can exist does exist’, Lewis 2000; see also Gardner 2003). This vague prescription actually allows a vast variety of different realizations with differing properties, leading to major problems in the definition of the ensembles and in averaging, due to the lack of a well-defined measure and the infinite character of the ensemble itself. Furthermore, it is not at all clear that we shall ever be able to accurately delineate the class of all possible universes.

The aim of this paper is to help clarify what is involved in a full description of an ensemble of universes. Our first contribution is clarifying what is required in order to describe the space of possible universes, where much hinges on what we regard as ‘possible’. However, that is only part of what is needed. It is crucial to recognize that anthropic arguments for existence based on ensembles of universes with specific properties require an actually existing ensemble with all the required properties. For the purposes of providing an explanation of existence, simply having a conceptually possible ensemble is not adequate — we need a link to objects or things that actually exist, or to mechanisms that make them exist.

The second contribution of this paper is to show how an actually existing ensemble may be described in terms of a space of possible universes, by defining a distribution function (discrete or continuous) on the space of possible universes. This characterizes which of the theoretically possible universes have been actualized in the ensemble — it identifies those that have actually come into existence. This leads us to our third point: the problems arising when it is claimed that there is an actually existing ensemble containing an infinite number of universes or of expanding universe regimes. Actually existing infinities are very problematic.

There are fundamental issues that arise in considering ensembles of actually existing universes. What would explain the existence of an ensemble, and its specific properties? Why should there be this particular ensemble, rather than some other one? Why should there be any regularity at all in its properties? The fourth point we make is that if all the universes in an ensemble show regularities of structure, then that implies some common generating mechanism. Some such structuring is necessary if we are to be able to describe a multiverse with specified properties — a coherent description is only possible through the existence of such regularities. Hence, a multiverse consisting of completely causally disconnected universes is a problematic concept.

The issue of testability is a further important consideration. Is there any conceivable direct or indirect way of testing for existence of an ensemble to which our Universe belongs? Our fifth point is that there is no way we can test any mechanism proposed to impose such regularities; they will of necessity always remain speculative. The sixth point (see Leslie 1996, Gardner 2003) is to argue that the existence of multiverses or ensembles is in principle untestable by any direct observations, and the same applies to any hypothesized properties we may suppose for them. However, certain observations would be able to disprove the existence of some multidomain ensembles. It is only in that sense that the idea is a testable proposition.

It is clear that in dealing with multiverses we inevitably run up against philosophical and metaphysical issues, for example concerning the ability to make scientific conclusions in the absence of observational evidence, and in pursuing the issue of realized infinities. A companion, more philosophically oriented paper will pursue those issues.

2 Describing Ensembles: Possibility

To characterize an ensemble of existing universes, we first need to develop adequate methods for describing the class of all possible universes. This requires us to specify, at least in principle, all the ways in which universes can be different from one another, in terms of their physics, chemistry, biology, etc.

2.1 The set of possible universes

The basis for describing ensembles or multiverses is contained in the structure and the dynamics of a space M of all possible universes m, each of which can be described in terms of a set of states s in a state space M. Each universe in M will be characterized by a set P of distinguishing parameters p, which are coordinates on S. Some will be logical parameters, some will be numerical constants, and some will be functions or tensor fields defined in local coordinate neighbourhoods for s. Each universe m will evolve from its initial state to some final state according to the dynamics operative, with some or all of its parameters varying as it does so. The course of this evolution of states will be represented by a path in the state space S, depending on the parametrization of S. Thus, each such path (in degenerate cases a point) is a representation of one of the universes m in M. The coordinates in S will be directly related to the parameters specifying members of M. The parameter space P has dimension N which is the dimension of the space of models M. The space of states S has _N_+ 1 dimensions, the extra dimension indicating the change of each model's states with time, characterized by an extra parameter, e.g. the Hubble parameter H which does not distinguish between models but rather determines what is the state of dynamical evolution of each model. Note that N may be infinite, and indeed will be so unless we consider only geometrically highly restricted sets of universes.

It is possible that with some parameter choices the same physical universe m will be multiply represented by this description; thus, a significant issue is the equivalence problem — identifying which different representations might in fact represent the same universe model. In self-similar cases we obtain a single point in S described in terms of the chosen parameters P; the state remains unchanged in terms of the chosen variables. However, we can always obtain such variables for any evolution, as they are just comoving variables, not necessarily indicating anything interesting is happening dynamically. The interesting issue is if this invariance is true in physically defined variables, e.g. expansion normalized variables; then physical self-similarity is occurring.

The very description of this space M of possibilities is based on an assumed set of laws of behaviour, either laws of physics or meta-laws that determine the laws of physics, which all universes m have in common; without this, we have no basis for setting up its description. The detailed characterization of this space, and its relationship to P, will depend on the matter description used and its behaviour. The overall characterization of M therefore must incorporate a description both of the geometry of the allowed universes and of the physics of matter. Thus, the set of parameters P will include both geometric and physical parameters.

The space M has a number of important subsets, for example as follows.

• (i)
  _M_FLRW — the subset of all possible exactly Friedmann–Lemaître–Robertson–Walker (FLRW) universes, described by the state space _S_FLRW (in the case of dust plus non-interacting radiation a careful description of this phase space has been given by Ehlers & Rindler (1989).
• (ii)
  _M_almost—FLRW — the subset of all perturbed FLRW model universes. These need to be characterized in a gauge-invariant way (see, for example, Ellis & Bruni 1989) so that we can clearly identify those universes that are almost-FLRW and those that are not.
• (iii)
  _M_anthropic — the subset of all possible universes in which life emerges at some stage in their evolution. This subset intersects _M_almost-FLRW, and may even be a subset of _M_almost-FLRW but does not intersect _M_FLRW (realistic models of a life-bearing universe like ours cannot be exactly FLRW, for then there is no structure).
• (iv)
  _M_Observational — the subset of models compatible with current astronomical observations. Precisely because we need observers to make observations, this is a subset of _M_anthropic.

If M truly represents all possibilities, we must have a description that is wide enough to encompass all possibilities. It is here that major issues arise. How do we decide what all the possibilities are? What are the limits of possibility? What classifications of possibility are to be included?‘All that can happen, happens’ must imply all possibilities, as characterized by our description in terms of families of parameters; all allowed values must occur, and they must occur in all possible combinations. The full space M must be large enough to represent all of these possibilities, along with many others we cannot even conceive of, but which can nevertheless in principle also be described by such parameters. An interesting related point has been pointed to us by Jean-Phillipe Uzan; it may be that the larger the possibility space considered, the more fine-tuned the actual universe appears to be — for with each extra possibility that is included in the possibility space, unless it can be shown to relate to already existing parameters, the actual universe and its close neighbours will live in a smaller fraction of the possibility space. For example, if we assume general relativity then there is only the parameter G to measure; but if we consider scalar-tensor theories, then we have to explain why we are so close to general relativity now. Hence there is a tension between including all possibilities in what we consider, and giving an explanation for fine-tuning.

From these considerations we have the first key issue.

Issue 1. What determines M? Where does this structure come from? What is the meta-cause that delimits this set of possibilities? Why is there a uniform structure across all universes m in M?

The meta-question is whether any of these questions can be answered scientifically. We return to this at the end.

2.2 Adequately specifying possible anthropic universes

When defining any ensemble of universes, possible or realized, we must specify all the parameters which differentiate members of the ensemble from one another at any time in their evolution. The values of these parameters may not be known or determinable initially in many cases — some of them may only be set by transitions that occur via processes such as symmetry breaking within given members of the ensemble. In particular, some of the parameters whose values are important for the origination and support of life may only be fixed later in the evolution of universes in the multiverse.

We can separate our set of parameters P for the space of all possible universes M into different categories, beginning with the most basic or fundamental, and progressing to more contingent and more complex categories. Ideally they should all be independent of one another, but we will not be able to establish that independence for each parameter, except for the most fundamental cosmological ones. In order to categorize our parameters, we can doubly index each parameter p in P as pj(i) such that those for _j_= 1–2 describe basic physics, those for _j_= 3–5 describe the cosmology (given that basic physics), and those for _j_= 6–7 pertain specifically to emergence and life (we must include the latter if we seriously intend to address anthropic issues). Our characterization is as follows.

• _p_1(i) are the basic physics parameters within each universe, excluding gravity — parameters characterizing the basic non-gravitational laws of physics in action, related constants such as the fine-structure constant α, and including parameters describing basic particle properties (masses, charges, spins, etc.). These should be logical parameters or dimensionless parameters, otherwise we may be describing the same physics in other units.
• _p_2(i) are basic parameters describing the nature of the cosmological dynamics, e.g. _p_2(1) = 1 indicates that Einstein gravity dominates, _p_2(1) = 2 indicates that Brans–Dicke theory dominates, _p_2(1) = 3 indicates that electromagnetism dominates, etc. Associated with each choice are the relevant parameter values, e.g. _p_2(2) =G, _p_2(3) =Λ, and in the Brans–Dicke case _p_2(4) =ω. If gravity can be derived from more fundamental physics in some unified fundamental theory, these will be related to the parameters _p_1(i); for example, the cosmological constant may be determined from quantum field theory and basic matter parameters.
• _p_3(i) are cosmological parameters characterizing the nature of the matter content of a universe. These parameters encode whether radiation, baryonic matter, dark matter, neutrinos, scalar fields, etc., occur, in each case specifying the relevant equations of state and auxiliary functions needed to determine the physical behaviour of matter (e.g. a barotropic equation of state for a fluid and the potential function for a scalar field). These are characterizations of physical possibilities for the macro-states of matter arising out of fundamental physics, so the possibilities here will be related to the parameters in _p_1(i). Realistic representations of the universe will include all the above, but simplified ensembles considered for exploratory purposes may exclude some or many of them.
• p_4(i) are physical parameters determining the relative amounts of each type of matter present in the specific cosmological solutions envisaged, for example, the density parameters Ω_i of various components at some specific stage of its evolution (which then, for example, determine the matter to antimatter ratio and the entropy to baryon ratio). The matter components present will be those characterized by _p_3(i).
• _p_5(i) are geometrical parameters characterizing the space–time geometry of the cosmological solutions envisaged, for example, the scalefactor a(t), Hubble parameter H(t), and spatial curvature parameter k in FLRW models. These will be related to _p_4(i) by the gravitational equations set in _p_2(i), for example, the Einstein field equations.
• _p_6(i) are parameters related to the functional emergence of complexity in the hierarchy of structure, for example allowing the existence of chemically complex molecules. Thus, _p_6(1) might be the number of different types of atoms allowed (as characterized in the periodic table), _p_6(2) the number of different states of matter possible (crystal, glass, liquid, gas, plasma, for example), and _p_6(3) the number of different types of molecules characterized in a suitable way. These are emergent properties arising out of the fundamental physics in operation, and so are related to the parameters set in _p_1(i).
• _p_7(i) are biologically relevant parameters related specifically to the functional emergence of life and of self-consciousness, for example _p_7(1) might characterize the possibility of supra-molecular chemistry and _p_7(2) that of living cells. This builds on the complexity allowed by _p_6(i) and relates again to the parameter set _p_1(i).

It is important to note that these parameters will describe the set of possibilities we are able to characterize on the basis of our accumulated scientific experience. The limits of our understanding are relevant here, in the relation between what we conceive of as this space of possibilities, and what it really is. There may be universes that we believe are possible on the basis of what we know of physics, which may in fact not be possible. There may also be universes that we conceive of as being impossible for one reason or another, which turn out to be possible. Also, it is very likely that we simply may not be able to imagine or envisage all the possibilities. However, this is by no means a statement that ‘all that can occur’ is arbitrary. On the contrary, specifying the set of possible parameters determines a uniform high-level structure that is obeyed by all universes in M.

We see, then, that a possibility space M is the set of universes (one-parameter sets of states S) obeying the dynamics characterized by a parameter space P, which may be considered to be the union of all allowed parameters pj(i) for all i, j as briefly discussed above:

Because the parameters P determine the dynamics, the set of paths in S characterizing individual universes m are determined by this prescription. In some particular envisaged ensemble, some of these parameters (‘class parameters’) may be fixed across the ensemble, thus defining a class of universes considered, while others (‘member parameters’) will vary across the ensemble, defining the individual members of that class. Thus

As we consider more generic ensembles, class parameters will be allowed to vary and so will become member parameters. In an ensemble in which all that is possible happens, all parameters will be member parameters; however, this is so hard to handle that we usually analyse subspaces characterized by particular class parameters.

2.3 Describing the geometry of possible universes

Cosmological models are characterized by a preferred time-like vector field u:_uaua_=−1, usually the fluid flow vector (Ellis 1971), but sometimes chosen for other reasons, e.g. to fit local symmetries. To describe a cosmological space–time locally we must give a description of its (generally inhomogeneous and anisotropic) geometry via suitable parameters _p_5(i). This description may be usefully given in terms of a tetrad basis as follows (see Ellis & van Elst 1999; Wainwright & Ellis 1996; Uggla et al. 2003).

Feature 1. A set of local coordinates X = {x i} must be chosen in each chart of a global atlas. This will, in particular, have a time coordinate t, which will be used to characterize evolution of the universe. This should be chosen in as uniform a way as possible across all the universes considered; for example, it may be based on surfaces of constant Hubble parameter H for the preferred vector field u.

Feature 2. In each chart, to determine the geometry we must be given the components ɛ = [_e_ia (_x_j)] of an orthonormal tetrad with the fluid flow vector chosen as the time-like tetrad vector (a, b, _c_… are tetrad indices; four of these components can be set to zero by suitable choice of coordinates). Together the coordinates and the tetrad form the reference frame

1

The metric tensor is then

where η_ab_ is the Minkowski metric

(because the tetrad is orthonormal) and e b j(xj) are the inverse of e i a(xj)

Thus, the metric is given by

2

The basic geometric quantities used to determine the space–time geometry are the rotation coefficients Γ_a_ bc of this tetrad, defined by

These may conveniently be given in terms of geometric quantities

3

characterized as follows:

Here, is the acceleration of the fluid flow congruence, θ is its expansion, σαβ=σ(αβ) is its shear (σ_b_ _b_= 0), and ωαβ=ω[αβ] its vorticity, while _n_αβ=n(αβ) and _a_α determine the spatial rotation coefficients (see Wainwright & Ellis 1996; Ellis & van Elst 1999). Greek indices (with range 1–3) indicate that all these quantities are orthogonal to ua. They are space–time fields, although in particular high-symmetry cases they may be independent of many or of all the coordinates. The Jacobi identities, Bianchi identities and Einstein field equations can all be written out in terms of these quantities, as can the components _E_αβ, _H_αβ of the Weyl tensor (see Ellis & van Elst 1999). Except in the special cases of isotropic space–times and locally rotationally symmetric space–times (see Ellis 1967; van Elst & Ellis 1996), the basis tetrad can be chosen in an invariant way so that three of these quantities vanish and all the rest are scalar invariants.

Thus, the geometry is determined by the 36 space–time functions in the combined set (ɛ, _P_geometry) with some chosen specification of coordinates X, with the metric then determined by equation (2). For detailed dynamical studies it is often useful to rescale the variables in terms of the expansion; see Wainwright & Ellis (1996) and Uggla et al. (2003) for details. Note that the same universe may occur several times over in this space; the equivalence problem determines when such multiple representations occur. We do not recommend going to a quotient space where each universe occurs only once, as for example in the dynamical studies of Fischer & Marsden (1979), for the cost of doing so is to destroy the manifold structure of the space of space–times. It is far better to allow multiple representations of the same universe (for example, several representations of the same Bianchi I universe occur in the Kasner ring in the space of Bianchi models; see Wainwright & Ellis 1996) both to keep the manifold structure intact and because then the dynamical structure becomes clearer.

Feature 3. To determine the global structure, we need a set of composition functions relating different charts in the atlas where they overlap, thus determining the global topology of the universe.

Together these are the parameters _p_5(i) needed to distinguish model states. A particular model will be represented as a path through those states. The nature of that evolution will be determined by the matter present.

2.4 Describing the physics of possible universes

Feature 4. To determine the matter stress-energy tensor we must specify the quantities

4

for all matter components present, where μ is the energy density, q_α is the momentum flux density, p is the pressure, π_ab_=π(ab) is the anisotropic pressure (π_b b_= 0) and Φ_A (_A_= 1 …_A_max) is some set of internal variables sufficient to make the matter dynamics deterministic when suitable equations of state are added (for example, these might include the temperature, the entropy, the velocity vi of matter relative to the reference frame, some scalar fields and their time derivatives, or a particle distribution function). These are parameters _p_4(i) for each type of matter characterized by _p_3(i). Some of these dynamical quantities may vanish (for example, in the case of a ‘perfect fluid’, _q_α= 0, π_ab_= 0) and some of those that do not vanish will be related to others by the equations of state (for example, in the case of a barotropic fluid, _p_=p(μ)) and dynamic equations (for example, the Klein–Gordon equation for a scalar field). These equations of state can be used to reduce the number of variables in _P_matter; when they are not used in this way, they must be explicitly stated in a separate parameter space _P_eos in _p_3(i). In broad terms

5

Given this information, the equations become determinate and we can obtain the dynamical evolution of the models in the state space; see, for example, Wainwright & Ellis (1996), Hewitt et al.(2003) and Horwood et al. (2003) for the case of Bianchi models (characterized by all the variables defined above depending on the time only) and Uggla et al. (2003) and Lim et al. (2003) for the generic case.

Feature 5. However, more general features may vary: the gravitational constant, the cosmological constant, etc., and even the dimensions of space–time or the types of forces in operation. These are the parameters _P_physics comprising _p_1(i) and _p_2(i). What complicates this issue is that some or many of these features may be emergent properties, resulting for example from broken symmetries occurring as the universe evolves. Thus, they may come into being rather than being given as initial conditions that then hold for all time.

Initially we might think that considering all possible physics simply involves choices of coupling constants and perhaps letting some fundamental constant vary. However, the issue is more fundamental than that. Taking seriously the concept of including all possibilities in the ensembles, the space of physical parameters _P_physics used to describe M, the parameters _p_2(i) might, for example, include a parameter _p_grav(i) such that: for _i_= 1 there is no gravity; for _i_= 2 there is Newtonian gravity; for _i_= 3 general relativity is the correct theory at all energies — there is no quantum gravity regime; for _i_= 4 loop quantum gravity is the correct quantum gravity theory; for _i_= 5 a particular version of superstring theory or M-theory is the correct theory.

Choices such as these will arise for all the laws and parameters of physics. In some universes there will be a fundamental unification of physics expressible in a basic ‘theory of everything’, in others this will not be so. Some universes will be realized as branes in a higher dimensional space–time, others will not.

2.5 The anthropic subset

We are interested in the subset of universes that allow intelligent life to exist. This means that we need a function on the set of possible universes that describes the probability that life may evolve. An adaptation of the Drake equation (Bennett et al. 2002) gives for the expected number of planets with intelligent life in any particular universe m in an ensemble

6

where _N_g is the number of galaxies in the model and _N_S the average number of stars per galaxy. The probability that a star provides a habitat for life is expressed by the product

7

and the probability of coming into existence of life, given such a habitat, is expressed by the product

8

Here, _ƒ_S is the fraction of stars that can provide a suitable environment for life (they are ‘Sun-like’), _ƒ_p is the fraction of such stars that are surrounded by planetary systems, _n_e is the mean number of planets in each such system that are suitable habitats for life (they are ‘Earth-like’), _ƒ_l is the fraction of such planets on which life actually originates, and _ƒ_i represents the fraction of those planets on which there is life where intelligent beings develop. The anthropic subset of a possibility space is that set of universes for which _N_life(m) > 0.

The quantities {_N_g, _N_S, _ƒ_S, _ƒ_p, _n_e, _ƒ_l, _ƒ_i} are functions of the physical and cosmological parameters characterized above, so there will be many different representations of this parameter set depending on the degree to which we try to represent such interrelations.

The astrophysical issues expressed in the product Π are the easier ones to investigate. We can, in principle, make a cut between those consistent with the eventual emergence of life and those incompatible with it by considering each of the factors in _N_g, _N_S and Π in turn, taking into account their dependence on the parameters _p_1(i) to _p_5(i), and only considering the next factor if all the previous ones are non-zero (an approach that fits in naturally with Bayesian statistics and the successive allocation of relevant priors). In this way, we can assign ‘bio-friendly intervals’ to the possibility space M. If _N_g*_N_S*Π is non-zero, we can move on to considering similarly whether F is non-zero, based on the parameters _p_6(i) to _p_7(i) determining if true complexity is possible, which in turn depends on the physics parameters _p_1(i) in a crucial way that is not fully understood. It will be impossible at any stage to characterize that set of the multiverse in which all the conditions necessary for the emergence of self-conscious life and its maintenance have been met, for we do not know what those conditions are (for example, we do not know if there are forms of life possible that are not based on carbon and organic chemistry). Nevertheless, it is clear that life demands unique combinations of many different parameter values that must be realized simultaneously. When we look at these combinations, they will span a very small subset of the whole parameter space (Davies 2003; Tegmark 2003).

If we wish to deal with specifically human life, we need to make the space M large enough to deal with all relevant parameters for this case, where free will arises. This raises substantial extra complications, discussed in the companion (more philosophical) paper.

2.6 Parameter space revisited

It is now clear that some of the parameters discussed above are dependent on other ones, so that while we can write down a more or less complete set at varying levels of detail it will in general not be an independent set. There is a considerable challenge here — to find an independent set. Inter alia this involves solving both the initial value problem for general relativity and the way that galactic and planetary formation depend on fundamental physics constants (which, for example, determine radiation transfer properties in stars and in protoplanetary gas clouds), as well as relations there may be between the fundamental constants and the way the emergent complexity of life depends on them. We are a long way from understanding all these issues; allowing the cosmological parameters to vary simultaneously can change the predictions enormously (Aguirre 2001). This means we can provide necessary sets of parameter values but cannot guarantee completeness or independence.

3 The Set of Realized Universes

We have now characterized the set of possible universes. However, in any given ensemble, they may not all be realized, and some may be realized many times. The purpose of this section is to set up a formalism making clear which of the possible universes (characterized above) occur in a specific realized ensemble.

3.1 A distribution function describing an ensemble of realized universes

In order to select from M a set of realized universes, we need to define on M a distribution function ƒ(m) specifying how many times each type of possible universe m in M is realized. The function ƒ(m) expresses the contingency in any actualization — the fact that not every possible universe has to be realized, and that any actual universe does not have to be realized as a matter of necessity. Things could have been different! Thus, ƒ(m) describes the ensemble of universes or multiverse envisaged as being realized out of the set of possibilities. If these realizations were determined by the laws of necessity alone, they would simply be the set of possibilities described by M. In general, they include only a subset of possible universes, and multiple realizations of some of them. This is the way in which chance or contingency is realized in the ensemble.2

The class of models considered is determined by all the parameters held constant (‘class parameters’). Considering the varying parameters for the class (‘member parameters’), some will take only discrete values, but for each one allowed to take continuous values we need a volume element of the possibility space M characterized by parameter increments d_pj_(i) in all such varying parameters pj(i). The volume element will be given by a product

9

where the product Π_i_,j runs over all continuously varying member parameters i, j in the possibility space, and mij weight the contributions of the different parameter increments relative to each other. These weights depend on the parameters pj(i) characterizing the universe m. The number of universes corresponding to the set of parameter increments d_pj_(i) will be d_N_ given by

10

for continuous parameters. For discrete parameters, we add in the contribution from all allowed parameter values. The total number of universes in the ensemble will be given by

11

(which will often diverge), where the integral ranges over all allowed values of the member parameters and we take it to include all relevant discrete summations. The probable value of any specific quality p(m) defined on the set of universes will be given by

12

Such integrals over the space of possibilities give numbers, averages and probabilities.

Hence, a (realized) ensemble E of universes is described by a possibility space M, a measure π on M, and a distribution function ƒ(m) on M:

13

The distribution function ƒ(m) might be discrete (e.g. there are three copies of universe _m_1 and four copies of universe m_2, with no copies of any other possible universe), or continuous (e.g. characterized by a given distribution of densities Ω_i). In many cases, a distribution function will exclude many possible universes from the realization it specifies.

Now it is conceivable that all possibilities are realized — that all universes in M exist at least once. This would mean that the distribution function

However, there are an infinite number of distribution functions which would fulfil this condition, and so a really existing ‘ensemble of all possible universes’ is not unique. In such ensembles, all possible values of each distinguishing parameter would be predicted to exist in different members of the multiverse in all possible combinations with all other parameters at least once, but they may occur many times. One of the problems is that this often means that the integrals associated with such distribution functions would diverge, preventing the calculation of probabilities from such models (see our treatment of the FLRW case below).

From this consideration we have the second key issue, as follows.

Issue 2. What determines ƒ(m)? What is the meta-cause that delimits the set of realizations out of the set of possibilities?

The answer to this question has to be different from the answer to Issue 1, precisely because here we are describing the contingency of selection of a subset of possibilities from the set of all possibilities, determination of the latter being what is considered in Issue 1. Again, the meta-question is whether this can be answered scientifically.

3.2 Measures and probabilities

It is clear that ƒ(m) will enable us to derive numbers and probabilities relative to the realization it defines only if we also have determined a unique measure π on the ensemble, characterized by a specific choice of the weights mij(m) in equation (9), where these weights will depend on pj(i). There are three issues here.

First, what may seem a ‘natural’ measure for M in one set of coordinates will not be natural in another set of coordinates. Hence, the concept of a measure is not unique, as is illustrated below in the FLRW case. This is aggravated by the fact that the parameter space will often contain completely different types of quantities (density parameters and the values of the gravitational constant and the cosmological constant, for example), and assigning the weights entails somehow assigning a relative weighting between these quite different kinds of quantities.

Secondly, it is possible that we might be able to assign probabilities χ(m) to points of M from some kind of physical argument, and then predict ƒ(m) from these, following the usual line of argument for determining entropy in a gas. However, we then have to determine some reason why χ(m) is what it is and how it then leads to ƒ(m). In the entropy case, we assume equal probability in each phase space volume; why should that hold for an ensemble of universes? Realizing such probabilities seems to imply a causal mechanism relating the created members of the multiverse to one another so they are not, in fact, causally disjoint. Otherwise, there is no reason why any probability law (Gaussian normal, for example) should be obeyed. We will return to this point later.

Finally, the relevant integrals may diverge. In that case, assigning mean values or averages for physical quantities in an ensemble of universes is problematic (see Kirchner & Ellis 2003, and references therein).

3.3 The anthropic subset

The expression (6) can be used in conjunction with the distribution function ƒ(m) of galaxies to determine the expected number of civilizations arising in the whole ensemble:

14

(which is a particular case of equation 12 based on equation 6). An anthropic ensemble is one for which _N_life(E) > 0. If the distribution function derives from a probability function, we may combine the probability functions to obtain an overall anthropic probability function; for an example, see Weinberg et al. (2000a,b) discussed below, where it is assumed that the probability for galaxy formation is the only relevant parameter for the existence of life. This is equivalent to assuming that _N_S*_ƒ_S*_ƒ_p*_n_e*_ƒ_l*_ƒ_i > 0.

This assumption might be acceptable in our physically realized Universe, but there is no reason to believe it would hold generally in an ensemble because these parameters will depend on other ensemble parameters, which will vary.

3.4 Problems with infinity

When speaking of multiverses or ensembles of universes — possible or realized — the issue of infinity often crops up. Researchers often envision an infinite set of universes, in which all possibilities are realized. Can there really be an infinite set of really existing universes? We suggest that, on the basis of well-known philosophical arguments, the answer is no.

The common perception that this is possible arises from not taking seriously enough the difficulties associated with this profoundly difficult concept. Because we can assign a symbol to represent ‘infinity’ and can manipulate that symbol according to specified rules, we assume corresponding entities can exist in practice. This is highly questionable.

There is no conceptual problem with an infinite set — countable or uncountable — of possible or conceivable universes. However, as stressed by Hilbert (1964), it can be argued that a really existing infinite set is not possible. As he points out, following many others, the existence of the actually infinite inevitably leads to well-recognized unresolvable contradictions in set theory, and thus in definitions and deductive foundations of mathematics itself (Hilbert 1964 pp. 141–142). His basic position therefore is that ‘just as operations with the infinitely small were replaced by operations with the finite which yielded exactly the same results …, so in general must deductive methods based on the infinite be replaced by finite procedures which yield exactly the same results’ (Hilbert 1964 p. 135). He concludes: ‘Our principle result is that the infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought … The role that remains for the infinite to play is solely that of an idea … which transcends all experience and which completes the concrete as a totality …’ (Hilbert 1964 p. 151). Others (see Spitzer 2000; Stoeger 2003, and references therein) have further pointed out that realized infinite sets are not constructible — there is no procedure we can in principle implement to complete such a set — they are simply incompletable. However, if that is the case, then ‘infinity’ cannot be arrived at, or realized. On the contrary, the concept itself implies its inability to be realized in a real physical setting! This is why, for example, a realized past infinity in time is not considered possible from this standpoint — because it involves an infinite set of completed events or moments. There is no way of constructing such a realized set, or actualizing it.3

Thus, it is important to recognize that infinity is not an actual number we can ever specify or reach — it is simply the code word for ‘it continues without end’. Whenever infinities emerge in physics, such as in the case of singularities, we can be reasonably sure, as is usually recognized, that there has been a breakdown in our models. An achieved infinity in any physical parameter (temperature, density, spatial curvature) is almost certainly not a possible outcome of any physical process — simply because it means traversing in actuality an interval of values which never ends. We assume space extends forever in Euclidean geometry and in many cosmological models, but we can never prove that any realized three-space in the real universe continues in this way — it is an untestable concept, and the real spatial geometry of the universe is almost certainly not Euclidean. Thus, Euclidean space is an abstraction that is probably not realized in physical practice. In the physical universe, spatial infinities can be avoided by compact spatial sections, either resultant from positive spatial curvature or from choice of compact topologies in universes that have zero or negative spatial curvature, (for example FLRW flat and open universes can have finite rather than infinite spatial sections). We argue that the theoretically possible infinite space sections of many cosmologies at a given time are simply unattainable in practice — they are a theoretical idea that cannot be realized. Future infinite time also is never realized; rather, the situation is that whatever time we reach, there is always more time available. Much the same applies to claims of a past infinity of time; there may be unbounded time available in the past in principle, but in what sense can it be attained in practice? The arguments against an infinite past time are strong — it is simply not constructible in terms of events or instants of time, besides being conceptually indefinite.4

We emphasize that the problem with infinity is not primarily physical, in the usual sense — it is primarily a conceptual or philosophical problem with the idea of ‘realized infinity’. Infinity as it is mathematically conceived and used is not the sort of property that can be physically realized, like a definite number can. As emphasized above, it means ‘indefinitely large’, or ‘continues without limit’. The same problem of a realized infinity may be true in terms of the supposed ensembles of universes. It is difficult enough conceiving of an ensemble of many ‘really existing’ universes that are totally causally disjoint from our own, and how that could come into being, particularly given two important features. First, specifying the geometry of a generic universe requires an infinite amount of information because the quantities in _P_geometry are fields on space–time, in general requiring specification at each point (or equivalently, an infinite number of Fourier coefficients) — they will almost always not be algorithmically compressible. This greatly aggravates all the problems regarding infinity and the ensemble. Only in highly symmetric cases, such as the FLRW solutions, do these data reduce to a finite number of parameters. We can suggest that a statistical description would suffice, where a finite set of numbers describe the statistics of the solution, rather than giving a full description. Whether this suffices to adequately describe an ensemble where ‘all that can happen, happens’ is a moot point. We suggest not, for the simple reason that there is no guarantee that all possible models will obey any known statistical description. That assumption is a major restriction on what is assumed to be possible.

Secondly, many universes in the ensemble may themselves have infinite spatial extent and contain an infinite amount of matter, with the paradoxical conclusions that entails (Ellis & Brundrit 1979). To conceive of physical creation of an infinite set of universes (most requiring an infinite amount of information for their prescription, and many of which will themselves be spatially infinite) is at least an order of magnitude more difficult than specifying an existent infinitude of finitely specifiable objects.

The phrase ‘everything that can exist, exists’ implies such an infinitude, but glosses over all the profound difficulties implied. We should note here particularly that problems arise in this context in terms of the continuum assigned by classical theories to physical quantities and indeed to space–time itself. Suppose, for example, that we identify corresponding times in the models in an ensemble and then assume that all values of the density parameter occur at each spatial point at that time. Because of the real number continuum, this is an uncountably infinite set of models — and genuine existence of such an uncountable infinitude is highly problematic. However, on the other hand, if the set of realized models is either finite or countably infinite, then almost all possible models are not realized — the ensemble represents a set of measure zero in the set of possible universes. Either way the situation is distinctly uncomfortable. However, we might try to argue around this by a discretization argument; maybe differences in some parameter of less than say 10−10 are unobservable, so we can replace the continuum version by a discretized one, and perhaps some such discretization is forced on us by quantum theory. If this is the intention, then that should be made explicit. That solves the ‘ultraviolet divergence’ associated with the small-scale continuum, but not the ‘infrared divergence’ associated with supposed infinite distances, infinite times, and infinite values of parameters describing cosmologies.

4 Ensembles of Flrw Universes

Having established the broad set of issues concerning multiverses that we believe need to be addressed, we shall for the remainder of this paper limit ourselves to the FLRW sector _M_FLRW of the ensemble of all possible universes M in order to illustrate these issues.5 We assume the family considered is filled with matter components characterized by a γ-law equation of state, and mainly restrict our attention to their cosmological parameters, although full consideration of anthropic issues would be characterized by including all the other parameters. Our descriptive treatment will consider FLRW universe domains (whether a true multiverse or separate domains in a single space–time) as distinct but with common physical characteristics.

4.1 Properties of FLRW models

FLRW models are homogeneous and isotropic models described by the metric

15

where dΩ2= dϑ2+ sin2(ϑ) dϕ2 denotes the line element on the two-dimensional unit sphere, a(t) is the scalefactor, and

is the normalized curvature. The FLRW model is completely determined by k and the scalefactor a(t), which incorporates the time evolution and is obtained from the Einstein field equations together with the matter description.6

Assuming gravity is described by the Einstein field equations, the evolution of FLRW models is described by the Friedmann equation

16

where (a dot denotes differentiation with respect to proper time) is the Hubble parameter and Ω is the density parameter. We restrict our discussion to models with only a cosmological constant Λ and one comoving matter component which obeys a γ-law equation of state, i.e. its pressure p and density ρ are related by _p_= (γ− 1)ρ, where γ is constant. This specification of parameters _p_3(i) includes in particular the case of dust (γ= 1) and radiation (γ= 4/3). The total density parameter is

17

where the matter density parameter is and the vacuumenergy density parameter is (representing a cosmological constant). These form the parameters _p_4(i).

The second time derivative of the scalefactor is determined by the Raychaudhuri equation

18

where is the dimensionless deceleration parameter. The matter evolution is given by the energy-conservation equation

19

or equivalently by

20

Besides the normalized curvature k there are two constants of motion, χ≡κρ_a_3γ/3 and the cosmological constant Λ. Given these parameters, the dynamical evolution is determined from the implied initial conditions: {a(t_0), Ω_i_0, γ_i, k}⇒a(t).

4.2 Parametrizing FLRW models

In order to define a FLRW ensemble, we need a set of independent parameters which uniquely identify all possible FLRW models. We want to consider all possible FLRW models with the same physical laws as in our Universe, but possibly different coupling constants. There is then one set of parameters _p_2(i) which defines the gravitational ‘physics’ of the model in terms of the coupling constants — for simplicity let us only consider the gravitational constant G here — and further sets _p_5(i) and _p_4(i) which identify the geometry and matter content of the actual model, and which are related to each other via the Einstein field equations.

Among the various options there are two particularly useful parametrizations. Ehlers & Rindler (1989) developed a parametrization in terms of the observable density parameters (they also include a radiation component) and the Hubble parameter. With , the Friedmann equation becomes

21

The curvature parameter Ω_k_ determines k_= sgn(Ω_k). For _k_≠ 0 the scalefactor, and hence the metric (15), is determined by

while for _k_= 0 its value is unimportant because of scale-invariance in that case. Hence, any state is completely described by Ωm, ΩΛ and H.

In order to parametrize the models rather than the states, we need to select one particular time _t_0 for each model at which we take the above parameters as representative parameters Ωm0, ΩΛ0 and _H_0 for this model.7 We note that this time _t_0 can be model-dependent because not all models will reach the age _t_0.

All big-bang FLRW models start with an infinite positive Hubble parameter whose absolute value reaches or approaches asymptotically a minimum value _H_min. Hence, we could define the time _t_0 as the time when the model first takes a certain value _H_0(_p_I) > _H_min(_p_I), where _p_I represents the model parameters. One particular choice of _H_0(_p_I) is given by

On the other hand, by setting _H_0(_p_I) = const and excluding all models which never reach this value, we easily find a parametrization of all models which reach this Hubble value during their evolution.

While the above choice of parameters gives a convenient parametrization in terms of observables which covers closed, flat and open models, it is disturbing that for each model an arbitrary time has to be chosen. This also leads to a technical difficulty, because the parameters {_H_0, Ωm0, ΩΛ0} are subject to the constraint _H_0=_H_0(_p_I).

For these reasons it is often convenient to use a set of parameters which are comoving in the state space, i.e. parameters which are constants of motion. As mentioned above, for open and closed models such a set is given by the matter constant χ, the cosmological constant Λ and the normalized curvature constant k. For flat models we can rescale the scalefactor, which allows us to set χ= 1.

These parameters are related to the observational quantities by (for _k_=±1)

The evolution of these models through state space is illustrated here in terms of two different parametrizations of the state space, see Figs 1(a) and 1(b). For a detailed investigation of these evolutions for models with non-interacting matter and radiation see Ehlers & Rindler (1989).

Figure 1.

Phase-space diagrams showing the evolution of FLRW models (as given by the Friedmann equation) in the Ωm–H and ΩΛ–Ωm plane for Λ= 1. Models can evolve in both directions along the lines. Models which start with a big bang originate on the right in (a) (corresponding to _H_=∞). Models which reach _H_= 0 (in a finite time) reverse their direction of evolution, i.e. they follow the same line backwards. These are the recollapsing models. The limiting case to the forever-expanding models is given by the model which approaches asymptotically the Einstein static universe. The evolution of life should only be possible in sufficiently flat models — otherwise the universes either recollapse too early or are too empty — at not too early and not too late times. This life-allowing region is indicated schematically.

4.3 The possibility space

The structures defined so far are the uniform structures across the class of models in this possibility space, characterized both by laws of physics (in particular, general relativity) and by a restricted class of geometries. It is clear that universes in a multiverse should be able to differ in at least some properties from each other. We have just characterized the geometrical possibilities we are considering. The next question is which physical laws and parameters can vary within the ensemble, and which values can they take? For this simplified discussion let us just assume that only the gravitational constant G and the cosmological constant Λ (which also qualifies as a model parameter) are variables, with the ranges8_G_∈[0, ∞) and Λ∈ (−∞, ∞). However, if we consider ‘all that is possible’ within this restricted class of FLRW models, maybe we should consider _G_∈ (−∞, ∞). There is still considerable uncertainty as to the nature of an ensemble even within this restricted context. Whatever is chosen here defines the set of possibilities that can arise.

4.4 The measure

For a complete probabilistic description of an ensemble we need not only a distribution function P, but also a measure π=μ(x) d_x_ for each parameter space with parameter x (see Section 3.1). The information entropy

22

can then be maximized for the probability distribution equal to this measure, representing the state of minimal knowledge.

Without knowledge of the creation mechanism, it is impossible to determine this measure with certainty. Nevertheless, we might ask what our best guess for such a measure should be in a state of minimal information, where only a certain set of independent parameters, describing the ensemble, and their ranges are known.

The only known method for constructing such a measure is Jaynes' principle. Its application to FLRW models with γ-law equation of state has been discussed in Kirchner & Ellis (2003). One identifies a set of transformations _x_′(x) in possibility space which leaves the mathematical structure invariant, and demands that the measure is invariant under these transformations. The two most important cases are given by parameters which can take all real values, and those that take on all positive real values.

In the first case, if z is a valid parameter value, so is _z_′=z+α for all real α. According to Jaynes' principle, the measure should obey π=μ(z) d_z_=μ(_z_′) d_z_′ and hence μ(z) =μ(_z_′), i.e. the measure is constant.

If, on the other hand, a parameter u only takes non-zero positive values, we can generate another valid parameter value by u_′=λ_u, where λ∈R+. Demanding invariance of the measure yields .

These transformations are not unique, and hence it is possible to find many different measures. Nevertheless, in the state of minimum information we do not know what the natural parametrization is for the possibility space and different measures correspond to different guesses. Surprisingly Jaynes' principle is ‘relatively invariant’ under simple parametrization changes. For example, introducing a new parametrization for a positive quantity G by _G_=mn for positive m, or by _G_= exp(λ) for real λ, will give the same measure.

There are two important points to note. First, the measure is derived from the chosen set of parameters. Generally a different choice of parameters yields a different minimum-information measure, predicting another maximum-entropy distribution function. Let us consider the example of an ensemble of dust-FLRW models. The different open and closed models are most conveniently parametrized by the constants of motion, which are given by the cosmological constant Λ and matter constant χ≡_a_ρ3γ, where ρ is the energy density. This leads to the minimum-information measure (Kirchner & Ellis 2003):

23

Considering dust models (γ= 1) and the subset of all big-bang models which reach a certain Hubble parameter _H_0 at a time _t_0 during their evolution, this measure becomes

with ΩΛ0≤ 1 +Ωm0/2. On the other hand, as mentioned above, there is a convenient parametrization for this particular subset of models (Ehlers & Rindler 1989) in terms of the observables Ωm0 and ΩΛ0 (in Ehlers & Rindler 1989 an additional radiation component was also included). Using this parametrization yields the minimum information measure

which is clearly different from the above result.

Secondly, the measure is in general non-normalizable and hence there is no normalizable maximum-entropy distribution. Without additional information we are not able to calculate definite probabilities. Because it seems questionable whether there will ever be additional information about the ensemble of universes available, we have to accept that certain questions will have no well-defined probabilities.

It should be mentioned that we encounter similar problems when we want to find a probability measure for physical parameters such as the gravitational constant G. Let us assume that G can take any non-zero positive value. Jaynes' principle then suggests the probability measure ΩΛ0 ≤ 1 + Ωm0/2. On the other hand, if we decide to use _m_= sinh(G) as our parameter, then we find the different measure9

4.5 Distribution functions on _M_FLRW

Now, having properly parametrized _M_FLRW and defined a measure on it, we can represent particular multiverses by giving distribution functions over the parameter space (as discussed in Section 3.4). Given a distribution function ƒ it determines the number of universes in a small parameter interval by

which is invariant under a change of parametrization. Hence, it is the combination of measure and distribution function which is of importance.

While distribution functions can be parametrized by any set of coordinates over the possibility space, we need different distribution functions for different possibility spaces. For example, if universes in a multiverse must have a common value for the gravitational constant G then distribution functions must not depend on G.

It is clear that a particular distribution function can be expressed in any set of coordinates. Obviously there is a vast set of possible distribution functions. We want to examine some particular examples.

First, we could have a distribution function which is constant over the parameter space. The actual ensemble then really depends on the measure and is the maximum-entropy distribution (which maximizes equation 2210). If we choose the observational quantities H, Ωm, ΩΛ to represent the model and allow for different values of the gravitational coupling constant G this would be

for all allowed values of the stated parameters. On the other hand, if we choose the constants of motion as coordinates in possibility space

The probability P A to find a universe in a certain parameter region A is given by

where the integral in the denominator extends over the whole possibility space. For many distribution functions, such as for the above constant distribution function together with equation (23), this expression is not well defined. Let us assume that the measure is non-integrable, i.e. non-normalizable, and that we have a constant distribution function. If the set A does not include any point of non-integrability of the measure then P A = 0, if it includes all points of non-integrability then P A = 1, but if it includes only some of the non-integrabilities then P A is not well defined.

Of course, the above expression might be integrable for all sets A given a good distribution function. For instance, the distribution function

together with the measure (23) is integrable everywhere. On the other hand

diverges for χ→∞. A distribution function can also introduce an additional divergence, for instance

is non-integrable at χ= 0 and χ→∞.

A multiverse might contain a finite or infinite countable number of universes. In these cases, the distribution function contains Dirac δ-functions, e.g.

which represents a multiverse which contains five copies of closed FLRW models with χ= 3 and Λ= 0.5, etc. The distribution function

represents an ensemble with a countably infinite number of universes — all are closed with Λ= 1 and one for each χ= 1/i. Similarly, we could imagine an ensemble of 107 copies of our universe, which would be represented by the distribution function

where _k_0, χ0, Λ0 represent the parameter values for our universe. This is very unlikely in terms of a generating mechanism, but for ensembles without generating mechanisms it is as likely as any other possibility. If a multiverse is ‘tested’ by its prediction that our universe is a likely member, then such an ensemble should be the most satisfying one — but then we might just as well be happy with one copy, i.e. just our universe.

Similar distribution functions determine the distribution of physical parameters such as the gravitational constant G. For example, with G_∈R the minimum information measure is dμ_G_= d_G and

gives a Gaussian distribution around _G_= 0. If, on the other hand, G_∈ (0, ∞) the measure is d_G/G and ƒ(g) =G_exp(−_G) would be an example of a distribution function.

We can imagine various types of distributions, e.g. a Gaussian distribution in G or in _H_0, or in the other parameters. However, in order to establish these in a non-arbitrary way, we need a theory of how this particular ensemble is selected for from all the other possible ones.

4.6 The problem of infinities again

Even within the restricted set of FLRW models, one of the most profound issues is the problem of realized infinities; if all that is possible in this restricted subset happens, we have multiple infinities of realized universes in the ensemble. First, there are an infinite number of possible spatial topologies in the negative curvature case (see, for example, Lachieze-Rey & Luminet 1995), so an infinite number of ways that universes which are locally equivalent can differ globally. Secondly, even though the geometry is so simple, the uncountable continuum of numbers plays a devastating role locally. Is it really conceivable that FLRW universes actually occur with all values independently of both the cosmological constant and the gravitational constant, and also all values of the Hubble constant, at the instant when the density parameter takes the value 0.97? This gives three separate uncountably infinite aspects of the ensemble of universes that is supposed to exist. The problem would be allayed if space–time is quantized at the Planck level, as suggested for example by loop quantum gravity. In that case we can argue that all physical quantities also are quantized, and the uncountable infinities of the real line become transmuted into finite numbers in any finite interval — a much better situation. We believe that this is a physically reasonable assumption to make, thus softening a major problem for many ensemble proposals. However, the intervals are still infinite for many parameters in the possibility space. Reducing the uncountably infinite to countably infinite does not in the end resolve the problem of infinities in these ensembles. It is still an extraordinarily extravagant proposal.

4.7 The anthropic subset

We can identify those FLRW universes in which the emergence and sustenance of life is possible at a broad level11– the necessary cosmological conditions have been fulfilled allowing the existence of galaxies, stars and planets if the universe is perturbed, so allowing a non-zero factor _N_g*_N_S*Π as discussed above. These are indicated in Fig. 1 (anthropic universes are those intersecting the regions labelled ‘life allowing’). The fraction of these that will actually be life-bearing depends on the fulfilment of a large number of other conditions represented by the factor _F_=_ƒ_l*_ƒ_i, which will also vary across a generic ensemble, and the above assumes this factor is non-zero.

5 On The Origin of Ensembles

Ensembles have been envisaged both as resulting from a single causal process, and as simply consisting of discrete entities. We discuss these two cases in turn, and then show that they are ultimately not distinguishable from each other.

5.1 Processes naturally producing ensembles

Over the past 15 or 20 years, many researchers investigating the very early Universe have proposed processes at or near the Planck era which would generate a really existing ensemble of expanding universe domains, one of which is our own observable Universe. In fact, their work has provided both the context and stimulus for our discussions in this paper. Each of these processes essentially selects a really existing ensemble through a generating process from a set of possible universes, and often leads to proposals for a natural definition of a probability distribution on the space of possible universes. Here we briefly describe some of these proposals, and comment on how they fit within the framework we have been discussing.

Andrei Linde's chaotic inflationary proposal (Linde 1983, 1990, 2003) is one of the best-known scenarios of this type. The scalar field (inflaton) in these scenarios drives inflation and leads to the generation of a large number of causally disconnected regions of the universe. This process is capable of generating a really existing ensemble of expanding FLRW-like regions, one of which may be our own observable universe region, situated in a much larger universe that is inhomogeneous on the largest scales. No FLRW approximation is possible globally; rather there are many FLRW-like subdomains of a single fractal universe. These domains can be very different from one another, and can be modelled locally by FLRW cosmologies with different parameters.

Linde and others have applied a stochastic approach to inflation (Starobinsky 1986; Linde, Linde & Mezhlumian 1994; Vilenkin 1995; Garriga & Vilenkin 2001; Linde 2003), through which probability distributions can be derived from inflaton potentials along with the usual cosmological equations (the Friedmann equation and the Klein–Gordon equation for the inflaton) and the slow-roll approximation for the inflationary era. A detailed example of this approach, in which specific probability distributions are derived from a Langevin-type equation describing the stochastic behaviour of the inflaton over horizon-sized regions before inflation begins, is given in Linde & Mezhlumian (1993) and in Linde et al. (1994). The probability distributions determined in this way generally are functions of the inflaton potential.

This type of scenario suggests how overarching physics, or a ‘law of laws’ (represented by the inflaton field and its potential), can lead to a really existing ensemble of many very different FLRW-like regions of a larger universe. However, these proposals rely on extrapolations of presently known physics to realms far beyond where its reliability is assured. They also employ inflaton potentials which as yet have no connection to the particle physics we know at lower energies. Also, these proposals are not directly observationally testable — we have no astronomical evidence the supposed other FLRW-like regions exist. Thus they remain theoretically based proposals rather than established fact. There additionally remains the difficult problem of infinities; eternal inflation with its continual reproduction of different inflating domains of the Universe is claimed to lead to an infinite number of universes of each particular type (Linde, private communication). How can we deal with these infinities in terms of distribution functions and an adequate measure? As we have pointed out above, there is a philosophical problem surrounding a realized infinite set of any kind. In this case, the infinities of really existent FLRW-like domains derive from the assumed initial infinite flat (or open) space sections — and we have pointed out above the problems in assuming such space sections are actually realized.

Finally, from the point of view of the ensemble of all possible universes often invoked in discussions of multiverses, all possible inflaton potentials should be considered, as well as all solutions to all those potentials. They should all exist in such a multiverse, which will include chaotic inflationary models which are stationary as well as those which are non-stationary. Many of these potentials may yield ensembles which are uninteresting as far as the emergence of life is concerned, but some will be bio-friendly. The price of this process for creating anthropically favourable universe regions is the multiplication of realized infinities, most of which will be uncountable (for example, the parameters in any particular form of inflaton potential will take all possible values in an interval of real numbers).

5.2 Probability distributions for the cosmological constant

Weinberg (2000a,b) and Garriga & Vilenkin (2000) derive a probability distribution for the cosmological constant in the context of an ensemble of regions generated in the same inflationary sequence via the action of a given inflaton potential, where the cosmological constant is given by the potential energy of a scalar field. In multidomain universes, where spatial variations in a scalar field cause different regions to inflate at different rates, the cosmological constant would be distributed according to some probability distribution P(ρΛ). During inflation the scalar field undergoes randomization by quantum fluctuations, such that later on its values in different regions are distributed according to ‘the length’ in field space (Garriga & Vilenkin 2002). This leads to a probability distribution (or distribution function — the probability distribution is just the normalized distribution function) of values of the vacuum-energy density ρΛ in these regions given by

where V(φ) is the inflaton potential, and the prime signifies differentiation with respect to the inflaton φ.

It has been suggested (Vilenkin 1995; Weinberg 1997) that the way the probability distribution for the existence of galaxies depends on the cosmological constant can be approximated by

24

where N(ρΛ) is the fraction of baryons that form galaxies. The requirement of structure formation as a prerequisite for life places strong anthropic constraints on the domains in which observers could exist; these constraints must be satisfied in the really existing universe.

Let us first note that galaxy formation is only possible for a narrow range around ρΛ= 0 (Weinberg 2000a,b). It has been shown that anthropic restrictions demand ρΛ≤ 10−28 g cm−3 (Kallosh & Linde 2002; Garriga & Vilenkin 2002). Consequently, the anthropic selection factor N(ρΛ) is sharply peaked and vanishes for |ρΛ| > ρΛmax for some ρΛ max, which is of the same order of magnitude as the observed cosmological constant. In scalar-field models P(ρΛ) is in direct relation to the a priori distribution of the scalar-field fluctuations and it has been argued (Weinberg 2000a,b) that for a wide class of potentials the variations of P(ρΛ) over the anthropically allowed range (where N(ρΛ) ≠ 0) should be negligible. Nevertheless, as has been shown in Garriga & Vilenkin (2002), this is not always the case; in particular, for power-law potentials V(φ) =φ_n_ with n > 1 we find an integrable divergence at ρΛ= 0.

It is clear that a similar relation to equation (24) should hold for multiverses in the wider sense. Nevertheless, we could imagine multiverses containing universes with and without scalar fields, or with different potentials. Hence, we cannot link the distribution of the cosmological constant to that of the scalar field in a unique way, and there is a vast choice for possible a priori probability distributions for the cosmological constant. Let us assume that the cosmological constant is a remnant of some underlying (unknown) theory and as such might be restricted to some domain of values. Depending on this domain we find different possible minimum information measures, which result from Jaynes' principle. If the domain is given by all real numbers then the (non-normalizable) measure will be constant. If, on the other hand, the domain is given by all positive real numbers then the minimum-information measure gives an (non-normalizable) a priori probability distribution proportional to 1/ρΛ (Kirchner & Ellis 2003). In this case the divergence is located inside the anthropically allowed region and is non-integrable. For this case, the expectation value vanishes, i.e.

and we fail to explain the observed non-zero value of the cosmological constant.

An interesting alternative is given by allowing the cosmological constant to take values in the domain R+∪{0} (e.g. if the cosmological constant prediction is given by a quadratic term). The minimum information-measure is then proportional to . Again, there is a divergence in the anthropically allowed region, but this time it is integrable. The expectation value becomes

5.3 The existence of regularities

Consider now a genuine multiverse. Why should there be any regularity at all in the properties of universes in such an ensemble, where the universes are completely disconnected from each other? If there are such regularities and specific resulting properties, this suggests a mechanism creating that family of universes, and hence a causal link to a higher domain which is the seat of processes leading to these regularities. This in turn means that the individual universes making up the ensemble are not actually independent of each other. They are, instead, products of a single process, as in the case of chaotic inflation. A common generating mechanism is clearly a causal connection, even if not situated in a single connected space–time — and some such mechanism is needed if all the universes in an ensemble have the same class of properties, e.g. being governed by the same physical laws or meta-laws.

The point then is that, as emphasized when we considered how we can describe ensembles, any multiverse with regular properties that we can characterize systematically is necessarily of this kind. If it did not have regularities of properties across the class of universes included in the ensemble, we could not even describe it, much less calculate any properties or even characterize a distribution function.

Thus, in the end the idea of a completely disconnected multiverse with regular properties but without a common causal mechanism of some kind is not viable. There must necessarily be some pre-realization causal mechanism at work determining the properties of the universes in the ensemble. What are claimed to be totally disjoint universes must in some sense indeed be causally connected together, albeit in some pre-physics or meta-physical domain that is causally effective in determining the common properties of the universes in the multiverse.

Related to this is the issue that we have emphasized above, namely where does the possibility space come from and where does the distribution function come from that characterizes realized models? As emphasized above, we have to assume that some relevant meta-laws pre-exist. We now see that we need to explain also what particular meta-laws pre-exist. If we are to examine ‘all that might be, exists’, then we need to look at the ensemble of all such meta-laws and a distribution function on this set. We seem to face an infinite regress as we follow this logic to its conclusion, and it is not clear how to end it except by arbitrarily calling a stop to this process. However, then we have not looked at all conceivable possibilities.

6 Testability and Existence

Finally, the issue of evidence and testing has already been briefly mentioned. This is at the heart of whether an ensemble or multiverse proposal should be regarded as physics or as metaphysics.

6.1 Evidence and existence

Given all the possibilities discussed here, which specific kind of ensemble is claimed to exist? Given a specific such claim, how can we show that this is the particular ensemble that exists rather than all the other possibilities?

There is no direct evidence of existence of the claimed other universe regions in an ensemble, nor can there be any, for they lie beyond the visual horizon; most will even be beyond the particle horizon, so there is no causal connection with them. In the case of a true multiverse, there is not even any possibility of any indirect causal connection of any kind — the universes are then completely disjoint and nothing that happens in any one of them is causally linked to what happens in any other one.

What weight does a claim of such existence carry, in this context when no direct observational evidence can ever be available? The point is that there is not just an issue of showing a multiverse exists — if this is a scientific proposition we need to be able to show which specific multiverse exists; but there is no observational way to do this. Indeed, if you cannot show which particular one exists, it is doubtful you have shown any one exists. What does a claim for such existence mean in this context?Gardner (2003) puts it this way: ‘There is not the slightest shred of reliable evidence that there is any universe other than the one we are in. No multiverse theory has so far provided a prediction that can be tested. As far as we can tell, universes are not even as plentiful as even two blackberries.’ This contrasts strongly for example with Deutsch's and Lewis's defence of the concept (Deutsch 1998; Lewis 2000).

These issues are discussed in more depth in the companion philosophical paper, where we consider the various ways one may claim entities exist even when there is no direct or even indirect evidence for such existence. It is possible to make such claims, but they must be very carefully supported, for one ends up in deep philosophical waters in making such claims. Thus, philosophical argumentation is unavoidable if one is to seriously argue the claim for existence of a multiverse. Even the concept of what ‘existence’ might mean in this context needs careful consideration.

6.2 Observations and physics

The one way we might make a reasonable claim for the existence of a multiverse would be if we could show its existence was a more or less inevitable consequence of well-established physical laws and processes. Indeed, this is essentially the claim that is made in the case of chaotic inflation. However, the problem is that the proposed underlying physics has not been tested, and indeed may be untestable. There is no evidence that the postulated physics is true in this universe, much less in some pre-existing metaspace that might generate a multiverse. Thus, belief in the validity of the claimed physics that could lead to such consequences is just that, a belief — it is based on unproved extrapolation of established physics to vastly beyond where it has been tested. The issue is not just that the inflaton is not identified and its potential untested by any observational means. It is also that, for example, we are assuming quantum field theory remains valid far beyond the domain where it has been tested, and we have faith in that extreme extrapolation despite all the unsolved problems at the foundation of quantum theory, the divergences of quantum field theory, and the failure of that theory to provide a satisfactory resolution of the cosmological constant problem.

6.3 Observations and probabilities

The ‘doomsday argument’ has led to a substantial literature on relating the existence of universe models to evidence, based on analysis of probabilities, often using a model of choosing a ball randomly from an urn and of associated selection effects (see, for example, Bostrom 2002). However, usually these models either in effect assume an ensemble exists, or else are content to deal with potentially existing ensembles rather than actually existing ones (see, for example, Olum 2002). This does not deal with the case at hand. We would have to extend those arguments to trying to decide, on the basis of a single ball drawn from the urn, as to whether there was one ball in the urn or an infinite number. It is not clear to us that the statistical arguments used in those papers lead to a useful conclusion in this singular case, which is the case of interest for the argument in this paper.

In any case, in the end those papers all deal just with observational probabilities, which are never conclusive. Indeed, the whole reason for the anthropic literature is precisely the fact that biophilic universes are clearly highly improbable within the set of all possible universes (see, for example, the use of anthropic arguments as regards the value of Λ discussed in Section 5.2). We are working in a context where large improbabilities are the order of the day. That is why multiverse concepts were introduced in the first place — to try to introduce some form of scientific explanation into a context where the probabilities of existence of specific universe models preferred by observation are known to be very small.

6.4 Observations and disproof

Despite the gloomy prognosis given above, there are some specific cases where the existence of a chaotic inflation (multidomain) type scenario can be disproved. These are when we live in a ‘small universe’ where we have already seen right round the universe (Ellis & Schrieber 1986; Lachieze-Rey & Luminet 1995) for then the universe closes up on itself in a single FLRW-like domain and so no further such domains that are causally connected to us in a single connected space–time can exist.

This ‘small universe’ situation is observationally testable, and indeed it has been suggested that the cosmic background radiation power spectrum might already be giving us evidence that this is indeed so, because of its lack of power on the largest angular scales (Luminet et al. 2003). This proposal can be tested in the future by searching for identical circles in the cosmic microwave background sky. This would disprove the usual chaotic inflationary scenario, but not a true multiverse proposal, for that cannot be shown to be false by any observation. Neither can it be shown to be true.

7 Conclusion

The introduction of the multiverse or ensemble idea is a fundamental change in the nature of cosmology, because it aims to challenge one of the most basic aspects of standard cosmology, namely the uniqueness of the Universe (see Ellis 1991, 1999, and references therein). However, previous discussions have not made clear what is required in order to define a multiverse, although some specific physical calculations have been given based on restricted low-dimensional multiverses. The aim of this paper is to make clear what is needed in order to properly define a multiverse, and then to examine some of the consequences that flow from this.

Our fundamental starting point is the recognition that there is an important distinction to be made between possible universes and realized universes, and our main conclusion is that a really existing ensemble or multiverse is not a priori unique, nor uniquely defined. It must somehow be selected for. We have pointed out a clear distinction between an ensemble of possible universes M, and an ensemble of really existing universes, which is envisioned as generated by the given primordial process or action of an overarching cosmic principle. These effectively select a really existing multiverse from the possibilities in M, and, as such, effectively define a distribution function over M. Thus, there is a definite causal connection, or ‘law of laws’, relating all the universes in these multiverses. It is this really existing ensemble of universes, not the ensemble of all possible universes, which provides the basis for anthropic arguments. Anthropic universes lie in a small subset of M, whose characteristics we understand to some extent. It is very likely that the simultaneous realization of all the conditions for life will pick out only a very small sector of the parameter space of all possibilities; anthropic universes are fine-tuned.

The fine-tuning problem is very controversial. Two counter-attacks maintain that there is no fine-tuning problem, so it is not necessary to construct solutions to it by employing the multiverse idea. The first promotes the view that whatever happens will always be unlikely (any hand of cards is as unlikely as any other). Thus, because it is just an example of chance, there is nothing special about a universe that admits life. The counter response is that the existence of life is quite unlike anything else in the physical world — its coming into being is not just like choosing one out of numerous essentially identical hands of cards. It is like being transformed into an entirely different higher level game, and so does indeed require explanation. The second counter-attack argues that inflation explains the current state of the universe, making its apparently unlikely state probable. However, this move is only partially successful, because very anisotropic or inhomogeneous models may never inflate. The counter response is that this does not matter; however small the chances are, if it works just once then that is sufficient to give a model close enough to the standard FLRW cosmological models to be friendly to life. However, this does not account for the rest of the coincidences enabling life, involving particle masses and the values of the fundamental constants. Perhaps progress in quantum cosmology will in the future lead to some unique theory of creation and existence that will guide the discussion. At present, uniqueness eludes us.

Among those universes in which the necessary cosmic conditions for life have been fulfilled is the subset of almost-FLRW universes which are good models of our own observable Universe, given the precision of the observational data we have at present. They are a small subset of all possible universes, and it is conceivable they are part of a multiverse or ensemble of universes. It is, however, abundantly clear that ‘really existing multiverses’ which can be defined as candidates for the one to which our Universe belongs are not unique, and neither their properties nor their existence are directly testable. The only way in which arguments for the existence of one particular kind of multiverse would be scientifically acceptable is if, for instance, there would emerge evidence (either direct or indirect) for the existence of a specific inflaton potential which would generate one particular kind of ensemble of expanding universe domains, as in chaotic inflation. We have no such evidence.

Despite these problems, the idea of a multiverse is probably here to stay with us — it is an important concept that needs exploration and elucidation. Does the idea that ‘all that can exist, exists’ in the ensemble context provide an explanation for the anthropic puzzles? Yes it does do so. The issue of fine-tuning is the statement that the biophilic set of universes is a very small subset of the set of possible universes; but if all that can exist exists then there are universe models occupying this biophilic subspace. However, there are the following problems: (i) the issue of realized infinities discussed above; (ii) the problem of our inability to describe such ensembles because we do not know what all the possibilities are, so our solution is in terms of a category we cannot fully describe; (iii) the multiverse idea is not testable or provable in the usual scientific sense — existence of the hypothesized ensemble remains a matter of faith rather than proof. Furthermore, in the end it simply represents a regress of causation. Ultimate questions remain. Why this multiverse with these properties rather than others? What endows these with existence and with this particular type of overall order? What are the ultimate boundaries of possibility — what makes something possible, even though it may never be realized? In our view these questions — Issues 1 and 2 discussed in this paper — cannot be answered scientifically because of the lack of any possibility of verification of any proposed underlying theory. They will of necessity have to be argued in philosophical terms.

The concept of a multiverse raises many fascinating issues that have not yet been adequately explored. The discussion given here on how they can be described should be useful in furthering this endeavour.

Acknowledgments

We thank A. Linde, A. Lewis, A. Malcolm, C. Harper, A. Aguirre and J. P. Uzan for helpful comments and references related to this work. GFRE and UK acknowledge financial support from the University of Cape Town and the NRF (South Africa).

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‘Connected’ implies ‘Locally causally connected’, i.e. all universe domains are connected by _C_0 time-like lines which allow any number of reversals in their direction of time, as in Feynman's approach to electrodynamics. Thus, it is a union of regions that are causally connected to each other, and transcends particle and event horizons; for example, all points in de Sitter space–time are connected to each other by such lines.

2

It has been suggested to us that in mathematics terms it does not make sense to distinguish identical copies of the same object; they should be identified with each other because they are essentially the same. However, we are here dealing with physics rather than mathematics, and with real existence rather than possible existence, and then multiple copies must be allowed (for example, all electrons are identical to each other; physics would be very different if there were only one electron in existence).

3

When contemplating mathematical concepts, it is debatable as to whether a procedure for construction is needed. But we are talking physics, and the issue is precisely whether the concept is realizable.

4

One way out would be, as quite a bit of work in quantum cosmology seems to indicate, to have time originating or emerging from the quantum-gravity dominated primordial substrate only ‘later’. In other words, there would have been a ‘time’ or an epoch before time as such emerged. Past time would then be finite, as seems to be demanded by philosophical arguments, and yet the timeless primordial state could have lasted ‘forever’, whatever that would mean. This possibility avoids the problem of constructibility.

5

Many discussions implicitly suggest that this is the whole possibility space, as they only consider FLRW models as possibilities. However, these clearly form a very small subspace of all geometrical possibilities.

6

The way the tetrad description given above relates to FLRW universes is described in detail in Ellis & MacCallum (1969). The standard coordinates given here are more convenient if we discuss only the FLRW models.

7

Hence, in general, different models correspond to the same values of Ωm and ΩΛ, depending on the value of H.

8

It is worth noting that when _G_= 0, we do obtain FLRW solutions to the Einstein field equations: the Milne universe which is effectively empty or, equivalently, without gravity.

9

However, it should be noted that any power (_m_=Gn, _m_∈+, _n_≠ 0) and logarithmic relationship (_m_= ln(G), m_∈) leads to the same measure dμ_G.

10

In equation (22)P(x) d_x_ corresponds to ƒ(_p_I)π.

11

More accurately, perturbations of these models can allow life — the exact FLRW models themselves cannot do so.

Author notes

†

Permanent address: Vatican Observatory Research Group, Steward Observatory, University of Arizona, Tucson, Arizona 85721, USA.

© 2004 RAS