Multiverses and Cosmology: Philosophical Issues W. R. Stoeger1,2, G. F. R. Ellis1, and U. Kirchner1. 6 0 May 15, 2006 0 2 n 1 DepartmentofMathematicsandAppliedMathematics,UniversityofCape a J Town, 7700 Rondebosch, South Africa. 9 1 2 Permanent Address: Vatican Observatory Research Group, Steward Ob- servatory, The University of Arizona, Tucson, Arizona 85721,USA. 2 v Abstract 9 2 The idea of a multiverse – an ensemble of universes or universe do- 3 mains – has received increasing attention in cosmology, both as the out- 7 come of the originating process that generated our own universe, and as 0 an explanation for why our universeappears to befine-tunedfor life and 4 consciousness. Here we review how multiverses should be defined,stress- 0 ing the distinction between the collection of all possible universes and / h ensembles of really existing universes, which distinction is essential for p anthropic arguments. We show that such realised multiverses are by no - means unique, and in general require the existence of a well-defined and o physicallymotivateddistributionfunctiononthespaceofallpossibleuni- r t verses. Furthermore, a proper measure on these spaces is also needed, so s thatprobabilities canbecalculated. Wethendiscussseveralotherphysi- a : calandphilosophical problemsarising inthecontextofensemblesofuni- v verses, including realized infinities and the issue of fine-tuning– whether i X veryspecialorgenericprimordialconditionsaremorefundamentalincos- mology. Thenwebrieflysummarisescenarioslikechaoticinflation,which r a suggest how ensembles of universe domains may be generated, and point out that the regularities underlying any systematic description of truly disjoint multiverses must imply some kind of common generating mech- anism, whose testability is problematic. Finally, we discuss the issue of testability, which underlies the question of whether multiverse proposals are really scientific propositions rather than metaphysical proposals. Key Words: Cosmology;inflation; multiverses; anthropic principle 1 Introduction Overthepasttwentyyearstheproposalofareallyexistingensembleofuniverses – a ‘multiverse’ – has gained prominence in cosmology, even though there is 1 so far only inadequate theoretical and observational support for its existence. The popularity of this proposal can be traced to two factors. The first is that quite a few promising programsof researchin quantumand veryearly universe cosmologysuggestthattheveryprocesseswhichcouldhavebroughtouruniverse orregionoftheuniverseintoexistencefromaprimordialquantumconfiguration, wouldhavegeneratedmanyotheruniversesoruniverseregionsaswell. Thiswas first modelled in a specific way by Vilenkin (1983)and was developed by Linde (Linde 1983, 1990) in his chaotic cosmology scenario. Since then many others, e. g. Leslie (1996), Weinberg (2000), Sciama (1993), Deutsch (1998), Tegmark (1998, 2003), Smolin (1999), Lewis (2000), Weinberg (2000), and Rees (2001) have discussed ways in which an ensemble of universes or universe domains might originate physically. More recently specific impetus has been given to this possibility by superstring theory. It is now claimed by some that versions of these theories provide “landscapes” populated by a large number of vacua, * each of which could occur in or initiate a separate universe domain,* with different values of the physical parameters, such as the cosmological constant, the masses of the elementary particles and the strengths of their interactions (Kachru, et al. 2003;Susskind 2003, 2005,and references therein). So far, none of these proposals 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 im- ply that there is only one possible multiverse, characterised by “all that can exist does exist” (Lewis 2000, see also Gardner 2003). This vague prescription actually allows a vast variety of different realisations 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 en- semble itself. Furthermore, it is not at all clear that we shall ever be able to accurately delineate the class of all possible universes. The second factor stimulating the popularity of multiverses is that it is the only scientific way of avoiding the fine-tuning seemingly required for our universe. This applies firstly to the cosmological constant, which seems to be fine-tuned by 120 orders of magnitude relative to what is expected on the basis of quantum field theory (Weinberg 2000, Susskind 2005). If (almost) all values of the cosmological constant occur in a multiverse, then we can plausibly live in one with the very low observed non-zero value; indeed such a low value is required in order that galaxies, stars, and planets exist and provide us with a suitable habitat for life. Thisisanexampleofthesecondmotivation,namelythe‘anthropicprinciple’ connection: If any of a large number of parameters which characterize our universe–includingfundamentalconstants,thecosmologicalconstant,andand initialconditions–wereslightlydifferent,ouruniversewouldnotbesuitablefor complexity or life. What explains the precise adjustment of these parameters so that microscopic and macroscopic complexity and life eventually emerged? One can introduce a “Creator”who intentionally sets their values to assure the eventual development of complexity. But this move takes us beyond science. The existence of a large collection of universes, which represents the full range 2 of possible parameter values, though not providing an ultimate explanation, would provide a scientifically accessible way of avoiding the need for such fine- tuning. If physical cosmogonic processes naturally produced such a variety of universes, one of which was ours, then the puzzle of fine- tuning is solved. We simply find ourselves in one in which all the many conditions for life have been fulfilled. Ofcourse,throughcosmologywemustthendiscoveranddescribetheprocess by which that collection of diverse universes, or universe domains, was gener- ated,oratleastcouldhavebeengenerated,withthefullrangeofcharacteristics theypossess. Thismaybepossible. Itisanalogousto thewayinwhichwelook upon the special character of our Solar System. We do not agonize how initial conditionsfortheEarthandSunwerespeciallysetsothatlifewouldeventually emerge – though at some level that is still a mystery. We simply realize that our SolarSystem in one of hundreds of billions of others in the Milky Way,and accept that, though the probability that any one of them is bio-friendly is very low, at least a few of them will naturally be so. We have emerged as observers in one of those. No direct fine-tuning is required, provided we take for granted both the nature of the laws of physics and the specific initial conditions in the universe. The physical processes of stellar formation throughout our galaxy naturallyleadstothegenerationofthefullrangeofpossiblestellarsystemsand planets. Before going on, it is necessary to clarify our terminology. 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 sin- gle spacetime. In our view (as ‘uni’ means ‘one’) the Universe is by definition the one unique connected1 existing spacetime of which our observedexpanding cosmological domain is a part. We will refer to situations such as in chaotic inflation as a Multi-Domain Universe, as opposed to a completely causally dis- connected 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 Multi-Domain Universe, we shall for simplic- ity simply use the word “ensemble”. When the universes of an ensemble are all sub-regions of a larger connected spacetime - the “Universe as a whole”- we have the multi-domain situation, which should be described as such. Then we canreserve“multiverse”forthecollectionofgenuinelydisconnected“universes” – those which are not locally causally related. In this article, we shall critically examine the concept of an ensemble of universes or universe domains, from both physical and philosophical points of view, reviewing how they are to be defined physically and mathematically in cosmology (Ellis, Kirchner and Stoeger 2003, hereafter referred to as EKS), how their existence could conceivably be validated scientifically, and focusing 1“Connected” implies“Locally causally connected”, that isall universe domains are con- nectedbyC0 timelikelineswhichallowanynumberofreversalsintheirdirectionoftime,as in Feynman’s approach to electrodynamics. Thus, it is a union of regions that are causally connectedtoeachother,andtranscendsparticleandeventhorizons;forexampleallpointsin deSitterspacetimeareconnected toeachother bysuchlines. 3 upon some of the key philosophical problems associated with them. We have already addressed the physics and cosmology of such ensembles in a previous paper (EKS), along with some limited discussion of philosophical issues. Here we shall summarize the principal conclusions of that paper and then discuss in detail the more philosophical issues. Firstofall,wereviewthedescriptionofthethesetofpossibleuniversesand sets of realised (i. e., really existing) universes and the relationship between these two kinds of sets. It is fundamental to have a general provisionally ade- quateschemetodescribethesetofallpossibleuniverses. Usingthiswecanthen moveforwardtodescribepotentialsetsofactuallyexistinguniversesbydefining distributionfunctions(discreteorcontinuous)onthespaceofpossibleuniverses. A given distribution function indicates which of the theoretically possible uni- verseshave been actualized to give us a really existing ensemble of universes or universedomains. Itisobviouslycrucialtomaintainthedistinctionbetweenthe setofallpossibleuniverses,andthesetofallexistinguniverses. Foritistheset ofallexistinguniverseswhichneedstobeexplainedbycosmologyandphysics– thatis,byaprimordialoriginatingprocessorprocesses. Furthermore,itisonly an actually existing ensemble of universes with the required range of properties which can provide an explanation for the existence of our bio-friendly universe without fine-tuning (see also McMullin 1993, p. 371). A conceptually possible ensemble is not sufficient for this purpose – one needs universes which actually exist, along with mechanisms which generate their existence. We consider in some depth how the existence of such an actually existing ensemble might be probed experimentally and observationally - this is the key issue determining whether the proposal is truly a scientific one or not. Thoughtheensembleofallpossibleuniversesisundoubtedlyinfinite,having an infinite ensemble of actually existing universes is problematic – and further- moreblocksourabilitytoassignstatisticalmeasurestoit,asweshalldiscussin some detail later. For all these reasons, any adequate cosmological account of the originofouruniverseasone ofacollectionofmanyuniverses–orevenasa single realised universe – must include a process whereby the realisedensemble isselectedfromthespaceofallpossibleuniversesandphysicallygenerated. But it must also providesome metaphysicalview on the originof the set ofpossible universes as a subset of the set of conceivable universes - which is itself a very difficult set to define2. 2 Describing Ensembles: Possibility To characterize an ensemble of existing universes, we first need to develop ad- equate methods for describing the class of all possible universes. This itself is philosophically controversial, as it depends very much on what we regard as ”possible.” At the very least, describing the class of all possible universes re- quires us to specify, at least in principle, all the ways in which universes can 2Sciencefictionandfantasyprovidearichtreasuryofconceivableuniverses,manyofwhich willnotbe”possibleuniverses”asoutlinedabove. 4 be different from one another, in terms of their physics, chemistry, biology, etc. We have done this in EKS, which we shall review here. 2.1 The Set of Possible Universes Ensembles of universes, or multiverses, are most easily represented classically by the structure and the dynamics of a space M of all possible universes, each of which can be described in terms of a set of states s in a state space S (EKS).EachuniverseminMwillbe characterisedbyasetP ofdistinguishing parameters p, which are coordinates on S (EKS). Each m will evolve from its initial state to some final state accordingto the operative dynamics, with some or all of its parameters varying as it does so. The course of this evolution of states will be representedby a path in the state space S. Thus, each such path (indegeneratecases,apoint)isarepresentationofoneoftheuniversesminM. 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 indicatingthechangeofeachmodel’sstateswithtime,characterisedbyanextra 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. This classical, non-quantum-cosmological formulation of the set of all pos- sible universes is obviously provisional and not fundamental. Much less should it provide the basis for adjudicating the ontology of these ensembles and their components.3 It provides us with a preliminary systematic framework, con- sistent with our present limited understanding of cosmology, within which to begin studying ensembles of universes and universe domains. It is becoming very clear that, from what we are beginning to learn from quantum cosmology, a more fundamental framework will have to be developed that takes seriously quantumissuessuchasentanglement. Additionally,thereareseriousunresolved problems concerning time in quantum cosmology. Already at the level of gen- eralrelativity itself, as everyonerecognizes,time loses its fundamental, distinct character. What is given is space-time, not space and time. Time is now in- trinsicto agivenuniversedomainandits dynamicsandthereis nopreferredor uniquewayofdefiningit(Isham1988,1993;Smolin1991;Barbour1994;Rovelli 2004; and references therein). When we go to quantum gravity and quantum cosmology, time, while remaining intrinsic, recedes further in prominence and even seems to disappear. The Wheeler-de Witt equation for “wave function of the universe,” for example, does not explicitly involve time – it is a time- independent equation. However, our provisional classical formulation receives support from the fact that dynamics and an intrinsic time appear to emerge from it as the universe expands out of the Planck era (see, for instance, Isham 1988 and Rovelli 2004, especially pp. 296-301). Furthermore, as yet there is 3Wethankandacknowledgethecontributionofananonymousrefereewhohaspointedthis outtous,andhasstimulatedthisbriefdiscussionoftheimportantroleofquantumcosmology indefiningmultiverses. 5 no adequate quantum gravity theory nor quantum cosmological resolution to this issue of the origin and the fundamental character of time – just tantaliz- ing pieces of a much larger picture. The only viable approach at present is to proceed on the basis of the emergent classical description. And then there are related issues connected with decoherence – how is the transition from “the wave function of the universe” to the classicaluniverse, or anensembleofuniversedomains,effected, andwhatemergesinthis transition? Whatiscrucialhereisthatasthewavefunctiondecoheresanentireensembleof universes or universe domains may emerge. These would all be entangled with one another. This would provide the fundamental basis for the quantum ontol- ogy of the ensemble.4 Furthermore, it would provide a fundamental connection among a large number of the members of our classically defined M above. We havealreadystressedthedifferencebetweenamulti-domainuniverseandatrue multiverse. An entangled ensemble of universe domains decohering from a cos- mological wave function would be an important example of that case. This process of cosmological decoherence, which we as yet do not understand and have not adequately modelled , may turn out to be a key generating mecha- nism for a really existing multiverse. In that case we would want to define a muchmorefundamentalspaceofallpossiblecosmologicalwavefunctions. Each of these would generate an ensemble of classical universes or universe domains which we have represented individually in M. We could then map the wave- functions in that more fundamental space into the m of M. As yet, however, wedonothaveevenaminimallyreliablequantumcosmologythatwouldenable us to implement that. Despiteourlackofunderstandingatthequantumcosmologicallevel,andthe lessthanfundamentalcharacterofourspaceM,it enablesusproceedwithour discussion of cosmological ensembles at the non-quantum level - which is what cosmologicalobservationsrelateto. While doingso,wemustkeepthe quantum cosmological perspective in mind. Though we are without the resources to elaborateit morefully, itprovidesavaluable contextwithin whichto interpret, evaluate and critique our more modest classical discussion here. Returning to our description of the space M of possible universes m, we must recognize that it 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 m have in common. Without this, we have no basis for defining it. Its overall characterisationmust therefore incorporate a description both of the geometry ofthealloweduniversesandofthephysicsofmatter. Thusthesetofparameters P will include both geometric and physical parameters. AmongtheimportantsubsetsofthespaceMare(EKS):M ,thesubset FLRW ofallpossibleFriedmann-Lemaˆıtre-Robertson-Walker(FLRW)universes,which are exactly isotropic and spatially homogeneous; Malmost−FLRW, the subset of all universes which deviate from exact FLRW models by only small, linearly growinganisotropiesand inhomogeneities; M , the subset of allpossible anthropic universes in which life emerges at some stage in their evolution. This subset 4Again,wethankthesamerefereeforemphasizingtheimportanceofthepossibility. 6 intersectsMalmost−FLRW,andmayevenbeasubsetofMalmost−FLRW,butdoes not intersect M , since realistic models of a life-bearing universe like ours FLRW cannot be exactly FLRW, for then there is no structure. If M truly represents all possibilities, as we have already emphasized, one 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? From these considerations we have the first key issue (EKS): Issue 1: WhatdeterminesM? Wheredoesthis structurecomefrom? Whatis the meta-cause, or ground, that delimits this set of possibilities? Why is there a uniform structure across all universes m in M? It should be obvious that these same questions would also have to be ad- dressed with regard to the more fundamental space of all cosmological wave functions we briefly described earlier, which would probably underlie any en- sembles of universes or universe domains drawn from M. It is clear, as we have discussed in EKS, that these questions cannot be answered scientifically, though scientific input is necessary for doing so. How can we answer them philosophically? 2.2 Adequately Specifying Possible Anthropic Universes Whendefininganyensembleofuniverses,possibleorrealised,wemustspecifyall 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 like symmetry breakingwithin 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 ensemble. WecanseparateoursetofparametersP forthespaceofallpossibleuniverses Minto differentcategories,beginning withthe mostbasic orfundamental,and progressingto morecontingentandmorecomplex categories(see EKS).Ideally they should all be independent of one another, but we will not be able to es- tablishthat independence foreachparameter,exceptfor the mostfundamental cosmological ones. In order to categorise our parameters, we can doubly index each parameter p in P as p (i) such that those for j = 1−2 describe basic j physics, for j = 3−5 describe the cosmology (given that basic physics), and j = 6−7 pertain specifically to emergence of complexity and of life (see EKS for further details). Though we did not do so in our first paper EKS, it may be helpful to add a separate category of parameters p (i), which would relate directly to the 8 emergenceofconsciousnessandself-consciouslife, as wellasto the causaleffec- tiveness of self-conscious (human) life – of ideas, intentions and goals. It may turn out that all such parameters may be able to be reduced to those of p (i), 7 7 just as those of p (i) and p (i) may be reducible to those of physics. But we 6 7 also may discover, instead, that such reducibility is not possible. Alltheseparameterswilldescribethe setofpossibilitiesweareabletochar- acterise on the basis of our accumulated scientific experience. This is by no meansastatementthat“allthatcanoccur”isarbitrary. Onthecontrary,spec- ifying the set of possible parameters determines a uniform high-level structure that is obeyed by all universes in M. In the companion cosmology/physics paper to this one (EKS), we develop in detail the geometry, parameters p (i), and the physics, parameters p (i) to 5 1 p (i), of possible universes. There we also examine in detail the FLRW sector 4 M of the ensemble of all possible universes M to illustrate the relevant FLRW mathematicalandphysicalissues. Weshallnotrepeatthosediscussionshere,as theydonotdirectlyimpactourtreatmentofthephilosophicalissuesuponwhich we are focusing. However, since one of the primary motivations for developing the multiverse scenario is to provide a scientific solution to the anthropic fine- tuning problem, we need to discuss briefly the set of ”anthropic” universes. 2.3 The Anthropic subset The subset of universes that allow intelligent life to emerge is of particular interest. That means we need a function on the set of possible universes that describes the probability that life may evolve. An adaptation of the Drake equation (Drake and Shostak 1998) gives for the expected number of planets with intelligent life in any particular universe m in an ensemble (EKS), N (m)=N ∗N ∗Π∗F, (1) life g S where N is the number of galaxies in the model and N the average number g S of stars per galaxy. The probability that a star provides a habitat for life is expressed by the product Π=f ∗f ∗n (2) S p e and the probability of the emergence of intelligent life, given such a habitat, is expressed by the product F =f ∗f . (3) l i Here f is the fraction of stars that can provide a suitable environment for life S (they are ‘Sun-like’), f is the fraction of such stars that are surrounded by p planetary systems, n is the mean number of planets in each such system that e are suitable habitats for life (they are ‘Earth-like’), f is the fraction of such l planets on which life actually originates,and f represents the fraction of those i 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 (m)>0. life The quantities {N ,N ,f ,f ,n ,f ,f } are functions of the physical and g S S p e l i cosmological parameters characterised above. So there will be many different representations of this parameter set depending on the degree to which we try to represent such interrelations. 8 InEKS,followinguponourdetailedtreatmentofM weidentifythose FLRW FLRW universes in which the emergence and sustenance of life is possible on a broadlevel5 –thenecessarycosmologicalconditionshavebeenfulfilledallowing existence ofgalaxies,stars,andplanets if the universeis perturbed, soallowing a non-zero factor N ∗N ∗Π as discussed above. The fraction of these that g S willactuallybelife-bearingdependsonthefulfilmentofalargenumberofother conditions represented by the factor F = f ∗f , which will also vary across a l i generic ensemble, and the above assumes this factor is non-zero. 3 The Set of Realised Universes We have now characterised the set of possible universes. But in any given ex- isting ensemble, many will not be realised, and some may be realised many times. The purpose of this section is to review our formalism (EKS) for speci- fying whichofthe possible universes(characterisedabove)occurin aparticular realised ensemble. 3.1 A distribution function for realised universes In order to select from M a set of realised universes we need to define on M a distribution function f(m) specifying how many times each type of possible universe m in M is realised6. The function f(m) expresses the contingency in any actualisation – the fact that not every possible universe has to be realised. Things could have been different! Thus, f(m) describes the ensemble of uni- verses or multiverse envisaged as being realised out of the set of possibilities. In general, these realisations include only a subset of possible universes, and multiple realisation of some of them. Even at this early stage of our discussion wecansee thatthe reallyexisting ensembleofuniversesis byno meansunique. From a quantum cosmology perspective we can consider f(m) as given by anunderlying solutionof the Wheeler-de Witt equation, by agivensuperstring model, or by some other generating mechanism, giving an entangled ensemble of universes or universe domains. The class of models considered is determined by all the parameters which are held constant (‘class parameters’). Considering the varying parameters for a given class (‘member parameters’), some will take only discrete values, but for eachone allowedto take continuous values we need a volume elementof the possibility space M characterised by parameter increments dp (i) in all such j 5Moreaccurately, perturbations of these models can allow life– the exact FLRW models themselvescannot doso. 6Ithasbeensuggestedtousthatinmathematicaltermsitdoesnotmakesensetodistin- guish identical copies of the same object: they should be identified with each other because they are essentially the same. But we are here dealing with physics rather than mathemat- ics,andwithrealexistence rather thanpossibleexistence, andthen multiplecopies mustbe allowed(forexampleallelectrons areidenticaltoeachother; physicswouldbeverydifferent iftherewereonlyoneelectroninexistence). 9 varying parameters p (i). The volume element will be given by a product j π =Π m (m)dp (i) (4) i,j ij j where the product Π runs over all continuously varying member parameters i,j i,j in the possibility space, and the m weight the contributions of the dif- ij ferent parameter increments relative to each other. These weights depend on the parameters p (i) characterising the universe m. The number of universes j corresponding to the set of parameter increments dp (i) will be dN given by j dN =f(m)π (5) for continuous parameters; for discrete parameters, we add in the contribution fromallallowedparametervalues. The totalnumber ofuniversesinthe ensem- ble will be given by N = f(m)π (6) Z (whichwilloftendiverge),wheretheintegralrangesoverallallowedvaluesofthe member parameters and we take it to include all relevant discrete summations. Theprobablevalue ofanyspecific quantityp(m)definedonthe setofuniverses will be given by p(m)f(m)π P = (7) R f(m)π R Such integrals over the space of possibilities give numbers, averages,and prob- abilities. Now it is conceivable that all possibilities are realised– that alluniverses in M exist at least once. This would mean that the distribution function f(m)6=0 for all m∈M. Butthereareaninfinitenumberofdistributionfunctionswhichwouldfulfilthis condition. So not even a really existing ‘ensemble of all possible universes’ is unique. In such ensembles, all possible values of each distinguishing parame- ter would be represented by its members in all possible combinations with all other parameters at least once. One of the problems is that this means that the integrals associated with such distribution functions would often diverge, preventing the calculation of probabilities. From these considerations we have the second key issue: Issue 2: What determines f(m)? What is the meta-cause that delimits the set of realisations out of the set of possibilities? The answer to this question has to be different from the answer to Issue 1, preciselybecauseherewearedescribingthe contingencyofselectionofasubset of possibilities for realisation from the set of all possibilities – determination of 10