We do not know whether there are other universes. Perhaps we never shall. But I want to respond to Paul Davies' questions  by arguing that "do other universes exist?" can be a genuine scientific question. Moreover, I shall outline why it is an interesting question; and why, indeed, I already suspect that the answer may be "yes".
First, a pre-emptive and trivial comment: if you define the universe as "everything there is", then by definition there cannot be others. I shall, however, follow the convention among physicists and astronomers, and define the "universe" as the domain of space-time that encompasses everything that astronomers can observe. Other "universes", if they existed, could differ from ours in size, content, dimensionality, or even in the physical laws governing them.
It would be neater, if other "universes" existed, to redefine the whole enlarged ensemble as "the universe", and then introduce some new term — for instance "the metagalaxy" — for the domain that cosmologists and astronomers have access to. But so long as these concepts remain so conjectural, it is best to leave the term "universe" undisturbed, with its traditional connotations, even though this then demands a new word, the "multiverse", for a (still hypothetical) ensemble of "universes."
Ontological Status Of Other Universes
Science is an experimental or observational enterprise, and it's natural to be troubled by assertions that invoke something inherently unobservable. Some might regard the other universes as being in the province of metaphysics rather than physics. But I think they already lie within the proper purview of science. It is not absurd or meaningless to ask "Do unobservable universes exist?", even though no quick answer is likely to be forthcoming. The question plainly can't be settled by direct observation, but relevant evidence can be sought, which could lead to an answer.
There is actually a blurred transition between the readily observable and the absolutely unobservable, with a very broad grey area in between. To illustrate this, one can envisage a succession of horizons, each taking us further than the last from our direct experience:
(i) Limit of present-day telescopes
There is a limit to how far out into space our present-day instruments can probe. Obviously there is nothing fundamental about this limit: it is constrained by current technology. Many more galaxies will undoubtedly be revealed in the coming decades by bigger telescopes now being planned. We would obviously not demote such galaxies from the realm of proper scientific discourse simply because they haven't been seen yet. When ancient navigators speculated about what existed beyond the boundaries of the then known world, or when we speculate now about what lies below the oceans of Jupiter's moons Europa and Ganymede, we are speculating about something "real" — we are asking a scientific question. Likewise, conjectures about remote parts of our universe are genuinely scientific, even though we must await better instruments to check them.
(ii) Limit in principle at present era
Even if there were absolutely no technical limits to the power of telescopes, our observations are still bounded by a horizon, set by the distance that any signal, moving at the speed of light, could have travelled since the big bang. This horizon demarcates the spherical shell around us at which the redshift would be infinite. There is nothing special about the galaxies on this shell, any more than there is anything special about the circle that defines your horizon when you're in the middle of an ocean. On the ocean, you can see farther by climbing up your ship's mast. But our cosmic horizon can't be extended unless the universe changes, so as to allow light to reach us from galaxies that are now beyond it. If our universe were decelerating, then the horizon of our remote descendants would encompass extra galaxies that are beyond our horizon today. It is, to be sure, a practical impediment if we have to await a cosmic change taking billions of years, rather than just a few decades (maybe) of technical advance, before a prediction about a particular distant galaxy can be put to the test. But does that introduce a difference of principle? Surely the longer waiting-time is a merely quantitative difference, not one that changes the epistemological status of these faraway galaxies?
(iii) Never-observable galaxies from "our" Big Bang,
But what about galaxies that we can never see, however long we wait? It's now believed that we inhabit an accelerating universe. As in a decelerating universe, there would be galaxies so far away that no signals from them have yet reached us; but if the cosmic expansion is accelerating, we are now receding from these remote galaxies at an ever-increasing rate, so if their light hasn't yet reached us, it never will. Such galaxies aren't merely unobservable in principle now — they will be beyond our horizon forever. But if a galaxy is now unobservable, it hardly seems to matter whether it remains unobservable for ever, or whether it would come into view if we waited a trillion years. (And I have argued, under (ii) above, that the latter category should certainly count as "real".)
(iv) Galaxies in disjoint universes
The never-observable galaxies in (iii) would have emerged from the same Big Bang as we did. But suppose that, instead of causally-disjoint regions emerging from a single Big Bang (via an episode of inflation) we imagine separate Big Bangs. Are space-times completely disjoint from ours any less real than regions that never come within our horizon in what we'd traditionally call our own universe? Surely not — so these other universes too should count as real parts of our cosmos, too.
This step-by-step argument (those who don't like it might dub it a slippery slope argument!) suggests that whether other universes exist or not is a scientific question. But it is of course speculative science. The next question is, can we put it on a firmer footing? What could it explain?
Scenarios For A Multiverse
At first sight, nothing seems more conceptually extravagant — more grossly in violation of Ockham's Razor — than invoking multiple universes. But this concept is a natural consequence of several different theories ( albeit all speculative). Andrei Linde, Alex Vilenkin and others have performed computer simulations depicting an "eternal" inflationary phase where many universes sprout from separate big bangs into disjoint regions of spacetimes. Alan Guth and Lee Smolin have, from different viewpoints, suggested that a new universe could sprout inside a black hole, expanding into a new domain of space and time inaccessible to us. And Lisa Randall and Raman Sundrum suggest that other universes could exist, separated from us in an extra spatial dimension; these disjoint universes may interact gravitationally, or they may have no effect whatsoever on each other.
There could be another universe just a few millimetres away from us. But if those millimetres were measured in some extra spatial dimension then to us (imprisoned in our 3-dimensional space) the other universe would be inaccessible. In the hackneyed analogy where the surface of a balloon represents a two-dimensional universe embedded in our three-dimensional space, these other universes would be represented by the surfaces of other balloons: any bugs confined to one, and with no conception of a third dimension, would be unaware of their counterparts crawling around on another balloon. Variants of such ideas have been developed by Paul Steinhardt, Neil Turok and others. Guth and Edward Harrison have even conjectured that universes could be made in some far-future laboratory, by imploding a lump of material to make a small black hole. Could our entire universe perhaps then be the outcome of some experiment in another universe? If so, the theological arguments from design could be resuscitated in a novel guise. Smolin speculates that the daughter universe may be governed by laws that bear the imprint of those prevailing in its parent universe. If that new universe were like ours, then stars, galaxies and black holes would form in it; those black holes would in turn spawn another generation of universes; and so on, perhaps ad infinitum.
Parallel universes are also invoked as a solution to some of the paradoxes of quantum mechanics, in the "many worlds" theory, first advocated by Hugh Everett and John Wheeler in the 1950s. This concept was prefigured by Olaf Stapledon, in his 1937 novel, as one of the more sophisticated creations of his Star Maker: "Whenever a creature was faced with several possible courses of action, it took them all, thereby creating many ... distinct histories of the cosmos. Since in every evolutionary sequence of this cosmos there were many creatures and each was constantly faced with many possible courses, and the combinations of all their courses were innumerable, an infinity of distinct universes exfoliated from every moment of every temporal sequence". None of these scenarios has been simply dreamed up out of the air: each has a serious, albeit speculative, theoretical motivation. However, one of them, at most, can be correct. Quite possibly none is: there are alternative theories that would lead just to one universe. Firming up any of these ideas will require a theory that consistently describes the extreme physics of ultra-high densities, how structures on extra dimensions are configured, etc. But consistency is not enough: there must be grounds for confidence that such a theory isn't a mere mathematical construct, but applies to external reality. We would develop such confidence if the theory accounted for things we can observe that are otherwise unexplained. As the moment, we have an excellent framework, called the standard model, that accounts for almost all subatomic phenomena that have been observed. But the formulae of the "standard model" involve numbers which can't be derived from the theory but have to be inserted from experiment.
Perhaps, in the 21st-century theory, physicists will develop a theory that yields insight into (for instance) why there are three kinds of neutrinos, and the nature of the nuclear and electric forces. Such a theory would thereby acquire credibility. If the same theory, applied to the very beginning of our universe, were to predict many big bangs, then we would have as much reason to believe in separate universes as we now have for believing inferences from particle physics about quarks inside atoms, or from relativity theory about the unobservable interior of black holes.
Universal Laws, Or Mere Bylaws?
"Are the laws of physics unique?" is a less poetic version of Einstein's famous question "Did God have any choice in the creation of the Universe?" The answer determines how much variety the other universes — if they exist — might display. If there were something uniquely self-consistent about the actual recipe for our universe, then the aftermath of any big bang would be a re-run of our own universe. But a far more interesting possibility (which is certainly tenable in our present state of ignorance of the underlying laws) is that the underlying laws governing the entire multiverse may allow variety among the universes. Some of what we call "laws of nature" may in this grander perspective be local bylaws, consistent with some overarching theory governing the ensemble, but not uniquely fixed by that theory.
As an analogy (one which I owe to Paul Davies) consider the form of snowflakes. Their ubiquitous six-fold symmetry is a direct consequence of the properties and shape of water molecules. But snowflakes display an immense variety of patterns because each is moulded by its micro-environments: how each flake grows is sensitive to the fortuitous temperature and humidity changes during its downward drift. If physicists achieved a fundamental theory, it would tell us which aspects of nature were direct consequences of the bedrock theory (just as the symmetrical template of snowflakes is due to the basic structure of a water molecule) and which are (like the distinctive pattern of a particular snowflake) the outcome of accidents. The accidental features could be imprinted during the cooling that follows the big bang — rather as a piece of red-hot iron becomes magnetised when it cools down, but with an alignment that may depend on chance factors. It may turn out (though this would be a disappointment to many physicists if it did) that the key numbers describing our universe, and perhaps some of the so-called constants of laboratory physics as well, are mere "environmental accidents", rather than being uniquely fixed throughout the multiverse by some final theory. This is relevant to some now-familiar arguments (explored further in my book Our Cosmic Habitat) about the surprisingly fine-tuned nature of our universe.
Fine Tuning — A Motivation For Suspecting That Our "Universe" Is One Of Many.
The nature of our universe depended crucially on a recipe encoded in the big bang, and this recipe seems to have been rather special. A degree of fine tuning — in the expansion speed, the material content of the universe, and the strengths of the basic forces — seems to have been a prerequisite for the emergence of the hospitable cosmic habitat in which we live. Here are some prerequisites for a universe containing organic life of the kind we find on Earth:
First of all, it must be very large compared to individual particles, and very long-lived compared with basic atomic processes. Indeed this is surely a requirement for any hypothetical universe that a science fiction writer could plausibly find interesting. If atoms are the basic building blocks, then clearly nothing elaborate could be constructed unless there were huge numbers of them. Nothing much could happen in a universe that was was too short-lived: an expanse of time, as well as space, is needed for evolutionary processes. Even a universe as large and long-lived as ours, could be very boring: it could contain just black holes, or inert dark matter, and no atoms at all; it could even be completely uniform and featureless. Moreover, unless the physical constants lie in a rather narrow range, there would not be the variety of atoms required for complex chemistry.
If our existence depends on a seemingly special cosmic recipe, how should we react to the apparent fine tuning? There seem three lines to take: we can dismiss it as happenstance; we can acclaim it as the workings of providence; or (my preference) we can conjecture that our universe is a specially favoured domain in a still vaster multiverse. Some seemingly "fine tuned" features of our universe could then only be explained by "anthropic" arguments, which are analogous to what any observer or experimenter does when they allow for selection effects in their measurements: if there are many universes, most of which are not habitable, we should not be surprised to find ourselves in one of the habitable ones.
Testing Specific Multiverse Theories Here And Now
We may one day have a convincing theory that tells us whether a multiverse exists, and whether some of the so called laws of nature are just parochial by-laws in our cosmic patch. But while we're waiting for that theory — and it could be a long wait — the "ready made clothes shop" analogy can already be checked. It could even be refuted: this would happen if our universe turned out to be even more specially tuned than our presence requires. Let me give two quite separate examples of how this style of reasoning can be used to refute specific hypotheses.
(i) Ludwig Boltzmann argued that our entire universe was an immensely rare "fluctuation" within an infinite and eternal time-symmetric domain. There are now many arguments against this hypothesis, but even when it was proposed one could already have noted that fluctuations in large volumes are far more improbable than in smaller volumes.
So, it would be overwhelmingly more likely, if Boltzmann were right, that we would be in the smallest fluctuation compatible with our existence (Indeed, the most probable fluctuation would be a disembodied brain that merely simulated the sensations of the external world.) Whatever our initial assessment of Boltzmann's theory, its probability would plummet if we came to accept the extravagant scale of the cosmos.
(ii) Even if we knew nothing about how stars and planets formed, we would not be surprised to find that our Earth's orbit wasn't highly eccentric: if it had been, water would boil when the Earth was at perihelion and freeze at aphelion — a harsh environment unconducive to our emergence. However, a modest orbital eccentricity (certainly up to 0.1) is plainly not incompatible with life. If it had turned out that the earth moved in a near-perfect circle (with eccentricity, say, less than 0.00001) , this would be a strong argument against a theory that postulated anthropic selection from orbits whose eccentricities had a "Bayesian prior" that was uniform in the range from zero to one.
We could apply this style of reasoning to the important numbers of physics (for instance, the cosmological constant lambda) to test whether our universe is typical of the subset that that could harbour complex life. Lambda has to be below a threshold to allow protogalaxies to pull themselves together by gravitational forces before gravity is overwhelmed by cosmical repulsion (which happens earlier if lambda is large). An unduly fierce cosmic repulsion would prevent galaxies from forming.
Suppose, for instance, that (contrary to current indications) lambda was thousands of times smaller than it needed to be merely to ensure that galaxy formation wasn't prevented. This would raise suspicions that it was indeed zero for some fundamental reason. (Or that it had a discrete set of possible values, and all the others were well about the threshold).
The methodology requires us to decide what values of a particular physical parameter are compatible with our emergence. It also requires a specific theory that gives the relative Bayesian priors for any particular value. For instance, in the case of lambda, are all values equally probable? Are low values favoured by the physics? Or is there a finite number of discrete possible values, depending on how the extra dimensions "roll up"? With this information, one can then ask if our actual universe is "typical" of the subset in which we could have emerged. If it is a grossly atypical member even of this subset (not merely of the entire multiverse) then we would need to abandon our hypothesis. By applying similar arguments to the other numbers, we could check whether our universe is typical of the subset that that could harbour complex life. If so, the multiverse concept would be corroborated.
As another example of how "multiverse" theories can be tested, consider Smolin's conjecture that new universes are spawned within black holes, and that the physical laws in the daughter universe retain a memory of the laws in the parent universe: in other words there is a kind of heredity. Smolin's concept is not yet bolstered by any detailed theory of how any physical information (or even an arrow of time) could be transmitted from one universe to another. It has, however, the virtue of making a prediction about our universe that can be checked. If Smolin were right, universes that produce many black holes would have a reproductive advantage, which would be passed on to the next generation. Our universe, if an outcome of this process, should therefore be near-optimum in its propensity to make black holes, in the sense that any slight tweaking of the laws and constants would render black hole formation less likely. (I personally think Smolin's prediction is unlikely be borne out, but he deserves our thanks for presenting an example that illustrates how a multiverse theory can in principle be vulnerable to disproof.) These examples show that some claims about other universes may be refutable, as any good hypothesis in science should be. We cannot confidently assert that there were many big bangs — we just don't know enough about the ultra-early phases of our own universe. Nor do we know whether the underlying laws are "permissive": settling this issue is a challenge to 21st century physicists. But if they are, then so-called anthropic explanations would become legitimate — indeed they'd be the only type of explanation we'll ever have for some important features of our universe.
A Keplerian Argument
The multiverse concept might seem arcane, even by cosmological standards, but it affects how we weigh the observational evidence in some current debates. Our universe doesn't seem to be quite as simple as it might have been. About 5 percent of its mass is in ordinary atoms; about 25 percent is in dark matter (probably a population of particles that survived from the very early universe contains atoms, and dark matter; and the remaining 70 percent is latent in empty space itself.
Some theorists have a strong prior preference for the simplest universe and are upset by these developments. It now looks as thought a craving for such simplicity will be disappointed. Perhaps we can draw a parallel with debates that occurred 400 years ago. Kepler discovered that planets moved in ellipses, not circles. Galileo was upset by this. In his "Dialogues concerning the two chief systems of the world" he wrote "For the maintenance of perfect order among the parts of the Universe, it is necessary to say that movable bodies are movable only circularly".
To Galileo, circles seemed more beautiful; and they were simpler — they are specified just by one number, the radius, whereas an ellipse needs an extra number to define its shape (the "eccentricic"). Newton later showed, however, that all elliptical orbits could be understood by a single unified theory of gravity. Had Galileo still been alive when Principia was published, Newton's insight would surely have joyfully reconciled him to ellipses.
The parallel is obvious. A universe with at least three very different ingredients low may seem ugly and complicated. But maybe this is our limited vision. Our Earth traces out just one ellipse out of an infinity of possibilities, its orbit being constrained only by the requirement that it allows an environment conducive for evolution (not getting too close to the Sun, nor too far away). Likewise, our universe may be just one of an ensemble of all possible universes, constrained only by the requirement that it allows our emergence. So I'm inclined to go easy with Occam's razor: a bias in favour of "simple" cosmologies may be as short-sighted as was Galileo's infatuation with circles.
What we've traditionally called "the universe" may be the outcome of one big bang among many, just as our Solar System is merely one of many planetary systems in the Galaxy. Just as the pattern of ice crystals on a freezing pond is an accident of history, rather than being a fundamental property of water, so some of the seeming constants of nature may be arbitrary details rather than being uniquely defined by the underlying theory. The quest for exact formulas for what we normally call the constants of nature may consequently be as vain and misguided as was Kepler's quest for the exact numerology of planetary orbits. And other universes will become part of scientific discourse, just as "other worlds" have been for centuries. We may one day have a convincing theory that accounts for the very beginning of our universe, tells us whether a multiverse exists, and (if so) whether some so called laws of nature are just parochial by-laws in our cosmic patch. may be vastly larger than the domain we can now (or, indeed, can ever) observe. Most physicists hope to discover a fundamental theory that will offer unique formulae for all the constants of nature. But perhaps what we've traditionally called our universe is just an atom in an ensemble — a multiverse punctuated by repeated big bangs, where the underlying physical laws permit diversity among the individual universes.
Even though some physicists still foam at the mouth at the prospects of be being "reduced" to these so-called anthropic explanations, such explanations may turn out to be the best we can ever discover for some features of our universe (just as they are the best explanations we can offer for the shape and size of Earth's orbit). Cosmology will have become more like the science of evolutionary biology. Nonetheless (and here physicists should gladly concede to the philosophers), any understanding of why anything exists — why there is a universe (or multiverse) rather than nothing — remains in the realm of metaphysics.