Level I Multiverse: Infinite Regions and Your Cosmic Twin

Let us revert to the matter of your distant double. Supposing that space extends ad infinitum, and that the distribution of matter maintains a sufficient uniformity upon a grand scale, then even the most improbable of occurrences must, perforce, transpire somewhere. To wit, there exist an infinitude of inhabited planets, including not merely one, but an endless array, populated by individuals bearing the same likeness, cognomen, and recollections as yourself. Indeed, there are infinitely many other regions commensurate in size with our observable universe, wherein every conceivable cosmic narrative unfolds. This constitutes the Level I multiverse.

Albeit the implications may strike one as fantastical and counter-intuitive, this spatially infinite cosmological model doth in fact represent the simplest and most favoured amongst those presently available. It forms an integral component of the cosmological concordance model, which aligns with all current observational evidence and is employed as the bedrock for the majority of computations and simulations presented at gatherings of cosmologists. Contrariwise, alternatives such as a fractal universe, a closed universe, and a multiply connected universe, have been gravely impugned by observation. Yet the notion of a Level I multiverse has been met with contention (indeed, an assertion along analogous lines constituted one of the heresies for which Giordano Bruno was consigned to the flames by decree of the Vatican in the year of our Lord 1600† ); ergo, let us examine the standing of the two suppositions (infinite space and “sufficiently uniform” distribution).

Of what magnitude is space? Observationally, the lower bound has expanded markedly, with no indication of an upper limit. We all grant credence to the existence of entities that elude our present gaze, yet might be rendered visible were we to shift our vantage or tarry, such as vessels beyond the horizon's reach. Objects lying beyond the cosmic horizon hold a similar status, given that the observable universe expands by a light-year each annum, as light from more remote locales hath time to reach our purview‡ . Inasmuch as we are all instructed in the rudiments of Euclidean space at school, it may prove arduous to conceive of space as anything but infinite — for what, pray tell, would lie beyond the sign proclaiming “SPACE ENDS HERE — MIND THE GAP”? Nonetheless, Einstein’s theory of gravitation admits the possibility of finite space, by virtue of connections disparate from those of Euclidean space, such as a four-dimensional sphere or a torus, whereby protracted travel in one direction might precipitate a return from the opposite quarter. The cosmic microwave background facilitates sensitive tests of such finite models, yet thus far hath yielded no corroboration thereof — flat infinite models accord well with the data, and stringent limits have been imposed upon both spatial curvature and multiply connected topologies. Moreover, a spatially infinite universe is generically predicted by the cosmological theory of inflation (Garriga & Vilenkin 2001b). The signal triumphs of inflation, enumerated below, furnish further succour to the notion that space is, after all, simple and infinite, precisely as we were taught in our scholastic days.

To what degree does the matter distribution exhibit uniformity upon a grand scale? In an “island universe” model, wherein space is infinite, yet all matter is confined to a finite region, nigh all denizens of the Level I multiverse would be deceased, consisting solely of empty space. Such models have enjoyed historical favour, initially with Earth serving as the island, together with the celestial objects visible to the naked eye, and in the early 20th century, with the known portion of the Milky Way Galaxy fulfilling that role. Another nonuniform alternative is a fractal universe, wherein the matter distribution is self-similar, and all coherent structures in the cosmic galaxy distribution represent merely a minor facet of even grander coherent structures. Both the island and fractal universe models have been decisively refuted by recent observations, as recounted in Tegmark (2002). Maps of the three-dimensional galaxy distribution have demonstrated that the spectacular large-scale structure observed (galaxy groups, clusters, superclusters, etc.) yields to a drab uniformity upon a grand scale, with no coherent structures exceeding approximately 1024 m. More precisely, let us imagine placing a sphere of radius R at various random locations, gauging the amount of mass M enclosed each time, and computing the variation between the measurements, as quantified by their standard deviation ∆M . The relative fluctuations ∆M/M have been measured to be of order unity on the scale R ∼ 3 × 1023m, and diminishing on larger scales. The Sloan Digital Sky Survey has found ∆M/M as small as 1% on the scale R ∼ 1025 m, and cosmic microwave background measurements have established that the trend towards uniformity persists all the way to the edge of our observable universe (R ∼ 1027 m), where ∆M/M ∼ 10−5 . Conspiracy theories aside, wherein the universe is purportedly designed to deceive us, the observations thus speak plainly and unequivocally: space, as we apprehend it, continues far beyond the edge of our observable universe, teeming with galaxies, stars, and planets.

The physics description of the world is traditionally bifurcated into two components: initial conditions and laws of physics specifying how the initial conditions evolve. Observers residing in parallel universes at Level I observe the self-same laws of physics as we do, albeit with initial conditions disparate from those prevalent in our Hubble volume. The prevailing theory posits that the initial conditions (the densities and motions of diverse types of matter early on) were engendered by quantum fluctuations during the inflationary epoch (vide section 3). This quantum mechanism begets initial conditions that are, for all practical purposes, random, yielding density fluctuations described by what mathematicians term an ergodic random field.§ Ergodic signifies that, should one envision generating an ensemble of universes, each with its own random initial conditions, then the probability distribution of outcomes in a given volume is identical to the distribution obtained by sampling diverse volumes in a single universe. In other words, it denotes that everything that could, in principle, have occurred here, did in fact occur somewhere else.

Inflation, in sooth, engenders all possible initial conditions with non-zero probability, the most probable being nearly uniform, with fluctuations at the 10−5 level, amplified by gravitational clustering to give rise to galaxies, stars, planets, and other structures. This implies both that nigh all imaginable matter configurations transpire in some Hubble volume far removed, and also that we should anticipate our own Hubble volume to be reasonably typical — at least typical amongst those harbouring observers. A rough calculation suggests that the nearest identical copy 29 91 of you is approximately ∼ 1010 m distant. At a remove of about ∼ 1010 m, there ought to exist a sphere of radius 100 light-years, identical to the one centered here, such that all perceptions experienced by us during the ensuing century will be identical to those of our 115 counterparts over there. At a distance of approximately ∼ 1010 m, there ought to subsist an entire Hubble volume, indistinguishable from our own.∗∗ This prompts an intriguing philosophical consideration, which shall resurface and bedevil us in Section V B: should there indeed be numerous copies of “you”, with identical past lives and memories, you would be incapable of computing your own future, even were you possessed of complete knowledge of the entire state of the cosmos! The rationale being that there exists no means by which you may ascertain which of these copies constitutes “you” (they all perceive themselves to be). Yet their lives shall typically diverge eventually, such that the utmost you may accomplish is to predict probabilities for what you shall experience from this juncture forward. This annihilates the traditional notion of determinism.

Is a multiverse theory more properly categorised as metaphysics, rather than physics? As accentuated by Karl Popper, the distinction between the two resides in whether the theory is empirically testable and falsifiable. The inclusion of unobservable entities doth not, per se, render a theory untestable. For instance, a theory asserting the existence of 666 parallel universes, all devoid of oxygen, makes the testable prediction that we should observe no oxygen here, and is therefore refuted by observation.

As a more substantive illustration, the Level I multiverse framework is routinely employed to discount theories in modern cosmology, albeit this is rarely articulated explicitly. For example, cosmic microwave background (CMB) observations have recently demonstrated that space exhibits nigh no curvature. Hot and cold spots in CMB maps possess a characteristic size, contingent upon the curvature of space, and the observed spots appear excessively large to accord with the previously favoured “open universe” model. However, the average spot size randomly fluctuates slightly from one Hubble volume to another, rendering statistical rigour of paramount importance. When cosmologists declare that the open universe model is refuted at 99.9% confidence, they signify, in truth, that were the open universe model veracious, fewer than one in every thousand Hubble volumes would exhibit CMB spots as large as those we observe — consequently, the entire model, with all its infinitely many Hubble volumes, is refuted, notwithstanding the fact that we have, of course, only mapped the CMB in our own particular Hubble volume.

The lesson to be gleaned from this example is that multiverse theories are amenable to testing and falsification, but solely if they predict the nature of the ensemble of parallel universes, and specify a probability distribution (or, more broadly, what mathematicians term a measure) over it. As we shall witness in Section V B, this measure problem can be quite acute, and remains unresolved for certain multiverse theories.