Level II Multiverse: Chaotic Inflation and Post-Inflation Bubbles

Should one find the Level I multiverse capacious and challenging to comprehend, one might attempt to envisage an infinite assemblage of disparate entities, some, perchance, exhibiting dimensionalities and physical constants distinct from our own. This is consonant with the pronouncements of the chaotick theory of inflation, currently enjoying considerable vogue, which we shall designate the Level II multiverse. These disparate domains reside at a distance exceeding infinity, insofar as traversal thereto, even at the velocity of light, would prove infeasible in perpetuity. The rationale lies in the incessant inflation of the intervening space betwixt our Level I multiverse and its adjacent counterparts, engendering an ever-expanding volume at a rate surpassing any potential rate of traverse. Conversely, access to a Level I universe, howsoever distant, remains tenable given sufficient patience and a deceleration of cosmic expansion. (Astronomical evidence intimates that cosmic expansion is presently accelerating. Should this acceleration persist, even the Level I parallel universes shall endure in perpetual separation, the intervening expanse expanding at a velocity exceeding the velocity of light. The matter remains sub judice, however, with prevailing models prognosticating an eventual cessation of cosmic acceleration, possibly culminating in recollapse.)

By the 1970s, the Big Bang model had evidenced itself a highly efficacious explication for the greater part of our universe's history. It had elucidated the manner in which a primordial conflagration expanded and cooled, synthesising Helium and other light elements during the initial moments, attaining transparency after 400,000 years, thereby releasing the cosmic microwave background radiation, and gradually accreting clumpiness through gravitational clustering, thus giving rise to galaxies, stars, and planets. Nevertheless, vexing questions persisted regarding the genesis of all things. Did existence arise ex nihilo? Wherefore the absence of superheavy particles, designated magnetic monopoles, whose creation in the nascent universe is predicted by particle physics (the “monopole problem”)? Wherefore the vastness, antiquity, and flatness of space, when generic initial conditions presage the growth of curvature over time and a convergence of density towards either zero or infinity within a timeframe of approximately 10−42 seconds (the “flatness problem”)? What contrivance engendered the 10−5 level seed fluctuations from which all structure burgeoned (the “horizon problem”)? What agency precipitated the 10−5 level seed fluctuations from which all structure arose?

A process known as inflation possesses the capacity to resolve all these issues with a single stroke (vide reviews by Guth & Steinhardt 1984 and Linde 1994), and has, consequently, emerged as the prevailing theory concerning the primordial epochs of the universe. Inflation constitutes a rapid distension of space, thereby diluting monopoles and other detritus, rendering space flat and uniform akin to the surface of an expanding balloon, and stretching quantum vacuum fluctuations into macroscopically vast density fluctuations capable of seeding galaxy formation. Since its inception, inflation has withstood additional trials: observations of the CMB have corroborated the extreme flatness of space and have measured the seed fluctuations, evincing an approximately scale-invariant spectrum devoid of a substantial gravity wave component, all in impeccable concordance with inflationary predictions.

Inflation is a ubiquitous phenomenon, manifest in a broad spectrum of theories pertaining to elementary particles. In the popular model known as chaotick inflation, inflation concludes in certain regions of space, thereby permitting life as we apprehend it, whereas quantum fluctuations induce other regions to inflate at an even more accelerated rate. Essentially, an inflating bubble engenders other inflationary bubbles, which, in turn, beget others in a ceaseless chain reaction. The bubbles wherein inflation has ceased constitute the elements of the Level II multiverse. Each such bubble possesses infinite dimensions (Surprisingly, it has been demonstrated that inflation can engender an infinite Level I multiverse even within a bubble of finite spatial volume, owing to an effect whereby the spatial directions of spacetime curve towards the (infinite) temporal direction (Bucher & Spergel 1999).), yet there exist infinitely many bubbles, given the perpetual nature of the chain reaction. Indeed, should this exponential proliferation of bubbles have persisted ad infinitum, there shall exist an uncountable infinity of such parallel universes (equalling the infinity ascribed to the set of real numbers, which surpasses that of the [countably infinite] set of integers). In this scenario, there exists neither a beginning of time nor an absolute Big Bang: there is, was, and ever shall be an infinite number of inflating bubbles and post-inflationary regions akin to our own, forming a fractal pattern.

The prevailing perspective posits that the physics we observe to-day is merely a low-energy approximation of a far more symmetrical theory that manifests itself at exceedingly high temperatures. This underlying fundamental theory may exist in 11 dimensions, exhibit supersymmetry, and encompass a grand unification of the four fundamental forces of nature. A common attribute of such theories lies in the presence of several distinct minima (occasionally denoted “vacuum states”) in the potential energy of the field(s) driving inflation, corresponding to disparate modes of symmetry breaking and, consequently, to divergent low-energy physics. For instance, all save for three spatial dimensions could be curled up (“compactified”), resulting in an effectively three-dimensional space akin to our own, or fewer could undergo curling, leaving a 7-dimensional space. The quantum fluctuations driving chaotick inflation could precipitate disparate symmetry breaking in distinct bubbles, thereby resulting in the Level II multiverse encompassing members of differing dimensionality. Many symmetries observed in particle physics also stem from the specific mode of symmetry breaking; consequently, there could exist Level II parallel universes wherein there are, say, two rather than three generations of quarks.

In addition to such discrete properties as dimensionality and fundamental particles, our universe is distinguished by a set of dimensionless numbers designated physical constants. Examples encompass the electron/proton mass ratio mp /me ≈ 1836 and the cosmological constant, which appears to approximate 10−123 in so-called Planck units. There exist models wherein even such continuous parameters can vary from one post-inflationary bubble to another. (Although the fundamental equations of physics remain invariant throughout the Level II multiverse, the approximate effective equations governing the low-energy world that we observe shall diverge. For instance, transitioning from a three-dimensional to a four-dimensional (non-compactified) space alters the observed gravitational force equation from an inverse square law to an inverse cube law. Similarly, disparate breaking of the underlying symmetries of particle physics shall modify the arrangement of elementary particles and the effective equations that delineate them. However, we shall reserve the terms “different equations” and “different laws of physics” for the Level IV multiverse, wherein it is the fundamental, rather than effective, equations that undergo change.)

The Level II multiverse is, therefore, likely to exhibit greater diversity than the Level I multiverse, encompassing domains wherein not only the initial conditions differ, but also the dimensionality, elementary particles, and physical constants may vary.

Prior to proceeding, let us briefly comment upon several closely related multiverse notions. Firstly, if one Level II multiverse can subsist, eternally self-replicating in a fractal pattern, then there may well exist infinitely many other Level II multiverses that are entirely disconnected. However, this variant appears untestable, as it would neither introduce any qualitatively distinct worlds nor alter the probability distribution for their properties. All conceivable initial conditions and symmetry breakings are already realised within each one.

An idea propounded by Tolman and Wheeler, and recently elaborated upon by Steinhardt & Turok (2002), posits that the (Level I) multiverse is cyclical, undergoing an infinite succession of Big Bangs. Should it exist, the ensemble of such incarnations would also constitute a multiverse, arguably exhibiting a diversity akin to that of Level II.

An idea advanced by Smolin (1997) entails an ensemble exhibiting a diversity similar to that of Level II, yet mutating and sprouting new universes through black holes rather than during inflation. This prognosticates a form of natural selection favouring universes with maximal black hole production.

In braneworld scenarios, another 3-dimensional world could exist quite literally parallel to our own, merely offset in a higher dimension. However, it remains uncertain whether such a world (“brane”) merits designation as a parallel universe separate from our own, given our potential for gravitational interaction therewith, akin to our interaction with dark matter.

Physicists evince a disinclination towards unexplained coincidences, interpreting them as evidence that models are untenable. In Section I C, we observed how the open universe model was refuted at a 99.9% confidence level, owing to its implication that the observed pattern of CMB fluctuations is exceedingly improbable, a one-in-a-thousand coincidence occurring in a mere 0.1% of all Hubble volumes.

Suppose one checks into a hotel, is assigned room 1967, and notes, with surprise, that this coincides with one's year of birth. Upon reflection, one concludes that this is not particularly surprising, given the multiplicity of rooms within the hotel and the improbability of such thoughts arising had one been assigned a different room. One then realises that, even in the absence of prior knowledge of hotels, one could have inferred the existence of other hotel rooms, since the singularity of a room number in the entire universe would constitute an inexplicable coincidence.

As a more pertinent example, consider M , the mass of the Sun. M influences the luminosity of the Sun, and utilizing fundamental physics, one can compute that life, as we comprehend it on Earth, remains viable solely if M resides within the narrow range of 1.6 × 1030 kg − 2.4 × 1030 kg — otherwise, Earth’s climate would be colder than Mars or hotter than Venus. The measured value is M ∼ 2.0 × 1030 kg. This apparent congruence of habitable and observed M -values may provoke unease, given calculations indicating the potential existence of stars within the broader mass range of M ∼ 1029 kg − 1032 kg. However, as in the hotel example, we can elucidate this apparent coincidence through recourse to an ensemble and a selection effect: should there exist numerous solar systems with a range of central star sizes and planetary orbits, we should, ipso facto, expect to inhabit one of the habitable ones.

More generally, the apparent congruence of habitable and observed values of some physical parameter can be construed as evidence for the existence of a more expansive ensemble, of which our observation constitutes merely one member among many (Carter 1973). Whilst the existence of other hotel rooms and solar systems remains uncontroversial and observationally validated, that of parallel universes does not, given their unobservability. Nevertheless, should fine-tuning be observed, one can advocate for their existence employing the same logic as above. Indeed, there exist numerous instances of fine-tuning suggesting parallel universes with divergent physical constants, albeit the degree of fine-tuning remains under active debate and requires clarification through additional calculations — vide Rees (2002) and Davies (1982) for popular accounts and Barrow & Tipler (1986) for technical elaborations.

For instance, should the electromagnetic force be weakened by a mere 4%, the Sun would forthwith explode (the diproton would exhibit a bound state, thereby augmenting the solar luminosity by a factor of 1018 ). Should it be strengthened, there would subsist fewer stable atoms. Indeed, most, if not all, parameters influencing low-energy physics appear fine-tuned at some level, in the sense that modest alterations thereto yield a qualitatively distinct universe.

Should the weak interaction be substantially weaker, hydrogen would be absent, having been converted to helium shortly after the Big Bang. Should it be either much stronger or much weaker, the neutrinos from a supernova explosion would fail to dispel the outer components of the star, and it remains dubious whether life-sustaining heavy elements would ever escape the stars wherein they were produced. Should protons be 0.2% heavier, they would decay into neutrons incapable of retaining electrons, thereby precluding the existence of stable atoms. Should the proton-to-electron mass ratio be significantly smaller, stable stars would be unattainable, and should it be substantially larger, ordered structures such as crystals and DNA molecules would be precluded.

Discussions pertaining to fine-tuning frequently become heated upon the mention of the “A-word”, anthropic. The author apprehends that deliberations concerning the so-called anthropic principle have generated more heat than illumination, with a multiplicity of divergent definitions and interpretations of its import. The author remains unaware of any dissent from what might be termed MAP, the minimalistic anthropic principle: • MAP: When scrutinizing fundamental theories with observational data, neglecting selection effects can yield erroneous conclusions.

This is manifest from the examples above: were we to disregard selection effects, we should be astonished to orbit a star as massive as the Sun, given the greater abundance of lighter and dimmer stars. Likewise, MAP asserts that the chaotick inflation model is not refuted by the fact that we inhabit the minuscule fraction of space wherein inflation has ceased, given the uninhabitable nature of the inflating portion. Fortunately, selection effects cannot salvage all models, as elucidated a century ago by Boltzmann. Were the universe in classical thermal equilibrium (heat death), thermal fluctuations could still induce atoms to assemble at random, briefly creating a self-aware observer such as yourself, once in a blue moon; consequently, the fact of your present existence does not invalidate the heat death cosmological model. However, statistically, one should anticipate encountering the remainder of the world in a high-entropy mess, rather than in the ordered low-entropy state observed, thereby refuting this model.

The standard model of particle physics encompasses 28 free parameters, and cosmology may introduce additional independent parameters. Should we, in verity, inhabit a Level II multiverse, then, for those parameters exhibiting variability across the parallel universes, we shall remain incapable of predicting our measured values ab initio. We can merely compute probability distributions for what we should anticipate encountering, accounting for selection effects. We should anticipate encountering everything that can vary across the ensemble as being as generic as is compatible with our existence. As detailed in Section V, this question of what constitutes “generic” and, more specifically, how to compute probabilities in physics, is emerging as an embarrassingly thorny problem.