possible multiverses
There are many proposed multiverses, multiple of which may be true at once. I consider “multiverse” to mean any model that implies we’re living in a Big World - that is, a reality where all possible structures are realized - where, for each value of each free parameter of that multiverse, that specified structure exists.
The “physical multiverse” class, as I call it, contains all multiverses which are implied by or required for physical theories as we understand them. The first and most obvious - which someone might take a shot at me for even terming a “multiverse” - is a spatially infinite universe with an ergodic distribution of matter. Every possible structure with our laws of physics is realized, because the initial conditions vary across space.
Another prime example is quantum mechanics, the most straightforward interpretation of which implies a universal wavefunction that evolves according to the Schrödinger equation, itself implying (without any extra magic) that every outcome of a quantum measurement is realized - or, to put it more accurately, the measured system becomes entangled with its environment (including its observers), and as such the outcomes decohere into distinct “branches”. Note that while this multiverse is restricted to the same fundamental laws of physics, certain other things might vary like physical constants, particles, etc.
The last physical multiverse I’ll cover is the one implied by eternal inflation, wherein the universe continually and eternally expands, with quantum fluctuations creating asymmetric regions and spawning patches with differing physical properties (but likely the same laws of physics). This gives us the same stuff as our “multiverse” implied by quantum mechanics.
The other class of multiverses, according to me, is the “abstract multiverse” class. Herein I group multiverses that are not implications of actual physical theories, but philosophical proposals. (And if I’m being honest, these are more fun.) To be clear, “abstract” is more relating to the model of the multiverse itself, rather than its contents; after all, each and every multiverse mentioned in this post contains our world, as physical and real as it seems to us.
One that is near and dear to my heart is the computational multiverse. There are many takes on this idea, but they all basically propose a multiverse that is the set of all computable programs. Seeing as these can get weird, most propose to augment it with the notion of the “universal distribution” or “universal prior”, which effectively assigns each program’s probability as inversely exponentially proportional to its length in bits (such that for each extra bit, the probability is halved). Juergen Schmidhuber goes further with the speed prior, which makes not only shorter programs more likely, but faster ones.
A variant of the computational multiverse is Wolfram physics, a project by Stephen Wolfram to model physics as iterated rewriting rules on a hypergraph; he suspects our universe might be such a hypergraph produced by one such rule (perhaps even a simple one). He also conceives of the notion of “rulial space”, the space of all possible hypergraph rules, wherein the one which produced our universe is but a point. This is Wolfram’s multiverse: the set of all possible rules, each of which gives rise to a distinct universe (though, conceivably, rules could “arise” within others - i.e. like how, in our universe, Wolfram is simulating others). It’s deeply intriguing, though seeing as these rules are all computable, I wonder if rulial space constitutes a subset of the aforementioned computational multiverse or, if this computation-by-graph-rewriting is Turing complete, is equivalent to it. Still, it has certain implications that the computational multiverse might not - for instance, it implies the whole universe is doing a vast amount of compute all the time, which we (or our superintelligent progeny) might harness for our own purposes (i.e. surviving heat death).
The last multiverse I’ll cover for now is Max Tegmark’s mathematical universe. This is, in a sense, simpler than all the others, and yet contains far more. Though the computational multiverse (and possibly Wolfram’s) contain all computable programs, what is computable is really a very tiny subset of all math. Tegmark proposes that all mathematical objects exist - they’re all equally as real as the one we inhabit, and the only reason we think they aren’t is that we aren’t inside them. I like this multiverse a lot, if only because it allows all logically possible (read: mathematically consistent) things to exist. (Note that it also contains the computational multiverse, as per the MRDP theorem, which proved the correspondence between Diophantine sets and Turing machines.)
multiversal scaling laws
There’s a certain trend across all these models - can you spot it? There’s an inverse relationship between the amount of information to specify a multiverse’s structure and diversity of stuff that it contains. This might seem counterintuitive at first - if it takes somewhere between 10^90 bits (entropy of all known matter) and 10^123 bits (given by the holographic principle) to describe our universe, and to encode two universes takes double that number, and so on… then surely an infinite set of universes should take an infinite number of bits to describe!
Except, not really. Information is subtractive: it specifies subsets or elements, much like a sculptor chiseling away at a block of marble until he “finds” the right statue inside. The more information that goes into specifying this subset, the smaller that subset is. In the first multiverse - a flat, infinite universe with an ergodic distribution of matter - the laws of physics, constants, dimensions, et cetera are already specified, so all configurations which don’t match them are ruled out. In the next two - quantum and inflationary - the constants might vary (perhaps), but most of the other properties remain fixed. The computable universe, and likely Wolfram’s, rule out anything and everything incomputable. Tegmark’s multiverse of all math is the only one which describes every logical possibility - and because it allows for all possibilities, it takes no information to describe. This is most metaphysically sound, since it means we don’t need to ponder the idea of where did all of this come from at all - because, in this case, we would live on the inside of Nothing.
tldr
I weakly classify multiverses as either “physical” (implied by existing physical theories) or “abstract” (more philosophical in nature, but still giving rise to our physics). Also, the more specific your multiverse, the smaller it is - meaning the ultimate multiverse is one containing every logical possibility, and thus requiring nothing at all to exist.