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After reading numerous articles about Observable universe, Universe and Hubble volume, I wondered: is there a formal, rigorous definition in physics of what we normally call a Universe or a cosmos, namely a particular spacetime continuum within (behold a speculation) the postulated Multiverse?

I seek a sound definition in terms of mathematics (topology) or at least an accurate physical definition. All the definitions I've found so far are highly abstract, speculative or philosophical and metaphysical up to a certain extent.

I did find one but unfortunately it is not complete and is not fully formalised:

Universe - is a region of spacetime, larger than the Hubble volume

Can someone provide other formal definitions assuming they do actually exist?

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    $\begingroup$ What do you mean by "multiverse"? $\endgroup$
    – hft
    Mar 1 '15 at 3:19
  • $\begingroup$ @hft I'm late, but by multiverse I mean hypothetical set of infinite or finite possible universes (including our). Look here: en.wikipedia.org/wiki/Multiverse. $\endgroup$ Oct 27 '15 at 9:22
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If the question is asking whether there is a definition that encapsulates our universe, then I believe the answer is No. This is because encapsulating a "space" into a formal system requires defining bounds. However, we don't know the bounds of our own universe--let alone what bounds might be possible for any universe.

We can only describe what we can detect in the observable universe. From this we can immediately introduce 2 problems to creating a definition of the universe.

  1. We have a limited range of the universe that we have the ability to analyze.
  2. We can only analyze things that we have the ability to detect.

As a bizarre example: If the universe is like a balloon and the walls of the balloon were made of a different substance than everything inside of the balloon, would we be able to detect the walls if we don't know what to look for? You might think we'd see an "edge" or a "boundary", but not necessarily, if the universe is say, a torus.

Then we can ask questions about the properties of a physical boundary. Is it permeable? What is on the other side? Does there need to be an "other side"? And these questions don't even require defining whether or not a multiverse is possible.

In mathematics, topology is often brought up when formally describing spaces:

Topological spaces are of analytic nature. Open sets, given in a topological space by definition, lead to such notions as continuous functions, paths, maps; convergent sequences, limits; interior, boundary, exterior.

The only one of those last 3 that we can measure in our universe is the interior. Ultimately it may be the only relevant one when related to a universe. But then again, we don't have further proof to say "there is no boundary to our universe", lending more credence to no formal definition existing.

Others might argue that we can define "enough" of the universe by putting values to the physical constants we observe. These constants combine to create the shapes and interactions of the observable pieces of our universe.

There are also alternative theories about how the universe might have been created, which would radically change our definition of the universe. From M-theory, for example

Heterotic M-theory has been used to construct models of brane cosmology in which the observable universe is thought to exist on a brane in a higher dimensional ambient space. It has also spawned alternative theories of the early universe that do not rely on the theory of cosmic inflation.

And you could throw yet another wrench into the mix by adding time. What was the universe at time=0? Could it collapse (even if it is found to be speeding up right now)?

Or get really whacky and ask whether or not our universe could split like a biological cell (link) (link). What would the formal definition of our universe look like then? I don't want to go into such ideas, but they do show why adding the "time" of spacetime can drastically change your definition, since we can't observe it from beginning to end (that we know of).

But as an optimist, maybe my answer should be "not yet", or "you can get close for practical purposes". Any field might create a "good enough" version of a definition for solving problems in that field. If creating a formal definition requires omniscience, then I'd move my answer back to a firm "No".

Just for completeness, I feel the need to add a few pieces from Dictionary.com's definitions:

  1. the totality of known or supposed objects and phenomena throughout space; the cosmos; macrocosm.
  2. Also called universe of discourse. Logic. the aggregate of all the objects, attributes, and relations assumed or implied in a given discussion.
  3. Also called universal set. Mathematics. the set of all elements under discussion for a given problem.
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Sometimes the word universe is just used colloquially and can just refer to everything on some side of a horizon (an event horizon, a causality horizon, etc.) But when used precisely, I'm sure different definitions are used in different fields.

For instance, in mathematical general relativity, you assume that your universe is a connected four dimensional spacetime. If it were disconnected, then there would be two parts neither of which can affect each other, or even mutually affect something or be mutually affected by something, even time travel or travel at infinite speed couldn't make one part affect the other. So it makes sense to say they aren't in the same universe. This can still allow rather strange things, since it is only the connection as 4d things that matters. So they are still connected as 4d objects if the spatial connection between two parts isn't going to happen yet for a billion years (you could have mergers or splits but if they ever were or ever will be connected spatially then as 4d things they are connected, so they are parts of the same universe). It's also important to mention that in standard general relativity there is no larger space in which curved 4d spacetimes live as hyperdimensional surfaces. So in mathematical general relativity, the multiverse is just disconnected spacetimes, that have nothing to do with each other, they aren't near each other or touching each other or anything they are just completely distinct as if they were the only universe.

But there are nonstandard theories (or maybe proto-theories, this latter part of my answer is not my field) where our universe is a brane in some larger than 4D thing. In which case a universe could be a connected 4D surface in that larger thing, or maybe all the parts that mutually interact. People make definitions to facilitate the conversations they want to have. In regular GR, there isn't a larger thing in which curved spacetimes are embedded as hyperdimensional surfaces, so the definitions in regular GR aren't made to talk about them since they aren't considered. But a theory that considers them could call them branes or something that is meaningfully describes (within that theory) exactly what properties it have or doesn't have. But they might also call them universes.

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In physics, the universe may be defined as the smallest system containing the earth and not interacting with something outside it. This is in agreement with the common usage of the term in astronomy and cosmology. It has a precise meaning in any concrete mathematically formulated theory of physics. In each such theory, there is only a single such system.

General relativity is the study of the classical aspects of this universe (and variants of it that may be easier or more interesting to study), while quantum gravity is the study of its quantum aspects.

In a simplified version of the universe where gravitation is eliminated, the standard model covers all aspects of the universe.

(In less conventional areas of physics, where one speculates about many worlds, a universe is something different, not really well-defined.)

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