Is the Universe Precisely Flat and Infinite?

[BurtJordaan]

The short answer to this question is, we do not know. The bad part is that if space is precisely flat, we can never experimentally confirm it, simply because we cannot observe anything with infinite accuracy. We will always have some error bars on all observations.

As it happens, all validated observational evidence on the large scale spatial curvature sits so close to zero that the error bars straddle the zero. Hence, closed, flat and open are all on the cards. Ironically, although we use the flat case for LCDM, because it is the simplest to work with, it is the other two possibilities that we might be able to observationally confirm, e.g. if the error bars would ever fall on one side or the other, excluding zero, we would have shown that the universe is not flat.

If that ever happens to be the case, one can only hope that the curvature falls just-just on the positive (closed) side, because it removes the problems of wrapping ones head around infinite space. And if it is very large, we can still use the LCDM model with exquisite precision due to the negligible curvature.

Please let it be extremely large, but closed! :-)

[dandelion]

:)

[Faradave]

A positive spatial curvature model accommodates cosmic expansion (as per balloon analogy) with a limited past and unlimited future. It also simplifies "dimension" to mean just one kind: radial unidirectional time, with "space" merely a bidirectional derivative (that 3-surface enclosing a 4D temporal field).

It also happens to predict a dimensionless constant of nature, observed as speed limit c, with an inherent value of one (and like the fine structure constant, having no real units). This value is precisely finite, constant, universal and invariant as we observe c to be.

Further, a variable rate of cosmic expansion can be accommodated by the forces of nature without necessarily invoking "dark energy". Gravity for example, can be locally attractive and yet cosmically expanding at the current cosmic age. Strong and EM forces would have accomplished this, to a much greater degree, very early in cosmic evolution.

Flat geometry leaves a speed limit inexplicable, and a negative curvature may provide a limit to how slowly anything can travel.

[charon]

Why on earth (no pun intended) should space be 'flat'? It's a many-dimensional affair. 'Flat' is counter-intuitive. I don't know why these things are complicated.

[BurtJordaan]

Why on earth (no pun intended) should space be 'flat'? It's a many-dimensional affair. 'Flat' is counter-intuitive. I don't know why these things are complicated.

We are talking here about the global curvature of space, which, as far as we can tell is zero, meaning "flat like a tabletop". Unlike a tabletop, where we can measure it from the outside and determine if it is flat or curved, we don't have that luxury with the universe. We have to measure it from within, and it seems not to be curved at all.

The problem is that we found it to be so close to flat that we cannot tell, within experimental uncertainties, if it is not precisely flat, on which side of flat it lies.

It may pay off if you would Google "spatial curvature" to find out more.

[BurtJordaan]

A positive spatial curvature model accommodates cosmic expansion (as per balloon analogy) with a limited past and unlimited future.

Well, the flat and negatively curved Friedmann cosmology also accommodate all of that en a lot more.

It also simplifies "dimension" to mean just one kind: radial unidirectional time, with "space" merely a bidirectional derivative (that 3-surface enclosing a 4D temporal field).

You know the arguments against this and we are not going to debate it again here.

It also happens to predict a dimensionless constant of nature, observed as speed limit c, with an inherent value of one (and like the fine structure constant, having no real units). This value is precisely finite, constant, universal and invariant as we observe c to be.

I disagree that your toy-model "predicts" it, for the simple reason that the one-way speed of light is simply a matter of choice of coordinate system - flat Minkowski spacetime coordinates. It does not hold generally and not for any of the usual cosmological coordinate systems.

Your "prediction" is nothing more than a pictorial representation of what is generally known as "space-propertime coordinates" (a.k.a. Epstein coordinates), first publicized in 1981 by Lewis Carroll Epstein as "Relativity Visualized".

[Faradave]

We have to measure it from within

You know the arguments against this [curved-space, radial-time] and we are not going to debate it again here.

Fair enough. No reply required.
I'm not trying to antagonize but I thought we had agreed that in the coordinates you presented in cosmic heart the worldline of an observer at rest points radially outward toward the future (i.e. toward future spatial "foliations" or "simultaneities").



Of course, such an observer would find that his watch tics normally and might reasonably devise a coordinate system to reflect his experience. That is curved-space, radial-time coordinates devoid of scale factor. Aging is time experienced on the way to the future.

Well, the flat and negatively curved Friedmann cosmology also accommodate [cosmic expansion from a limited past and unbounded future, and] a lot more.

Agreed, but limit c remains inexplicable magic in those geometries.

Your "prediction" [of speed limit c] is nothing more than a pictorial representation of what is generally known as "space-propertime coordinates" (a.k.a. Epstein coordinates), first publicized in 1981 by Lewis Carroll Epstein as "Relativity Visualized".

You are confusing Epstein's space-proper time for my Euclidean interval-time coordinates as they naturally derive from the tangent and radius (respectively) at any location (except the exact center) of a curved-space, radial-time model. Speed limit c corresponds to the tangent direction, enforced by time's fundamental unidirectionality.


A 2D slice from a temporal 4-field, emanating from the Big Bang (BB), reveals interval-time coordinates at any event (p). Time is radially outward, indicating rest (v₀), while increasing speeds reach a maximum (v_max) tangent to curved space. This is maintained in the rest frame of v₁ (right), exhibiting its the invariance of c.

So they aren't at all the same. Epstein, as so many others, ridiculously adheres to space as a dimension. And Epstein admitted, with some embarrassment, that he completely fumbled representation of a photon.

A peculiar feature of the Epstein diagram - the lightcone is replaced by a light-sphere that "looks the same" for all inertial observers, but they observe different portions of it. Lewis Carroll Epstein has cautioned against taking this concept too literally - he actually called it "a myth" in his book, albeit a very useful one.

- 2013

My model explicitly reveals light to be interval contact, entirely consistent with its definition (i.e. "zero interval separation"). Epstein and Minkowski (and everyone else) had no such luck depicting it. This indicates a mismatch between the diagram and the region it represents.

the one-way speed of light is simply a matter of choice of coordinate system - flat Minkowski spacetime coordinates.

In 4D it's better to think of limit c as a ratio than a speed. Structurally (i.e. geometrically) time and space are both separators of events. Forget human units, nature is oblivious to them. In the lightlike limit as space and time become equal seperators, the interval goes to zero. c is a fundamental constant of Nature with an inherent value of exactly 1 (to any number of decimal places). Consistent with such dimensionless constants, the units for a ratio of separation/separation naturally cancel.

[charon]

Burt -

the global curvature of space, which, as far as we can tell is zero, meaning "flat like a tabletop"

Which is a bit like talking about the redness of a strawberry which, as far as we can tell is twelve, meaning 'wet like water'.

Obviously if you measure something from within it's impossible to tell what shape it is in its entirety (if it has one) or very much of any other quality it may have. Like trying to understand what a house is - and beyond that everything else in the universe - from inside a single room. It's pure guesswork. You probably wouldn't know you were in a house. The word 'house' may not even exist for you.

Plato's cave comes to mind.

It may pay off if you would Google "spatial curvature" to find out more.

Here are two apparently learned articles. I wasn't going to trawl through them but I did do a word search on 'flat' and 'zero'.

Try it, see you what you find. It won't take long :-)

http://www.math.ucr.edu/home/baez/einstein/node9.html

https://aapt.scitation.org/doi/10.1119/1.4955154

[BurtJordaan]

Burt wrote:
the global curvature of space, which, as far as we can tell is zero, meaning "flat like a tabletop

Which is a bit like talking about the redness of a strawberry which, as far as we can tell is twelve, meaning 'wet like water'.

We can measure the spectrum of the light from a strawberry in a laboratory, but that's irrelevant. ;)

The flatness of space is much more difficult to measure on large scales, but we can do it from within. Just check if the inside angles of triangles add up to 180 degrees in all orientations - then the spatial curvature is zero. More than 180 and the space is negatively curved; less than 180 and it is positively curved. Obviously it is not as simple as it sounds here in an expanding universe, but it is done.

The two articles that you referred to are purposely chosen to be about closed, static cosmologies, which is easier for people to wrap their heads around. As I said, I also hope that it eventually turns out to be closed and expanding, but based on evidence, cosmologists still consider flat as the most likely scenario.

"If it walks like a duck and quacks like a duck - it probably is a duck"!

The other thing is that the flat scenario makes the math much easier than a closed scenario, but I don't think that the universe cares about our preferences, or our difficulties...

[BurtJordaan]

I'm not trying to antagonize but I thought we had agreed that in the coordinates you presented in cosmic heart the worldline of an observer at rest points radially outward toward the future (i.e. toward future spatial "foliations" or "simultaneities").

Nope, outward indicates the scale factor and its vagaries, and it can actually be pointing inwards if we had a collapsing universe. I have tried to tell you that to be compatible with Friedman-cosmology, you should portray time going perpendicular to your radial, in order to produce the conventional Friedmann spatial slicing over time, e.g.

I think you can still do your "Epstein-like" stuff if portrayed like that.

If you insist on viewing it directly into the large end, you will be forced to have a nonlinear timescale, like my 2-billion-year rings.

[charon]

Burt -

Thanks for your reply. I'm not a troll, I think I'm a fairly sensible person, but things have got to make sense for me. To be honest, this doesn't.

Presumably, if the universe is flat (or indeed any other shape), it exists in space. Everything exists in space. It must exist in something.

The universe (and apparently we only know about 4% of it or something) is tangible otherwise we couldn't measure it at all, so in some respects it is a 'thing'.

What's the relation between the universe and what we call space? Most of the universe is what we call 'space'. After all, that's what we know, lots of things, from atoms to giant stars, existing in space.

But you're saying this universe, which is a Very Big Thing Indeed, is flat. That means a surface. Actually I've no idea what that means either.

I'd say everything was multi-dimensional. In other words it's all around us on many levels. We're in it, of it, surrounded by it. And you're saying beyond that, however big it is, it's just something flat.

Okay, let's say it's flat. How long is it? It's got to have more than one aspect, it can't just be flat. A table-top is flat but it's also part of a table which is not flat.

Sorry to be insistent. If you wouldn't mind explaining all that in simple language that would be great. And I've found that when someone really understands something complex they can always explain it in simple terms. Like explaining to a child what a cloud is, or an elephant. It can be done.

Here's a quote from an article:

'If you're in a square room and walk around the corners, you'll return to your starting point having made four 90-degree turns. You can say that your room is flat. This is Euclidian geometry.'

https://phys.org/news/2017-06-universe- ... ology.html

But a room is not flat, it's a box. I don't understand the language these people use. It's just not normal English!

[Positor]

Charon,

You may wish to read the following article, which is aimed at beginners:

http://curious.astro.cornell.edu/about-us/103-the-universe/cosmology-and-the-big-bang/geometry-of-space-time/606-how-can-the-universe-be-flat-we-re-3d-beginner

[davidm]

Charon, “flat” in the context that Burt Jordaan is using the word, and in the context that cosmologists use it, does not mean “two-dimensional.” It is referring to the topology of 3D space.

If 3D space goes on forever, with no global curvature, it is said to be “flat” — though of course it is not 2D.

3D space could be curved and closed on itself, such that if you traveled far enough, for a long enough time, you could, in principle, return to where you started, just like someone circumnavigating the earth. I say in principle because, in practice, even if the universe is curved and closed like this, it could not be circumnavigated because the expansion of the universe rules out circumnavigation, which would require superluminal velocities to achieve.

Alternatively, space could be negatively curved, like a saddle, but it’s still all 3D space — not “flat” in the sense of being only 2D. But, as in the previous example, it is not “flat” in the topological sense.

The room example you gave is “flat.” Yes, it’s a box, but it obeys Euclidean geometry — hence, flat. Positively and negatively curved 3D spaces do not obey such geometry, though they may do so locally, in the same way that the earth seems flat locally.

[charon]

Ah, so flat doesn't mean flat at all. What other things don't mean what they say? Does universe really mean the totality of existence (the dictionary definition) or something else?

Why are they borrowing words that mean something quite different to everybody else? Why don't they invent their own words, then it might clearer all round!

What about my other points? Why do we think we can measure something from 'within' if we've never been outside it? It's like trying to guess what the world is like beyond my little village in the jungle, which I've never left. Without information I'd say that was impossible.

I think I know what you'll say, that it can be mathematically inferred. Possibly, but we still don't really know, do we?

And what about space? If universe means the totality of existence then it must include space. But the universe exists IN space. The universe must occupy space like any other thing. Which means it's not the same as space. So universe, as you're using it, isn't the totality of existence.

[davidm]

Charon, I refer you to the link that Positor gave you. From that article:

…when we say the universe is flat it is not in the same sense that a piece of paper is flat, but rather means that the geometry of the universe is such that parallel lines will never cross, the angles in a triangle will always add up to 180 degrees, and the corners of cubes will always make right angles. We call this kind of geometry (the kind you learned in school) Euclidean geometry.

And yes, we can measure the geometry of the universe from within it, at least in principle. In principle it is possible to measure whether the universe is globally positively or negatively curved. If the former, the angles in a triangle will add up to more than 180 degrees, if the latter less than 180 degrees. In either case, Euclidean geometry will not describe our universe.

See here for more.

[charon]

at least in principle

Didn't the quantum business upset the Newtonian ideas? Why isn't Euclidean geometry affected too? Or is it?

[BurtJordaan]

Didn't the quantum business upset the Newtonian ideas? Why isn't Euclidean geometry affected too? Or is it?

Not quite, because Newton's and Einstein's theories are about he larger scales where gravity rules, while quantum physics is about the very-very small scale where nuclear forces rule. Unfortunately we have not found a fully workable theory of quantum gravity yet.

Listen to this lecture by Dr. Don Lincoln, one of our specialist members, sadly too busy to be active here any more.

He talks about the reconciliation of quantum theory and special relativity to get quantum-electrodynamics (QED)

This one is about the attempts to marry quantum theory and general relativity to get loop-quantum-gravity (LQG), just one of a number of possible paths towards the 'marriage'.

To get a full list of talks by Don on the subject, google 'Don Lincoln quantum theory'

[charon]

Again, thanks for the reply. But the 'See here for more' link from Davidm to 'Geometry Of The Universe' comes up with three possible versions, positive, negative, and flat. It also says:


'To date all these methods have been inconclusive because the brightest, size and number of galaxies changes with time in a ways that we have not figured out. So far, the measurements are consistent with a flat Universe, which is popular for aesthetic reasons.'

Since when were aesthetic reasons considered good science?!

[davidm]

Actually, aesthetic reasons are often considered good science, though that is a separate discussion.

In any case, WMAP surveys show that the universe is flat with only a 0.4% margin of error, so we almost certainly live in a spatially infinite universe. It cannot be proven beyond any doubt whatsoever, though, but then again, science never proves anything beyond any doubt whatsoever.

[charon]

Oh, David -

we almost certainly live in a spatially infinite universe.

But I could have told you that.

You see, this is far from the first discussion I've had with scientists about these issues. In fact, it goes back years. And been told that the universe is limited. Or limited yet unbounded. Or something else.

If one asks, if it's limited, what is beyond the limitation - seeing as there's no such thing as literally nothing - they waffle. But now you've said unlimited. Excellent.

Unless, as I asked before, you're differentiating between the universe and space. If the universe exists IN space then it may be the space which is unlimited but the universe not. In that case, of course, what is called the universe can be measured, it can be expanding, there may be more than one, it could have a definite beginning - all the things that could be applied to a discrete object in space.

But, if by universe we mean the totality of existence (dictionary definition) then it must surely mean unlimited. In which case...!

[BurtJordaan]

Since when were aesthetic reasons considered good science?!

Flat also happens to be the simplest of the 3 possible options, mathematically speaking. So if the uncertainty is spread equally to each side of flat, why won't we choose flat?

'Occam's razor' and all that, you know...

[davidm]

If one asks, if it's limited, what is beyond the limitation - seeing as there's no such thing as literally nothing - they waffle.

But do they waffle? Or perhaps you misunderstood what they were saying?

Yes, one can waffle whether the universe is flat, or positively or negatively curved. But as noted in my last post, the empirical evidence overwhelming favors a flat (spatially infinite) universe.

If you say they are waffling about what is “beyond” the universe, I seriously doubt that. You may have, again, misunderstood them.

The universe is, all that there is, so nothing is beyond it. This fact can, however, become a terminological dispute, which is of no great moment. Some people speak, theoretically, of a “multiverse” — more than one universe. And there are several ways in which a multiverse could be true.

However, if one defines “universe” as all that exists, then there is only one universe by definition. In that case the supposed “multiverse” represents subsets of the “universe” — all that there is.

A finite but unbounded universe, with positive curvature, is one that can, in principle (though not in practice) be circumnavigated. “Unbounded” means that there is no edge — no frontier beyond which space somehow “stops.”

However, if the universe is like this (and, as noted, the evidence shows that it isn’t) it does not follow that there is something beyond the universe, or that it is expanding “into” anything.

OTOH, brane theory holds that there is a higher-dimensional 4-space and that our universe is a 3-space brane expanding into that 4-space, and that there are other branes that may collide. This could be true, but there is no evidence (that I know of) that it is true, and more important, there is no need for such a 4-space, to explain the expansion of a finite but unbounded 3-space. But, again, although a finite but unbounded 3-space is possible, the evidence shows that the universe is flat — spatially infinite.

[A_Seagull]

Physics is all about measurement and relating those measurements in a mathematical framework.

It is not possible to measure any quantity as being infinite.

Therefore it is not meaningful to refer to any aspect of the physical universe as being infinite.

[BurtJordaan]

Physics is all about measurement and relating those measurements in a mathematical framework.

This exactly what is happening. The mathematical framework that fits the measurements best says that the universe is very likely to be spatially infinite.

[charon]

Since when were aesthetic reasons considered good science?!

Flat also happens to be the simplest of the 3 possible options, mathematically speaking. So if the uncertainty is spread equally to each side of flat, why won't we choose flat?

'Occam's razor' and all that, you know...

Yes, I know, but choice implies confusion... There's no choice when the issue is clear.

[charon]

David -

But do they waffle? Or perhaps you misunderstood what they were saying?

Oh, yes, they waffled! I was there, I'm not making it up. If we could access those discussions now you'd see it.

Yes, one can waffle whether the universe is flat, or positively or negatively curved. But as noted in my last post, the empirical evidence overwhelming favors a flat (spatially infinite) universe.

They didn't waffle about that, it wasn't brought up.

If you say they are waffling about what is “beyond” the universe, I seriously doubt that. You may have, again, misunderstood them.

I don't think so. It was in quite clear English. They didn't admit anything beyond it, as I recall. Yet they claimed it was measurable and expanding. I said not possible (see below).

The universe is, all that there is, so nothing is beyond it. This fact can, however, become a terminological dispute, which is of no great moment.

That's what I'm saying, but I'd disagree it's merely a terminological dispute. We're talking about the reality we live in. It's not some childish dispute over words (I hope).

Some people speak, theoretically, of a “multiverse” — more than one universe. And there are several ways in which a multiverse could be true.

However, if one defines “universe” as all that exists, then there is only one universe by definition. In that case the supposed “multiverse” represents subsets of the “universe” — all that there is.

You're saying 'the universe' may mean several - or at least more than one - universes. Forgive me, I'm not creating this confusion! I mean, you must see the point and not just brush it off. From now on, whenever the term 'universe' is used I'm going to ask which one. Would that be fair?

A finite but unbounded universe, with positive curvature, is one that can, in principle (though not in practice) be circumnavigated. “Unbounded” means that there is no edge — no frontier beyond which space somehow “stops.”

I know, but that's all nonsense if 'universe' means an infinite, limitless totality. It can't be finite and infinite both. So which one are you talking about?

However, if the universe is like this (and, as noted, the evidence shows that it isn’t) it does not follow that there is something beyond the universe, or that it is expanding “into” anything.

It can only expand if it has a certain size or quality to begin with. Something can only get bigger when it's smaller first. But if it's infinite and limitless then the idea of expansion doesn't arise because it doesn't apply.

OTOH, brane theory holds that there is a higher-dimensional 4-space and that our universe is a 3-space brane expanding into that 4-space, and that there are other branes that may collide. This could be true, but there is no evidence (that I know of) that it is true, and more important, there is no need for such a 4-space, to explain the expansion of a finite but unbounded 3-space. But, again, although a finite but unbounded 3-space is possible, the evidence shows that the universe is flat — spatially infinite.

I see no reason why something limitless could not also be multi-dimensional, whether 3D, 4D or a billionD. Why not? In fact, as it's limitless, it's more likely to be a billionD than 3 or 4 :-)

[A_Seagull]

Physics is all about measurement and relating those measurements in a mathematical framework.

This exactly what is happening. The mathematical framework that fits the measurements best says that the universe is very likely to be spatially infinite.

It may well be that the universe is locally flat.

Mathematical models can be extrapolated beyond the data set for which they are formed, but this does not mean that the extrapolation is valid.

[BurtJordaan]

Mathematical models can be extrapolated beyond the data set for which they are formed, but this does not mean that the extrapolation is valid.

In the case of cosmology, the model was developed without any data on expansion and then the expansion was observed by Edwin Hubble. The spatial curvature term was in the model, but the values of the free parameters were poorly known (e.g. the rate of expansion and energy density, read together meaning the curvature density).

Very soon after that, the observed value for the curvature constant was settling around zero, but with large error bands, indicating that the total universe could be either closed, flat or open. As @davidm also indicated and gave references for, the present error bands are now so small, that the probability that the universe is infinite is huge. There is still a chance that it is closed and finite at any specific time, but so huge and expanding so rapidly that we can never observe its circumference, with whatever technology to come.

The lucky thing is, it makes no difference to our cosmological equations, because the curvature terms vanish in the noise...

[davidm]

Oh, yes, they waffled! I was there, I'm not making it up. If we could access those discussions now you'd see it.

I can’t comment further on this, without seeing the discussions. Are they no longer online? You can’t link to them?

You're saying 'the universe' may mean several - or at least more than one - universes. Forgive me, I'm not creating this confusion! I mean, you must see the point and not just brush it off. From now on, whenever the term 'universe' is used I'm going to ask which one. Would that be fair?

It is indeed a terminological dispute, of no great consequence except, perhaps, to linguists.

There is an interpretation of quantum theory, a meta-theory, which holds that there is no state vector reduction, no so-called wavefunction collapse. It is popularly known as the Many Worlds interpretation of QM. Apply this to Schrödinger’s famous cat, and you obtain a version of reality in which an observer is quantum-entangled with a live cat, and his doppelgänger observer is quantum-entangled with a dead cat. Both are real. Are these now two different universes, or the same universe with two different versions of the cat and the observer, one cat dead, the other alive? Does it really matter? Again, if you wish to define “universe” as all that there is, anywhere, at any time, ever, then the Many Worlds are, indeed, different “worlds” within the same universe (all that there is, or ever could be). Alternatively, one may argue that because the live-cat, dead-cat “worlds” are now spatiotemporally disconnected, and cannot again interact in any way (as the Many Worlds meta-theory maintains), they are effectively two different “universes.” I say again, this is purely a terminological difference. It makes no difference to the science whether you say that the two different cats, one dead and the other alive, are in two different “universes,” or that they now occupy two different “worlds” within the same universe. Who cares?

Consider, for example, that there are many different locations in the universe, and many different moments. Are each of these different spatial and temporal locations just subsets of a singe universe? Or would you wish to define them each as an individual universe? You can do the latter if you wish, but it just makes no difference to empirical facts. It’s just a matter of how people prefer to employ language.

We know that locations in space and time are indexicals — basically, point-of-view dependent. Wherever I am, I call it “here.” Whenever I am, I call it “now.” There is a philosophical idea (not scientific!) originated by the late analytic philosopher David K. Lewis, which holds that actuality, like locations in time in space, is indexical. If true, this would mean that all counterfactuals are, in fact, actual — but only to their own observers. There is a “world,” according to Lewis, actual to its own inhabitants but not actual to us, in which pigs actually fly. There is another in which donkeys actually talk. There is another in which the ancient Greek gods are literally real. Basically, on this doctrine, everything that is logically possible, is actual. (There is no Lewisian world with four-sided triangles, for example, since that is not logically possible and hence cannot be actual.)

Now whether you think this is all hooey or not is beside the point. (Lewis said that whenever it sank in to people what he was saying, they invariably gave him “an incredulous stare.”)

The point is, Lewis specifies that these actualities are spatiotemporally isolated from one another, and cannot communicate even in principle — no different, really, from the QM Many Worlds idea. Are these, then, different universes, or different versions of one universe (all that there is, anywhere, anytime, or ever could be)? Why does it even matter? It’s purely a terminological dispute, a matter of how one wishes to define certain words.

The upshot is, there cannot both be one universe, and many universes, at the same time. If one wishes to define universe as all that there is, ever was, or ever could be, then it encompasses different regions — different regions in space and time, different actualities if Lewis is right, different quantum branches if Many Worlds is right, and etc. However, if one wishes to define these different regions as independent universes, then there is more than one universe.

I know, but that's all nonsense if 'universe' means an infinite, limitless totality. It can't be finite and infinite both. So which one are you talking about?

Right, it can’t be both infinite and finite — I never said that it could! If the universe is flat it is infinite, if it’s curved it’s finite (but unbounded). On or the other, not both!

It can only expand if it has a certain size or quality to begin with. Something can only get bigger when it's smaller first. But if it's infinite and limitless then the idea of expansion doesn't arise because it doesn't apply.

Yes, it does apply. Expansion means that cosmic distances — basically, distances between galaxies — increase over time. This can happen perfectly well whether the universe is spatially infinite or spatially finite but unbounded (curved). In the deep past, all objects in a spatially infinite universe could have been infinitely close together. The universe would still be spatially infinite.

[Faradave]

I thought we had agreed that in the coordinates you presented in cosmic heart the worldline of an observer at rest points radially outward toward the future (i.e. toward future spatial "foliations" or "simultaneities").

Nope, outward indicates the scale factor and its vagaries, you should portray time going perpendicular to your radial...

As I view the diagram, Time is the big red arrow emanating from the Big Bang (left). That may be an oversimplification but it is how any observer at rest with respect to background radiation (including the cosmos, which is always at rest with respect to itself) would experience time, regardless of scaling. Looking into the trumpet (from right), the opening is a 3 surface corresponding to one of you cosmic heart's outer rings. They put the scale function a(t) as the curve of the trumpets long cross section (i.e. varying perpendicular to time's arrow).

The trumpet represents the observable universe, I fill in the rest of it as a trumpet bouquet to generate a curved-space, radial-time model (with space given as a 3-ring analogous to the rings in the cosmic heart diagram).

A spatial 3-ring is equivalent to a 3-sphere enclosing a temporal 4-field centered on the Big Bang (BB). Thus, curved-space, radial-time is modeled analogous to the cosmic heart depiction.

If you insist on viewing it directly into the large end, you will be forced to have a nonlinear timescale, like my 2-billion-year rings.

I don't have a problem with that, but again we still experience time linearly. You might complain that adjacent trumpets in the bouquet don't fit perfectly together because of their curved profiles. That can be fixed with a spatial scale factor. Allowing time to be linear, that varies the hash marks along each circumferential ring. Concentric rings can thus be evenly spaced. What's good for the goose (time) is good for the gander (space).

I think you can still do your "Epstein-like" stuff if portrayed like that.

Yes. I believe that in either the cosmic heart or my curved-space, radial-time speed limit c (including outgoing light) would be tangent to any spatial curve. Along with radial time, that yields interval-time coordinates.