B Big Bang: Size of the Universe at Different Epochs

[KingGambit]

TL;DR Summary

The size of the universe after
Inflationary epoch
Quark epoch
Baryogenesis
Photon epoch

Dear PF Forum,

I've been searching what was the size of the early univese, but I think I might have hit the wall.
I read that in the end of inflationary epoch, 10^-32 seconds, the size of the universe was as big as a grape fruit?
And after that, it seems that I can't find a reference to the size of the universe.
For example in Baryogenesis epoch, 10^-11 seconds, if all atoms were created at that point, then what was the size of the universe then?
The mass of the Observable Universe is about... 10²³ solar mass. And we know that an average neutron star (1.5 solar mass) size is 20 km in diameter. I know this is wrong if I am to extrapolate that number, the size of the universe at around Baryogenesis would be... 10⁷ solar diameter or about 1 light year? I know this is completely wrong.
I am wondering if someone can help me with the size of the universe at
- Quark epoch (10^-5 seconds)
- Photon epoch (380,000 years)

Thank you very much.

I have read this:
https://en.wikipedia.org/wiki/Chronology_of_the_universe
But I found no information about the size of the universe.

 

[anorlunda]

I read that in the end of inflationary epoch, 10-32 seconds, the size of the universe was as big as a grape fruit?

You made a very common mistake missing one key word.

"in the end of inflationary epoch, 10-32 seconds, the size of the visible universe was as big as a grape fruit"

The visible universe is the portion that we can see. The entire universe is believed to be infinite in size today. Therefore it must have been infinite in size before the big bang.

This PF Insights article may help you.
https://www.physicsforums.com/insights/brief-expansion-universe/

 

[kimbyd]

Nit: we don't and can't know if the universe is infinite. But it's generally expected to be at least much larger than the observable portion that was the size of a grapefruit in that early epoch.

 

[pinball1970]

Nit: we don't and can't know if the universe is infinite. But it's generally expected to be at least much larger than the observable portion that was the size of a grapefruit in that early epoch.

I have read a few threads on this and when one comes round I always try and think and see if I am understanding what is being said.

It is thought the whole universe is infinite in size and always has been, it is not thought to be temporally infinite yes?

The observable universe began very small hot and dense and expanded.

So a question.

Was the region “outside” our part, the infinite part, also thought to be in the same/similar condition?

So if I could travel beyond the observable universe I would find more of the same? Galaxies with stars?

Or is this not really discussed that much since it can never be detected? Not science?

Also if there are threads/insights I should read on pf? Specifically on this

There are lots of posts like this https://www.physicsforums.com/threa...rse-and-infinity-according-to-physics.639254/

I would like a recent, relevant one if poss. Thanks

 

[Drakkith]

It is thought the whole universe is infinite in size and always has been, it is not thought to be temporally infinite yes?

It is unknown whether the universe has existed in some form for an infinite or finite amount of time. Our models predict a t=0, but only because that's when the math breaks down and we get a singularity. It is entirely possible this is an artifact of our incomplete knowledge of physics at the extremely high energy and density ranges found during this period.

It's also thought that the universe is infinite in size. Mainly because we haven't seen an edge or boundary and because these things are somewhat problematic to have in any model. Oddly enough, the simplest model is one where the universe is infinite in size, as it doesn't require any of the special shapes or conditions that the universe must have in order to be spatially finite.

The observable universe began very small hot and dense and expanded.

That's right.

Was the region “outside” our part, the infinite part, also thought to be in the same/similar condition?

Yes. The entire universe is presumed to have been in a near-identical state.

So if I could travel beyond the observable universe I would find more of the same? Galaxies with stars?

That is correct.

 

[Ibix]

Was the region “outside” our part, the infinite part, also thought to be in the same/similar condition?

The FLRW cosmological model says the universe can be infinite in extent and pretty much the same everywhere, or it can be finite and closed (like the surface of a sphere - finite area but no edges). Observations match the universe being infinite in extent to our best precision, but we can't rule out a REALLY big closed universe, so big our entire observable universe is like your kitchen floor compared to the Earth.

Either way, we predict everything to be more or less the same everywhere even in the bits we can't see - but we can't actually go and check.

 

[anorlunda]

it is not thought to be temporally infinite yes?

If you think it through, it is not possible for anything (universe or not universe) to grow from finite size to infinite size. To do so would require infinite speed of growth. So the word temporary can not be paired with finite or infinite.

 

[pinball1970]

If you think it through, it is not possible for anything (universe or not universe) to grow from finite size to infinite size. To do so would require infinite speed of growth. So the word temporary can not be paired with finite or infinite.

I put temporally not temporary.

I had never thought about this till reading about the subject on pf (within the last year possibly) and yes it makes sense.

If it is infinite now it always had to be and always will be (what if an finite volume expanded for an infinite amount of time? Joke)

One thing I cannot understand is how an infinite volume of space now could have a finite past?

That to me says there was an infinity at the beginning of time, apart from the fact that sounds like a Yes album, that completely blows my mind.

 

[Ibix]

One thing I cannot understand is how an infinite volume of space now could have a finite past?

In a vanilla FLRW model, the scale factor goes to zero 14bn years ago, which means that all points in space are distinct but the distance between any of them is zero. That's the Big Bang singularity. We don't like it because it's exactly as contradictory as it sounds. More modern models get rid of the singularity, and I think (and I'm sure @PeterDonis will correct me if I'm wrong) that they give an infinite past to spacetime.

 

[PeterDonis]

More modern models get rid of the singularity

Some do, and some don't. Whether or not there is a past singularity in our actual universe is still an open question.

 

[PeterDonis]

the scale factor goes to zero 14bn years ago, which means that all points in space are distinct but the distance between any of them is zero. That's the Big Bang singularity. We don't like it because it's exactly as contradictory as it sounds.

Strictly speaking, the singularity itself is not part of spacetime, so there is no actual part of spacetime where the contradictory-sounding thing is true. However, it is still true that you can only go a finite time into the past along any past-directed timelike or null geodesic if there is a past singularity, and many physicists believe that is not physically reasonable--that any geodesic should be extendible indefinitely. That is why many physicists believe that the presence of a singularity in a model is a signal that the model breaks down in that regime, and that we will need some better model, possibly using a more comprehensive theory (such as quantum gravity), for that regime.

 

[phinds]

I put temporally not temporary.

Ha. I also read it as temporarily.

 

[pinball1970]

Ha. I also read it as temporarily.

Good catch!

I could have said, 'please credit me with at least some intelligence,' but that's a bit of an ask on pf isn't it?
Vanadium would say. 'compared to what?'

@anorlunda scan read I think. Fair play, I'm asking basics

 

[anorlunda]

In a vanilla FLRW model, the scale factor goes to zero 14bn years ago, which means that all points in space are distinct but the distance between any of them is zero.

Is that true in the limit? If two points are infinitely far apart in space, does the distance between them approach zero as the scale factor approaches zero? I think not. GR is not needed, it's just the definition of infinity.

 

[Ibix]

Is that true in the limit? If two points are infinitely far apart in space, does the distance between them approach zero as the scale factor approaches zero? I think not. GR is not needed, it's just the definition of infinity.

I don't think you can really discuss it. As Peter reminds me, a singularity is not part of spacetime so it's not clear that "distance between points" makes sense since that's a notion provided by the metric of spacetime which the singularity isn't in. I was trying to express that the singularity is a mathematical self-contradiction and I think I probably over-stepped what you can say about it.

 

[kimbyd]

Whether the universe is infinite in space or not is, in my view, a distraction. The spatial infinity that exists in most cosmological models is a simplification: it's there not because it may or may not reflect reality, but because it's easier to write the equations.

What you should read that infinity as actually meaning about our universe is that it's big enough that its size is likely irrelevant to any observations we might make. Certainly this is the case so far: no observations to date have shown a hint of a finite size. That means the actual size is really really big. How big? Nobody knows.

I don't think it will ever be possible to conclude that the universe is infinite in space, as our observations will always be finite.

 

[pinball1970]

Whether the universe is infinite in space or not is, in my view, a distraction. The spatial infinity that exists in most cosmological models is a simplification: it's there not because it may or may not reflect reality, but because it's easier to write the equations.

What you should read that infinity as actually meaning about our universe is that it's big enough that its size is likely irrelevant to any observations we might make. Certainly this is the case so far: no observations to date have shown a hint of a finite size. That means the actual size is really really big. How big? Nobody knows.

I don't think it will ever be possible to conclude that the universe is infinite in space, as our observations will always be finite.

Does the infinite volume help with the flatness problem?
Measurements look flat because the volumes are so huge?
Also should the flatness look flatter as time goes on? The larger a sphere (for example) becomes the flatter the region you happen to be in looks? Geometrically?

 

[jbriggs444]

Is that true in the limit? If two points are infinitely far apart in space,

The assumption that the universe is spatially infinite does not entail the existence of two points with infinite separation.

 

[DaveC426913]

I don't get how to reconcile these two ideas:
- the universe may be infinite in size
- the observable universe grew from a very small volume to a very large volume without having a centre.

If the universe is/was infinite in size, and our observable universe was once a very small volume in that universe, then does that not mean the observable universe was/is in a specific location? That makes one pat of the universe special.

I've always thought that the BB spawned a universe that was closed - if you go in any direction for long enough, you will end up back where you started - which is how I can see the idea that there is no centre (or everywhere is the centre). The BB cannot have occurred an as expansion in space because there's nowhere you could have stood outside it and meausre any distance - all locations are within the volume of the expanding BB.

I guess my model of the universe is egregiously outdated now.

 

[jbriggs444]

If the universe is/was infinite in size, and our observable universe was once a very small volume in that universe, then does that not mean the observable universe was/is in a specific location? That makes one pat of the universe special.

We are at the center of our observable universe. Some alien 22 billion light years away is at the center of his. Neither location is special.

 

[phinds]

I don't get how to reconcile these two ideas:
- the universe may be infinite in size
- the observable universe grew from a very small volume to a very large volume without having a centre.

There is nothing remotely contradictory in those two statements EXCEPT that you have misrepresented the observable universe as not having a center. It has and always has had. It's exactly at your left eyeball when your right eye is closed. It's the UNIVERSE, not the observable universe, that has no center.

EDIT: And, yes, I have a different observable universe than you do (but not by much in the overall scheme of things)

 

[phinds]

The BB cannot have occurred an as expansion in space

Exactly. The BB was an expansion OF space, not IN space. The universe does not exist inside anything, it's all there is.

 

[PeterDonis]

I've always thought that the BB spawned a universe that was closed

This is one possibility, but our best current model says it is unlikely.

which is how I can see the idea that there is no centre (or everywhere is the centre)

This will be true in any homogeneous space, which is to say any space with constant curvature. The 3-sphere (constant positive curvature, closed) is one possibility, but not the only one. The other two are infinite flat space (zero curvature) and infinite hyperbolic space (constant negative curvature).

there's nowhere you could have stood outside it and meausre any distance

This is true whether the universe is spatially closed or not.

 

[DaveC426913]

We are at the center of our observable universe. Some alien 22 billion light years away is at the center of his. Neither location is special.

Yes. Of course.
I think I may have derailed my own idea, talking about the observable universe. Made a fool of myself.
Let's drop the observable universe component completely.So: does the hypothesis of an infinite universe still reconcile with a universe that expanded from a small, dense volume? How?

 

[KingGambit]

You made a very common mistake missing one key word.

"in the end of inflationary epoch, 10-32 seconds, the size of the visible universe was as big as a grape fruit"

The visible universe is the portion that we can see. The entire universe is believed to be infinite in size today. Therefore it must have been infinite in size before the big bang.

This PF Insights article may help you.
https://www.physicsforums.com/insights/brief-expansion-universe/

Yess..., the observable universe. I miss that word. I just realized the day after I post this question.
Thank you very much.

 

[KingGambit]

Thanks for all responses.
I really appreciate it.

 

[Drakkith]

So: does the hypothesis of an infinite universe still reconcile with a universe that expanded from a small, dense volume? How?

No, but that's not what happened. If our universe is infinite in size then the volume was never small. It was simply more dense in the past.

 

[OCR]

Made a fool of myself.

If our universe is infinite in size then the volume was never small.

And. . . light travels faster than sound.

This is why some people appear bright until they speak. . . . . Lol. . . . j/k .

.

 

[PeroK]

So: does the hypothesis of an infinite universe still reconcile with a universe that expanded from a small, dense volume? How?

A simple mathematical analogy is to consider the one-dimensional number line, with the integers (postive and negative) one unit apart. Now, imagine the effect of $f(n) = 2n$ for any integer $n$. Each number is now twice the distance from any other number than it was originally.

Note that although we might imagine $0$ at the centre of the number line, geometrically the number line looks the same from any position, so there is no geometric centre.

To take this analogy further, we could imagine a "time" parameter $t$ and the function: $f(n, t) = tn$. At time $t = 1$ we have the regular number line. At time $t > 1$, the number lines expands. And, as we run the time parameter back towards $t = 0$, the integer points become more dense. Note that at any finite time $t$ we have an infinite number line with finite density. For $t > 0$ this process is fully reversible.

If, however,we try $t = 0$, we have all integers mapped to a single point. The number line has gone and we have a singularity. Not least, because there is now no inverse process to get us from a single point to an infinite number line.

 

[anorlunda]

The assumption that the universe is spatially infinite does not entail the existence of two points with infinite separation.

Uh oh. That, I don't understand. Because of curvature? Where can I learn about that?

 

[PeroK]

Uh oh. That, I don't understand. Because of curvature? Where can I learn about that?

The number line is infinite, but the difference of any two numbers is finite.

There's a difference between saying that an integer can be arbitrarily large and saying that an integer can be infinite.

 

[jbriggs444]

The assumption that the universe is spatially infinite does not entail the existence of two points with infinite separation.

Uh oh. That, I don't understand. Because of curvature? Where can I learn about that?

It is a bit of pure mathematics. It sounds contradictory but it isn't. It may require adjusting your intuition of what is meant by an infinite set.

[It took a week for me to grasp and wrap an intuition around. It was second year of University and we were being exposed to an axiomatic description of the natural numbers using the Peano axioms]

The example given by @PeroK (the number line populated with integers) is very much on point.

So you have this number line. Every position on the number line corresponds to a finite integer. But there are infinitely many such positions. It is tempting to try to weasel out by considering the number line as some sort of incomplete process or potential infinity. But you need to accept it as a completed whole.

That is an uncomfortable intuition. But can you find an actual contradiction? Can you explain that apparent contradiction clearly enough that we can discuss it?

 

[DaveC426913]

No, but that's not what happened. If our universe is infinite in size then the volume was never small. It was simply more dense in the past.

Ah. Got it. PeroK's number line analogy helps.

 

[pinball1970]

The assumption that the universe is spatially infinite does not entail the existence of two points with infinite separation.

Even if it is flat?

Using the number line analogy what about plus and minus infinity?

If plus and minus infinity can be a physical thing then points in those places should also be infinitely apart?

If they are not infinitely apart because the number line is a concept rather than a physical thing then how can the physical universe be infinite?

 

[jbriggs444]

Using the number line analogy what about plus and minus infinity?

Neither are positions on the number line.

 

[PeterDonis]

If plus and minus infinity can be a physical thing

They can't be. There are no spatial points in the universe with location infinity.

 

[anorlunda]

The number line is infinite, but the difference of any two numbers is finite.

It is a bit of pure mathematics. It sounds contradictory but it isn't. It may require adjusting your intuition of what is meant by an infinite set.

Thanks guys. I thought that I had fully internalized the idea that infinity is not a big number. But you've made my understand that my concept is still flawed.

 

[ergospherical]

It is however possible to conformally compactify the metric $g \rightarrow \Omega^2(x) g$ so that points "at infinity" w.r.t. $g$ end up at finite distances w.r.t. $\Omega^2(x) g$. (The classic example is the compactification of Minkowski spacetime to a subset of the Einstein static universe (ESU), within which there is a point $i_0$ (spatial infinity) where radial spacelike geodesics start and end.)

A pictorial analogy of a conformal map:

 

[KingGambit]

So...,
What was the size of the observable universe at Baryogenesis epoch?
And what was its size during Photon (380 thousands years) epoch?

 

[DaveC426913]

So...,
What was the size of the observable universe at Baryogenesis epoch?

I'm not sure "observable" makes sense during a time when
- the universe was opaque to light, and
- the universe was expanding much faster than the speed of light.

 

[ergospherical]

I'm not sure "observable" makes sense during a time when
- the universe was opaque to light, and
- the universe was expanding much faster than the speed of light.

The size of the observable universe is set by the particle horizon, which is as far away as light could have travelled, in principle, between a suitable spacetime origin and the current cosmic time $t$. So I don't believe the initial "opacity" of the universe is relevant to this definition.

It's particularly straightforward in terms of conformal time $\eta$ (defined by $c dt = a d\eta$) and co-moving distance $\chi$ (defined by $r = a\sin{\chi}$ for positive curvature or $r= a\sinh{\chi}$ for negative curvature), in which case the particle horizon is defined by a line where $\chi$ varies in direct proportion to $\eta$.

It is also not relevant to this definition that co-moving bodies were initially undergoing superluminal recession.

 

[kimbyd]

Does the infinite volume help with the flatness problem?
Measurements look flat because the volumes are so huge?
Also should the flatness look flatter as time goes on? The larger a sphere (for example) becomes the flatter the region you happen to be in looks? Geometrically?

1) No
2) No
3) In our universe at present, yes, but it didn't have to be that way. And it wasn't always.

The flatness problem can be seen to be solved by inflation: the rapid, exponential expansion during inflation drove the universe towards extreme flatness in a very short period of time.

The size actually has very little to do with it. It's the expansion history that matters there. One way to see this is to look at the first Friedmann equation, which can be written as:
$$H^2 = {8 \pi G \over 3} \rho - {k \over a^2}$$

The critical thing to look at is the fact that the effect of the spatial curvature on the expansion scales as $1/a^2$. What matters for whether curvature will become important later on is whether $\rho$ dilutes faster or slower than $1/a^2$. Normal matter dilutes more rapidly: $1/a^3$. So if you have a universe with only normal matter and no cosmological constant, then as it expands, the curvature becomes more consequential over time. Bigger, in such a universe, results in more relative curvature.

Inflation dilutes out the curvature because the inflaton acts very much like a cosmological constant with a high energy density. Because the inflaton doesn't dilute but undergoes rapid expansion, the curvature quickly becomes negligible.

And remember, at the end of inflation we're talking about a not terribly huge volume which ends up making up our entire observable universe. The universe expands dramatically after that point, and the effect of curvature gets larger for much of that time (because the matter and radiation densities dilute much faster than the effect of the curvature). Only more recently, as dark energy has come to dominate the expansion, has the effect of curvature started to drop again.

 

[Jorrie]

It's particularly straightforward in terms of conformal time $\eta$ (defined by $c dt = a d\eta$) and co-moving distance $\chi$ (defined by $r = a\sin{\chi}$ for positive curvature or $r= a\sinh{\chi}$ for negative curvature), in which case the particle horizon is defined by a line where $\chi$ varies in direct proportion to $\eta$.

Also, in straightforward proper distance against cosmological time, the early epochs show linearly increasing proper distance over time for both the Hubble radius (R) and the particle horizon (Dpar). So until the standard cosmological model fails, one can extrapolate backwards for the size of the observable universe easily.

The size of the observable universe is set by the particle horizon, which is as far away as light could have travelled, in principle, between a suitable spacetime origin and the current cosmic time $t$. So I don't believe the initial "opacity" of the universe is relevant to this definition.

Not restricted to light. Neutrinos and gravitational waves may also be suitable observational media in principle. For very early times, one can linearly extrapolate backwards using the LCDM model to estimate the sizes of the Hubble radius (R) and the particle horizon radius (Dpar), in e.g.

 

[Torbjorn_L]

First, the grapefruit volume estimate of the visible universe after inflation ends locally (as it should when it exits its slow roll phase, considering we see it has quantum fluctuations) is a bit dated. If you take the Planck 2018 observations of low constraint on inflation energy density the best we can say is that it could have been at most a few meters across, unless I'm mistaken in my back-of-the-envelope estimate.

Second, the singularity idea is also showing its old age since it is an expectation of using a de Sitter map of the universe and connect it with Planck energy densities. But again, unless you mistake a convenient map for the territory, there seems to be no singularity or need for space curvature flattening in a cosmologically flat universe of an eternally inflating universe (which is the natural state). Expansion of an essentially constant energy vacuum state, which it looks to be, is then exponential and not singularity causing super-exponential.

It would by the way square well with the recent BICEP3/Keck data that prefers a Higgs like hilltop scalar potential for the inflation vacuum state. But by the same token we should soon see a smidgen of gravitational backreaction - tensor components in the cosmic background radiation - so the promise was that it can be observed or rejected within a decade.

Speaking of tests, it took me a long while to hunt up the references that the prediction of late Steven Weinberg's anthropic multiverse theory underwent a bona fide later test in observing the current vacuum energy density. To be fair, and take the opportunity of blatant namedrop, it was a comment from the also late Joe Polchinski under an article of Sean Carroll that put me on the right track, he remarked that the result was impressive to him. Most descriptions imply the reverse and devalues the result as inconsequential. (It reminds me how Crick's Dogma of sequence information *never* getting out of proteins has been replaced in name by the Watson's later textbook standard transcript-to-protein pathway which has more exceptions than you can shake your finger at. Most biologists and bioinformaticians gets it backwards during early studies and that too takes going back to the references to dig out.)

Whether or not that means space flatness is a symmetry can be discussed along the lines of need for flattening. It could be needed, but it appears it would mean some form of finetuning to make it so.

Physicists suspect that eternal inflation is generic, meaning a consequence of most, if not all, models of inflation. So, following this suspicion, if inflation is correct, then eternal inflation is also likely correct, and the multiverse might be real.

[ https://www.livescience.com/how-real-is-the-multiverse ; includes recent updates on the theory.]

At this point it is tempting to inject - not equations, since we can dig up more references later, but - a dated aphorism: "Everything Should Be Made as Simple as Possible, But Not Simpler" ["Einstein may have crafted this aphorism, but there is no direct evidence in his writings" - https://quoteinvestigator.com/2011/05/13/einstein-simple/ ].

 

[StandardsGuy]

The universe does not exist inside anything, it's all there is.

What kind of proof do have for that statement?

 

[jbriggs444]

What kind of proof do have for that statement?

It is a definition.

 

[Ibix]

What kind of proof do have for that statement?

Perhaps a more precise statement would be that every phenomenon that we know of can be modeled without embedding spacetime in something else. Thus Occam's Razor currently disfavours anything "beyond" or "outside" spacetime.

 

[kimbyd]

Perhaps a more precise statement would be that every phenomenon that we know of can be modeled without embedding spacetime in something else. Thus Occam's Razor currently disfavours anything "beyond" or "outside" spacetime.

From that statement alone, the mediocrity principle (basically "we are not special") would suggest that there are things beyond or outside of our space-time.

But thinking about it in that simplistic way is nevertheless misleading. Take the expansion itself. Often people wonder what space is expanding into. This model presupposes some kind of static background with a defined volume. In this naïve view, the universe occupies more and more of this static background as it expands.

This naïve view is categorically incorrect. It can't even be correct, because the imagined static background is impossible. It derives from the difficulty in understanding that space-time itself is a mutable construct, imagining that the curvature of space-time must be relative to some "uncurved" outside description. For an example of how this might occur, consider the curvature of the Earth's surface. This curvature exists because the Earth's surface is a two-dimensional surface embedded in three-dimensional universe. The question arises: can the curvature of General Relativity be described by imagining it embedded in some larger number of dimensions? The answer is most definitely no.

You can "flatten" certain geometries in GR by describing them in a higher number of (flat) dimensions. But I don't think this is something you can do generally. And even if you could, it would require a large number of large extra dimensions, something which I'm pretty sure is ruled out by observations.

All this is to say, the only way that curvature even makes sense in General Relativity is to describe the curvature internally only, without reference to some kind of uncurved background. This extends to our concepts of the universe as well: even if there are things "outside" of our space-time, the relationship between those structures and our own is completely different from what you'd expect from the naïve view.

For example, if there was a "big bang" caused by a quantum fluctuation within our own universe, from the perspective of an observer in our universe it would look like a microscopic black hole which would fluctuate into existence then rapidly evaporate. Internally there might be an entire universe that behaves very much like our own. But that universe is disconnected entirely from its "parent".

So yes, there might be stuff that isn't described by the same Big Bang expansion we observe. But it's not as simple as "outside".

 

[PAllen]

But thinking about it in that simplistic way is nevertheless misleading. Take the expansion itself. Often people curvature of space-time must be relative to some "uncurved" outside description. For an example of how this might occur, consider the curvature of the Earth's surface. This curvature exists because the Earth's surface is a two-dimensional surface embedded in three-dimensional universe. The question arises: can the curvature of General Relativity be described by imagining it embedded in some larger number of dimensions? The answer is most definitely no.

Actually, the answer is definitely yes. There are analogs of the Nash embedding theorems for pseudo-Riemannian manifolds.

You can "flatten" certain geometries in GR by describing them in a higher number of (flat) dimensions. But I don't think this is something you can do generally. And even if you could, it would require a large number of large extra dimensions, something which I'm pretty sure is ruled out by observations.

It does require a large number of dimensions for the general case ( for a smooth embedding; for a C1 embedding, very few extra dimensions are needed) but these dimensions would be unobservable, just as a being within a 2-sphere is unaware of an embedding in 3-space. Embedding doesn’t change the geometry of a manifold, it just provides a different way of describing it. It also allows definition of things which are presumably unobservable and embedding dependent like extrinsic curvature.

All this is to say, the only way that curvature even makes sense in General Relativity is to describe the curvature internally only, without reference to some kind of uncurved background. This extends to our concepts of the universe as well: even if there are things "outside" of our space-time, the relationship between those structures and our own is completely different from what you'd expect from the naïve view.

Quite a few authors for GR first introduce embedding and extrinsic curvature first, then derive intrinsic curvature. Personally, I always found this a waste of time, but some teachers prefer it pedogogically. But, hey, this is the way Gauss went about it. Who am I to question Gauss?

 

[PeroK]

Quite a few authors for GR first introduce embedding and extrinsic curvature first, then derive intrinsic curvature. Personally, I always found this a waste of time, but some teachers prefer it pedogogically. But, hey, this is the way Gauss went about it. Who am I to question Gauss?

Where and when did Gauss learn GR?!