B Big Bang: Size of the Universe at Different Epochs

[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?!

 

[PAllen]

Where and when did Gauss learn GR?!

Gauss developed theory of extrinsic and intrinsic curvature of surfaces and showed equivalence of embedding approach and intrinsic approach.

 

[PeroK]

Gauss developed theory of extrinsic and intrinsic curvature of surfaces and showed equivalence of embedding approach and intrinsic approach.

A man ahead of his time!

 

[Hornbein]

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

The idea is that the volume has always been infinite. So it isn't possible to compare two sizes directly. We CAN however measure and compare the densities at two different times.

The density of the universe is believed to be always an infinite amount of matter divided by an infinite volume. Just the sort of operation they told you not to do in high school. But the infinite pound gorilla does as it pleases.

Intuitively one would think that if the amount of matter remains the same and density decreases then the volume must be larger. But we have not been able to assign any number to this volume.

I don't know, but I've been told, that if the curvature is non-zero that it increases over time. So for it to be so close to zero now it must have been much closer to zero back then. Or so they say.

 

[DaveC426913]

Just one nitpick:

I don't know, but I've been told, that if the curvature is non-zero that it increases over time.

Does not rhyme. Also, meter is off.

Spoiler

 

[PAllen]

The idea is that the volume has always been infinite. So it isn't possible to compare two sizes directly. We CAN however measure and compare the densities at two different times.

What you can say is that given the volume enclosing some amount of matter (galaxies) in the present, then the volume enclosing that same matter decreases as you go back in time to the initial state we call the big bang. In a pure FLRW solution (i.e. no inflation, etc.) there is no lower bound to this volume. Despite this, the universe as a whole can still be infinite at all times. It's just a feature of 'infinite' - no matter how many times you cut it in half, it is still infinite.

I don't know, but I've been told, that if the curvature is non-zero that it increases over time. So for it to be so close to zero now it must have been much closer to zero back then. Or so they say.

This is not generally true. In a wide range of pure FLRW solutions, the curvature scalar R grows without bound as time is followed backwards. I am not aware of any general tendency for curvature to increase over time.

[edit: In fact it is quite easy to have constant curvature scalar throughout - just have flat spatial slices with exponential scale factor. This produces a constant value for R]

 

[PeterDonis]

I've been told, that if the curvature is non-zero that it increases over time.

This is referring to a particular definition of spatial curvature (and in a particular type of model, see below), not spacetime curvature. As @PAllen posted, the spacetime curvature of any expanding FRW solution will decrease with time.

In matter and radiation dominated FRW spacetimes, if the density parameter $\Omega$ (the ratio of actual density to critical density) is not exactly $1$, then it will diverge from $1$ over time. In models with zero cosmological constant, $\Omega - 1$ can also be treated as a (sort of) spatial curvature parameter; if it is zero (i.e., $\Omega = 1$, density exactly critical), the universe is spatially flat; positive $\Omega - 1$ means positive spatial curvature (closed universe), and negative $\Omega - 1$ means negative spatial curvature (open universe). A universe with positive $\Omega - 1$ will have increasing $\Omega - 1$ over time, whicn can (sort of) be thought of as increasingly positive spatial curvature, and a universe with negative $\Omega - 1$ will have decreasing $\Omega - 1$ over time, which can (sort of) be thought of as increasingly negative spatial curvature.

(Note that I put in the qualifier "sort of" each time I mentioned spatial curvature in the previous paragraph. That's because $\Omega - 1$ is not the actual geometric quantity that would normally be referred to as the "spatial curvature" of a spacelike slice of constant FRW coordinate time in these spacetimes. Sources that talk about it as related to "spatial curvature" without any qualification are being sloppy.)

The reason cosmologists are interested in all this is that, for most of our universe's history, it was either matter or radiation dominated, and yet our current universe's $\Omega$ is extremely close to $1$. This requires extremely sensitive fine-tuning of initial conditions, which is known as the "flatness problem":

https://en.wikipedia.org/wiki/Flatness_problem

 

[kimbyd]

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

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.

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?

I see. So it isn't so much that it's impossible, but rather that it's ridiculously complicated, and disfavored for that reason. At least when plain GR is concerned.

I'd have to guess that attempting to quantize the embedded space-time would break that model entirely.

Edit: A quick search confirms my intuition that QM would probably muck this up:
https://academic.oup.com/ptp/article/83/5/894/1875237

 

[JimJCW]

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)

Using Jorrie’s calculator and PLANCK Data (2015), the size of the observable universe is calculated to be

R = 0.87 Mly at t = 0.38 Myr​

From the extrapolation of a linear log(R) vs. log(t) plot, the size of the observable universe is estimated to be

R = 3 km at t = 10 µs​

@Jorrie

 

[KingGambit]

Using Jorrie’s calculator and PLANCK Data (2015), the size of the observable universe is calculated to be

R = 0.87 Mly at t = 0.38 Myr​

From the extrapolation of a linear log(R) vs. log(t) plot, the size of the observable universe is estimated to be

R = 3 km at t = 10 µs​

@Jorrie

Thank you very much @JimJCW