Changing curvature as a universe evolves

[Will Learn]

TL;DR Summary

(There doesn't seem to be an easy way to find similar threads without creating one and posting it, so I'll just post a question).
Standard texts exist that describe the time evolution of a FLRW universe given parameters such as density. Can the curvature parameter, k, change as the universe evolves? Can curvature change so much that the geometry of the spatial hypersurface changes character (spherical <--> flat <--> hyperbolic)?

Hi again. I'm still off work and struggling to learn some physics. I'm searching for discussion about the possibility of a universe changing its curvature as it evolves.

I'm still new here and so I'd be grateful for advice either about: (i) How to search for past discussions, or else, (ii) any discussion or pointers.

The sumary should describe the situation but I'll write it out again here, just in case:

Modelling the universe as a Robertson-Walker universe - can the spacetime change its curvature (as time progresses) so that the geometry of the spatial hypersurface changes from one type (e.g. spherical) to another (e.g. flat) over its course of evolution?

I can post more info if required.

 

[Will Learn]

Found some threads listed in the "related threads" section now appearing at the bottom. Skim reading, I haven't found much of relevance.
Space/time curvature of the young universe - by EskWired in 20124 looks most relevant from about post no.5 but the info there is old and there's probably not much hope of active discussion with those people.

 

[PeterDonis]

Modelling the universe as a Robertson-Walker universe - can the spacetime change its curvature (as time progresses) so that the geometry of the spatial hypersurface changes from one type (e.g. spherical) to another (e.g. flat) over its course of evolution?

To be clear, the curvature parameter $k$ describes spatial curvature, not spacetime curvature. The spacetime curvature does vary in an FRW spacetime--it's not the same everywhere (though it is the same everywhere at some instant of comoving coordinate time)--but that doesn't appear to be what you are asking about.

What you appear to be asking about is not just a change in the geometry of a spacelike hypersurface of constant comoving coordinate time, but a change in its topology. A spatially flat universe ($k = 0$) has spatial topology $R^3$, but a spherical universe ($k > 0$) has topology $S^3$. I'm not aware of any classical GR model that allows a topology change of this kind, so as far as classical GR is concerned, I think the answer to your question is no.

It might be possible that quantum gravity will change this; speculations that quantum gravity effects might allow the topology of spacetime to change (though not always on the scale of the entire universe) are common in the literature. However, unless and until we discover and experimentally verify a theory of quantum gravity, those speculations will remain only speculations.

 

[Will Learn]

To be clear, the curvature parameter describes spatial curvature, not spacetime curvature. The spacetime curvature does vary in an FRW spacetime--it's not the same everywhere (though it is the same everywhere at some instant of comoving coordinate time)--but that doesn't appear to be what you are asking about.

Hi and thanks. Sorry for the ambiguity. I think I understand what you're saying here. The curvature parameter, k, does indeed describe the curvature of a (3-dimensional) spatial hypersurface (at a fixed co-ordinate time) exactly as you said.

I should re-write the original question as follows:
... can the spacetime change its curvature parameter (as time progresses) so that the geometry of the spatial hypersurface changes from one type (e.g. spherical) to another (e.g. flat) over its course of evolution?..

The curvature parameter is a property of the entire spacetime (in addition to being a property of the spatial hypersurfaces). It is the curvature of the spatial hypersurfaces. It is not the curvature of the spacetime but it is a characterisation of the spacetime as being Open, Flat or Closed (depending on the sign of k). By adjustment of the scale factor (and one co-ordinate), it is possible to normalise the curvature parameter so that k = +1, 0 or -1. With a normalised curvature parameter, we see that the nature of the spatial curvature determines the nature of the spacetime (and vice.versa.). So I have tended to blur the distinction between them and I can only apologise for that.

A spatially flat universe () has spatial topology , but a spherical universe () has topology .

The mathematical symbols seem to be missing from that quote, sorry. In any case I agree with what you said. Although I believe the topology of the 3-dimensional flat (k=0) space is not limited to the Euclidean plane R^3. It could include a 3-torus like S^1 x S^1 x S1, for example. (That's important for later, I think, because it means that a flat space does not have to be infinite in extent).

The topology of a spatial hypersurface with k=-1 is usually a generalisation of a hyperboloid and would be infinite in extent. However, I think it could also be an unusual compact space and could then be finite in extent.

The topology of the spatial hypersurface with k=+1 is usually the 3-sphere, S^3, but it could be another non-orientable space. Whatever the case, I think this one always has to be finite in extent.

Checking to see if the curvature parameter can change is obviously the first step, before worrying about a full description of the global topology.

I'm not aware of any classical GR model that allows a topology change of this kind, ...

I'm not aware of one either but you are absolutely correct - this sort of fundamental topology change is exactly the sort of thing I was asking about. Thank you for identifying that in the original post and I can only apologise for my mistakes and misuse of terminology.
Can a universe change from being of finite extent to being of infinite extent? Although, perhaps it doesn't even have to considering the various topologies of the spatial hypersurfaces that are possible.

All the older cosmologies I have seen imply that the classification of the spacetime as open, closed or flat is synonymous with a description of its ultimate fate. As if knowing what the curvature is now will establish the nature of the universe for-ever after. However, I can't see any reason why the curvature parameter couldn't evolve. I can write more about that but I'm sure you (Peter Donnis) don't need it.

I don't know. I'm keen to discuss with anyone or find any text describing a situation where the curvature parameter changes as a universe evolves and that is what the OP was about.

Thank you for your time and attention.

 

[PeterDonis]

The mathematical symbols seem to be missing from that quote

Yes, for some reason PF's quoting feature does not copy LaTeX, at least not if it's inline.

Although I believe the topology of the 3-dimensional flat (k=0) space is not limited to the Euclidean plane R^3.

I suppose one could consider possibilities like the flat 3-torus, yes. But whatever the topology is, it can't be $S^3$.

The topology of a spatial hypersurface with k=-1 is usually a generalisation of a hyperboloid

That's geometry, not topology. Topology doesn't care about the metric. The topology of a $k < 0$ universe, or at least the topology that is usually assumed, is $R^3$, just like the $k = 0$ case. I don't know whether any other topologies are possible for the $k < 0$ case.

Can a universe change from being of finite extent to being of infinite extent?

I would give the same answer to this as I gave in post #3, and for the same reason.

All the older cosmologies I have seen imply that the classification of the spacetime as open, closed or flat is synonymous with a description of its ultimate fate.

That's true for the case of zero cosmological constant, but our best current model of our universe has a small positive cosmological constant. For that case, the ultimate fates are somewhat different, and in particular the $k > 0$ case (spherical) will not necessarily recollapse to a Big Crunch.

I can't see any reason why the curvature parameter couldn't evolve.

For it to evolve, there would have to be a solution of the Einstein Field Equation that had this property. I'm not aware of any such solution. Just saying you can't see any reason why not is not suffcient; you would actually need to find a solution that has the property.

 

[Orodruin]

It could include a 3-torus like S^1 x S^1 x S1, for example.

This is incorrect. The 3-torus, while flat, is not isotropic, which violates the basic assumptions behind the Robertson-Walker spacetime.

 

[Will Learn]

Thanks Orodruin and PeterDonis.
If I may reply to Orodruin just because it's shorter.

@Orodruin
I was pretty blinkin' well certain of this possibility.

My main source of information was from the book Spacetime and Geometry, Sean Carroll, Chapter 8, page 331 in the discussion between equations 8.34 and 8.36.
If I had a scanner I could scan that page - but I don't. Perhaps you have access to that text on your own.
It is directly stated that for the k=0 case, the spatial hypersurface is flat Euclidean space but that ...Globally, it could describe R³ or a more complicated manifold, such as the three torus S¹ X S¹ X S¹...

However, I've taken some time to consider your answer. Despite my inclination to go with that textbook, I think you may actually be correct.

You'll understand if I take a day to dismiss the proposition presented in that textbook. That chapter in my 3rd printing of that book is significantly different from what appears in the original lecture notes which I believe formed the original skeleton of the book. So, it's possible that Sean Carroll may have edited the section and put things together in a slightly disjointed way. Taken in isolation each one of his sentences makes sense. Thus, the flat Euclidean metric could certainly support a topology like the 3-torus but the additional requirement of maximal symmetry, which we must have in this context, would seem to directly eliminate that possibility.

Thank you for taking the time to discuss the topic.

 

[Orodruin]

Locally, there is just no way of telling the torus and the Euclidean space apart. The cosmological principle of isotropy is also an assumption and there is really no way to test it if the torus is assumed big enough for the horizon not to start overlapping. However, the torus generally is not a maximally symmetric space.

 

[Will Learn]

That's geometry, not topology. Topology doesn't care about the metric. The topology of a $k < 0$ universe, or at least the topology that is usually assumed, is $R^3$, just like the $k = 0$ case. I don't know whether any other topologies are possible for the $k < 0$ case.

Thanks for continuing to discuss the topic. Once again I can only apologise for where I may have mixed up terms and been extremely sloppy with terminology. I'm self-taught.
I hope you won't be offended but I'm not sure I can go along with all of your statement. I think there is a connection between geometry and topology. At the very least we seem to agree that the curvature of the spatial hypersurface does influence the usual representation of that space. Exactly as you stated, the k>0 is associated with a 3-sphere, while k=0 is usually associated with Euclidean space R³. While the k<0 case (hyperbolic spatial geometry) is usually assumed to be a manifold denoted as H³. This being a 3-dimensional generalisation of H² which is usually represented as the Poincare half-plane.
In any case, as mentioned in an earlier post, I was checking to see if the curvature parameter can change to begin with. The possible change in global topology follows as a result but it can wait. Checking to see if the curvature parameter changes first is obviously the better place to start.

I would give the same answer to this as I gave in post #3, and for the same reason.

So, if I've understood correctly, your answer is -

I'm not aware of any classical GR model that allows a topology change of this kind, so as far as classical GR is concerned, I think the answer to your question is no.

Thank you. I'm very grateful for your time and attention. You do not need to spend more time.

I should explain that, I'm not trying to propose any new physics or present an article of pure speculation here. I am just trying to learn and I have no-where else to discuss science. Part of the learning process (for me) involves trying to understand why-on-earth things have to be "this way" and can't be "that way". I've bought a textbook at great expense but it won't talk back to me. I have no realistic opportunity to attend a university and discuss anything with other students or staff.
You are, of course, free to close down this thread if you wish and I acknowledge that an answer to the OP has been given. I was going to continue taking quotes from your post, in case there is anyone who may be able or willing to continue a discussion. It is not an attempt to be disrespectful or challenge your expert opinion, it's just that I must continue to probe the limits of why things have to be this way to continue learning.

(About Open, Flat, Closed being synonymous with the ultimate fate of the universe) .. That's true for the case of zero cosmological constant, but our best current model of our universe has a small positive cosmological constant. For that case, the ultimate fates are somewhat different, and in particular the case (spherical) will not necessarily recollapse to a Big Crunch.

That's one of the reasons I thought the unchanging nature of the curvature parameter could have been re-examined. We know of far more components of the cosmological fluid (like vacuum energy density) that can be included in a Robertson-Walker Universe today then when those earlier texts were written.

For it to evolve, there would have to be a solution of the Einstein Field Equation that had this property. I'm not aware of any such solution. Just saying you can't see any reason why not is not suffcient; you would actually need to find a solution that has the property.

Finding a solution doesn't seem impossible. We are taking a FRW universe to begin with. By my understanding, the requirement that the scale factor satisfies the Friedmann equations together with the requirement of having a metric in the form of the FLRW metric should guarantee that it satisfies the Einstein Field Equations where the stress-energy tensor is specified as that for a cosmological fluid as usual.
We are free to specify the density parameters of the various components in our cosmological fluid at an (initial) time. The Friedmann equations will ensure evolution that remains consistent with the Field equations.
1. We have a lot of components now available, like a non-zero vacuum energy density etc.
2. We know that different components have densities that evolve differently. For example matter density ~ a^-3 but radiation ~ a^-4.
3. We also know that the curvature parameter is determined by comparing the density to the critical density and we know precisely how that critical density evolves.

Just avoid having the blend of components in your cosmological fluid to make the evolution of the total density match exactly with the evolution of the critical density and then the curvature parameter is bound to evolve. Rather than believing the curvature parameter is a fixed parameter of the universe I have to take the alternative viewpoint: How could we (an arbitrary person studying GR) NOT think that the curvature parameter would evolve?
The above is easily seen to apply to the un-normalised curvature parameter, so that a closed universe could start with a spatial curvature that is negative but close to zero (so that it was almost flat) and evolve to a state with a more severe negative spatial curvature. It's not too interesting because the magnitude of curvature is quite arbitrary depending on the co-ordinates or units you choose to use. You could easily absorb such changes into the evolution of the scale factor, for example.
What would be more interesting is if the change could be so great that the sign of the curvature parameter changes.

Thanks, best wishes and bye.

 

[Will Learn]

Locally, there is just no way of telling the torus and the Euclidean space apart. The cosmological principle of isotropy is also an assumption and there is really no way to test it if the torus is assumed big enough for the horizon not to start overlapping. However, the torus generally is not a maximally symmetric space.

Yep, I've got it. Thanks.

 

[PeterDonis]

I think there is a connection between geometry and topology.

In the sense that some topologies are not compatible with some geometries, yes.

At the very least we seem to agree that the curvature of the spatial hypersurface does influence the usual representation of that space.

I assume that by "the usual representation" you mean "the usual geometry that is assumed". Yes, the usual geometry assumed for the three cases is the geometric 3-sphere for $k > 0$, flat Euclidean 3-space for $k = 0$, and 3-dimensional hyperbolic space for $k < 0$. But the topology for these cases is $S^3$ for the first and $R^3$ for the second two; here these only refer to the topological spaces, not the geometry. It is perfectly possible to have a manifold with topology $S^3$ that is not a geometric 3-sphere, or a manifold with topology $R^3$ that is not flat Euclidean space (indeed, the $k < 0$ hyperbolic space is just such a manifold), just as it is possible to have a 2-surface with topology $S^2$ that is not a geometric 2-sphere (such as the surface of the Earth, which is an oblate spheroid with minor irregularities, not a perfect 2-sphere, geometrically speaking, even though its topology is $S^2$).

That's one of the reasons I thought the unchanging nature of the curvature parameter could have been re-examined.

Depending on how you define "the curvature parameter", its magnitude can change (if it's nonzero). What can't change is its sign. The three possible signs (positive, zero, negative) are discrete alternatives, at least as far as the FRW spacetimes are concerned; there is no continuous way to change one into another. But any spacetime model that had a curvature parameter with changing sign would have to have it change continuously. So it doesn't seem like any such spacetime model would be possible, at least not if we use the FRW spacetimes as the basis for our model (but try to, say, "glue together" FRW regions with different signs for the curvature parameter).

We know of far more components of the cosmological fluid (like vacuum energy density) that can be included in a Robertson-Walker Universe today then when those earlier texts were written.

No, we "know of" no components that we didn't know of when those earlier texts were written. The possibility of a nonzero cosmological constant was already known then; it was just not considered in any detail because there was no evidence for one. When evidence came to light in the 1990s, cosmologists had no trouble at all adding a nonzero cosmological constant to their models because the models had already been constructed to allow for that possibility. No new "discovery" on the theoretical side was required.

Other than the cosmological constant, there are no "more components" to consider.

We also know that the curvature parameter is determined by comparing the density to the critical density

Sort of. What you are failing to realize is that, for a model that includes all of the known "components", including a cosmological constant, it can be proven that the sign of the curvature parameter can never change. So once you have determined the sign of that parameter by compring the density to the critical density at one instant, you know the sign of the curvature parameter for the entire spacetime. It can't change.

Just avoid having the blend of components in your cosmological fluid to make the evolution of the total density match exactly with the evolution of the critical density and then the curvature parameter is bound to evolve.

Nope, not possible. See above.

How could we (an arbitrary person studying GR) NOT think that the curvature parameter would evolve?

By having a proper understanding of how the FRW spacetime models work. See above.

The above is easily seen to apply to the un-normalised curvature parameter, so that a closed universe could start with a spatial curvature that is negative but close to zero (so that it was almost flat) and evolve to a state with a more severe negative spatial curvature.

A closed universe has positive spatial curvature, not negative.

 

[Buzz Bloom]

I am surprised that no on has suggested that instead of k as a parameter for spatial curvature, the variable (in the Friedmann equation of https://en.wikipedia.org/wiki/Friedmann_equations#Detailed_derivation ) Ω_k appears in the equation with coefficient 1/a². This gives the detailed manner of change of curvature since the curvature is proportional to Ω_k^1/2/a.

I appologize for all the edits to get the formula correct, particularly after @PeterDonis responded.

 

[PeterDonis]

I am surprised that no on has suggested that instead of k as a parameter for spatial curvature, the variable (in the Friedmann equation of https://en.wikipedia.org/wiki/Friedmann_equations#Detailed_derivation ) Ω_k appears in the equation with coefficient 1/a². This gives the detailed manner of change of curvature since the curvature is proportional to 1/(Ω_k[/SUB)^1/2.

Um, both @valenumr and myself have already referred to this. I referred to it here:

Depending on how you define "the curvature parameter", its magnitude can change (if it's nonzero). What can't change is its sign.

@valenumr referred to it here:

We also know that the curvature parameter is determined by comparing the density to the critical density and we know precisely how that critical density evolves.
<snipped>
The above is easily seen to apply to the un-normalised curvature parameter

 

[Will Learn]

Thanks @PeterDonis for your reply.

About your early Geometry / Topology comments:
1. Yes that's the sort of connection I had in mind.
2. No-one's interested in what I meant by a representation. It's not important.

The bit about changing curvature parameter (depending on how you define it):
1. We seem to be in agreement on the first part.
2. About continuity. You said -

The three possible signs (positive, zero, negative) are discrete alternatives, at least as far as the FRW spacetimes are concerned; there is no continuous way to change one into another.

I agree, continuity and the discrete nature of the sign seems most relevant. This also links with a later statement you made:

What you are failing to realize is that, for a model that includes all of the known "components", including a cosmological constant, it can be proven that the sign of the curvature parameter can never change.

It's not that I'm failing to realize it. I'm failing to find it in my textbook. If anyone can cite a reference to that proof, great, else I'll get around to trying to derive the result myself.
Nature doesn't try to normalise parameters, we do. I would just like to check the proof to see that no artificial discrete character has been introduced.About components of a cosmological fluid:
Thanks for the history and congratulations to those cosmologists.
I only meant that there was more willingness to use anything and everything. Various combinations of negative pressure and negative energy were taken seriously for some components so it wasn't just like one thing. It was also possible to use more of the old stuff than ever before - dark matter as a way of pushing up the density of matter etc. You could be seriously underestimating my age and the age of the texts that were around in my day.

The bit about my typing errors or muddled up words:
I do make a lot of these mistakes, sorry.

Thanks again for your time, it's been a pleasure talking to you.

 

[Will Learn]

I am surprised that no on has suggested that instead of k as a parameter for spatial curvature, the variable (in the Friedmann equation of https://en.wikipedia.org/wiki/Friedmann_equations#Detailed_derivation ) Ω_k appears in the equation with coefficient 1/a². This gives the detailed manner of change of curvature since the curvature is proportional to Ω_k^1/2/a.

I appologize for all the edits to get the formula correct, particularly after @PeterDonis responded.

Hi and thanks for your reply.

I appreciate how long it takes to type and I haven't even really bothered to use LaTex here yet. It may be just as fast to create a formula outside with some other software and paste it in here (even if it has to be a picture).

I'm going to be away doing stuff so I probably won't be engaging much more with this thread for a while. Obviously I don't own this thread and you can do whatever you want and may be able to engage with others.

Best wishes to you, bye for now.

 

[PeterDonis]

I'm failing to find it in my textbook.

Most textbooks on cosmology (or indeed most textbooks on GR in general) probably don't go into this because it's too technical. They're more concerned with using the standard FRW models than with mathematically proving theorems about their properties.

I believe there is a proof in Hawking & Ellis of a more general proposition of which the one we are discussing is a special case, but I don't have my copy handy right now to check. (Roughly, the more general proposition, as I remember it, is a continuity constraint on the behavior of the extrinsic curvature of spacelike slices that satisfy a symmetry criterion. But I may be misremembering the details, it's been quite a while since I last read Hawking & Ellis.)

Nature doesn't try to normalise parameters, we do.

The issue isn't normalization; AFAIK there is the same lack of continuity if you use an un-normalized curvature parameter, because the $k = 0$ case still is not a limit as $k \rightarrow 0$ of either the $k > 0$ case or the $k < 0$ case. For the $k > 0$ case this is easily seen as the topology would have to change; the limit as $k \rightarrow 0$ of a 3-sphere is not Euclidean 3-space, topologically (it's something more like Euclidean 3-space with a point at infinity added, which is topologically $S^3$ instead of $R^3$).

For the hyperbolic $k < 0$ case the topology at least can match ($R^3$ for both), but if I'm remembering what's in Hawking & Ellis correctly, the $k \rightarrow 0$ limit of that case still doesn't give you the flat $k = 0$ case. IIRC, a rough heuristic analogy using 1-dimensional curves in a plane goes something like this: for any hyperbola, if you try to match its points against the points of a straight line, one point of the hyperbola will be picked out as "special"--the one that lies closest to the line. But the "hyperbolic space" itself has to be homogeneous--all points the same. So there can't be any valid way to continuously convert the hyperbola to a straight line without violating the homogeneity property. Again, though, I may be misremembering the details.

I only meant that there was more willingness to use anything and everything.

"Anything and everything" only as regards to considering all possible values of $w$ in the standard equation of state for the cosmological fluid, $p = w \rho$. But that's still an extremely narrow range of possible equations of state. The reason a wider range of equations of state is not considered is simple: it would violate the homogeneity and isotropy assumptions that are basic to this family of models. But in modeling stars, for example, one has to consider equations of state that can't be expressed in that form. (Still less can you restrict yourself that way when modeling matter for purposes other than astronomy and cosmology.)

It was also possible to use more of the old stuff than ever before - dark matter as a way of pushing up the density of matter

Dark matter can't really be called "old stuff" in the sense of "a kind of matter we knew about before but didn't include in our cosmological models". Cosmologically speaking, dark matter is just matter--more precisely, it is anything that has $w = 0$ in the equation of state (i.e., zero pressure). And since we don't know what it is (we can't match up its properties with any known particle or field), we have no way of linking up the cosmological properties with anything else (the way, for example, that we can justify the zero pressure idealization for ordinary matter, or the way we derive $w = 1/3$ for the equation of state used in the cosmological models for radiation).

 

[kimbyd]

Summary:: (There doesn't seem to be an easy way to find similar threads without creating one and posting it, so I'll just post a question).
Standard texts exist that describe the time evolution of a FLRW universe given parameters such as density. Can the curvature parameter, k, change as the universe evolves? Can curvature change so much that the geometry of the spatial hypersurface changes character (spherical <--> flat <--> hyperbolic)?

Hi again. I'm still off work and struggling to learn some physics. I'm searching for discussion about the possibility of a universe changing its curvature as it evolves.

I'm still new here and so I'd be grateful for advice either about: (i) How to search for past discussions, or else, (ii) any discussion or pointers.

The sumary should describe the situation but I'll write it out again here, just in case:

Modelling the universe as a Robertson-Walker universe - can the spacetime change its curvature (as time progresses) so that the geometry of the spatial hypersurface changes from one type (e.g. spherical) to another (e.g. flat) over its course of evolution?

I can post more info if required.

The curvature in the FLRW universe is a constant value set by the initial conditions. It's determined by the relationship between the density of the universe and how fast it's expanding. The behavior of the universe with various curvatures is really easy to understand for a matter-dominated universe, because it relates closely to how more familiar gravity behaves.

If the rate of expansion is too slow to begin with compared to the density, then it's like throwing a ball on Earth, which goes up and comes back down. Such a universe eventually stops expanding and collapses back on itself.

Similarly, if the rate of expansion is really fast, the matter-dominated universe will go on expanding forever. This is directly analogous to throwing a ball fast enough for it to leave Earth's orbit.

Now, our universe doesn't behave like this. But that's because there's stuff in the universe that is extremely unfamiliar. In this case, dark energy.

How this relationship between expansion rate and density shows up as a spatial curvature is much more technical, and I don't know of a way to explain that beyond what the equations say. But General Relativity does say that a mismatch between the rate of expansion and density manifests as spatial curvature. Which can't change because it's just a constant set by the initial conditions.

All that said, the effect of the spatial curvature on the universe does change over time. Specifically, it scales as $1/a^2$, where $a$ is the scale factor. If the universe is dominated by normal matter, whose density scales as $1/a^3$, curvature becomes important only after the universe has expanded enough (how much depends upon the degree of curvature). If the universe is instead dominated by a cosmological constant, as ours appears to be, whose impact is independent of scale factor, curvature becomes less impactful at late times.

Our universe started out dominated by radiation, then matter, and now dark energy. Given our inability to measure a non-zero spatial curvature, this means that the spatial curvature of our observable universe has never been a significant factor in the way the universe expands (at early times, matter and radiation densities overwhelmed the effect of curvature, and now the cosmological constant prevents it having an impact in the future either).

 

[Will Learn]

@kimbyd
First and foremost, that was a remarkably beautiful and eloquent piece of writing. Thank you for your time and I'm sorry it's taken me a while to reply. That text could stand alone as a discussion of cosmology and if you haven't thought of writing for something like a science magazine then you should.

I especially like the acknowledgment that just Newtonian mechanics and Newtonian gravity is sufficient to derive the Friedmann equation for the simple case of a matter dominated universe. This sets a background against which the effects of things like dark energy can be understood and you've done that well.

I'm going to comment on a few sections later (maybe tomorrow) but I don't have enough time at the moment.

 

[kimbyd]

I'm glad it was useful! And yes, a matter-dominated universe is very much Newtonian. You can even derive the Friedmann equations in the exact same form.

As for writing, well, I have a well-paying day job. But thanks for the thought!

 

[Orodruin]

And yes, a matter-dominated universe is very much Newtonian. You can even derive the Friedmann equations in the exact same form.

I always found this a somewhat misleading statement. Even in the matter only Universe, the Universe is far from Newtonian with fundamentally different cosmological assumptions than what goes into the Newtonian ”derivation” of the Friedmann equations. The form of the equation is essentially given by dimensional analysis and I have yet to see a compelling argument that obtaining the correct numerical constant is anything but a happy coincidence.

 

[kimbyd]

I always found this a somewhat misleading statement. Even in the matter only Universe, the Universe is far from Newtonian with fundamentally different cosmological assumptions than what goes into the Newtonian ”derivation” of the Friedmann equations. The form of the equation is essentially given by dimensional analysis and I have yet to see a compelling argument that obtaining the correct numerical constant is anything but a happy coincidence.

I'm curious what fundamentally different assumptions you're referring to. As I understand it, the basic assumption at play is the universe being filled with a uniform fluid at some time-dependent density with no pressure and no cosmological constant.

Beyond that, the differences, as I understand them, are all about the fundamental differences between GR and Newtonian physics. Which results in things like the curvature parameter in the FLRW metric being about spatial curvature, and the equivalent parameter in the Newtonian derivation being the total kinetic + potential energy in a co-moving volume. And the Newtonian view certainly won't give a good answer for what an expanding universe actually looks like because there is no concept of curvature and it gets the behavior of light wrong.

But yes, I do think the fact that the equations match is probably a happy accident. One that I find both surprising and interesting. They certainly didn't need to match at all.

Edit: Interesting. The Newtonian derivation works for a cosmological constant as well, using the Newtonian limit described here: https://arxiv.org/abs/gr-qc/0004037.

 

[Orodruin]

As I understand it, the basic assumption at play is the universe being filled with a uniform fluid at some time-dependent density with no pressure and no cosmological constant.

The basic assumption in the Newtonian case is typically based on a spherically symmetric setting and collapse of that spherical system. It is worth noting that the Newtonian gravitational problem with a constant infinitely extended matter density generally does not allow an isotropic and homogeneous solution for the gravitational field.

 

[Will Learn]

To Kimbyd,

Thanks again and I meant what I said. There's nothing wrong with what you've written. I'm just a stubborn student trying to understand why things have to be a certain way.

The curvature in the FLRW universe is a constant value set by the initial conditions

Why? This is an assertion not a proof.
You say something similar again later:

But General Relativity does say that a mismatch between the rate of expansion and density manifests as spatial curvature. Which can't change because it's just a constant set by the initial conditions.

Yes to the first, my understanding of what G.R. says is summarised below. The second sentence is just an assertion not a proof. The purpose of the OP is to test that and understand why.

What does determine the curvature parameter?

It's determined by the relationship between the density of the universe and how fast it's expanding.

Yes exactly. If I can get LaTeX working properly then these expressions apply.
The density parameter is given by [Equation 1]:
$$\Omega = \frac {8 \pi G} {3 H^2} \rho = \frac {\rho} {\rho_{crit}}$$
and then the curvature*, $\kappa$ is found from [Equation 2]:
$$\kappa = ( \Omega - 1 ) (H^2a^2)$$

^{[Reference: Page 337, Spacetime and Geometry, Sean Carroll]} ​

Where: $\rho$ = Density ; $\rho_{crit}$ = Critical density ; H = Hubble parameter ; a = Scale factor

​

* This "Curvature" in [Equation 2] has been described as the "un-normalised curvature parameter" in previous posts. It is the curvature of a spatial hypersurface at fixed co-ordinate time.

The Hubble parameter and the density are involved, so your statement makes sense. In previous posts I said the curvature is determined by comparing the density to the critical density and we can use [Equation1] and [Equation 2] to describe the full set of relationships between these things that we are interested in.

We have the following relationships:
$\rho < \rho_{crit} \iff \Omega < 1 \iff \kappa <0 \iff \text {The universe is Open}$
$\rho = \rho_{crit} \iff \Omega = 1 \iff \kappa =0 \iff \text {The universe is Flat}$
$\rho > \rho_{crit} \iff \Omega > 1 \iff \kappa >0 \iff \text {The universe is Closed}$

The density, $\rho$ depends on the scale factor. You gave the example of matter density ~ a^-3 and there are other components of the cosmological fluid with different power laws relating their density to the scale factor, as usual.

In the general case, the scale factor varies with time. You can construct exceptions (like the Einstein static universe) if you want but the case with a constant scale factor is uninteresting and won't be pursued here.
In the general case, we can be confident that the total density evolves and by choosing a mixture of species in our cosmological fluid we even have considerable control over that change.

The critical density also depends on time.
$$\rho_{crit} = \frac {3 H^2} {8 \pi G}$$
and the Hubble parameter is seen to depend on time (in the general case).

So the comparison of density to critical density is dynamic. Their ratio, $\Omega$, varies with time. This doesn't seem to be in dispute in the texts I have read.

This is a slightly more simplified version of [Equation 2]:
$$\kappa = ( \Omega - 1 ) \dot a^2$$
Looking at the above or [Equation 2] we expect $\kappa$ to evolve because just about everything on the RHS does vary with time.

That the un-normalised curvature parameter can change with time doesn't seem to be in dispute from any of the previous posts in this thread, although I must state that it is not directly mentioned in any of the Cosmology texts I've seen. However, it's not that interesting provided the sign of the curvature doesn't change. The magnitude of the curvature always was arbitrary depending on the co-ordinates or units of length you choose to use, if you want to look at it that way.

Summary:
I believe that $\Omega$ and $\kappa$ certainly can change with time as the universe evolves. Therefore, I am seeking the proof that $\Omega$ cannot cross that boundary at $\Omega = 1$, or equivalently that the curvature cannot change its sign.
@PeterDonis has suggested a proof may be found in Hawking and Ellis but isn't sure where. I haven't had access to that book yet.

You went on to say:

All that said, the effect of the spatial curvature on the universe does change over time. Specifically, it scales as a^-2, where a is the scale factor.

That all seems reasonable, a quantity $\rho_c = \frac {- 3 \kappa} {8 \pi G a^2}$ can be introduced and is usually described as a fictitious energy density for the curvature. Re-writing the (first) Friedmann equation with this density contribution, the effect of curvature $\kappa$ falls off as ~a². That's what I have understood by your statement.

That describes the effect of curvature on the expansion of the universe. I'm not too concerned about the expansion of the universe, only on what the curvature is. So your statement, while correct, doesn't seem to relate to the OP.

An old "folk belief":
There was a time when the curvature parameter was synonymous with the ultimate fate of the universe. So talking about the expansion of the universe was exactly the same as talking about the curvature. However, modern cosmology has adapted since a vacuum energy component has been more widely accepted. This point was made earlier by @PeterDonis.

but our best current model of our universe has a small positive cosmological constant. For that case, the ultimate fates are somewhat different, and in particular the case k > 0 (spherical) will not necessarily recollapse to a Big Crunch.

I believe it is actually far more general then just applying to the case k>0. If I had a scanner I would copy figure 8.4 from Sean Carroll. I don't have a scanner, here's a quote:

Traditional disdain for the cosmological constant has lead to a folk belief that this [..the old correspondance between curvature and ultimate expansion fate..] is a necessary correspondance; once the possibility of vacuum energy is admitted, however, any combination of spatial geometry and eventual fate is possible.

^{[Page 343, Spacetime and Geometry, 3rd printing, Sean Carroll]}​

Of course, I understand that there is some interest in the expansion of the universe and its ultimate fate and I enjoyed reading your description of the situation. However, for the moment I'm just being a purist and wondering if the intrinsic geometry of the universe can change.

You also went on to say:

Given our inability to measure a non-zero spatial curvature, this means that the spatial curvature of our observable universe has never been a significant factor in the way the universe expands (at early times, matter and radiation densities overwhelmed the effect of curvature, and now the cosmological constant prevents it having an impact in the future either).

Yes, I agree with the essence of what is said. Just to clarify:
1. There ARE ways to measure the spatial curvature BUT it does seem to be very close to zero (flat) at the current time. An experiment that I'm interested in is a proposal to "draw" a triangle in space using lasers from a triangle of satellites.
2. I'm not too concerned about how important curvature is or was for expansion. I'm interested in the evolution of curvature just for the impact it has on geometry. For example, is a real triangle drawn in space going to have more or less than 180 degrees of internal angles and can that change with time? More-over, it doesn't even have to be our real universe but just any FRW universe.

Back to the OP:
I'm making good progress answering my own question by looking at the process of normalising the curvature parameter carefully. The continuity requirements seem to be the key. I've bored everyone else enough already and probably won't write anymore about that. I'm very grateful for all the time, attention and replies I have received on Physics Forums.

Best wishes, bye for now.

 

[kimbyd]

The basic assumption in the Newtonian case is typically based on a spherically symmetric setting and collapse of that spherical system. It is worth noting that the Newtonian gravitational problem with a constant infinitely extended matter density generally does not allow an isotropic and homogeneous solution for the gravitational field.

A homogeneous and isotropic universe is spherically-symmetric, at every location. We know from that symmetry and Gauss's Law that whatever sphere of the Universe we're looking at at any given time, the behavior of the system outside has no impact on the dynamics inside the sphere. From this we get a result which states how the density of that sphere will change over time. Since we could have used any origin and any size sphere to get this result, it applies to the entire universe.

With regard to the gravitational field, I'm not sure that's particularly relevant. It certainly is weird compared to how we use Newtonian gravity in other contexts, because the gravitational field is dependent upon our choice of origin. Usually the gravitational field in Newtonian mechanics is considered to be a fixed thing independent of the choice of origin.

But the gravitational field isn't something that is actually observable in the first place. It's more a mathematical tool that is useful because of the equivalence principle: the acceleration due to gravity of an object is independent of its mass. The actual observable things are those accelerations themselves. And every co-moving observer sees the same pattern accelerations of the matter around them.

To see this, consider an observer A who is sitting at the origin, and an observer B who is far-away and moving along with the expansion. Our gravitational field here is centered on A. What does B see? Well, B is in an accelerated frame, so they'll see the accelerations of matter calculated by A with their own acceleration subtracted out. The resulting accelerations are exactly the same as if we'd centered our gravitational field on B instead. This is a way of confirming what I said above about being able to pick any origin to get the same dynamics.

 

[Will Learn]

I always found this a somewhat misleading statement. Even in the matter only Universe, the Universe is far from Newtonian with fundamentally different cosmological assumptions than what goes into the Newtonian ”derivation” of the Friedmann equations. The form of the equation is essentially given by dimensional analysis and I have yet to see a compelling argument that obtaining the correct numerical constant is anything but a happy coincidence.

I can only point to the lectures that are available online from Leonard Susskind. In those Cosmology lectures, the Friedmann equations were derived entirely using Newtonian physics. GR was only used at the very end and just to show that the same equations appear. I think @kimbyd has made this point.



The ability to derive the Friedmann equations from Newtonian physics may be a little surprising but it's not unlike the derivation of an object that was called a "dark star" and has many of the same properties for the object we now describe as a Black hole using GR.

 

[PeterDonis]

In those Cosmology lectures, the Friedmann equations were derived entirely using Newtonian physics.

"Derived" is too strong a term, as it implies a rigorous proof. You can't rigorously derive any GR solution based on Newtonian physics. A better term would be "gave a heuristic plausibility argument".

 

[Will Learn]

@PeterDonis
Umm.. OK.
If the aim isn't to derive a GR solution but just to derive a set of formulae that does seem to describe the expansion of the universe, then you CAN derive that from Newtonian physics.

What is more surprising is that I don't believe anyone did derive those equations from Newtonian physics until AFTER the results were becoming apparent from G.R. and the work of Friedmann.

 

[PeterDonis]

This is an assertion not a proof.

It's an assertion based on understanding the properties of FRW spacetime. Those properties are well known and understood and have been for decades.

Basically, you are saying you, personally, don't understand those properties well enough to see why @kimbyd's statement is true. So to you, it's "an assertion not a proof". But just saying that as a bald statement isn't going to make much progress in this discussion. Nor is continuing to just say you don't understand why certain things are the case, when you haven't taken the time to go build a better understanding of how FRW spacetime works on your own. If you are expecting to get all the information you need to improve your understanding from this thread, you are expecting too much. None of us get paid for this, and anyway just being told things won't make you understand them. You have to do the hard work of building your own understanding. The best we can do is try to point the way.

The purpose of the OP is to test that and understand why.

Perhaps that was the purpose of your OP in this thread, but it's not what the purpose of this thread has evolved into.

The simple fact is that, if we consider a single spacetime model from the class of FRW spacetimes, the normalized spatial curvature parameter (i.e., either $+1$, $0$, or $-1$) is a constant for the model. If you want a "proof" of that simple statement, you will need to take some time to consult a textbook on cosmology that derives the FRW solutions from first principles, so you can understand how that derivation makes the statement true. Doing that explicitly here in this thread is well beyond the scope of what a PF discussion is supposed to be. I suggest Sean Carroll's online lecture notes on GR as a starting point; IIRC he does the derivation in a fairly straightforward manner.

So the only way to construct a model in which the normalized curvature parameter can change is to take portions of two different FRW spacetime models that have two different parameters, and "glue" them together along some boundary. The obvious choice of boundary would be some particular spacelike hypersurface: roughly speaking, such a model would have curvature parameter $k_1$ up to some cosmological time $t$, and then would have curvature parameter $k_2 \neq k_1$ after that time. In post #16, for example, that was the kind of model I was thinking of when I talked about whether it's possible to match the two parameters continuously.

 

[PeterDonis]

If the aim isn't to derive a GR solution

The Friedmann equations are a GR solution.

just to derive a set of formulae that does seem to describe the expansion of the universe

Only in a very restricted sense, and only if you are willing to ignore the issues that have been raised about this already in this thread.

 

[PeterDonis]

A homogeneous and isotropic universe is spherically-symmetric, at every location.

But the converse is not the case: not every spherically symmetric solution is homogeneous and isotropic. The point @Orodruin was making is that in Newtonian physics, it is impossible to have a homogeneous and isotropic solution. It is of course possible to have a spherically symmetric solution in Newtonian physics, but only about one point, not about every point.

 

[Orodruin]

A homogeneous and isotropic universe is spherically-symmetric, at every location. We know from that symmetry and Gauss's Law that whatever sphere of the Universe we're looking at at any given time, the behavior of the system outside has no impact on the dynamics inside the sphere. From this we get a result which states how the density of that sphere will change over time. Since we could have used any origin and any size sphere to get this result, it applies to the entire universe.

This common belief is wrong as the requirements for only the inside the sphere to govern the dynamics are not satisfied. The requirement is that the boundary conditions at infinity do not spoil the symmetry - which is true when you consider a potential that tends to zero at infinity. However, the Poisson equation with a constant density term is incompatible with any solution where the symmetry is maintained. This implies external fields - givenby the boundary conditions - also exist inside the sphere. Compare with adding an external electric field to the field of a soherical charge distribution. The external field will certainly affect the dynamics of a test charge inside the sphere.

I can only point to the lectures that are available online from Leonard Susskind. In those Cosmology lectures, the Friedmann equations were derived entirely using Newtonian physics. GR was only used at the very end and just to show that the same equations appear. I think @kimbyd has made this point.



The ability to derive the Friedmann equations from Newtonian physics may be a little surprising but it's not unlike the derivation of an object that was called a "dark star" and has many of the same properties for the object we now describe as a Black hole using GR.

Those are usually heuristic arguments. Regardless, it is not the same Friedmann equation as it is based on Newtonian physics. There is no a priori reason to expect the exact same result in a FLRW universe.

 

[kimbyd]

To Kimbyd,

Thanks again and I meant what I said. There's nothing wrong with what you've written. I'm just a stubborn student trying to understand why things have to be a certain way.Why? This is an assertion not a proof.

This is very true. The full derivation is very complicated, and I haven't done it for a number of years. But basically when you try to describe a homogeneous and isotropic universe, you perform an integral at some point which results in a constant of integration. That constant turns out to be the spatial curvature.

If you tried to make it not constant, you wouldn't have a valid solution to Einstein's equations any longer.

 

[kimbyd]

This common belief is wrong as the requirements for only the inside the sphere to govern the dynamics are not satisfied. The requirement is that the boundary conditions at infinity do not spoil the symmetry - which is true when you consider a potential that tends to zero at infinity. However, the Poisson equation with a constant density term is incompatible with any solution where the symmetry is maintained. This implies external fields - givenby the boundary conditions - also exist inside the sphere. Compare with adding an external electric field to the field of a soherical charge distribution. The external field will certainly affect the dynamics of a test charge inside the sphere.

I seem to remember this. My take-away is generally just that the application of Gauss's Law might not have worked in this situation, due to the diverging gravitational potential at infinity. But it does: Newtonian gravity with this construction produces the exact same result as General Relativity. Thus the effects at infinity didn't spoil the solution when using a spherical construction.

My guess is that it's probably down to the fact that the spherical construction respects the homogeneity and isotropy conditions.

 

[PeterDonis]

the application of Gauss's Law might not have worked in this situation, due to the diverging gravitational potential at infinity. But it does

Doesn't the Gauss's Law integral for this case diverge?

 

[Orodruin]

I seem to remember this. My take-away is generally just that the application of Gauss's Law might not have worked in this situation, due to the diverging gravitational potential at infinity. But it does: Newtonian gravity with this construction produces the exact same result as General Relativity. Thus the effects at infinity didn't spoil the solution when using a spherical construction.

My guess is that it's probably down to the fact that the spherical construction respects the homogeneity and isotropy conditions.

The condition broken is homogeneity. By construction the spherical construction is isotropic. It is also not Gauss law that breaks down but the spherical shell theorem. That it by chance reproduces the correct result is not necessarily a sign of the derivation showing the same.

It may be possible to derive the same dynamic due to tidal forces regardless of the boundary condition at infinity (I have not had the time to look into this). This seems plausible to me as the volume change due to tidal forces should be proportional to a second derivative of the potential and the only such derivative is the Laplace operator. However, this is still a heuristic argument for obtaining the result, it does not show that the same equation is obtained in GR, and the solution is not necessarily isotropic and homogeneous (even if the density is).