B How can the Universe be infinite and yet have a finite age?

[RingNebula57]

How can the universe be infinite and yet have a finite age?

 

[phinds]

How can the universe be infinite and yet have a finite age?

It could have started off infinite. We don't know whether it is finite or infinite but whichever it is, that's how it started off.

 

[bapowell]

How can the universe be infinite and yet have a finite age?

The observable universe has a finite age, and it has a finite size. We know nothing of the universe outside what we can observe.

 

[RingNebula57]

Ok ,thank you! I will do more research on this topic.

 

[laymanB]

I thought I would resurrect ones of these infinite universe threads to make some points and hear feedback.

Is it not the case in physics that when we get infinities as a solution to our equations, it usually indicates a problem? For instance, we say that GR cannot be the whole story because the equations give us singularities with infinite densities at the beginning of the universe and in the center of black holes. Why do we think that this indicates a problem with the theory and that we need new physics, as opposed to saying that there is indeed infinite density there?

Let me take this idea into the question of whether the universe is spatially infinite or finite. We have confirmed that the local geometry is flat with small error bars. We don't observe anything that would lead us to believe that the local topology is anything besides Euclidean space. So, we plug these observations into a FRW model and the solution that we get out is a spatially infinite universe.

Now my question is: why don't we think this points to something wrong with the model, instead of embracing this infinity and trying to get rid of the other infinities (like infinite densities). It seems to me in the history of physics, that trying to get rid of infinities has led to meaningful progress.

 

[phinds]

Is it not the case in physics that when we get infinities as a solution to our equations, it usually indicates a problem?

Usually, yes

For instance, we say that GR cannot be the whole story because the equations give us singularities with infinite densities at the beginning of the universe and in the center of black holes. Why do we think that this indicates a problem with the theory and that we need new physics, as opposed to saying that there is indeed infinite density there?

Because infinite density makes no physical sense.

Now my question is: why don't we think this points to something wrong with the model, instead of embracing this infinity and trying to get rid of the other infinities (like infinite densities).

Because infinite extent does not seem to be physically impossible.

 

[PeroK]

I thought I would resurrect ones of these infinite universe threads to make some points and hear feedback.

Is it not the case in physics that when we get infinities as a solution to our equations, it usually indicates a problem? For instance, we say that GR cannot be the whole story because the equations give us singularities with infinite densities at the beginning of the universe and in the center of black holes. Why do we think that this indicates a problem with the theory and that we need new physics, as opposed to saying that there is indeed infinite density there?

Let me take this idea into the question of whether the universe is spatially infinite or finite. We have confirmed that the local geometry is flat with small error bars. We don't observe anything that would lead us to believe that the local topology is anything besides Euclidean space. So, we plug these observations into a FRW model and the solution that we get out is a spatially infinite universe.

Now my question is: why don't we think this points to something wrong with the model, instead of embracing this infinity and trying to get rid of the other infinities (like infinite densities). It seems to me in the history of physics, that trying to get rid of infinities has led to meaningful progress.

You are confusing two different concepts of infinity. Let's look at the mathematics. There is nothing inherently problematic with an "infinite" set - or, more precisely, an "unbounded" set. The simplest example is:

$\mathbb{N} = \{1, 2, 3 \dots \}$

In mathematical terms, that set not only has an infinite number of elements, but the distance between any pair of elements is unbounded.

Note that when it comes to cosmology, physicists tend to use "infinite" or "of infinite extent" to mean "unbounded" in the mathematical sense.

Infinitity, however, turns up again in mathematics in other cases. E.g.:

$\lim_{t \rightarrow 0^+} \frac{1}{t} = +\infty$

This is problematic if you want to assign a value at $t = 0$. And, if that limit came from a model of a physical process, then there is a "singularity" at the point $t = 0$. In the sense that you cannot assign a finite value at that point.

It is these "singularities" that indicate a problem with the model.

 

[Chronos]

Infinite density creates an issue because the volume of a body with a non-zero finite can only be zero. That strikes most scientists as unphysical and just plain wrong. Since all known black hole have a finite mass, it is difficult to avoid concluding they must also occupy a non-zero volume. It's not just black holes that have this problem, it also pops up all the time in particle physics. As soon as you wedge a finite amount of anything into zero volume you get an infinite density of whatever it is you just tried to stuff into a size zero sack, be it bull droppings or the charge of an electron. Strictly speaking mathematically the rule is the result of division by zero is undefined [i.e., nonsense]. So the best answer for the density of matter in a black hole singularity is "undefined". Undefined is not a problem until you try to impose zero as the only allowable option for volume. For a black hole the obvious choice for volume is defined by its event horizon - considering, by definition, a black hole is any region of space where escape velocity equals or exceeds c. The same thing applies to the volume of the universe. We do not kinow if it truly goes on forever. Fortunately, that's only a guess [its hard to imagine what would constitute a spatial boundary on the universe] The easy way out of this mess is to assume it only extends as far as we can see - and we already know that cannot exceed the distance light could have traveled since the universe began. Answers are easier when you poae the right questions.

 

[laymanB]

In mathematical terms, that set not only has an infinite number of elements, but the distance between any pair of elements is unbounded.

Note that when it comes to cosmology, physicists tend to use "infinite" or "of infinite extent" to mean "unbounded" in the mathematical sense.

This is part of what I'm trying to make sense of. The place where mathematical abstraction maps onto physical realities. Let's say that we consider a square with sides of length 1. We know that the length of the diagonals are $\sqrt{2}$. We know that $\sqrt{2}$ is an irrational number and that the decimal places never end nor repeat. But don't we conclude that the length of the diagonal is a discrete length with end points in reality when we build this square out of wood or some other material?

We can ask a mathematician to show us different types of infinities, discuss the Hilbert Hotel, and talk about the continuum hypothesis, but do any of these mathematics give us justification for discussing infinities in physics? I'm thinking along the lines of Zeno's paradox, where only traveling half the distance between two points never gets us to the end point. While it is true if you can only travel by halving distances, real world particles are under no such constraint. They simply traverse the whole distance.

 

[Chris Miller]

I was going to begin a new thread, but seeing this, I think I'll just post my questions here:

According to Sylvia Nasar's biography of John Nash ("A Beautiful Mind"), Nash once presented Einstein with some mathematics theorizing a flat, non-expanding universe. Einstein seemed to patronize him, suggesting he "study physics." Now, from what I gather here, Nash's theory was closer to current cosmological thinking than Einstein's at the time. Am I mistaken in my understanding that the universe is now believed to have begun as an infinitely large, dense mass? That matter has fragmented and is separating according to Hubble's constant, but that space itself has always been and remains infinite? That only our infinitesimal observable portion of the universe is actually growing?

 

[PeroK]

This is part of what I'm trying to make sense of. The place where mathematical abstraction maps onto physical realities. Let's say that we consider a square with sides of length 1. We know that the length of the diagonals are $\sqrt{2}$. We know that $\sqrt{2}$ is an irrational number and that the decimal places never end nor repeat. But don't we conclude that the length of the diagonal is a discrete length with end points in reality when we build this square out of wood or some other material?

We can ask a mathematician to show us different types of infinities, discuss the Hilbert Hotel, and talk about the continuum hypothesis, but do any of these mathematics give us justification for discussing infinities in physics? I'm thinking along the lines of Zeno's paradox, where only traveling half the distance between two points never gets us to the end point. While it is true if you can only travel by halving distances, real world particles are under no such constraint. They simply traverse the whole distance.

You have infinite sets at the heart of physics, because calculus relies on the real numbers. You can't do (normal) calculus on finite sets of points. And, it is problematic to try to model spacetime as a discrete set (of points a non-zero distance apart).

I don't see the relevance of Zeno's Paradox. Which is fairly feeble in any case, IMO.

 

[laymanB]

You have infinite sets at the heart of physics, because calculus relies on the real numbers. You can't do (normal) calculus on finite sets of points. And, it is problematic to try to model spacetime as a discrete set (of points a non-zero distance apart).

Good point.

I don't see the relevance of Zeno's Paradox. Which is fairly feeble in any case, IMO.

Sorry, this was non-linear rambling on my part. Let me try again.

If we are talking about the escape velocity of an object, what is the physical meaning of the object slowing as it approaches infinity. In other words, does infinity have to exist to launch something with escape velocity? Sorry if this is still muddled.

 

[phinds]

Good point.Sorry, this was non-linear rambling on my part. Let me try again.

If we are talking about the escape velocity of an object, what is the physical meaning of the object slowing as it approaches infinity. In other words, does infinity have to exist to launch something with escape velocity? Sorry if this is still muddled.

As far as that sort of thing is concerned, infinity is just a mathematical fiction and does not need to exist to do the computation.

 

[PeroK]

If we are talking about the escape velocity of an object, what is the physical meaning of the object slowing as it approaches infinity. In other words, does infinity have to exist to launch something with escape velocity? Sorry if this is still muddled.

This is a good example, IMO, of how turning the question round to use finite quantities resolves the matter.

If you launch a rocket at a speed $v$ and it returns to Earth in a finite time, then $v$ is less than the escape velocity. Otherwise, $v$ is greater than or equal to the escape velocity.

Put another way, you can set up an equation for the time the rocket returns. If that equation has no solution, then the velocity is greater than or equal to the escape velocity, which then is the lowest velocity for which the equation has no solution.

No need to mention the "i" word!

 

[PeterDonis]

you can set up an equation for the time the rocket returns. If that equation has no solution, then the velocity is greater than or equal to the escape velocity, which then is the lowest velocity for which the equation has no solution.

No need to mention the "i" word!

Not explicitly, but it's still there implicitly, in the derivation of the equation you set up. That equation implicitly assumes that space is infinite. If you drop that assumption and substitute the assumption that space is a 3-sphere with some finite total 3-volume, you can find solutions with the same initial conditions where you couldn't before.

 

[laymanB]

What, if any, are the implications of a spatially infinite universe at t > 0? Let me elaborate.

Am I correct in saying that the energy density, the stress-energy tensor, and therefore the spacetime curvature was extreme in the first second of the Big Bang?

Assuming this is right, then what are the implications if that energy density is extended into infinity versus being finite? Does it put constraints on the scalar field for inflation? Does it make it much more likely that the spacetime curvature will result in an immediate collapse?

 

[PeterDonis]

from what I gather here, Nash's theory was closer to current cosmological thinking than Einstein's at the time

Why do you think this? Current cosmological thinking is that the universe is spatially flat and expanding, not spatially flat and non-expanding.

 

[PeterDonis]

Am I correct in saying that the energy density, the stress-energy tensor, and therefore the spacetime curvature was extreme in the first second of the Big Bang?

Yes.

what are the implications if that energy density is extended into infinity versus being finite? Does it put constraints on the scalar field for inflation?

AFAIK, no; any constraints on the inflaton field are local, not global; they don't "care" whether the universe is globally finite or infinite, spatially.

Does it make it much more likely that the spacetime curvature will result in an immediate collapse?

No. The equations governing this are also local, not global.

 

[PeroK]

What, if any, are the implications of a spatially infinite universe at t > 0? Let me elaborate.

Am I correct in saying that the energy density, the stress-energy tensor, and therefore the spacetime curvature was extreme in the first second of the Big Bang?

Assuming this is right, then what are the implications if that energy density is extended into infinity versus being finite? Does it put constraints on the scalar field for inflation? Does it make it much more likely that the spacetime curvature will result in an immediate collapse?

If you are already on Chapter 22 of Hartle, you must have bashed through Chapters 1-21 fairly quickly!

 

[laymanB]

If you are already on Chapter 22 of Hartle, you must have bashed through Chapters 1-21 fairly quickly!

Ha! I am much lazier than you think. I'm only about a quarter of the way through, and haven't touched it for the last month or two. I'm trying to get more big picture ideas while trying to learn the theories. Instead of going through the standard protocol of learning the math and foundations first, before moving on to more complex subjects. I would not advise it for a college curriculum.

 

[Chris Miller]

Why do you think this? Current cosmological thinking is that the universe is spatially flat and expanding, not spatially flat and non-expanding.

I think this, because I find it hard to conceive of infinite, flat space expanding. Really, as the poster here has noted very well, I think, the term "physical infinity" is an oxymoron. Certainly opens up some philosophical cans of worms.

 

[PeroK]

I think this, because I find it hard to conceive of infinite, flat space expanding. Really, as the poster here has noted very well, I think, the term "physical infinity" is an oxymoron. Certainly opens up some philosophical cans of worms.

In these matters you have to be objective. Even if you are appalled by the possibility that the universe is spatially infinite, that is no reason to oppose or doubt the idea.

 

[PeterDonis]

I think this, because I find it hard to conceive of infinite, flat space expanding.

To be blunt, so what? We have a perfectly consistent model of a flat, expanding universe. There is no requirement that the model be easy for you, or anyone, to conceive of, just that it be consistent and make correct predictions. Our current best fit model does.

Besides that, the claim of yours that I questioned was that Nash's proposed model was closer to current cosmological thinking than Einstein's. That has nothing to do with whether or not you personally find current cosmological thinking hard to conceive of. What is your basis for your claim about Nash's model vs. Einstein's?

 

[Chris Miller]

To be blunt, so what? We have a perfectly consistent model of a flat, expanding universe. There is no requirement that the model be easy for you, or anyone, to conceive of, just that it be consistent and make correct predictions. Our current best fit model does.

Besides that, the claim of yours that I questioned was that Nash's proposed model was closer to current cosmological thinking than Einstein's. That has nothing to do with whether or not you personally find current cosmological thinking hard to conceive of. What is your basis for your claim about Nash's model vs. Einstein's?

Thank you for addressing my concerns/confusion. Of course, what I think/believe/can or can't get my head around, is irrelevant. My assertion re Nash's model was based entirely on posts I've read here pertaining to an infinite, flat, expanding universe, by persons more educated than myself.

 

[PeterDonis]

My assertion re Nash's model was based entirely on posts I've read here pertaining to an infinite, flat, expanding universe, by persons more educated than myself.

Can you give some references? An infinite, flat, expanding universe is our current best thinking; but your previous post said that Nash's model postulated an infinite, flat, non expanding universe (and that's consistent with what little I know of the conversation you referenced between Nash and Einstein). But an infinite, flat, non expanding universe is inconsistent with the Einstein Field Equation: there is no such solution. That might possibly be what Einstein meant when he suggested that Nash needed to learn more physics; but I'm just speculating since I don't know the actual content of their conversation.

 

[kimbyd]

I think this, because I find it hard to conceive of infinite, flat space expanding. Really, as the poster here has noted very well, I think, the term "physical infinity" is an oxymoron. Certainly opens up some philosophical cans of worms.

One of the problems with this kind of question is that "physically infinite" and "very, very large" are indistinguishable when it comes to predictions about what we might be able to observe. That makes it the kind of question that most people working in physics aren't likely to be very interested in. Usually physicists work with models that are physically infinite because it makes the math simple, and avoid making strong claims about whether or not it's actually true. If they work with a finite universe model, it's usually to try to come up with ways of measuring the features of that model (e.g. if our universe was a 3-sphere with a sufficiently small radius, we might be able to measure its curvature).

Most theorists are really, really reluctant to say "X is impossible" based solely upon philosophical arguments. The reason for this is simple: maybe it is impossible, but maybe there's some loophole we haven't yet conceived of.

A physically-infinite universe is certainly very weird, but that's not necessarily a guide to truth. Quite a lot about the universe that has been verified experimentally is exceedingly weird.

For now, I think the best answer is, "We don't know. Maybe it's infinite. Maybe it's finite. We don't yet have solid models validated by observation which give us a strong indication one way or another."

Finally, let me just point out that we also don't know whether our universe is finite in time. Certainly there was an event nearly 14 billion years in our past which kicked off the region of the universe we can observe. But that doesn't mean that nothing came before. Time might certainly be infinite into the past.

 

[phinds]

A physically-infinite universe is certainly very weird,

No more weird than a finite universe and in some ways a lot more rational. I mean if it's finite, what's its shape? What's its extent?

 

[kimbyd]

No more weird than a finite universe and in some ways a lot more rational. I mean if it's finite, what's its shape? What's its extent?

This comment highlights the problem: without observation to guide us, what seems 'weird' often comes down to personal interpretation.

 

[laymanB]

For now, I think the best answer is, "We don't know. Maybe it's infinite. Maybe it's finite. We don't yet have solid models validated by observation which give us a strong indication one way or another."

So essentially we are left to conclude that the universe is much larger than the volume that we can observe, but the observational evidence gives greater credence to the hypothesis that the global geometry is flat and infinite in extent, assuming isotropy and homogeneity for the universe as a whole?

Do you foresee any future experiments or modifications to theory that would give greater confidence to this hypothesis?

 

[phinds]

Do you foresee any future experiments or modifications to theory that would give greater confidence to this hypothesis?

My understanding is that current measurements give a specific indication to within the tolerance of our measuring ability. It's likely that our measuring ability will increase at least somewhat which will likely just make the current beliefs more solid but COULD begin to lean the other way. "The other way" would be a bit of a surprise since out of all the values that "flatness" could have, it would just be downright weird that the actual value would be "really close to flat but not quite flat" instead of just flat.

 

[rootone]

Considering that only a century ago it became realized that there was more to the Universe than the MW galaxy, I expect there is still a lot to learn.

 

[timmdeeg]

A physically-infinite universe is certainly very weird, but that's not necessarily a guide to truth. Quite a lot about the universe that has been verified experimentally is exceedingly weird.

As far as I remember, but I have no reference, most cosmologists don’t think about weird, rather they tend to believe that the universe is infinite simply because a trivial topology seems more obvious than e.g. a 3-torus.

 

[timmdeeg]

Let me take this idea into the question of whether the universe is spatially infinite or finite. We have confirmed that the local geometry is flat with small error bars. We don't observe anything that would lead us to believe that the local topology is anything besides Euclidean space. So, we plug these observations into a FRW model and the solution that we get out is a spatially infinite universe.
.

This conclusion is not correct. If the universe is spatially flat it isn’t neccessarily spatially infinite, its topology can be compact also, e.g. a 3-torus is spatially flat and if large enough we will never be able to confirm it by observation.
The FRW model yields the dynamics of the universe, not its topology.

 

[PeterDonis]

they tend to believe that the universe is infinite simply because a trivial topology seems more obvious than e.g. a 3-torus.

That's not quite true. If the universe has flat 3-torus topology, there is a lower limit to the "size" of the 3-torus, because if it were small enough, we would see multiple images of distant galaxies coming from different directions. We don't. So the "size" of the 3-torus would have to be much larger than the size of the observable universe.

 

[timmdeeg]

That's not quite true. If the universe has flat 3-torus topology, there is a lower limit to the "size" of the 3-torus, because if it were small enough, we would see multiple images of distant galaxies coming from different directions. We don't. So the "size" of the 3-torus would have to be much larger than the size of the observable universe.

I have mentioned in post #33 that “a 3-torus is spatially flat and if large enough we will never be able to confirm it by observation” and was anticipating in my previous post that comologists would “prefer” a trivial topology as there are no oberservational indications for a compact topology. I’ve read this somewhere, but can’t find it. If I understand it correctly such reasoning makes sense because “trivial” in this case means less assumptions however would be glad to know your thoughts in this matter.

 

[laymanB]

This conclusion is not correct. If the universe is spatially flat it isn’t neccessarily spatially infinite, its topology can be compact also, e.g. a 3-torus is spatially flat and if large enough we will never be able to confirm it by observation.

I'm confused. I thought that a 3-torus was a closed manifold, meaning that it is spatially finite? And that the only FRW spacetimes with Euclidean space were spatially infinite? What am I missing here?

The FRW model yields the dynamics of the universe, not its topology.

Thanks. Yeah, I'm still learning and trying to get the correct terminology. So what yields the global geometry of the universe? Is it just the RW metric before using the Einstein Equations(EE)? Are topology and geometry synonyms?

 

[PeterDonis]

the only FRW spacetimes with Euclidean space were spatially infinite?

That's how it's often phrased, but it's not strictly correct. The strictly correct statement is that the only FRW spacetimes with flat (Euclidean) spacelike slices of constant FRW coordinate time with trivial topology (i.e., topology $R^3$) are spatially infinite.

what yields the global geometry of the universe?

Not geometry, topology. The Einstein Field Equation can't tell you the global topology of a spacetime, because the EFE is local; it relates spacetime curvature to stress-energy in a small local region of spacetime. If there are multiple topologically different ways of extending that local region into a global spacetime, the EFE cannot distinguish between them.

 

[PeterDonis]

“trivial” in this case means less assumptions

Not really less assumptions; assuming any global topology, including $R^3$ (for spacelike slices--$R^4$ for spacetime), is an assumption. I think the assumption of global $R^3$ spatial topology seems more parsimonious to cosmologists because any local region of spacetime that can be covered by a single coordinate chart without coordinate singularities or ad hoc restrictions on the ranges of the coordinates must have spacetime topology $R^4$, and any spacelike slice of it must have spatial topology $R^3$. Therefore those topologies are the obvious ones to assume for the global topology, except in particular cases in which a global $R^4$ topology is impossible (for example, maximally extended Schwarzschild spacetime); but the only FRW spacetime in which that is the case is the one with positive spatial curvature, which can only have global topology $S^3 \times R$.

 

[timmdeeg]

I'm confused. I thought that a 3-torus was a closed manifold, meaning that it is spatially finite? And that the only FRW spacetimes with Euclidean space were spatially infinite? What am I missing here?

Yes, a 3-torus is spatially finite and flat. “Compact” topology means spatially finite. And no, FRW spacetimes which are spatially flat (euclidean) can be spatially finite or infinite.

So what yields the global geometry of the universe? Is it just the RW metric before using the Einstein Equations(EE)? Are topology and geometry synonyms?

The global geometry or perhaps better the topology of the universe can not be calculated based on the knowledge of its ingredients (the various energy densities) which determine the local spatial curvature, spherical, euclidean or hyperbolic. One needs observational data, as mentioned by @PeterDonis in his previous post.
Topology and geometry are not synonymus. I think topology requires constant local curvature but am not sure if this is the whole story and would leave that to experts around here.

 

[PeterDonis]

I think topology requires constant local curvature

I'm not sure what you mean by this. Topology does not imply any particular curvature. A spacetime with global topology $R^4$, for example, could be flat Minkowski spacetime, or it could be a curved FRW spacetime with critical density matter and flat infinite spacelike slices, or it could be a different curved FRW spacetime with sub-critical density matter and open (hyperbolic) infinite spacelike slices, or it could be de Sitter spacetime, with just a positive cosmological constant everywhere and nothing else. Or it could be, as our best current model of the actual universe is, a combination of the second and fourth alternatives I gave just now.

 

[timmdeeg]

I'm not sure what you mean by this. Topology does not imply any particular curvature.

Hm, with “constant local curvature” I meant that the curvature is everywhere the same regardless the “particular” curvature.

 

[PeterDonis]

with “constant local curvature” I meant that the curvature is everywhere the same regardless the “particular” curvature.

Topology has nothing to say about that. A spacelike slice could have topology $R^3$, or $T^3$, the 3-torus, for that matter, and still not have the same curvature everywhere. There are some particular restrictions that topology can place on possible curvatures: for example, no sphere topology ($S^n$ for any $n$) can be flat (have zero curvature everywhere), and the 2-torus, $T^2$, also cannot be flat (but the 3-torus can). But those restrictions don't restrict very much--for example, there's nothing preventing a manifold with topology $S^3$ from having curvature that varies from point to point, or even from being flat in some finite region (just not everywhere).

 

[laymanB]

or it could be a curved FRW spacetime with critical density matter and flat infinite spacelike slices

If we take a spacelike slice of the universe now in this constant FRW coordinate time and at the time of the CMB, does this let us extrapolate anything about the global topology?

Or is it that we cannot fully determine whether the topology of these spacelike slices are trivial or not?

 

[PeterDonis]

If we take a spacelike slice of the universe now in this constant FRW coordinate time and at the time of the CMB, does this let us extrapolate anything about the global topology?

We can't "take a spacelike slice" because we can't observe an entire spacelike slice; we can only observe the portion of it that is in our past light cone. That is the fundamental restriction that prevents us from proving that the global topology is one thing or another. We can only collect data and place limits, such as, if the global topology of a spacelike slice is a 3-torus, or a 3-sphere, instead of $R^3$, its "size" (roughly the maximum possible distance between distinct points in the slice) must be much larger than the size of our observable universe (which is what we can see in our past light cone, and which looks like flat Euclidean space with no sign of non-trivial topology).

 

[laymanB]

We can't "take a spacelike slice" because we can't observe an entire spacelike slice; we can only observe the portion of it that is in our past light cone. That is the fundamental restriction that prevents us from proving that the global topology is one thing or another. We can only collect data and place limits, such as, if the global topology of a spacelike slice is a 3-torus, or a 3-sphere, instead of $R^3$, its "size" (roughly the maximum possible distance between distinct points in the slice) must be much larger than the size of our observable universe (which is what we can see in our past light cone, and which looks like flat Euclidean space with no sign of non-trivial topology).

I see. Thanks.

 

[kimbyd]

So essentially we are left to conclude that the universe is much larger than the volume that we can observe, but the observational evidence gives greater credence to the hypothesis that the global geometry is flat and infinite in extent, assuming isotropy and homogeneity for the universe as a whole?

The universe is much larger than the volume we observe (most likely a few hundred times larger at the minimum). There is no observational preference whatsoever given to "infinite" vs. "very big".

Do you foresee any future experiments or modifications to theory that would give greater confidence to this hypothesis?

Our only real chance is learning more about the event that kicked off our early universe. If learning more about that event doesn't tell us, then there's probably no way to know.

 

[laymanB]

If we can calculate the density parameter $\Omega$ was even closer to 1 in the very early universe than it is now, would this not rule out a finite, closed global topology? In other words, if the global topology was an spherical or hyperbolic topology now, shouldn't we be able to see that by calculating $\Omega$ for the early universe? Is this possible?

 

[PeterDonis]

If we can calculate the density parameter $\Omega$ was even closer to 1 in the very early universe than it is now, would this not rule out a finite, closed global topology?

No. The density parameter being 1, even if it is exactly 1, does not specify the global topology. It might rule out certain topologies (such as $S^3$), but does not narrow it down to only one.

 

[kimbyd]

If we can calculate the density parameter $\Omega$ was even closer to 1 in the very early universe than it is now, would this not rule out a finite, closed global topology? In other words, if the global topology was an spherical or hyperbolic topology now, shouldn't we be able to see that by calculating $\Omega$ for the early universe? Is this possible?

The topology isn't really measurable.

It could be measurable if the universe wrapped back on itself near the horizon, but current observations seem to rule this out.

In the end, the topology is just the overall connectedness of the universe. If you have a flexible surface, then anything you can do to that surface that doesn't cause a cut or tear (stretching, contracting, bulging outward, etc.) will not do anything to change the topology. If that surface is the surface of a sphere, then no matter what you do to it it will still be a spherical topology. If it's the surface of a torus, it will always have a toroidal topology.

This means that unless the universe wraps back on itself pretty close to (or within) the cosmological horizon, any local geometry will be consistent with any global topology. Since we don't see the impact of the topology within the observable universe, we can't say anything about it.

Our one hope of learning something about the topology would be learning more about the very early universe. But that may also tell us nothing.

 

[laymanB]

Thanks everybody.