# Shape of the universe

Hardly. The cosmological principle holding to infinity requires an infinite degree of fine-tuning: how did the universe, out to infinite distances, know to be the same density in all locations, with the appropriate time-slicing?

It's rather like the horizon problem, but expanded to infinite distances instead of merely being required to hold in our visible universe.

Another way of stating the problem is to look at the classic model of inflation. If inflation were extended infinitely into the past, then inflation could easily explain a global cosmological principle. However, we know that can't be the case: extending inflation infinitely into the past also requires infinite fine-tuning: inflation predicts a singularity somewhere in the finite past, and the further back you try to push that singularity, the more fine-tuning you need. And if inflation can only be extended a finite distance back into the past, then it isn't possible for the universe as a whole to have reached any sort of equilibrium density, as if you go far enough away, you'll eventually reach locations that have always, since the start of inflation, been too far for light to reach one another. Any regions of the universe that lie beyond this distance aren't likely to be remotely close to one another in density.

Of course, this argument is based upon the assumption that a simplistic model of inflation is true, but the argument is reasonably-generic among most inflation models.
If time to the singularity is considered finite it can never require an infinite degree of fine tuning, but I can see how this could conflict with a spatially infinite geometry together with a global CP as I mentioned in my previous post.
But I just can't understand how exactly the horizon problem came to be considered a problem in the first place(unless the above mentioned conflict was also evident at the time), the three possible space geometries allowed by the FRW model are both geometrically and physically homogeneous by definition, so if one is going to use this model one is assuming that homogeneity and shouldn't worry about causal contact.

Chalnoth
But I just can't understand how exactly the horizon problem came to be considered a problem in the first place(unless the above mentioned conflict was also evident at the time),
Because there's no a priori expectation of the cosmological principle necessarily being true: it only makes sense if the physics of the early universe set things up that way, and the horizon problem points out that if you just take GR with the observed components of the current universe, it is impossible for any physical process to set up a universe that is approximately homogeneous and isotropic.

This is one of the reasons why inflation was proposed, but inflation doesn't extend the distance at which we expect the cosmological principle to hold out to infinity.

the three possible space geometries allowed by the FRW model are both geometrically and physically homogeneous by definition, so if one is going to use this model one is assuming that homogeneity and shouldn't worry about causal contact.
I think you're too attached to the FRW model. It's just a model. It's not reality.

... There are no trivial questions, feel free to ask anything.
OK. Well, here goes ... I'm prepared for some scoffing.

The wiki article Flatness_problem proceeds to this equation
(Ω$^{-1}$ - 1)ρa$^{2}$ = -3kc$^{2}$/8∏G
with the claim that all of the terms on the RHS are constants.

Now, here comes the silly part ... are they? Constant, I mean. Really? Are we sure?

Take, for example, ∏. Is this supposed to be exactly and only the value of ∏ as calculated mathematically? Or is it supposed to be the ratio of circumference to radius of a circle in the universe in question AND at the time in question? So, if the Universe is spatially curved then wouldn't ∏ potentially be different to the value that we have calculated in a flat (Euclidean) geometry and use in our current equations? And, in particular, wouldn't it change value as the Universe expands causing the curvature to change?

So, take k as well. The wiki article says
k is the curvature parameter — that is, a measure of how curved spacetime is
but, if the Universe is finite and expanding then wouldn't that change the value of k over time as well?

And, then, onto the big one - c. Isn't it conceivable that the speed of light has changed as the Universe has expanded.
I completely accept the important position of the theories of relativity to modern physics (and I even understand the theories to a limited extent myself - particularly special relativity). But, as I understand it, the constancy of the speed of light as used in (special) relativity is with respect to different inertial frames of reference. But that doesn't mean that the speed of light measured in an earlier epoch(?) of the Universe would have to be the same value as now, does it? In one sense, it reads to me as different frames of reference in a 'static' universe.
Is it possible that the speed of light is a function of the curvature of space? So, that in the early Universe, when the curvature was extremely high the speed of light would have been much different (smaller?) to now. Obviously that would greatly affect our measures for the age and size of the Universe, but it might also provide an alternative to 'inflation' and it would also explain why the speed of light is a limit, as now the limit is actually imposed by the topology/structure of the Universe.
Is there any evidence to suggest that the speed of light is different for different values of curvature? For example, light is 'bent around' very massive objects such as galaxies - is this not the same as saying that light is refracted by very massive objects? Does such 'refraction' of light imply a velocity change in the region of the massive object, i.e. the region of (locally) different spatial curvature?

Chalnoth
Take, for example, ∏. Is this supposed to be exactly and only the value of ∏ as calculated mathematically? Or is it supposed to be the ratio of circumference to radius of a circle in the universe in question AND at the time in question?
The value of $\pi$ is independent of the universe. It's just a transcendental mathematical number, and is no less constant than the 3 or 8 in that formula.

The speed of light is, to the best of our knowledge, also constant.

The spatial curvature, k, is a constant that is a way of encapsulating the relationship between the expansion rate and the energy density of the universe. The value of k doesn't change because of how we define the term.

OK. Well, here goes ... I'm prepared for some scoffing.

The wiki article Flatness_problem proceeds to this equation
(Ω$^{-1}$ - 1)ρa$^{2}$ = -3kc$^{2}$/8∏G
with the claim that all of the terms on the RHS are constants.

Now, here comes the silly part ... are they? Constant, I mean. Really? Are we sure?

Take, for example, ∏. Is this supposed to be exactly and only the value of ∏ as calculated mathematically? Or is it supposed to be the ratio of circumference to radius of a circle in the universe in question AND at the time in question? So, if the Universe is spatially curved then wouldn't ∏ potentially be different to the value that we have calculated in a flat (Euclidean) geometry and use in our current equations? And, in particular, wouldn't it change value as the Universe expands causing the curvature to change?

So, take k as well. The wiki article says
but, if the Universe is finite and expanding then wouldn't that change the value of k over time as well?

And, then, onto the big one - c. Isn't it conceivable that the speed of light has changed as the Universe has expanded.
I completely accept the important position of the theories of relativity to modern physics (and I even understand the theories to a limited extent myself - particularly special relativity). But, as I understand it, the constancy of the speed of light as used in (special) relativity is with respect to different inertial frames of reference. But that doesn't mean that the speed of light measured in an earlier epoch(?) of the Universe would have to be the same value as now, does it? In one sense, it reads to me as different frames of reference in a 'static' universe.
Is it possible that the speed of light is a function of the curvature of space? So, that in the early Universe, when the curvature was extremely high the speed of light would have been much different (smaller?) to now. Obviously that would greatly affect our measures for the age and size of the Universe, but it might also provide an alternative to 'inflation' and it would also explain why the speed of light is a limit, as now the limit is actually imposed by the topology/structure of the Universe.
Is there any evidence to suggest that the speed of light is different for different values of curvature? For example, light is 'bent around' very massive objects such as galaxies - is this not the same as saying that light is refracted by very massive objects? Does such 'refraction' of light imply a velocity change in the region of the massive object, i.e. the region of (locally) different spatial curvature?
The three parameters are constant in that formula, c and $\pi$ are obviously constant, now k here is referring to the normalized curvature that is normally used in the Friedmann equations and in the FRW line element it can only be 1, 0, or -1. The evolution of positive or negative spatial curvature is then integrated in the scale factor a.

Is there any evidence to suggest that the speed of light is different for different values of curvature?
no.

locally the speed of light is always 'c'.

What HAS changed over the age of the universe is the rate of expansion. The scale factor
a[t] is a function of time determined from general relativity.

With all due respect, I was hoping for slightly less offhand answers.
The value of $\pi$ is independent of the universe. It's just a transcendental mathematical number, and is no less constant than the 3 or 8 in that formula.
So, you're saying that if you draw a circle on, say, a balloon and then measure its circumference and diameter and divide the one by the other you're going to get a value of 3.141529... ? Or, are you saying that if you draw a circle on a balloon then measure its circumference and diameter and divide the one by the other AND then inflate the baloon to double its size and remeasure the circle's circumference and diameter and divide the one by the other you will get the same result?

The speed of light is, to the best of our knowledge, also constant.
Haven't you been saying, about the Cosmological Principle, that just because it holds on the large in the observable universe (i.e. to the best of our knowledge it holds) that is no reason to believe that it holds out to infinity? In some cases you are prepared to consider (currently) unknown (unknowable) possibilities and in others you are not?
Is there some means of determining the speed of light as it was 5 billion years ago? 10 billion years ago? 14 billion years ago? 10$^{-30}$ seconds after the Big bang?
I'll ask my previous question again - does the bending (refraction?) of light by a massive object not imply a change in its velocity due to the change in local (at the massive object) curvature of space?

The spatial curvature, k, is a constant that is a way of encapsulating the relationship between the expansion rate and the energy density of the universe. The value of k doesn't change because of how we define the term.
I was under the impression that the relationship doesn't appear to be as expected - and an inflationary energy of some sort has had to be posited?

Chalnoth
The first point is that that isn't how $pi$ is defined. It is defined in such a way that the curvature of space simply doesn't matter, so that in curved space the ratio of the circumference of a circle to its diameter is no longer $pi$, but $pi$ times some measure of the enclosed curvature.

As for the speed of light, well, certainly you can come up with some different laws of physics that allow it to vary, but then the entire equation needs to be re-evaluated in that situation.

With regard to k, again, that is a constant based upon how the terms in FRW are defined. You could certainly re-define your terms such that the value that encapsulates the curvature isn't a constant (as is done routinely by using $\Omega_k$ to describe the curvature). You could also imagine a universe that doesn't follow the symmetries of FRW and thus doesn't have a single curvature term (and in that situation, like the above, you'd have to re-evaluate the entire equation, not just the one term).

A shorter way to argue this point is to just state that there are certain assumptions built into deriving the Friedmann equations in the first place. You can't break those assumptions after the fact and get something sensible: you have to re-derive the equations from scratch using the new set of assumptions.

The value of $\pi$ is independent of the universe. It's just a transcendental mathematical number, and is no less constant than the 3 or 8 in that formula.
The three parameters are constant in that formula, c and $\pi$ are obviously constant...

In fact, isn't testing for the interior sum of triangles just another form of looking for a value of $\pi$ that differs to that on a flat surface?

Chalnoth
In fact, isn't testing for the interior sum of triangles just another form of looking for a value of $\pi$ that differs to that on a flat surface?
Not the way $\pi$ is defined.

The first point is that that isn't how $pi$ is defined. It is defined in such a way that the curvature of space simply doesn't matter, so that in curved space the ratio of the circumference of a circle to its diameter is no longer $pi$, but $pi$ times some measure of the enclosed curvature.
Surprising! So, a mathematical symbol which has a clear universal definition (the ratio of circumference to diameter of a circle) is redefined for use here? And the symbol itself isn't even changed? I'd have expected something like $\pi$$_{0}$ or $\pi$$_{E}$ in such a case, to indicate that the equations were using the value of $\pi$ in Euclidean geometry.
I'd be interested in a reference to where this definition is formally made (preferably a layman understandable source, if possible).

As for the speed of light, well, certainly you can come up with some different laws of physics that allow it to vary, but then the entire equation needs to be re-evaluated in that situation.

...

A shorter way to argue this point is to just state that there are certain assumptions built into deriving the Friedmann equations in the first place. You can't break those assumptions after the fact and get something sensible: you have to re-derive the equations from scratch using the new set of assumptions.
Excellent! Let's do it. Are there any theoretical physicits on here up to such a challenge?

I'm not interested, per se, in the shape and state of the Universe only with regard to models that we currently have which have shortcomings and artificial constructs / ideas to get around these shortcomings. What I'm interested in is the discovery / exploration of a model which explains all the observations without any dissatisfying artificial additions. I recognise, of course, that such a model might not exist, but isn't it the fundamental goal of theoretical physics - to continue searching for such a model anyway?

Chalnoth
Surprising! So, a mathematical symbol which has a clear universal definition (the ratio of circumference to diameter of a circle) is redefined for use here? And the symbol itself isn't even changed? I'd have expected something like $\pi$$_{0}$ or $\pi$$_{E}$ in such a case, to indicate that the equations were using the value of $\pi$ in Euclidean geometry.
I'd be interested in a reference to where this definition is formally made (preferably a layman understandable source, if possible).
Well, it's just a convention. It doesn't really mean anything. But this is the way that General Relativity has been developed.

Excellent! Let's do it. Are there any theoretical physicits on here up to such a challenge?
It's profoundly difficult. There's a reason why FRW is so ubiquitous: it's basically the simplest possible universe you can think of without being completely trivial. Removing some assumptions ends up being incredibly complicated. Many physicists do still try, but it so far hasn't produced anything that matches reality any better than FRW.

What I'm interested in is the discovery / exploration of a model which explains all the observations without any dissatisfying artificial additions.
What dissatisfying artificial additions? Why are they dissatisfying, and why are they artificial?

Because there's no a priori expectation of the cosmological principle necessarily being true[...]
I think you're too attached to the FRW model. It's just a model. It's not reality.
It's profoundly difficult. There's a reason why FRW is so ubiquitous: it's basically the simplest possible universe you can think of without being completely trivial. Removing some assumptions ends up being incredibly complicated. Many physicists do still try, but it so far hasn't produced anything that matches reality any better than FRW.
Seems like I'm not the only one attached to the FRW model.
But you have some kind of double standard here, removing or changing some assumptions like for instance the cosmological principle seems ok to you.

Chalnoth
Seems like I'm not the only one attached to the FRW model.
But you have some kind of double standard here, removing or changing some assumptions like for instance the cosmological principle seems ok to you.
Let me be clear: Nobody, to my knowledge, has successfully solved the Einstein equations for a global space-time that does not follow FRW (except in the special case of spherical symmetry). I have no problem considering other space-times, and don't think that FRW is likely to be accurate at distances much larger than our horizon, but it turns out that doing it right is fantastically difficult.

It's profoundly difficult. There's a reason why FRW is so ubiquitous: it's basically the simplest possible universe you can think of without being completely trivial. Removing some assumptions ends up being incredibly complicated. Many physicists do still try, but it so far hasn't produced anything that matches reality any better than FRW.
Fair enough. That's understandable. So, maybe this is straying into another thread. I'd like to explore, from a semi-philosophical point of view (and, particularly, given that I'm not a physicist of any sort), without having to tie it down to a specific existing mathematical description, what might make sense in terms of an alternative model for the Universe. Conceivably, such an exploration might develop into a mathematical description (though I lack the math knowledge myself), but even if it didn't it might be interesting nonetheless.

What dissatisfying artificial additions? Why are they dissatisfying, and why are they artificial?
Well, two spring to mind immediately.
1. Cosmic Inflation. While it solves a lot of problems and there are observations that seem to confirm it, the fact that there's no real basis for it and no mechanism that we've observed seems to provide for it is dissatisfying. On a purely philosophical level, if you will, it's messy that for no apparent reason (other than the convenience of resulting in a universe that suits our observations) the Universe just suddenly underwent an inflationary period which then stopped, again for no apparent reason.
2. The universal constants - G and c (for a start). It is dissatisfying that they are not derivable. They should be a function of some aspect of the Universe. As I implied earlier, for example, isn't it conceivable that c might be related to the curvature of space(time?)?

Chronos
Gold Member
Inflation is science in its most fundamental form - a model that matches observational evidence. If a better model comes along, inflation will become an historical footnote. G and c are not the only universal constants that cannot be derived from something more fundamental. Some things can be measured, but, not explained. It's the nature of the universe.

Chalnoth
Well, two spring to mind immediately.
1. Cosmic Inflation. While it solves a lot of problems and there are observations that seem to confirm it, the fact that there's no real basis for it and no mechanism that we've observed seems to provide for it is dissatisfying. On a purely philosophical level, if you will, it's messy that for no apparent reason (other than the convenience of resulting in a universe that suits our observations) the Universe just suddenly underwent an inflationary period which then stopped, again for no apparent reason.
I don't see that it's messy at all. In its most basic form, it's nothing more than proposing a single field that, when excited in the right way, produces an inflating universe.

2. The universal constants - G and c (for a start). It is dissatisfying that they are not derivable. They should be a function of some aspect of the Universe. As I implied earlier, for example, isn't it conceivable that c might be related to the curvature of space(time?)?
This would require a more fundamental law of physics. And even then, it isn't clear that these constants could ever be derived from first principles (though that has been a goal of many theoretical physicists for decades), but at least having the more fundamental laws of physics would give us an answer to how the constants we measure got that way (and the answer may be, in part, by accident).

At any rate, the fact is that discovering a correct fundamental set of physical laws is profoundly difficult. String theory has been the primary proposal of such a set of physical laws for decades, but even now we don't yet know everything about string theory, let alone whether or not it applies to reality.

Inflation is science in its most fundamental form - a model that matches observational evidence. If a better model comes along, inflation will become an historical footnote.
I don't see that it's messy at all. In its most basic form, it's nothing more than proposing a single field that, when excited in the right way, produces an inflating universe.
I don't agree that this is science in its most fundamental form. Science in its most fundamental form, for me, is taking phenomena that occur (continuous) and proposing a theory to explain those phenomena with testable predictions and repeatable observations. There are no ongoing phenomena of 'inflation'. In fact there are no direct observations of 'inflation' at all (are there?). Essentially, there are a number of tricky little problems that seem to come out of a basic Big Bang model (e.g. flatness and fine tuning), and some forms of inflation would explain away those problems if it had happened.
So, we've accepted inflation for so long now it has become a core part of the standard Big Bang models.
But, as so nicely put, it's nothing more than proposing a single field that, when excited in the right way, produces an inflating universe. A field that we haven't managed to stimulate in any way, or even seen direct evidence of its stimulation. We've managed to recreate (certain) conditions as far back as milliseconds(?) from the Big Bang event and yet not seen any inflation type field stimulation in those experiments.

G and c are not the only universal constants that cannot be derived from something more fundamental. Some things can be measured, but, not explained. It's the nature of the universe.
Now, this is completely contradictory to the fundamental philosophy of science - you're not prepared to ask 'why'? Why is c such-and-such a value? Why is G such-and-such a value? Why are atoms the fundamental unit of matter that cannot be divided further? Oh, wait ... they're not!! One of the basic goals of science is to eliminate such constants by demonstrating how they emerge from the basic structure / make-up of the Universe. And, I see no reason why we can't explore ideas / theories (however difficult) that attempt to do just that.

This would require a more fundamental law of physics. And even then, it isn't clear that these constants could ever be derived from first principles (though that has been a goal of many theoretical physicists for decades), but at least having the more fundamental laws of physics would give us an answer to how the constants we measure got that way (and the answer may be, in part, by accident).

At any rate, the fact is that discovering a correct fundamental set of physical laws is profoundly difficult. String theory has been the primary proposal of such a set of physical laws for decades, but even now we don't yet know everything about string theory, let alone whether or not it applies to reality.
'Difficult' is irrelevant. If it was easy it would be boring - it would all be done by now and we could all sit around watching soaps and talking about psychology.

I'm interested in a discussion about the basic assumptions that are inherent in the theories we currently work with - what problems exist because of these assumptions and what alternatives look interesting in terms of overcoming those problems and moving to alternative testable models that might be worth developing. I guess I'll start another thread on this.

Fair enough. That's understandable. So, maybe this is straying into another thread. I'd like to explore, from a semi-philosophical point of view (and, particularly, given that I'm not a physicist of any sort), without having to tie it down to a specific existing mathematical description, what might make sense in terms of an alternative model for the Universe. Conceivably, such an exploration might develop into a mathematical description (though I lack the math knowledge myself), but even if it didn't it might be interesting nonetheless.

Well, two spring to mind immediately.
1. Cosmic Inflation. While it solves a lot of problems and there are observations that seem to confirm it, the fact that there's no real basis for it and no mechanism that we've observed seems to provide for it is dissatisfying. On a purely philosophical level, if you will, it's messy that for no apparent reason (other than the convenience of resulting in a universe that suits our observations) the Universe just suddenly underwent an inflationary period which then stopped, again for no apparent reason.

-------

How inflation MAY follow simply from quantum gravity:
http://eprints.port.ac.uk/6488/

Chalnoth
I don't agree that this is science in its most fundamental form. Science in its most fundamental form, for me, is taking phenomena that occur (continuous) and proposing a theory to explain those phenomena with testable predictions and repeatable observations. There are no ongoing phenomena of 'inflation'. In fact there are no direct observations of 'inflation' at all (are there?). Essentially, there are a number of tricky little problems that seem to come out of a basic Big Bang model (e.g. flatness and fine tuning), and some forms of inflation would explain away those problems if it had happened.
Yes, but the theory made testable predictions as to the pattern of structure in our universe, and those predictions were very strongly confirmed by WMAP (to some extent previous experiments as well, but not nearly as strongly), and have been further strengthened by Planck.

The next big step in confirming inflation would be to detect B-mode polarization in the CMB. This isn't easy, unfortunately, as we don't know how big the B-mode polarization is (it may, sadly, be undetectable). But if we do detect it in the next few years, that would likely be a pretty strong confirmation of inflation.

If we could also observe the primordial gravity wave background, that would likely give us even greater insight.

But, as so nicely put, it's nothing more than proposing a single field that, when excited in the right way, produces an inflating universe. A field that we haven't managed to stimulate in any way, or even seen direct evidence of its stimulation. We've managed to recreate (certain) conditions as far back as milliseconds(?) from the Big Bang event and yet not seen any inflation type field stimulation in those experiments.
It's not really expected that we could, considering the very large energies at which inflation occurs.

In short, you're prioritizing laboratory science over non-laboratory science in a very absurd way. The vast majority of science is done outside the laboratory, and we would know very little about the universe if we restricted ourselves only to what we can discover in a lab.