Exploring the Shape of a Flat Universe

[Holocene]

I just want to make sure my understanding is correct.

In a "flat" universe, the total density of the universe matches the "critical density". As a result, the universe will expand forever, and never "fall back" into itself.

The shape of a flat universe, as the name suggests, is flat like a piece of paper.

In a flat universe, the Pythagorean theorem hold true, even when applied to distances of many billions of light years. This is because a flat universe does not curve in any way, except of course for the areas where space-time is curved around massive objects, including black holes.

Now for a question:

Is it correct to think of the flat universe of a sphere that will never stop expanding? I ask this because, if this is true, then the shape of the universe at any given time would be a sphere, and therefore would in fact be curved.

Or should I think of the flat universe as a "cube" that will keep expanding?

 

[Garth]

Exotic topologies not withstanding, think of the flat - space universe as a flat sheet that expands.

It is infinite yet it keeps on expanding, i.e. representative galaxies embedded in it move apart from each other.

Garth

 

[Holocene]

Exotic topologies not withstanding, think of the flat - space universe as a flat sheet that expands.

It is infinite yet it keeps on expanding, i.e. representative galaxies embedded in it move apart from each other.

Garth

In that case, what happens if you travel in a "vertical" dirrection, perpendicular to the sheet?

 

[Garth]

The 'sheet' is an 2D analogy of 3D space.

You have to imagine the 3D volume expanding.

The point about it being flat is that if so the geometry of that 3D space would be Euclidean.

We can test for Euclidean space by various cosmological observations such as the angular size of the fluctuations in the first peak of the CMB power spectrum.

Garth

 

[George Jones]

I just want to make sure my understanding is correct.

In a "flat" universe, the total density of the universe matches the "critical density". As a result, the universe will expand forever, and never "fall back" into itself.

I'm not sure what you mean by "critical density", but some care is needed.

There is a critical density above which the curvature of (spatial sections) of the universe is positively curved (like the surface of a sphere) and below which the universe is negatively curved (like a saddle). However, unless the cosmological constant/dark energy is zero, this critical density is not the dividing line between universes that expand forwever and universes that eventually collapse.

Current observations indicate that:

1) the cosmological constant/dark energy is non-zero;

2) the density of the universe is close to the critical density for flatness;

3) the universe will expand forever even if it doesn't have critical density.

The shape of a flat universe, as the name suggests, is flat like a piece of paper.

In a flat universe, the Pythagorean theorem hold true, even when applied to distances of many billions of light years. This is because a flat universe does not curve in any way, except of course for the areas where space-time is curved around massive objects, including black holes.

Now for a question:

Is it correct to think of the flat universe of a sphere that will never stop expanding? I ask this because, if this is true, then the shape of the universe at any given time would be a sphere, and therefore would in fact be curved.

Or should I think of the flat universe as a "cube" that will keep expanding?

The latter.

In that case, what happens if you travel in a "vertical" dirrection, perpendicular to the sheet?

The universe is modeled by a 4-dimensional spacetime that is curved. A statement like "the universe is flat" doesn't mean that 4-dimensional spacetime is flat, it means that 3-dimensional spatial sections of the universe are flat.

We are traveling "perpendicular" (in a technical sense) to the the three dimensions of space, i.e., we're moving forward in time!

 

[Garth]

Just to open ourselves up to more 'exotic' possibilities and topologies, the surface of a torus (American donut) is locally flat though not globally so, so the universe could have a toroidal global topology and yet in the cosmological 'neighbourhood' be locally flat.

The universe might even be shaped like a soccer ball with our neighbourhood appearing nearly flat, but that now seems to be unlikely.

Garth

 

[Frans]

I understand that as early as 1969 the flatness problem was identified. That implies that by that time there were observations that showed the universe to be flat. Can anyone tell me what those observations were?

 

[Garth]

I understand that as early as 1969 the flatness problem was identified. That implies that by that time there were observations that showed the universe to be flat. Can anyone tell me what those observations were?

First Welcome to these Forums Frans!

In 1969 the universe was generally thought to have less density than required for spatial flatness, this was based on the amount of visible matter that had been observed. Adding together the visible mass in the form of stars and nebulae and invisible mass required to account for both galactic rotation rates, and to keep galactic clusters gravitationally bound, it was thought the total density had to be about 20 - 30 % of the critical density.

The measurements were so crude however that everybody realized that more mass might be hiding - as indeed has proved to be the case - and the flat or closed models were still thought to have been possibilities.

The flatness problem is something different.

The problem was that even if the visible mass was all that there was, still that density would have been almost the critical density - then thought to be about 1% critical density (now actually believed to be only 0.3%). This was a problem because in a universe that had decelerated in its expansion rate from Planck Time until now the actual density ought to have been far less, or far greater, than critical density, by a factor of the order of 10⁶⁰.

Over time from the Planck Era until now deceleration thrusts apart the value of the actual density from the critical value.

So for the actual density to be this close to the critical density then in that Planck Era it had to have been incredibly fine tuned to have been almost equal to the critical density to within one part ~ in 10⁶⁰.

The solution to the Flatness Problem, also called the Density Problem, was found in the Inflation Hypothesis, in which an explosive exponentially accelerating expansion by a factor of more than 10⁶⁰ rammed together the actual and critical densities so closely that the subsequent deceleration has not been able to separate them even in the present epoch.

The Inflation hypothesis, although explaining why the present and critical densities are so close, is not without its problems.

(Just as a sideline note that if the universe has not been decelerating over its early life then the Flatness Problem would not exist in the first place.)

The observation that nails the observed density to be nearer equal or equal to the critical density is the first maximum in the power spectrum of the Cosmic Microwave Background, as measured by WMAP amongst others.

Garth

 

[Frans]

First Welcome to these Forums Frans!

The problem was that even if the visible mass was all that there was, still that density would have been almost the critical density - then thought to be about 1% critical density (now actually believed to be only 0.3%). This was a problem because in a universe that had decelerated in its expansion rate from Planck Time until now the actual density ought to have been far less, or far greater, than critical density, by a factor of the order of 10⁶⁰.

OK, I now understand that 1% of the critical density (actual density of mass in the universe as observed in 1969), is compared to an order of 10⁶⁰, very close to the critical density and hence to flatness (dark energy didn't yet enter the equations in those days, right?)

The WMAP observations from the early 21st century show that the actual density (the sum of matter and dark energy density) is even closer to critical density than the 1%.

Am I right?

Frans.

 

[Garth]

OK, I now understand that 1% of the critical density (actual density of mass in the universe as observed in 1969), is compared to an order of 10⁶⁰, very close to the critical density and hence to flatness (dark energy didn't yet enter the equations in those days, right?)

The WMAP observations from the early 21st century show that the actual density (the sum of matter and dark energy density) is even closer to critical density than the 1%.

Am I right?

Frans.

Yes - WMAP shows that the geometry of space is flat or almost flat, therefore the actual total density is the critical density to within about 2% accuracy.
i.e. $\Omega$ is in the range 0.9929, 1.0181 at the 95% confidence level.

Garth

 

[sylas]

OK, I now understand that 1% of the critical density (actual density of mass in the universe as observed in 1969), is compared to an order of 10⁶⁰, very close to the critical density and hence to flatness (dark energy didn't yet enter the equations in those days, right?)

The WMAP observations from the early 21st century show that the actual density (the sum of matter and dark energy density) is even closer to critical density than the 1%.

Am I right?

Frans.

Current WMAP based estimates suggest that the universe is now very close to critical density, but you have to include dark energy in the energy density. The numbers are:

4% baryonic matter
23% dark matter
73% dark energy

The curvature of the universe is positive (closed universe) if these numbers add up to be [strike]less[/strike] more than 100%, and negative (open universe) if they add up to be [strike]more[/strike] less. These numbers add up to 100% of critical density, which implies that the universe is very close to flat. Or, conversely, given observations that the universe appears flat, we can use this to help constrain numbers knowing that they add up to 100%.

These numbers change over time. As the universe expands, the density of matter reduces, and also the Hubble constant changes, which alters the criticial density. The ratio of density of various kinds of energy to critical density alters over time. There is a nice "phase space" diagram showing how these ratios are expected to develop over time. A flat universe always remains flat, but the energy density becomes more and more dominated by dark energy.

Can anyone out there help? I want a diagram with Ω_m on the horizontal axis, and Ω_Λ on the vertical axis, and trajectories for the universe over time as these ratios change. The closest I have is this

from cosmology lectures by James Schombert at Uni of Oregon; it shows the eventual fate of the universe, but not the trajectories through the phase space. A flat universe simply moves along the horizonal blue line, to the right for positive dark energy. What is interesting is that other not quite flat universes move towards being flat as they expand. This is what may have happened during the inflationary epoch.

Cheers -- sylas

 

[George Jones]

The curvature of the universe is positive (closed universe) if these numbers add up to be less than 100%, and negative (open universe) if they add up to be more.

How about the other way round?

 

[sylas]

How about the other way round?

Argh. I always do that. Dyslexics of the world... UNTIE!

Thanks for picking it up -- sylas

 

[George Jones]

Can anyone out there help? I want a diagram with Ω_m on the horizontal axis, and Ω_Λ on the vertical axis, and trajectories for the universe over time as these ratios change.

I have a book open in front of me, General relativity: an introduction for physicists by Michael Paul Hobson, George Efstathiou, Anthony N. Lasenby, that, on page 416, has a couple of these diagrams with a few trajectories in each diagram. Unfortunately, I wasn't able to display page 416 using

http://books.google.com/books?id=xm...r&dq=hobson+lasenby&cd=6#v=onepage&q=&f=false.

I might look around some more.

 

[Chronos]

Dark energy is pretty weird stuff. As the universe expands, it appears to increases just enough to keep Omega almost exactly 1.000. I suspect this is a feedback effect.

 

[sylas]

Dark energy is pretty weird stuff. As the universe expands, it appears to increases just enough to keep Omega almost exactly 1.000.

Actually, that's not a property of dark energy, so much as a consequence of being flat. Whether you have dark energy, or matter, or radiation, as the major contribution for energy, you always have Ω = 1 as an invariant. Even quintessence does this. Note that as the universe expands, the Hubble "constant" changes, which in turn changes the actual energy density required to be critical. The critical density is in fact precisely the energy density that keeps the "trajectory" of the universe in its phase space at the point of being critical.

Cheers -- sylas

 

[SW VandeCarr]

If the universe is precisely asymptotically flat, than it should be infinite in three spatial dimensions, no?

If so, is this consistent with the Big Bang/inflation cosmology? Given flat infinite 3 space, is a finite amount of matter an "island" expanding within it? This seems to contradict inflation which states that space itself is expanding. So therefore it must be finite but unbounded (closed), no?.

It seems the Big Bang cosmology depends on 3 space having positive curvature. What if you don't find it?

 

[sylas]

If the universe is precisely asymptotically flat, than it should be infinite in three spatial dimensions, no?

If so, is this consistent with the Big Bang/inflation cosmology? Given flat infinite 3 space, is a finite amount of matter an "island" expanding within it? This seems to contradict inflation which states that space itself is expanding. So therefore it must be finite but unbounded (closed), no?.

It seems the Big Bang cosmology depends on 3 space having positive curvature. What if you don't find it?

As I understand it... if the universe is everywhere flat, and topologically simple, then it is infinite. However, you can still have a finite flat universe if it has a topology that connects back on itself. But otherwise, yes.

More crucially, the nature of expansion is that it is the same (or similar) everywhere. So in principle there is no problem with an infinite universe expanding everywhere; inflation included.

On the other hand, I think most cosmologists tend to suspect that the universe is finite; or else that we can speak of a finite fluctuation which inflated and kicked of the expansion of everything we can see and far beyond as well.

Even if expansion is not the same everywhere, I think it is still misleading to think of expansion of an island within a larger space. The expansion itself makes more space, in a sense.

Inflation flattens things out but if I understand it correctly, there's nothing wrong with having very slightly positive curvature in some regions and very slightly negative in others. Assume it was a finite seed or fluctuation of some kind which first inflated into our universe, I an inclined to expect that on sufficiently large scales the universe will be very slightly positive curved; but small inhomogeneities could mean that our observable universe within this is negatively curved.

I'm not completely sure this is right... comment from some of the more expert regulars would be appreciated.

Cheers -- sylas

 

[SW VandeCarr]

As I understand it... if the universe is everywhere flat, and topologically simple, then it is infinite. However, you can still have a finite flat universe if it has a topology that connects back on itself. But otherwise, yes.

More crucially, the nature of expansion is that it is the same (or similar) everywhere. So in principle there is no problem with an infinite universe expanding everywhere; inflation included.

On the other hand, I think most cosmologists tend to suspect that the universe is finite; or else that we can speak of a finite fluctuation which inflated and kicked of the expansion of everything we can see and far beyond as well.

Even if expansion is not the same everywhere, I think it is still misleading to think of expansion of an island within a larger space. The expansion itself makes more space, in a sense.

Inflation flattens things out but if I understand it correctly, there's nothing wrong with having very slightly positive curvature in some regions and very slightly negative in others. Assume it was a finite seed or fluctuation of some kind which first inflated into our universe, I an inclined to expect that on sufficiently large scales the universe will be very slightly positive curved; but small inhomogeneities could mean that our observable universe within this is negatively curved.

I'm not completely sure this is right... comment from some of the more expert regulars would be appreciated.

Cheers -- sylas

Thank Silas. I thought of ways where an infinite set can expand. For example simply systematically inserting rational fractions between all the natural numbers. So I understand how inflation can occur in an infinite 3 space. It's still not clear to me how a 'small' apparently finite universe can expand to become an infinite one. Also, would infinite space imply infinite matter?

 

[sylas]

Thank Silas. I thought of ways where an infinite set can expand. For example simply systematically inserting rational fractions between all the natural numbers. So I understand how inflation can occur in an infinite 3 space. It's still not clear to me how a 'small' apparently finite universe can expand to become an infinite one. Also, would infinite space imply infinite matter?

A finite universe is always finite and has always been finite; an infinite universe is always infinite and has always been infinite. Infinite space does mean infinite matter; I don't think anyone is proposing an infinite empty space. Expansion does not refer to expansion into a pre-existing space; it means that space itself increases in volume as everything flies apart from everything else.

Cheers -- sylas

 

[SW VandeCarr]

Thanks.