Meaning of a "Flat," "Open," and "Closed" Universe

dchartier

Hello all,

I'm reading up on cosmology and the potential shapes of the universe based on its density in relation to the critical density. When one says that a universe is "flat," what precisely does this mean? Does this mean that the space-time of the universe is described by Euclidean geometry? Similarly, what does it mean to say the universe is "open" or "closed"?

Thanks in advance!

Mordred

Open and closed is simply infinite or finite.
For the geometry its a factor of its critical density.
If the density parameter is greater than 1 then the universe is closed (finite) and positively curved like the surface of a ball.
If the density parameter is less than 1 then its open with negative curvature like a saddle. (open equals infinite)
If its exactly equals 1 then its flat and infinite
WMAP results show that its flat.

In the flat geometry this expansion is suppose to be at a steady rate. Dark energy is the reason its accelerating.

dchartier

Thanks, that's very helpful.

When speaking of flatness or curvature, I presume you mean flatness/curvature of space-time, not just space?

Also, what sort of physical effects would you expect to see from an open or closed universe due to its curvature? For example, if you created a huge triangle across a very large segment of space, would you expect the sum of its angles to be more or less than 180 degrees?

chill_factor

i am not an astrophysicist, but here's what i believe.

when we talk about flat or curved, we are actually talking about the metric of spacetime, which is related to the metric tensor G.

the metric tensor is a (covariant) rank 2 tensor whose elements are made up of the dot products of the direct basis vectors:

g[itex]_{uv}[/itex] = e[itex]_{u}[/itex][itex]\bullet[/itex]e[itex]_{v}[/itex]

the direct basis vectors are defined by e[itex]_{u}[/itex] = ds/dq[itex]^{u}[/itex] where q[itex]^{u}[/itex] is a variable in your coordinate system and s is the expression for a length element vector in your coordinate system.

for example: lets say we have a Cartesian coordinate system in 2-D. s = x x + y y. the 2 basis vectors would be then

e[itex]_{x}[/itex] = d(x,y)/dx = dx/dx x + dy/dx y = x
e[itex]_{y}[/itex] = d(x,y)/dy = dx/dy x + dy/dy y = y

as a rank 2 tensor we can conveniently represent it as a matrix. the Euclidian metric tensor that we are familiar with represents the graphs we are familiar with:

g[itex]_{xx}[/itex] = e[itex]_{x}[/itex][itex]\bullet[/itex]e[itex]_{x}[/itex]
= x[itex]\bullet[/itex]x = 1

g[itex]_{yy}[/itex] = e[itex]_{y}[/itex][itex]\bullet[/itex]e[itex]_{y}[/itex]
= y[itex]\bullet[/itex]y = 1

g[itex]_{xy}[/itex] = g[itex]_{yx}[/itex] = e[itex]_{x}[/itex][itex]\bullet[/itex]e[itex]_{y}[/itex] = x[itex]\bullet[/itex]y = 0

thus the entire metric tensor G is the following matrix:

[1 0]
[0 1]

note that it is a symmetric matrix. that means the basis vectors are orthogonal. also note that the magnitudes of the basis vectors are one. thus this is a coordinate system made from an orthonormal basis; just as we expect.

that seems too easy. lets do a nontrivial example: polar coordinate systems in 2-D.

x = rcosθ, y = rsinθ
s = rcosθ x + rsinθ y

e[itex]_{r}[/itex] = d(rcosθ x + rsinθ y)/dr = cosθ x + sinθ y

e[itex]_{θ}[/itex] = d(rcosθ x + rsinθ y)/dθ = r(-sinθ x + cosθ y)

g[itex]_{rr}[/itex] = e[itex]_{r}[/itex][itex]\bullet[/itex]e[itex]_{r}[/itex]
= cosθ x * cosθ x + 2 cosθ x * sinθ y + sinθ y * sinθ y = cos[itex]^{2}[/itex]θ + 0 (because x and y are orthogonal) + sin[itex]^{2}[/itex]θ = cos[itex]^{2}[/itex]θ + sin[itex]^{2}[/itex]θ = 1

g[itex]_{rθ}[/itex] = g[itex]_{θr}[/itex] = e[itex]_{r}[/itex][itex]\bullet[/itex]e[itex]_{θ}[/itex]
= 0

g[itex]_{θθ}[/itex] = e[itex]_{r}[/itex][itex]\bullet[/itex]e[itex]_{θ}[/itex]
= r(-)sinθ x * r(-)sinθ x -2 rsinθ x * cosθ y + rcosθ y * rcosθ y = r[itex]^{2}[/itex]cos[itex]^{2}[/itex]θ + r[itex]^{2}[/itex]sin[itex]^{2}[/itex]θ = r[itex]^{2}[/itex]

so G =

[1 0]
[0 r[itex]^{2}[/itex]]

this is obviously not a flat surface. this gets more complicated for other shapes. now we have set the stage to talk about curved space - as we see, space CAN be curved!

chill_factor

now we have the metric tensor. what do we do with it? well first we note that coefficients of the elements of the matrix are not necessarily 1. these coefficients represent magnitude. we can now introduce a concept called the metric coefficient, or in physics language, the scale factor.

the scale factor for an element of the matrix h[itex]_{i}[/itex] = [itex]\sqrt{g_{i}}[/itex]

note that for the polar coordinate, we have a scale factor with r. That means as r increases, the other basis vector increases! could spacetime behave the same way? sure!

we can also note that certain coordinates do not lead to a symmetric metric tensor. this would result in a non-orthogonal space. Sure, that sounds really complicated, why is it useful?

It turns out that in spacetime, things travel along their geodesics if you give no force input. A geodesic is the closest coordinate distance between 2 points; going against the geodesic requires a force.

If we have a 4-D spacetime that's Euclidian, the geodesic would be a straight line. But if it was not Euclidian, but rather defined by some other metric tensor, the geodesic might not be a straight line.

Indeed, it may lead towards something. Such as a gravity source. This is why we talk about spacetime being bent by mass - the metric tensor in presence of mass changes.

This actually leads to a scary thought: if it were not for the electromagnetic force, the degeneracy force... where does the geodesics lead? all particles are trying to follow the geodesics, to one point: their center of mass. all particles are trying to become black holes. Sleep on that.

However these problems are questions of local scale. The universe as a whole may also be defined by a metric tensor that is not Euclidian. That would mean geodesics would not be straight lines, which would mean triangles would not have their angles add up to 180 degrees.

detective

without getting all heavy on the math and terminology, EVERYTHING will naturally try to follow its geodesic path...

...you don't see an ocean going ship try to jump out of the water...it just follows a path around a curved surface...local direction changes notwithstanding...

...a force on that very ship MAY make it jump out of the water, but that is not its natural predilection ....and so the very same with all objects containing mass, whereby anything massive will ''fall'' and it is only because an initial force has pushed the massive object outward, its tendency will be to gradually fall back in to its more stable, if not placid, state..

...our physics on earth are just a mirror for the more volatile physics in the wider universe

...

bapowell

The 3 possible shapes of the universe refer to the spatial geometry. They refer to the way the universe looks to an observer at a given instant in time (you may hear these 'snapshots' of the universe referred to as spatial slices, or spacelike hypersurfaces.) I should correct something Mordred said: a flat universe needn't be infinite. In addition to geometry, the physical manifold making up the universe also has topological properties. There do exist geometrically flat surfaces that are bounded and finite as a result of nontrivial topology, like the torus, for example.

Indeed -- and that is precisely how the geometry of the universe is ascertained. If we have a distant structure aligned perpendicular to our line of sight (that we know the length of), and we know the distance to that structure, we can test the accuracy of the relation [itex]s=r\theta[/itex].

Mordred

Glad you caught that I had forgotten to add the finite portion

Chalnoth

There's been a lot of noise in this thread, so I thought I'd try to clarify somewhat. First, take the first Friedmann equation:

[tex]H^2 = \rho(a) - {k \over a^2}[/tex]

(Pedantic note: I have neglected the constants in an attempt to improve clarity)

Here we have four things that need to be considered. The first is [itex]H[/itex], the rate of expansion. The second is the total mass/energy density [itex]\rho[/itex]. This is a function of the scale factor [itex]a[/itex] because the density changes as the universe expands (and how it changes depends upon the sort of matter/energy). The scale factor is the amount by which things have expanded. We typically define the scale factor as being equal to one today, which means that when the scale factor was 0.5, things were, on average, half as far apart. The final parameter, [itex]k[/itex], is the spatial curvature: [itex]k=0[/itex] is flat, [itex]k > 0[/itex] is closed, [itex]k < 0[/itex] is open.

What does this mean? Well, to see that, look at the other terms in the equation: they are the expansion rate and the density of the universe. If you have a high energy/mass density, but a low expansion rate, then you have a large positive curvature [itex]k[/itex], and the universe is closed. As long as you just have normal matter, this expansion will be too slow for gravitational attraction of all that dense matter and it will collapse in on itself.

Similarly, if the expansion is much faster than the energy density, then this is balanced by having a large negative curvature, which is an open universe that tends to expand forever.

There is a middle point where the expansion rate exactly balances the energy density, and that is what is called a flat universe.

Because of the way General Relativity operates, this has geometric effects so that if you have a significant amount of spatial curvature, triangles drawn across large regions of the universe, e.g. by sending light across billions of light years, will tend to have angles which don't add up to 180 degrees.

I'll end on a pedantic point: the open/flat/closed bit doesn't actually say what the eventual fate of the universe is. The issue is with regard to how the effect of the curvature scales with the expansion:

[tex]H^2 = \rho(a) - {k \over a^2}[/tex]

If we have an expanding universe, and some of the stuff that makes up [itex]\rho(a)[/itex] dilutes more slowly than [itex]1/a^2[/itex], then if the universe is allowed to expand enough, that form of slowly-diluting energy will dominate, and the fact that there is curvature becomes less and less important. This is the case with dark energy, for example. So if this extra stuff that dilutes more slowly tends to make the universe expand faster, as in the case of dark energy, then a closed universe will expand forever anyway.

Similarly, one can have a negative dark energy that causes an open universe to recollapse.

In reality, our universe appears to be very close to flat, with [itex]|k| < 0.01[/itex] in some units, meaning that the expansion rate [itex]H^2[/itex] differs from the density [itex]\rho[/itex] by no more than one part in a hundred.

timmdeeg

Does this mean, that the triangle is measured by light rays? But then, we measure the curvature of space-time, not the spatial curvature, right?
I wonder, why the 1° angle corresponding to this certain peak in the CMBR, yields the spatial geometry. Because to my understanding the triangle measured by analysing the direction of incoming photons can't be a triangle "at a given instant in time". Any comment appreciated.

Chalnoth

Well, if you want to get into the details, it's about distance measures. See equation 16 of this paper:
http://arxiv.org/abs/astroph/9905116

This can be understood geometrically as the curvature affecting the angular size of objects we see which, in turn, impacts our estimate of their distance. The curvature estimate then comes from comparing the estimated distances between things that are very far and things that are closer. One of the most effective is comparing the typical separation between hot/cold spots on the CMB and the typical separation of galaxies in the relatively nearby universe (this is known as the Baryon Acoustic Oscillation measurement). The separation in either case is viewed as an angular diameter distance.

timmdeeg

Thanks. Equation 16 shows that in the case of spatial flatness (k=0) the transverse comoving distance equals the line-of-sight comoving distance. But, hmm, which conclusion should I draw?

If I understand your explanation correctly, the density spot on the CMB (angular size 1 degree, z = 1100) develops into a BAO (certain measurable angular size, small value of z).
Now, from page 5 "The angular diameter distance Da is defined as the ratio of an object's physical transverse size to it's angular size." I guess "physical transverse size" means a distance measured with a ruler. I have in mind that the size of the density spot on the CMB is around 150 Mpc (ruler!). So, with the known increase of z one should be able to calculate the size of the corresponding BAO.

I hope this is correct so far. But I still have no idea how to construct this very triangle "at a given instant in time" (which's sum of angles is 180°) by the comparison of the angular diameter distances of said objects.

Chalnoth

Basically that in the absence of spatial curvature, there is no angular distortion of objects we see. With spatial curvature, the universe as a whole acts as a sort of gravitational lens which distorts the apparent sizes of objects far away.

Not exactly. Baryon Acoustic Oscillations are what set up the hot and cold spots in the CMB in the first place. They also set up the matter density distribution, which later forms into galaxies. The two observations are not directly related, but were set up by the same early-universe physics, which allows us to compare them. It's a non-trivial exercise to show how to relate the hot and cold spots on the CMB to the typical separation between nearby galaxies.

Well, if you have the true size of the object, and that object has some angular size, then a triangle is formed by light rays coming from the edges of the object and meeting in the telescope.

timmdeeg

I have in my mind that while assuming spatial flatness the space-time still can be curved if the rate of expansion is not linear (it isn't, as we know). Provided that's correct, why wouldn't one expect to see an angular distortion of objects in that case? In other words, I don't understand why an angular size measurement based on light rays yields spatial curvature not space-time curvature information. Because during travelling the light "experiences" the expansion of the universe.
Okay, very interesting. Thanks for clarifying this issue.
Here I am facing the same problem already mentioned above. How can lightrays form a triangle "at a given instant in time" in order to yield the spatial geometry?

You mentioned the comparison of two "objects", the spot on the CMB and a characteristic galaxy separation. Does this mean, as in either case the true size is known, that one obtaines two values of the spatial geometry independent of each other?

Chalnoth

The expansion itself is space-time curvature. Well, a component of it anyway. If you work through the equations and compute the Ricci scalar curvature, you get two terms: one related to the expansion rate, and the other related to the spatial curvature.

I guess I'm not entirely certain, except to say that this is how the equations work out.

If you have spatial curvature, then the light rays from distant objects can bend towards or away from one another, modifying the angle that we observe.

For this observation, the more important point is just that the sizes are correlated with one another, not that their absolute sizes are known. A simple way of understanding it is that we know the ratio of the length scale given by the CMB and that given by the distribution of nearby galaxies.

bapowell

Is your question about what effect the passage of time has on the specific form of the triangle? As Chalnoth says, the spatial geometry determines the angular diameter of the object; the effect of time is to redshift the photon en route to Earth from the object.

timmdeeg

Okay, and as mentioned in this article, the comparison of these length scales provides information about the expansion of the universe and hence the dark energy. According to the authors the length of the BAO matter clustering today is 490 MLJ. With this "today" value and z = 1100 the corresponding length at the time of recombination should be 410000 LJ (?). And I guess this length then is a true length (measured with rulers), as it corresponds to that typical density fluctuation in the CMB, the angular size of which is 1 degree today.
Kindly comment and correct what's wrong. Having not studied physics, I appreciate any help very much.