# Open and closed universe

Hello,

I'm not sur if this is the right section. Please could a mod move it if it isn't.

I've only just got into physics and so i don't want anything tooo complicated

Could someone please explain the difference between an open and closed univers

Thanks

_Muddy_

Kurdt
Staff Emeritus
Gold Member
The expansion of the universe is governed by an equation called the Friedmann equation. It is given below:

$$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G \rho}{3} - \frac{kc^2}{a^2}$$

The final term with the constant k measures the curvature of space. If $k > 0$ then we have what is called a spherical geometry and the universe is closed. If $k < 0$ we have a hyperbolic geometry and the universe is open. The spherical case is called closed because a universe with this geometry must be finite. Hyperbolic geometry in a universe however would mean it would be infinite and thus open. There is a special case where $k = 0$ which gives us a flat Euclidean geometry. The universe would be infinite in this case as well. At present cosmologists are pretty sure $k=0$. Of course the universe could have a non-trivial topology in which case things would get a bit more complicated.

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The expansion of the universe is governed by an equation called the Friedmann equation. It is given below:

$$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G \rho}{3} - \frac{kc^2}{a^2}$$

The final term with the constant k measures the curvature of space. If $k > 0$ then we have what is called a spherical geometry and the universe is closed. If $k < 0$ we have a hyperbolic geometry and the universe is open. The spherical case is called closed because a universe with this geometry must be finite. Hyperbolic geometry in a universe however would mean it would be infinite and thus open. There is a special case where $k = 0$ which gives us a flat Euclidean geometry. The universe would be infinite in this case as well. At present cosmologists are pretty sure $k=0$. Of course the universe could have a non-trivial topology in which case things would get a bit more complicated.

Ok so a closed universe is when k(which measures the curvature of space) is greater than 0. And having a spherical Geometry means what? That the universe is a sphere? And that the universe is finite

And an open universe is when k(which measures the curvature of space) is less than 0. And then we'd have a hyperbolic geometry. What is a hyperbolic geometry? And an open universe is infinite.

What does a flat Eucildean geometry mean?

How is the curvature of space(k) measured?

In that Friedman equation what does the a stand for? And what does the Gp stand for?

If the universe is infinite will it ever stop expanding. I struggle to get my head around a universe being infinite.

Hey thanks so much for the help

_Muddy_

Kurdt
Staff Emeritus
Gold Member
Well lets first consider a flat Euclidean geometry. You are probably familiar with it since you learn it in school at an early age. It is the geometry in which triangles have internal angles that add to 180° and the circumference of a circle is $2\pi r$.

In spherical geometry the internal angles of a triangle are greater than 180° and the circumference of a circle is less than $2\pi r$. This corresponds to geometry on the surface of a sphere and hence the name. The main consequence of this type of geometry is that the universe will be finite.

In hyperbolic geometry the internal angles of a triangle will be less than 180° and the circumference of a circle will be greater than $2\pi r$. This is harder to visualise but is usually described as a saddle surface. A universe with hyperbolic geometry is infinite.

$a$ is the scale factor of the universe and it measures the expansion rate. $G$ is just the universal gravitational constant and $\rho$ is the density of material in the universe. There is a library entry on the Friedmann equation but it hasn't auto linked for some reason.

https://www.physicsforums.com/library.php?do=view_item&itemid=10

As for evidence for $k$ it mainly comes from measurements of the cosmic microwave background. Someone else will have to take over from there. :tongue:

So if the universe is closed and has spherical geometry does that mean that the universe is shaped like a sphere? And if i had a big spaceship and set of in one direction i would eventually get back to my starting point (probably after a long time)

I went to a lecture with my school a while ago and the guy doing the lecture showed us a diagram of the universe and said many scientists belived it to be "bell shaped". Which type of universe would that be?

If like you say many scientists belive that k=o and that the universe has a flat Euclidean geometry. Would'nt that mean that the universe was 2D?

Slightly unrelated but hwo do you say "Euclidean"? Is it (you-kle-dean)?

I've just a look on Google and found this picture, would I be correct in saying that $\Omega$ in this case of the same as the $k$ you have been referring to? The diagram below gives you a good visual concept of what is happening, I know I sometimes find things hard to work out if I don't have a visual idea. _Mayday_

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Kurdt
Staff Emeritus
Gold Member
This is where the conceptual difficulties come in. A sphere is 2D analogy of a positive curvature. Our universe is 3D but would have an intrinsic curvature.

I think what the guy at your lecture was referring to is the shape of the observable universe which traces the past light cone back to the big bang. The bell shape occurs because of the inflationary period where the universe expanded very rapidly.

The universe would not be 2D if k=0. All that would mean is 3D space has no intrinsic curvature.

Mayday, $\Omega$ is the density parameter and $\Omega =1$ is the same as k = 0.

remember those pictures are just 2D analogies.

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So if the universe is closed and has spherical geometry does that mean that the universe is shaped like a sphere?
What is closed is 4 dimensional spacetime, spacetime is "curved up" like a sphere.

Thanks for clearing that up kurdt, mejennifer and _Mayday_

How can an infinite universe expand? By infinite do we mean infinite in size?