# Is the flat curvature of the universe in the 5th dimension?

It is hard to imagine that the universe could have a flat curvature while I've read that it is at this moment the most likely.

When you look around the universe is expanding in all directions, so it looks like a spherical ball shaped universe. Nevertheless the curvature is until now measured flat.

But how to understand that something that has in all directions galaxies look flat. So I thought perhaps they mean that is in the 5th dimension, in analogy with an ant who can't see that the line he is walking on is actually 3 dimensions.

But I read that If the true value of the cosmological curvature parameter is larger than $10^{−3}$ we will be able to distinguish between the three models (spherical, hyperbolic and flat) even now. So I think how is something in the 5th dimension measured as it is beyond our capability of experiencing?

So somewhere I miss something but what?

Several topics here:

## No 5th dimension. Not even a 4th (spatial)

You're on the right track: The Universe being "flat" does not mean "like a table top", just like it being "spherical" and "hyperbolic" don't mean like a ball or a saddle. Those terms are just 2D analogies, whereas the Universe has three spatial dimensions. Like any analogy, they are great for understanding some things, but shouldn't be taken too far.

For instance, the sum of the angles of a triangle, the fate of parallel lines, and the volume of some region in space can be more easily visualized in 2D than in 3D. You can understand that a triangle going North Pole→Congo→Indonesia→North Pole has 270º rather than 180º. But whereas the 2D surface of Earth curves in 3D space, a 4th spatial dimension is not needed for our 3D Universe to curve (not to speak of a 5th). And imaging why a triangle consisting of what seems to be completely straight lines from Earth→GRB090423EGSY8p7→Earth should not have 180º is difficult, but nonetheless perfectly possible.

## The Universe is not a ball with you in the center

When you look around in the Universe, it looks like a spherical ball with you in the center, but that's only because what you see is everything from where light has had the time to reach you in 13.8 Gyr, i.e. since Big Bang. Everything inside that sphere is called the observable Universe, and it's bound by the so-called particle horizon. In the 2D analogy, you can say that Earth looks like a flat disk, but only because that's how far you can see. Again, don't take this analogy too seriously, since the reason here is just the curvature of Earth, and not that Earth has only existed for $10^{-4}\,\mathrm{s}$.

## Curvature

The dynamical and geometrical properties of the Universe depend on its expansion rate ($H_0$), the densities of its components (the "Omegas" of baryons, dark matter, dark energy, radiation, etc.), and its "intrinsic" curvature, which can also be written like a density parameter $\Omega_k$. The latter is found to be $0.000\pm0.005$ (Ade et al. 2015), but as you say, if it's less than $\sim10^{-4}$, we may never know (Vardanyan et al. 2009).

• Ok, the earth look in 2d flat, but it ís spherical more precise with a volume in 3d. Now to us the universe looks in 3d sperical with an expanding volume (like a rising currants bread) but what it really ís, that is the question. I've heard that the universe has a center but it is in the 4th spacial dimension. So than I think the shape of that 4th dimension could also be flat or curved (positive or negative). So can I say that? So it results in the conclusion that our universe looks spherical for us in 3d but it ís flat in the 4th spacial dimension. Feb 18 '16 at 9:42
• @Marijn: No, you can't really say that. I don't know where you've heard that the Universe has a center in a 4th dimension, but it is not a necessary condition, and definitely not part of the standard model. As I said, the "ball analogy" shouldn't be taken too far. The "geometrical rules" on the surface of Earth/in the volume of Space can be compared, but you must not consider what's outside Earth's surface. In this analog, the sky doesn't exist, and the underground doesn't exist; only the surface.
– pela
Feb 18 '16 at 12:19