B Big Bang: Size of the Universe at Different Epochs

[PAllen]

Where and when did Gauss learn GR?!

Gauss developed theory of extrinsic and intrinsic curvature of surfaces and showed equivalence of embedding approach and intrinsic approach.

 

[PeroK]

Gauss developed theory of extrinsic and intrinsic curvature of surfaces and showed equivalence of embedding approach and intrinsic approach.

A man ahead of his time!

 

[Hornbein]

I don't get how to reconcile these two ideas:
- the universe may be infinite in size
- the observable universe grew from a very small volume to a very large volume

The idea is that the volume has always been infinite. So it isn't possible to compare two sizes directly. We CAN however measure and compare the densities at two different times.

The density of the universe is believed to be always an infinite amount of matter divided by an infinite volume. Just the sort of operation they told you not to do in high school. But the infinite pound gorilla does as it pleases.

Intuitively one would think that if the amount of matter remains the same and density decreases then the volume must be larger. But we have not been able to assign any number to this volume.

I don't know, but I've been told, that if the curvature is non-zero that it increases over time. So for it to be so close to zero now it must have been much closer to zero back then. Or so they say.

 

[DaveC426913]

Just one nitpick:

I don't know, but I've been told, that if the curvature is non-zero that it increases over time.

Does not rhyme. Also, meter is off.

Spoiler

 

[PAllen]

The idea is that the volume has always been infinite. So it isn't possible to compare two sizes directly. We CAN however measure and compare the densities at two different times.

What you can say is that given the volume enclosing some amount of matter (galaxies) in the present, then the volume enclosing that same matter decreases as you go back in time to the initial state we call the big bang. In a pure FLRW solution (i.e. no inflation, etc.) there is no lower bound to this volume. Despite this, the universe as a whole can still be infinite at all times. It's just a feature of 'infinite' - no matter how many times you cut it in half, it is still infinite.

I don't know, but I've been told, that if the curvature is non-zero that it increases over time. So for it to be so close to zero now it must have been much closer to zero back then. Or so they say.

This is not generally true. In a wide range of pure FLRW solutions, the curvature scalar R grows without bound as time is followed backwards. I am not aware of any general tendency for curvature to increase over time.

[edit: In fact it is quite easy to have constant curvature scalar throughout - just have flat spatial slices with exponential scale factor. This produces a constant value for R]

 

[PeterDonis]

I've been told, that if the curvature is non-zero that it increases over time.

This is referring to a particular definition of spatial curvature (and in a particular type of model, see below), not spacetime curvature. As @PAllen posted, the spacetime curvature of any expanding FRW solution will decrease with time.

In matter and radiation dominated FRW spacetimes, if the density parameter $\Omega$ (the ratio of actual density to critical density) is not exactly $1$, then it will diverge from $1$ over time. In models with zero cosmological constant, $\Omega - 1$ can also be treated as a (sort of) spatial curvature parameter; if it is zero (i.e., $\Omega = 1$, density exactly critical), the universe is spatially flat; positive $\Omega - 1$ means positive spatial curvature (closed universe), and negative $\Omega - 1$ means negative spatial curvature (open universe). A universe with positive $\Omega - 1$ will have increasing $\Omega - 1$ over time, whicn can (sort of) be thought of as increasingly positive spatial curvature, and a universe with negative $\Omega - 1$ will have decreasing $\Omega - 1$ over time, which can (sort of) be thought of as increasingly negative spatial curvature.

(Note that I put in the qualifier "sort of" each time I mentioned spatial curvature in the previous paragraph. That's because $\Omega - 1$ is not the actual geometric quantity that would normally be referred to as the "spatial curvature" of a spacelike slice of constant FRW coordinate time in these spacetimes. Sources that talk about it as related to "spatial curvature" without any qualification are being sloppy.)

The reason cosmologists are interested in all this is that, for most of our universe's history, it was either matter or radiation dominated, and yet our current universe's $\Omega$ is extremely close to $1$. This requires extremely sensitive fine-tuning of initial conditions, which is known as the "flatness problem":

https://en.wikipedia.org/wiki/Flatness_problem

 

[kimbyd]

Actually, the answer is definitely yes. There are analogs of the Nash embedding theorems for pseudo-Riemannian manifolds.

It does require a large number of dimensions for the general case ( for a smooth embedding; for a C1 embedding, very few extra dimensions are needed) but these dimensions would be unobservable, just as a being within a 2-sphere is unaware of an embedding in 3-space. Embedding doesn’t change the geometry of a manifold, it just provides a different way of describing it. It also allows definition of things which are presumably unobservable and embedding dependent like extrinsic curvature.

Quite a few authors for GR first introduce embedding and extrinsic curvature first, then derive intrinsic curvature. Personally, I always found this a waste of time, but some teachers prefer it pedogogically. But, hey, this is the way Gauss went about it. Who am I to question Gauss?

I see. So it isn't so much that it's impossible, but rather that it's ridiculously complicated, and disfavored for that reason. At least when plain GR is concerned.

I'd have to guess that attempting to quantize the embedded space-time would break that model entirely.

Edit: A quick search confirms my intuition that QM would probably muck this up:
https://academic.oup.com/ptp/article/83/5/894/1875237

 

[JimJCW]

I am wondering if someone can help me with the size of the universe at
- Quark epoch (10^-5 seconds)
- Photon epoch (380,000 years)

Using Jorrie’s calculator and PLANCK Data (2015), the size of the observable universe is calculated to be

R = 0.87 Mly at t = 0.38 Myr​

From the extrapolation of a linear log(R) vs. log(t) plot, the size of the observable universe is estimated to be

R = 3 km at t = 10 µs​

@Jorrie

 

[KingGambit]

Using Jorrie’s calculator and PLANCK Data (2015), the size of the observable universe is calculated to be

R = 0.87 Mly at t = 0.38 Myr​

From the extrapolation of a linear log(R) vs. log(t) plot, the size of the observable universe is estimated to be

R = 3 km at t = 10 µs​

@Jorrie

Thank you very much @JimJCW