How can we calculate universe diameter at a given time

[nograviton]

Indeed it is! And that's just the part we can see!

Because World is non Realism. Phật pháp đã dạy !

 

[madness]

Why can't we estimate the size of the universe from the curvature of space? As far as I know, we have some empirical bounds on how flat the universe appears to be. From this, we could place bounds on how large it would be if it were a sphere, for example. Of course, we would need to make some assumptions, such as that the universe is globally similar to what we see locally.

I'm happy to be corrected by someone more knowledgeable here, but intuitively this seems to make sense to me.

 

[marcus]

Why can't we estimate the size of the universe from the curvature of space? As far as I know, we have some empirical bounds on how flat the universe appears to be. From this, we could place bounds on how large it would be if it were a sphere, for example. Of course, we would need to make some assumptions, such as that the universe is globally similar to what we see locally.

I'm happy to be corrected by someone more knowledgeable here, but intuitively this seems to make sense to me.

That's a clever observation. We can in fact estimate in the limited sense of giving a LOWER BOUND on the radius of curvature, or on the circumference.

As I recall this was done in one of the WMAP5 reports (the fifth year WMAP publications), as a kind of footnote. As you say it involves assumptions. You say "how large it would be if it were a sphere".
And "globally similar to what we see..."

Let's say we measure the spatial curvature and find a 95% confidence interval that Ω is between 0.99 and 1.01.
So we have an upper bound on the spatial curvature number |Ω_k| < 0.01

We take the current Hubble radius 14.4 Gly and we divide it by the square root of that number and that gives a 95% CONFIDENCE LOWER BOUND on the current radius of curvature of space.

144 Gly. The radius of curvature could be vastly bigger than that or it could be infinite. After all ZERO is in the confidence interval for the curvature number---zero curvature corresponds to infinite radius of curvature. but with 95% confidence the curvature is at most such and such so the radius is at least 144 billion light years.

The mental picture is of a 3D skin of a 4D ball except that the solid ball does not exist, is not actual. Only the 3D hypersphere exists. And it has a radius of curvature 144 Gly. So you can multiply by 2π and get its circumference. If you could pause expansion to make it possible to circumnavigate or send a flash of light around that circumference would indicate how long before the flash of light would return from the other direction.

 

[marcus]

Maybe someday a much more precise measurement of spatial curvature might be made and the 95% confidence interval for the curvature number might be all on the positive side of zero like 0.0001 < |Ω_k| < 0.0004
So then infinite radius of curvature would be excluded with 95% confidence and we would have an UPPER bound on it, as well as a lower bound.
The square root of 0.0001 is 0.01, so whatever the Hubble radius is at that time, say it is 14.6 Gly, we would divide that by 0.01 to get the upper bound estimate. And divide 14.6 Gly by 0.02 to get the lower bound estimate.

 

[madness]

Thanks Marcus. One question has been bugging me about this. As far as I'm aware, a popular viewpoint is that the universe is flat. To my knowledge, that requires a curvature of exactly zero (for a homogeneous universe). This seems incredibly unlikely, as any minute deviation below or above zero would lead to a hyperbolic or spherical universe, albeit very large. If this is true, how can the flat universe hypothesis be entertained, given that it requires a fine tuning to an infinitely precise degree?

I don't like the argument that the universe can be "approximately flat". Locally, maybe, but globally a sphere and a plane are different objects.

 

[julcab12]

One question has been bugging me about this. As far as I'm aware, a popular viewpoint is that the universe is flat. To my knowledge, that requires a curvature of exactly zero (for a homogeneous universe). This seems incredibly unlikely, as any minute deviation below or above zero would lead to a hyperbolic or spherical universe, albeit very large. If this is true, how can the flat universe hypothesis be entertained, given that it requires a fine tuning to an infinitely precise degree?

I don't like the argument that the universe can be "approximately flat". Locally, maybe, but globally a sphere and a plane are different objects.

In any form of precision(apparatus). We never get to zero. There is always a slight deviation and uncertainty principle takes over. Anything that involves with time is bound to deviations. We can only say approximately high percentage flat or approximately low percentage curved-- non zero curvature.

http://arxiv.org/abs/1502.01589
http://arxiv.org/pdf/gr-qc/0501061v1.pdf

 

[madness]

I'm not talking about measurement precision, I'm talking about the universe itself. There is exactly one parameter value out of an infinite number which yield a flat spacetime. Any perturbation to this parameter, however small, will yield a spherical or hyperbolic spacetime. Doesn't this effectively rule out the possibility that the universe is flat?

 

[marcus]

As far as I'm aware, a popular viewpoint is that the universe is flat.

..., how can the flat universe hypothesis be entertained, given that it requires a fine tuning to an infinitely precise degree?

I don't like the argument that the universe can be "approximately flat". Locally, maybe, but globally a sphere and a plane are different objects.

Part of the problem is careless use of words. Reporters should not imply that there is a consensus in favor of (perfect) flatness.

They could say nearly spatially flat. or near-zero curvature. or "observations are consistent with spatial flatness"...

As I recall the WMAP reports were always careful to use qualifications like this.

Still it is pretty impressive to be able to narrow Ω down to within 1% of 1, or narrow Ω_k down to within 1% of zero.
Speaking informally it certainly does seem to justify at least saying "nearly".

You'd be right to insist, though, that people should always be reminded that nearly flat does allow for space being finite and boundaryless like a 3D hypersphere---the 3D analog of the 2D surface of a balloon.

the problem is partly with language and our need to communicate quickly. You may be right that "approximately flat" encourages the mental image of infinite 3D space, just with a few little local humps and dips but overall flat. So what about "average curvature near zero"?

If the the average curvature were not exactly zero but were slightly positive wouldn't that suggest a (large) hypersphere?

It's hard to change how people talk. These are matters of nuance. I think all you can do is try to use language carefully and succinctly yourself. Bottom line, we don't know the overall topology of space---have to keep the mind open to different possibilities.

 

[julcab12]

I'm not talking about measurement precision, I'm talking about the universe itself. There is exactly one parameter value out of an infinite number which yield a flat spacetime. Any perturbation to this parameter, however small, will yield a spherical or hyperbolic spacetime. Doesn't this effectively rule out the possibility that the universe is flat?

Short answer is yes or no. Whatever you are comfortable with or how you evaluate flat in sense. A flat universe in a general relativistic term is one where the Reimann curvature is zero everywhere. It can only be possible unless the EM tensor is zero which is not the case for all we know right now. However on large scale FLRW metric k=0. K happens to be related to the spatial curvature of the universe on a 'fixed' time-slice. K=0 in the FLWR metric translate to the geometry of 3D space being Euclidean at any given time. Meaning space is flat but spacetime isn't.

"Experiments such as WMAP and Planck measure the Hubble parameter as well as the energy density of the universe, and the data obtained seems to strongly favour the energy density being exactly ρ=3H2(t)8πG. This automatically corresponds to k being zero in the Friedmann equation, a condition which some people refer to as "critically dense", instead of "flat"."

In a more shorter answer. The flatness of the universe is this sense has nothing to do with its shape. We simply do not know. What we observe is spherical, simply because the speed of light does not depend on direction, so looking in any direction the distance limit we can see is the same. -- Jerzy Pawlak PHD in HEP

 

[timmdeeg]

In case inflation holds the deviation from spatial flatness is less than $10^-30$, which means that there is still a very tiny chance that the sign of k is not zero.

I would be interested to learn whether cosmologists have any theoretical ansatz, which yields $k = 0$ for the pre-inflation epoch.

 

[nikkkom]

Subjective View # 1: I don’t accept ANYTHING as being “infinite.” Maybe I’m just being anal retentive (I’ve been called that more than once!), but I find the concept of an infinite universe as being unacceptable; I like “order” in my world, and an infinite cosmos (to ME, anyway) flies in the face of a clearly defined universe.

LOL...

Subjective View # 3: I do not accept the parallel universes concept (Everett’s many world’s interpretation of quantum physics), the term meaning that there are an infinite number of side-by-side universes with carbon copies of me in them, differing only in minor details (i.e. occupation, hair color, etc, ad infinitum). To me, hat’s just too “messy,” with a vast overabundance of realities!

With such an obsession of things matching your ad-hoc "feelings" how things "should be", I wonder how you managed to accept QM.

Over the last century, new discoveries of physics has shown us that we need to relax a lot and accept some quite unnatural laws of physics, if we want to have theories which explain observed experimental data.