How Can Redshift be Expressed as a Function of DM in an Empty Universe Model?

[Aztral]

Hi there!

So basically I'm trying to get the function which expresses redshift as a function of DM in the empty universe model (Omega=0).

I've downloaded the type 1A supernova data from http://www.astro.ucla.edu/~wright/sne_cosmology.html, now I'd like to plot the Omega=0 curve. My GR abilities are non-existant and I haven't what I'm looking for by searching

Any help would be appreciated!

 

[Chalnoth]

Hi there!

So basically I'm trying to get the function which expresses redshift as a function of DM in the empty universe model (Omega=0).

I've downloaded the type 1A supernova data from http://www.astro.ucla.edu/~wright/sne_cosmology.html, now I'd like to plot the Omega=0 curve. My GR abilities are non-existant and I haven't what I'm looking for by searching

Any help would be appreciated!

Maybe this will help?
http://arxiv.org/abs/astroph/9905116

 

[Aztral]

Hi there.

Thanks-think I almost have it. I also found http://www.ast.cam.ac.uk/~pettini/Physical%20Cosmology/lecture05.pdf & lecture05.pdf which helped alot.

Surprising-with the numerous milne space graphs all over the place you'd think somewhere that equation would be stated explicitly somewhere.

 

[Imax]

My guess is a closed Universe. A closed Universe doesn’t mean that the Universe can’t expand or shrink.

 

[Chalnoth]

My guess is a closed Universe. A closed Universe doesn’t mean that the Universe can’t expand or shrink.

The Milne cosmology is open, not closed. And there is zero acceleration in a Milne cosmology.

 

[Imax]

D'oh! I’m arguing semantics?

Closed Universe \neq Closed Manifold

You need a closed manifold to have a closed Universe, but a closed manifold doesn’t need to be static. If the Universe is a closed 4D manifold, then it doesn’t exclude the possibility that the Universe will end up in heat death, or a big freeze, rip, crunch, or bounce.

 

[Chalnoth]

D'oh! I’m arguing semantics?

Closed Universe \neq Closed Manifold

You need a closed manifold to have a closed Universe, but a closed manifold doesn’t need to be static. If the Universe is a closed 4D manifold, then it doesn’t exclude the possibility that the Universe will end up in heat death, or a big freeze, rip, crunch, or bounce.

I don't understand what you're getting at here.

But in the end, the spatial curvature of our observational universe doesn't necessarily say anything at all about whether the entire universe is closed or open. For example, if we measure our universe to be slightly open, it could just be because we are in a slightly underdense region of the whole universe, and the universe, when taken as a whole, may still be closed.

 

[Imax]

At some time point after the Big Bang, the universe could be considered as a closed manifold. If mass/energy distributions defines space-time, and mass/energy was very concentrated at the Big Bang singularity, then space-time could have been very concentrated, not much more than an atmosphere around a planet. As the Big Bang expanded, space-time itself began to expand.

 

[Chalnoth]

At some time point after the Big Bang, the universe could be considered as a closed manifold. If mass/energy distributions defines space-time, and mass/energy was very concentrated at the Big Bang singularity, then space-time could have been very concentrated, not much more than an atmosphere around a planet. As the Big Bang expanded, space-time itself began to expand.

I still don't see what you're trying to say here.

 

[Imax]

We live within a closed 4d space-time manifold.

 

[Chalnoth]

We live within a closed 4d space-time manifold.

Except the evidence is that we do not, because as long as it expands forever, as the evidence so far supports, then it isn't closed in four dimensions.

 

[Imax]

A closed manifold does not exclude the possibility that the universe expands forever.

 

[Chalnoth]

A closed manifold does not exclude the possibility that the universe expands forever.

Why not? A closed manifold is, by definition, finite. While an eternally-expanding universe is infinite.

 

[Imax]

A closed manifold is, by definition, finite, but it doesn't need to be static. A manifold can grow or shrink, changing with time.

 

[bcrowell]

Why not? A closed manifold is, by definition, finite. While an eternally-expanding universe is infinite.

I think you need to distinguish spatial infinity from temporal infinity. You can have cosmologies that are spatially finite but temporally infinite. See http://arxiv.org/abs/astro-ph/9812133 , figure 7.

A closed manifold is, by definition, finite, but it doesn't need to be static. A manifold can grow or shrink, changing with time.

The manifold we normally talk about in GR includes both space and time dimensions. I wouldn't refer to it as changing over time. There aren't different manifolds at different times. You could take the 4-dimensional manifold-with-metric that represents a cosmological solution and make a spacelike slice through it to form a 3-dimensional manifold, but that wouldn't be "the" manifold.

 

[Imax]

The manifold we normally talk about in GR includes both space and time dimensions. I wouldn't refer to it as changing over time. There aren't different manifolds at different times.

Agree. It's the same space-time manifold, but current cosmological data indicates that this manifold is growing with time, and that growth appears to be accelerating (something like a balloon analogy).

 

[bcrowell]

Agree. It's the same space-time manifold, but current cosmological data indicates that this manifold is growing with time, and that growth appears to be accelerating (something like a balloon analogy).

It doesn't make sense to say that "this manifold is growing with time." The manifold includes points at all times. I'm not disagreeing with cosmological expansion or the balloon analogy, I'm telling you that you're misusing the term "manifold." The balloon is not analogous to the manifold. The balloon is analogous to a spacelike surface, which is a slice through the manifold.

 

[Imax]

I'm telling you that you're misusing the term "manifold."

I can't use the term "metric." Isn't a metric a subset of a manifold?

 

[bcrowell]

I can't use the term "metric." Isn't a metric a subset of a manifold?

A manifold is a set of events with a topology but no other structure. A coffee cup is the same manifold as a doughnut. A metric is a separate piece of mathematical machinery that can be added onto a manifold, defining measurements of time and distance. If you want to say space is expanding, I would just use the word "space."

 

[Imax]

Ok. “Space” is expanding (i.e. Hubble redshift).

The spatial component of the manifold is expanding with time, and the distance between galaxies is increasing (but not necessarily for local clusters).

 

[Chalnoth]

I think you need to distinguish spatial infinity from temporal infinity. You can have cosmologies that are spatially finite but temporally infinite. See http://arxiv.org/abs/astro-ph/9812133 , figure 7.

Yes, but here I was responding to the statement of our universe being a closed space-time manifold, which I took to mean topologically closed in four dimensions, as opposed to simply talking about closed in the sense of having positive spatial curvature. Obviously a universe that has positive spatial curvature can, in the presence of a cosmological constant, expand forever. But a universe that is topologically closed in four dimensions, by definition, cannot.

 

[Imax]

I don’t mean to say there are different manifolds with time. Manifold is topology, and spacetime is a metric within that manifold. I’m wondering about metrics just after the Big Bang singularity. Could they be described within a compact Lorentz manifold?

 

[Chalnoth]

I don’t mean to say there are different manifolds with time. Manifold is topology, and spacetime is a metric within that manifold. I’m wondering about metrics just after the Big Bang singularity. Could they be described within a compact Lorentz manifold?

I have no idea what you are saying here. A manifold in General Relativity is not a three-dimensional object that evolves in time. A manifold is a fully four-dimensional object.

 

[Imax]

Spacetime just after the Big Bang singularity can describe as a metric within a compact Lorentz manifold?

 

[Chalnoth]

Spacetime just after the Big Bang singularity can describe as a metric within a compact Lorentz manifold?

a) There was no singularity, so I don't know what you're talking about.
b) Any small-enough region of space-time can be considered to be a Lorentz space-time, but it is the radius of curvature that determines how small you have to get before you can do that. Because of the expansion, at no time could you have considered the entire universe as being Lorentz.

 

[Imax]

Expansion does not exclude the possibility that the universe is Lorentz.

 

[Chalnoth]

Expansion does not exclude the possibility that the universe is Lorentz.

Yes, actually, it does. Except in the very special case of an empty universe (but then "expansion" is meaningless in that situation anyway). As long as the universe isn't empty, then you have space-time curvature. Lorentz space-time is flat.

 

[Imax]

Lorentz space-time is flat.

Are you saying a compact Lorentz manifold is flat?

 

[Chalnoth]

Are you saying a compact Lorentz manifold is flat?

If it's Lorentz, it's flat, period.

 

[George Jones]

Are you saying a compact Lorentz manifold is flat?

Do you mean "Lorentzian manifold"? All spacetimes in general relativity are 4-dimensional Lorentzian manifolds. All compact 4-dimensional Lorentzian manifolds have closed timelike curves; see

https://www.physicsforums.com/showthread.php?p=1254758#post1254758.

 

[Imax]

All compact 4-dimensional Lorentzian manifolds have closed timelike curves

A closed timelike geodesic is problematic. I’m thinking that any current spacetime manifold should be similar to a manifold near the Big Bang. Topology does not change with time.

 

[Imax]

Maybe time orientable is a locale property, which may not necessarily exclude the possibility of closed timelike geodesics?

 

[Chalnoth]

Maybe time orientable is a locale property, which may not necessarily exclude the possibility of closed timelike geodesics?

Well, the arrow of time is most likely a local property, but that doesn't make closed timelike curves any more sensible.

 

[Imax]

Well, the arrow of time is most likely a local property, but that doesn't make closed timelike curves any more sensible.

If we asume that the arrow of time is a local property within a compact Lorentzian manifold, then closed timelike geodesics may be alllowed, and events could repeat.

 

[Chalnoth]

If we asume that the arrow of time is a local property within a compact Lorentzian manifold, then closed timelike geodesics may be alllowed, and events could repeat.

Well, a compact Lorentzian manifold has closed timelike curves. That's one reason why our universe isn't one.

 

[Imax]

Well, a compact Lorentzian manifold has closed timelike curves. That's one reason why our universe isn't one.

I don’t necessarily agree, but I think Aztral already go the answer he/she was looking for. What I was trying to say is that maybe there is no such thing as an empty universe. Trying to calculate properties of such a universe may be a nice exercise in mathematics or “theoretical” physics, but it may have nothing to do with reality. CMB data seems to fit better with a finite model of the universe than with an infinite model, and that seems to make sense to me.

I can see two possibilities at or near the Big Bang:

1) Space-time was infinite and somewhere within that infinite space was a very small volume (a singularity?) that contained the mass/energy of the entire universe.

2) Space-time was so badly curved (i.e. mass can bend space-time in GR) that space-time itself was finite around this volume/singularity.

My $ is on #2. Excluding possible quantum effects, topology does not change with time.