Re: viewing light from the earliest stages of universe

From: N:dlzc D:aol T:com \(dlzc\) (net_at_nospam.com)
Date: 02/14/05 

Date: Mon, 14 Feb 2005 07:29:28 -0700

Dear Ben Rudiak-Gould:

"Ben Rudiak-Gould" <br276deleteme@cam.ac.uk> wrote in message
news:cuqas4$okh$1@gemini.csx.cam.ac.uk...
> N:dlzc D:aol T:com (dlzc) wrote:
>> Part of the misconception associated with the really bad term "Big Bang"
>> is that the Universe exploded from some sort of singularity. The
>> evidence is that the Universe was of some finite size (a few units to a
>> few tens of million light years "across") at the Big Bang, and so it
>> would take some time for light to cross it.
>
> I don't understand this claim. In the usual cosmological models, the size
> of
> the universe was infinite at the time of the big bang.

The CMBR filled the early Universe. The CMBR cannot have the spectrum it
does, unless it is a few tens of million light years thick, or less.
Therefore, based on this one set of *assumptions*, the Universe is
unbounded finite.

> In my opinion the easiest way to understand what's going on is in terms
> of
> the flat Robertson-Walker metric:
>
> ds^2 = dt^2 - R(t)(dx^2 + dy^2 + dz^2)
>
> This looks just like the SR metric, except for the factor R(t), which is
> the
> "scale factor" (not size) of the universe at the cosmological time t.
> R(t)
> is zero at the time t=0 (the big bang), and increases after that. In some
> models it eventually reverses and goes back to zero.
>
> The universe described by this solution is not expanding in any
> meaningful sense! It is always infinite in extent at every time t > 0.
> (We can't say anything about t = 0 because the metric is singular.)
> However, two galactic superclusters which are at rest with respect to the
> R-W coordinates will end up farther and farther apart in metric distance
> as R(t) increases. If you look backwards in time, the distance between
> them goes to zero as you approach the big bang. Furthermore they will
> each appear redshifted as seen by the other, because the metric distance
> between successive crests of the light wave will increase as it travels.
> The net cosmological redshift factor is R(t_now) / R(t_then), where
> t_then is the time of emission of the light and t_now is the time of
> detection.
>
> Now consider the question of which parts of this spacetime we can
> actually
> see, from our vantage point at t = t_now. To figure this out we draw a
> past-directed light cone from our vantage point. The stuff on the surface
> of
> this cone is what we can see. In principle we could see the stuff inside
> the
> cone too, but in practice we don't get many sub-light-speed signals from
> deep space.
>
> Drawing this cone is not trivial because the speed of light in R-W
> coordinates is not equal to c. Rather, it's 1/R(t). So to get the size of
> the visible universe at a time t_then in the past, we have to integrate
> dt/R(t) from t_now to t_then. In terms of working out a "size of the
> visible
> universe", the most sensible t_then is about 300000 yr, which is when the
> universe became transparent. By definition, we can't see anything before
> then. (We can see something *from* then, namely the cosmic microwave
> background.)
>
> Working out this integral gives us the radius of the light cone at t_then
> in
> the nonphysical R-W coordinates. We'd rather have a meaningful metric
> distance. But do we want the metric at t_then or t_now? There's no right
> answer to this question. If we choose t_now, we get a radius of about 47
> billion light years (according one one site I looked at) or 78 billion
> light years (according to another). If we choose t_then, we get 47 or 78
> *million* light years, because R(t_now)/R(t_then) is about 1000. Compare
> these figures to t_now - t_then, which is about 14 billion years. This
> last figure is the one usually quoted as the "radius of the visible
> universe" (as 14 billion light years), but it's completely wrong! It's
> based on a naive SR argument which just doesn't apply on cosmological
> scales.

The argument is based on GR, and the Universe may be flat in space, but is
not flat along the time axis (acceleration for example).

> Also, don't confuse the size of the visible universe with the size of the
> universe. There's no reason to expect any similarity between the two.
>
> I'm not sure where I'm going with this, but I hope it helps someone.

Agreed.

David A. Smith

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