Friday, April 25, 2008

Angular Correlation


Many things were known before we were born, but far more important is what will become known after we are born. Stirring the pot is useful, and the post "Inflation Sinking" has drawn lively comment. We have a great positive post from Carl Brannen, an entire post devoted to critique that Starts With A Bang, and positive review on the new Carnival of Space!

For nigel and others: here is the early crude version of the angular correlation function. David Cline lectured on this to us at Stanford back in 2003. "Something funny," Dr. Cline titles this page, "the angular correlation function is nearly zero for angles beyond 60 degrees." A student kept his notes, calculated the red prediction curve without computers, and wrote it with marker pen. The typeface has been cleaned up, the data curves untouched. (Dr. Cline may have confused TE with TT curves, but that too is untouched.)

This graph measures the scale of density fluctuations in the cosmic microwave background. The old inflationary paradigm states that the Universe is flat, like the Earth. Density fluctuations should exist on all scales. These fluctuations can be modelled using spherical harmonics, the same maths we students used to model the hydrogen atom. When those fluctuations are added together they produce the "Inflation" curve. This prediction is ruled out by both COBE and WMAP experiments.

Theory predicts that the Universe is curved with radius R = ct, therefore the biggest possible fluctuation has peak-trough distance of 180 degrees. This prediction is very robust. You can model this curve in a calculator using sine waves, sawtooth or even square waves because all repeating functions can be modelled as sums of sine waves. Given the limits of marker pen, prediction fits the data quite closely.

Responding to everyone will take some time, but more comments are welcome. Please note that nobody here hates inflation, it has been acknowledged many times as a useful step. This work has been discussed personally with Alan Guth, who is considered a friend. We loved the Titanic, but it too slipped beneath density waves.

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5 Comments:

Blogger CarlBrannen said...

Louise,

Regarding the angular correlation, I think when you do it with sine waves (which is how everyone does the calculation), it's not very intuitive because it becomes a Fourier space calculation, and I really couldn't see the 180 degree thing intuitively.

So that's why I went ahead and did it in the very obvious way in the untransformed domain. In that domain, the angular correlation becomes a simple matter of "was their already a causal connection between these two points in space at the time of recombination?"

But I do hope that you give a critique of my calculation. The central feature of it is to assume a choice of coordinates so that space is flat (and time is screwed up, more or less).

By the way, the whole concept of thinking of the variable speed of light as a modification of the passage of time appeals to me because we have the GR theory that already says that gravity acts by modifying the passage of time (and so, in a flat space choice of coordinates for a black hole such as Painleve coordinates this makes a variable speed of light required).

9:35 AM  
Blogger L. Riofrio said...

Carl: Your blog post was so wonderful that I am reluctant to offer any critique. Concerning distances, I would suggest putting scale factor R inside the integral sign. Then it all integrates to a multiple of t^{2/3}, even simpler. Measuring from the time of recombination, only 380,000 years after the BB, we can treat time as t = 0, whcih makes things even simpler.

1:44 PM  
Blogger Kea said...

Wonderful. The Carnival hosts always seem to have something nice to say. Clearly they are smart people.

4:26 PM  
Blogger CarlBrannen said...

Louise, I think I realize how to make my calculation fit into your notation exactly.

Let (x,y,z,t) define a point in spacetime. You have R and c as functions of t so I will write R(t) and c(t). They satisfy

R(t) = c(t) t.

I will use coordinates (x',y',z',t). To convert an event from one form to the other, you scale the (x,y,z) portion as follows:

(x',y',z') = (x,y,z) ( R_0 / R(t) )

These are perfectly valid coordinates. And in them, the speed of light depends on t only: c'(t) = R_0 / t. And furthermore, in these coordinates, space is flat, and so it is easy to compute angles using just trig. Distances are trickier but they don't matter, we are only interested in angles and times. And the times needed to follow a path are defined by c'(t), which is easy to integrate.

Now this all depends on the assumption that the scaling function preserves angles (and therefore maps null geodesics to null geodesics, and therefore the time calculations are correct). But since space is isotropic, I don't see how this could be wrong.

What this amounts to doing is doing the calculation in old fashioned Galilean coordinates rather than in the more usual coordinates that treat proper time as a coordinate.

If this seems right to you, I'll add something to this effect into the post so the calculation is exactly compatible to your papers.

7:50 PM  
Blogger L. Riofrio said...

Lo9oks good, Carl.

6:13 PM  

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