## Tuesday, February 16, 2010

### What Are These Correlation Functions Anyways?

I have, and will continue to discuss correlation functions so I might as well write a post describing what they are in more detail.

Two Point Correlation Function:
A two point correlation function describes how two different points in space are related.  For example: Think of the earth and what temperature it is right where you are standing.  What can you say about the temperature 10 feet away?  The odds are that it is close to being the same temperature.
Thus we say at small scales, or between two points close together, temperature is highly correlated.
Now, pretend you are in Irvine California.  If I ask this question: "Knowing it is 70 degrees outside in Irvine, what temperature would we expect it to be in Beijing Chaina?"  You would say "I have no way of knowing, the temperatures are not related." or in other words:
We say on large scales, or between two points far apart, temperature is hardly correlated.
Three (and higher) Point Correlation Function:

Now let's say you are in a small room where one wall is held at 100 degrees and the other held a 0 degrees.  Can you tell me anything about the temperature of a point in the middle of the room. Yes, it is probably some temperature between 0 and 100 degrees.  Knowing the temperature values of two points in a room can often tell you about temperature of a random third point.
Thus we say, in such a scenario, we have a non-zero three point function since each three points are somewhat correlated.
It should be obvious now that you can have higher n-point correlation functions.

Interactions Between Points:

In some sense, n-point correlation functions directly tell you something about interactions going on between n-points.

Two points close together that have different temperatures interact in a way that effects the temperature of the other point.  Two points very far away at different temperatures have a hard time interacting with each other in a way that effects the temperature of the other.
Thus, interactions are what gives rise to a non-zero two point function which therefore becomes a statistic quantifying the interactions.
I'm sure if you thought about it long enough you would realize in my three-point example above, correlations are again related to interactions between points.

The CMB.

Note in the figure how the CMB shows correlation between all scales. (Low l values represent correlation at large scales, high l values represent correlation at small scales.)   This could only happen if points at both small and large scales had ways of interacting with each other.  Given points at large scales in the sky are not in casual contact with other points, only one known thing can explain this:

A superluminal expansion or, in other words: Inflation!

1. Hey hey! As I was reading this I wondered if this could apply to inflation, and what do you know?! I think some of your smarty dust might be having an effect. =:)

2. Stan,

I'm glad the smarty dust is working.

3. I should also note, for those interested, the above 2 point correlation function, or power spectrum, looks different for small scales (large l) than large scales (small l).

This is a manifestation that different interactions dominate on different scales. Inflationary interactions with n_s < 1 (like the .96 we observe) should leave a flat line with a slight tilt downward. On large scales (smal l) inflation should dominate. We see this.

On small scales baryon acoustic oscillations govern so we should see oscillations at large l. We see this too!

Modern Cosmology is consistent!

4. I think the second indented point (just above the "Three (and higher) Point Correlation Function:" header) should read something like,

"We say on large scales, or between two points far apart, temperature is hardly correlated at all."

I appreciate the outreach effort that posts like these represent to those of us who haven't studied cosmology. Keep it up!

5. Ben,

Thank you. Typo fixed.

6. Currently in France the temperatures are quite low (unusually), and humorists are beginning to make some jokes about global warming :) .

7. Cartesian,

Yes, this seems to be the case in the US as well.