# Correlation Functions in Cosmology – What Do They Measure?

The cosmic microwave background (CMB) is a key observable in cosmology.  Experimentalists can precisely measure the temperature of microwave radiation left over from the big bang. The data shows very small differences in temperature across the sky. It’s up to theorists to figure out why!

The most popular explanation invokes a scalar field early in the universe. Quantum fluctuations in the field are responsible for the classical temperature distribution we see today. This argument, although naively plausible, requires some serious thought for full rigour.

Talks by cosmologists often parrot the received wisdom that the two-point correlation function of the scalar field can be observed on the sky. But how exactly is this done? In this post I’ll carefully explain the winding path from theory to observation.

First off, what really is a correlation function? Given two random variables $X$ and $Y$ we can (roughly speaking) determine their correlation as

$\mathbb{E}(XY)$

Intuitively this definition makes sense – in configurations where $X$ and $Y$ share the same sign there is a positive contribution to the correlation.

There’s another way of looking at correlation. You can think of it as a measure of the probability that for any random sample of $X$ there will be a value of $Y$ within some given distance. Hopefully this too feels intuitive. It can be proved more rigorously using Bayes’ theorem.

This second way of viewing correlation is particularly useful in cosmology. Here the random variables are usually position dependent fields. The correlation then becomes

$\langle \chi(x)\chi(y) \rangle$

where the average is over the whole sky with the direction of the vector $x- y$ fixed. The magnitude of this vector provides a natural distance scale for the probabilistic interpretation of correlation. We see that the correlation is an avatar for the lumpiness of the distribution at a particular distance scale!

Now let’s focus on the CMB. The temperature fluctuations are defined as the percentage deviation from the average temperature at each point on the sky. Mathematically we write

$\delta T / T (\hat{n})$

where $\hat{n}$ defines a point on the unit $2$-sphere. We want to relate this to theoretical predictions. Given our discussion above, it’s not surprising that our first step is to compute the correlation function

$C(\theta) = \displaystyle \langle \frac{\delta T}{ T}(\hat{n}_1) \frac{\delta T}{T}(\hat{n}_2)\rangle$

where the average is over the whole sky with the angle $\theta$ between $\hat{n}_1$ and $\hat{n}_2$ fixed. This average doesn’t lose any physical information since there’s no preferred direction in the sky! We can conveniently encode the correlation function using spherical harmonics

$\delta T / T = \sum a_{l,m} Y_{l,m}$

The coefficients $a_{l,m}$ are known as the multipole moments of the temperature distribution. Substituting this in the correlation function definition we obtain

$C(\theta) = \sum C_l P_l (\cos \theta)$

where $C_l = \sum_m |a_{l,m}|^2$. We’re almost finished with our derivation! The final step is to convert from the correlation function to it’s momentum space representation, known as the power spectrum. With a little work, you can show that the power at multipole number $l$ is given by

$l(l+1)C_l$

This is exactly the quantity we see plotted from sky map data on graphs comparing inflation theory to experiment!

From the theory perspective, this quantity is fairly easy to extract. We must compute the power spectrum of the primordial fluctuations of the inflation field. This is merely a matter of quantum field theory, albeit in de Sitter spacetime. Perhaps the most comprehensive account of this procedure is provided in Daniel Baumann’s notes.

Without going into any details, it’s worth mentioning a few theoretical models. The simplest option is to have a massless free inflaton field. This gives a scale-invariant power spectrum, which is almost correct! Adding mass corrects this result, providing fluctuations in the power spectrum. This is a better approximation, but has been ruled out by Planck data.

Clearly we need a more general potential. Here’s where the fun starts for cosmologists. The buzzwords are effective field theory, string inflation, non-Gaussianity and multiple fields! But that’ll have to wait for another blog post.

Written at the Mathematica Summer School 2015, inspired by Juan Maldecena’s lecture.