# 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.

# Double Whammy – New Evidence for Inflation and Gravitational Waves

Today the latest results from the BICEP2 telescope in Antarctica are out. And boy, are they exciting! They provide stark evidence for two widely believed theoretical predictions, namely inflation and gravitational waves. The authors are already being tipped for a Nobel prize.

So what’s the science behind this magnificent discovery? It’s easiest to start with the name of the telescope. BICEP stands for “Background Imaging of Cosmic Extragalactic Polarization”. That means looking for signals from the Big Bang. Cool, huh?

After the Big Bang  the universe was a hot dense soup of particles. Eventually (380,000 years later) things were cool enough for the universe to become transparent. Particles could bind together to form hydrogen atoms, emitting light in the process. Nowadays we see this ancient light as microwave radiation covering space.

This cosmic microwave background (CMB) has a particularly puzzling feature. It’s much more uniform than we should expect from a generic Big Bang explosion. Intuitively most explosions don’t generate exactly symmetrical outcomes!

What’s needed is some mechanism to smooth out the differences between different parts of space. Here’s where the idea of inflation comes in. A fraction of a second after the Big Bang we think that the universe blew up at an astonishing rate. This happened so fast that there was no time for inconsistencies to creep in. The result – a uniform cosmos.

It’s certainly an appealing explanation, but the problem is that there’s been little direct evidence. Until now, that is. Cosmologists on the BICEP project were looking for a particular signature from inflation, and it seems like they’ve found it!

To understand their method we need to know something about light. A wave of light can oscillate in different directions perpendicular to its path. A light wave coming into your eyes from your screen will oscillate somewhat up-down and somewhat left-right. These two options are known as polarizations of light.

It turns out that you can measure exactly how light in the CMB is polarized. This is useful because inflation produces a particular polarization pattern called a B mode. It’s taken decades to locate this smoking gun, but now the BICEP team have done it.

Hang on, couldn’t these B modes come about some other way? Probably not. The B mode pattern we observe seems to arise from the interaction of light with gravitational waves. And to get enough of these we need inflation. Or perhaps this effect is an observational fluke? According to the paper, we’re 99.999999% sure it isn’t.

It’s worth pointing out that this result is a double whammy. It confirms theories of inflation and gravity. Nobody has yet detected a gravitational wave, despite the fact they’re theoretically an easy consequence of Einstein’s general relativity. This latest development is further indirect evidence of their existence.