I just gave the first talk on my PhD work and I want to write about it here! The talk is very short but aims to motivate one of the main statistics I'm currently using to study the largest structures in the universe. [Thread]

We'll start with where we get information about the universe. I'll be breaking the universe into "early" and "late" universe. The early universe for our purposes means the cosmic microwave background (CMB), which is the earliest light we can measure.

Many of you will have seen images like this of the CMB (from the Planck telescope). The different colours represent slightly more or less dense regions of the universe. Quantifying these fluctuations statistically is how we extract information from the early universe.

The fluctuations in the early universe are VERY small, only about 10 parts per million. In contrast, the fluctuations in the "late" universe (when structures have collapsed under gravity) are large. Dark matter halos and galaxies are ~100 times the average density.

In addition to being larger, the fluctuations in the late universe are more complex than in the early universe. The small fluctuations in the CMB are about as simple as one could hope, being a nearly perfect Gaussian random field. In the late universe, we're less lucky.

Gravitational interaction causes small density fluctuations to grow, and these fluctuations cease being small, which means that linear theory predictions break down due to gravity. This turns nicely Gaussian fields into non-Gaussian fields over time.

However, we shouldn't give up! While large scale structure (LSS) statistics are harder than CMB statistics, the LSS holds more information than the CMB, as it is 3D data, tracing the state of the universe through time, while the CMB is 2D, just a snapshot at one time.

What sort of information can we hope to get out of studying these fluctuations? The first thing we can do is better measure current cosmology parameters, such as the rate of expansion, how much matter there is, and how quickly structures grow.

Additionally, and possibly more exciting is that we can measure or constrain fundamental physics parameters using cosmology, including the sum of the neutrino masses, modified gravity, the nature of dark energy, and primoridal non-Gaussianity.

However, it isn't obvious how to do this. For a non-Gaussian distribution, what summary statistics do you choose that might tell you about these things? Let's make a wishlist for our summary stats.

The first thing we should want is that our statistic should be sensitive to interesting physics. As a theorist, I would like to be able to predict or approximate our statistic from underlying physics. The statistic should also be easily measurable.

Enter our choice: the one point function. This is one of the simplest choices for summary statistic one could make. Let's see it at work on the CMB. First draw a bunch of circles of fixed size on the CMB calculate the mean density in each. Write that number down.

You now have a long list of numbers, the "count in cell" density for these spheres. Now make a histogram of all those numbers, and look at the resulting distribution. This should approximate the "probability distribution function" (PDF). For the CMB this is Gaussian.

This is what I mean by "simple statistics" in the early universe. This measurement is characterised by just one number, the width of the resulting Gaussian (we don't care about the mean, and normalise it to be 1).

However, we can do the same thing in the late universe now and see what happens. Laying down spheres in a cosmological simulation, we just average density then histogram. Here's the result.

Great! We have a statistic that we can measure from simulation easily (a bit more work has to be done to measure with real observables but it's possible). Can we predict this shape theoretically? How do we go from Gaussian to non-Gaussian?

It turns out that there is a way to do this analytically! Using some mild theoretical assumptions we can use large deviation theory (a branch of probability theory devoted to rare events) and spherical collapse dynamics to approximate the late time PDF.

(See e.g. https://arxiv.org/abs/1512.05793  for how to do this). We can compare these theoretical predictions (the solid lines) to data extracted from simulation and see they agree pretty well! (in fact, better than the empirical lognormal model that's often used).

What can we hope to measure with this statistic then? Is it sensitive to interesting physics? Again, the answer is yes. What we can do is forecast how well we could measure various quantities (neutrino mass for example) if they were present using this statistic.

These sorts of forecasts have been done for neutrino mass, primordial non-Gaussianity, and evolving dark energy (references at the end of the thread). The project I'm currently working on applies this statistic to modified gravity.

(In addition to forecasting for new parameters, this statistic also is useful for current parameters, and can break degeneracies between certain parameters measured using traditional 2 point statistics)

So there we have it. This "count in cells" statistic fulfils our wishlist! It's easy to measure (from sims at least), it's theoretically under control, and it's sensitive to interesting physics. [End Thread]

References for forecasting (non-extensive)
Neutrino mass: https://arxiv.org/abs/1911.11158 
Primordial non-Gaussianity: https://arxiv.org/abs/1912.06621 
Dark energy: https://arxiv.org/abs/1603.03347