Effective Field Theory and Large-Scale Structure

Been falling behind on my favorite thing to do on the blog: post summaries of my own research papers. Back in October I submitted a paper with two Caltech colleagues, postdoc Stefan Leichenauer and grad student Jason Pollack, on the intriguing intersection of effective field theory (EFT) and cosmological large-scale structure (LSS). Now’s a good time to bring it up, as there’s a great popular-level discussion of the idea by Natalie Wolchover in Quanta.

So what is the connection between EFT and LSS? An effective field theory, as loyal readers know, an “effective field theory” is a way to describe what happens at low energies (or, equivalently, long wavelengths) without having a complete picture of what’s going on at higher energies. In particle physics, we can calculate processes in the Standard Model perfectly well without having a complete picture of grand unification or quantum gravity. It’s not that higher energies are unimportant, it’s just that all of their effects on low-energy physics can be summed up in their contributions to just a handful of measurable parameters.

In cosmology, we consider the evolution of LSS from tiny perturbations at early times to the splendor of galaxies and clusters that we see today. It’s really a story of particles — photons, atoms, dark matter particles — more than a field theory (although of course there’s an even deeper description in which everything is a field theory, but that’s far removed from cosmology). So the right tool is the Boltzmann equation — not the entropy formula that appears on his tombstone, but the equation that tells us how a distribution of particles evolves in phase space. However, the number of particles in the universe is very large indeed, so it’s the most obvious thing in the world to make an approximation by “smoothing” the particle distribution into an effective fluid. That fluid has a density and a velocity, but also has parameters like an effective speed of sound and viscosity. As Leonardo Senatore, one of the pioneers of this approach, says in Quanta, the viscosity of the universe is approximately equal to that of chocolate syrup.

So the goal of the EFT of LSS program (which is still in its infancy, although there is an important prehistory) is to derive the correct theory of the effective cosmological fluid. That is, to determine how all of the complicated churning dynamics at the scales of galaxies and clusters feeds back onto what happens at larger distances where things are relatively smooth and well-behaved. It turns out that this is more than a fun thing for theorists to spend their time with; getting the EFT right lets us describe what happens even at some length scales that are formally “nonlinear,” and therefore would conventionally be thought of as inaccessible to anything but numerical simulations. I really think it’s the way forward for comparing theoretical predictions to the wave of precision data we are blessed with in cosmology.

Here is the abstract for the paper I wrote with Stefan and Jason:

A Consistent Effective Theory of Long-Wavelength Cosmological Perturbations
Sean M. Carroll, Stefan Leichenauer, Jason Pollack

Effective field theory provides a perturbative framework to study the evolution of cosmological large-scale structure. We investigate the underpinnings of this approach, and suggest new ways to compute correlation functions of cosmological observables. We find that, in contrast with quantum field theory, the appropriate effective theory of classical cosmological perturbations involves interactions that are nonlocal in time. We describe an alternative to the usual approach of smoothing the perturbations, based on a path-integral formulation of the renormalization group equations. This technique allows for improved handling of short-distance modes that are perturbatively generated by long-distance interactions.

As useful as the EFT of LSS approach is, our own contribution is mostly on the formalism side of things. (You will search in vain for any nice plots comparing predictions to data in our paper — but do check out the references.) We try to be especially careful in establishing the foundations of the approach, and along the way we show that it’s not really a “field” theory in the conventional sense, as there are interactions that are nonlocal in time (a result also found by Carrasco, Foreman, Green, and Senatore). This is a formal worry, but doesn’t necessarily mean that the theory is badly behaved; one just has to work a bit to understand the time-dependence of the effective coupling constants.

Here is a video from a physics colloquium I gave at NYU on our paper. A colloquium is intermediate in level between a public talk and a technical seminar, so there are some heavy equations at the end but the beginning is pretty motivational. Enjoy!