Avignon Day 1: Calculating Non-Gaussianities

By Sean Carroll | April 19, 2011 2:23 am

Greetings from Avignon, where I’m attending a conference on “Progress on Old and New Themes” in cosmology. (Name chosen to create a clever acronym.) We’re gathering every day at the Popes’ Palace, or at least what was the Pope’s palace back in the days of the Babylonian Captivity.

This is one of those dawn-to-dusk conferences with no time off, so there won’t be much blogging. But if possible I’ll write in to report briefly on just one interesting idea that was discussed each day.

On the first day (yesterday, by now), my favorite talk was by Leonardo Senatore on the effective field theory of inflation. This idea goes back a couple of years to a paper by Clifford Cheung, Paolo Creminelli, Liam Fitzpatrick, Jared Kaplan, and Senatore; there’s a nice technical-level post by Jacques Distler that explains some of the basic ideas. An effective field theory is a way of using symmetries to sum up the effects of many unknown high-energy effects in a relatively simple low-energy description. The classic example is chiral perturbation theory, which replaces the quarks and gluons of quantum chromodynamics with the pions and nucleons of the low-energy world.

In the effective field theory of inflation, you try to characterize the behavior of inflationary perturbations in as general a way as possible. It’s tricky, because you are in a time-dependent background with a preferred (non-Lorentz-invariant) frame provided by the expanding universe. But it can be done, and Leonardo did a great job of explaining the virtues of the approach. In particular, it provides a very nice way of calculating non-gaussianities.

At a first approximation, cosmological perturbations are gaussian: the fluctuations at every point are drawn from a normal (bell curve) distribution. That’s the basic prediction of inflation, and it’s consistent with what we observe. But a more careful calculation shows that perturbations from inflation can be slightly non-gaussian; the search for such a signal in the data is a primary goal of current cosmological observations.

What the EFT of inflation lets you predict is what form the non-gaussianities can take. One way of characterizing the deviation from gaussianity is using the bispectrum, the correlation between fluctuations (in Fourier space) with three different wave vectors.

$latex langlePhi_{vec{k}_1}Phi_{vec{k}_2}Phi_{vec{k}_3}rangle = 2pi^{(3)} delta^3left(sum vec{k_i}right)F(k_1, k_2, k_3) $

Although it looks like F depends on the three wave numbers, scale invariance implies that it really only depends on the ratios k2/k1 and k3/k1. (The directions don’t matter because of rotational invariance.) Long story short, you can plot the possibilities in terms of a function on a two-dimensional parameter space, like so:

The EFT of inflation lets you predict the shape of this function for any given inflationary model. Keep in mind: we haven’t yet observed any non-gaussianity, although we are trying. But it’s a reminder that there’s potentially a wealth of information about the early universe yet to be extracted from observable features today.

  • http://backreaction.blogspot.com/ Bee

    Yes, the non-gaussianity is getting a lot of attention recently. I wrote a (quite general) post on that some months back.

  • Daniel Baumann

    Hi Sean, I think you mean Leonardo Senatore, although Salvatore sounds very Italian too :) Unfortunately, I couldn’t make it to the meeting, but I look forward to hearing more about it.

  • cecil kirksey

    Can someone explain how exactly does a physical theory predict that a measured variable will have exactly a Gaussian distribution other than the central limit theorem which is a mathematical result not a physical result?

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    Daniel– Arrgh, thanks. I got it right one out of two times, so it was clearly a quantum fluctuation. I blame jet lag.

    cecil– The fluctuations originate as quantum fluctuations in the vacuum, which are Gaussian in the limit of free field theory. That’s just the lowest-energy state; each mode is like a harmonic oscillator.

  • cecil kirksey

    Yes I believe I understand the assumption of the “infinite ” harmonic oscillator. But is that physically realistic? Think of the predicted value of the cosmological constant based on this idea.

  • AnotherSean

    Sounds like a great conference. Put in a good word to the Pope for us.

  • Albert Zweistein

    Ah yes, those pesky over-idealizations of cosmology. They were introduced with such good intentions and led to such good initial progress in the early 20th century, only to come back and bite us in the hinder now that we, well most of us, have entered the 21st century.

    Exact homogeneity – I don’t think so.

    Exact isotropy – I don’t think so.

    Absolute acausal beginning of space-time – I don’t think so.

    Strict reductionism – I don’t think so.

    We need to move beyond the over-simplifications that made the math easier, and seriously explore models more like Szekeres’ inhomogeneous models that do not assume the idealized symmetries.

    Moreover, unbiased observational testing should guide the theoretical loose cannons who have run so amuck in recent decades.

    de Vaucouleurs was right on and quite prescient when he wrote “The Case For A Hierarchical Cosmology” way back in the 70s.

    Albert Z

  • http://www.damtp.cam.ac.uk/people/e.lim/ Eugene

    Hi Sean,

    I suppose I should wade in here and say a little bit more about calculating non-Gaussianities. This is not meant as a criticism of the EFT method (which I think is great), but just to address some fairly important but rather technical issues.

    Technically, there are some limitations on the EFT method — the original formalism works for a single adiabatic d.o.f., and for strictly slow-roll super smooth type inflation. The former means that you only have a single scalar field (Leonardo + friends have an extension to multifields, but there are extra rules on the theory space you need to impose to make those work in the EFT, which weakens its generality), the latter mean that you have drop all the H”, H”’, H”” etc terms (these are not mere nuisance, as large classes of popular models generates observable nonG use those terms — disclosure is that I wrote them but I think it’s fair to it is one of the four general main mechanisms usually known as “breaking slow roll”). Relaxing these conditions open up the full spectrum of possible nonGaussianities beyond those few “shapes” that EFT has access to. (Daniel probably can say more about additional shapes ala EFT!)

    Philosophically, the EFT “gives up”on knowing what the background dynamics are (it is an expansion around some fixed background dynamic). I don’t think there is anything wrong with this approach, however, but I think it’s fair to say that sometimes you do want to know what your background field theory is…


  • http://www.damtp.cam.ac.uk/people/e.lim/ Eugene

    (I should add that thinking about things in terms of an EFT is an awesome organizing principle, but it is not the only organizing principle out there.)

  • Pingback: Avignon Day 4: Dark Matter | Cosmic Variance | Discover Magazine()

  • A

    Sean, don’t you mean that *scale* invariance implies that F only depends on the ratios k2/k1 and k3/k1? In other words: k2/k1 and k3/k1 are the only quantities that matter for F if and only if the shape of the triangles — and not the overall scale of the triangles — is what matters. Rotational invariance won’t get you that; but you do need rotational invariance to be able to say that only the magnitudes of k1, k2, and k3 matter.

    I could be wrong here, but I’ve done some work in this area and I think this is true.

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    Oops, you’re certainly right. Fixed it now. Can I blame jet lag?


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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] cosmicvariance.com .


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