I wrote not so long ago about a possible observation non-Gaussianity in the CMB spectrum. Last Wednesday Tony Riotto gave an overwiew seminar explaining what that non-Gaussianity really is. I used to think that the issue was complicated. Tony explained very clearly and pedagogically that it is bloody damn complicated.

Unfortunately, it is also important. There is an overabundance of inflationary models and several mechanisms in which these models could produce the primordial density perturbations. The number of observables to discriminate between the models is less outgrown. What we've got so far is the amplitude and the spectral index of density perturbations. Applied to the minimal slow-roll scenario, these two provide useful constraints, but at the moment they do not clearly favour a particular class of models. One way to improve the situation is to measure the spectral index with greater precision. Detecting tensor perturbations would be fantastic, but many models (and most of those sensible from the effective field theory point of view) predict tensor perturbations well below the sensitivity of near-future experiments. Non-Gaussianity of the primordial perturbation spectrum provides another complementary handle on inflationary models. A mere detection of large primordial non-Gaussianity would flush many models down the drain, including the single field slow-roll inflation.

Gaussianity of the primordial perturbation spectrum arises when the inflaton behaves approximately as a free field in a curved background. This is never quite true: the inflaton potential introduces non-linear self-interactions and, more importantly, the inflaton is coupled to gravity which is itself non-linear. In simplest inflationary scenarios, however, these non-linear effects are negligible. In such a case, the equation governing the inflaton perturbations is just the harmonic oscillator equation with a time-varying frequency. Different oscillator frequencies are decoupled and they oscillate independently, seeding the universe with Gaussian perturbations. This means that the power spectrum of the two-point correlation function (that funny thing with peaks that CMB people show) carries all the statistical information about density fluctuations. In the presence of non-Gaussianity, the three- and higher-point correlation functions are independent from the two-point one. The only way to learn about non-Gaussianity is to perform statical tests on these higher-point functions.

Non-Gaussianity is conventionally parametrized as

$\zeta(x) = \zeta_{gauss}(x) - 3/5 f_{NL} (\zeta_{gauss}^2(x) - <\zeta_{gauss}^2>)$

Here $\zeta$ is a gauge-invariant characterization of curvature perturbations. The coefficient $f_{NL}$ is not a constant in general but, for simplicity, one often presents just a number corresponding to an overall amplitude of non-Gaussian contributions over the scales of interest. The current constraint for $f_{NL}$ is that it is at most of order 100. It follows that the second non-Gaussian term is a small perturbation (recall that $\zeta \sim 10^{-5}$) and that's what is meant by saying that the primordial perturbation spectrum is approximately Gaussian.

There are several inflationary models that could lead to primordial non-Gaussianity. Tony paused on the curvaton scenario that he seems to like. In the curvaton scenario, density fluctuations are produced by a different field than the inflaton, while perturbations from the inflaton field are negligible (which may well be, for example, when the scale of inflation is low enough). There is another field - the curvaton - who has a negligible impact on the dynamics of inflation, yet it acquires isocurvature perturbations during inflation. As it decays after the end of inflation, it transmits the fluctuation to the hot plasma in the early universe. Non-gaussianity of the primordial spectrum depends on the parameter $r$, which is a ratio of the curvaton energy density to the total energy density just before the curvaton decays. Lyth et al. found $f_{NL} \sim 5/(4r)$.

One thing that makes non-Gaussianity particularly complicated is that detecting it in the CMB does not imply detecting the primordial non-Gaussianity. Many phenomena contaminate the CMB during the evolution of the universe. First of all, gravity is non-linear, so that the second order perturbation theory introduces non-Gaussianity into the CMB at the level of $f_{NL} \sim 1$. This is the minimal amount of non-Gaussianity that we expect in the CMB, if our models of density perturbations evolution are correct. For comparison, the single field slow-roll inflation predicts $f_{NL} \sim (n - 1)$, which is drowned by the gravity contribution because the spectral index n is close to 1. On top of that, there is a whole lot of dirty astrophysics that contributes to non-Gaussianity: secondary anistropies from graviational lensing or Sunyayev-Zeldovich effect, radio point sources, instrument noise etc.

Concerning the possible signal I wrote about, Tony advised not to get excited. Many effects could bias the analysis of Yadav and Wandelt, one thing pointed out was the poorly known distribution of extragalactic radio sources, another was over-substraction of the noise. On the other hand, if the signal is real, Planck will confirm it at 8 sigma confidence level.

A few details or formulas omitted here can be found in this review by Tony et al.

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