The nature of cosmological initial conditions
What is the mechanism that generated the small primordial perturbations which grew under gravity to create the cosmological large-scale structures we see today?
The most widely accepted paradigm is inflation, a period of accelerated expansion a fraction of a second after the big bang.
The simplest inflationary models predict very nearly Gaussian, adiabatic perturbations. Inflation also predicts a stochastic background of gravity waves (tensors perturbations).
More about primordial gravity waves and their implication for inflation and cosmological observations can be found under the Cosmic Microwave Background section.
The field of primordial non-Gaussianity has received a renewed interest in the past few years. This is because it was realized that:
the simplest inflationary models (single field, slow roll, adiabatic vacuum) predict unmeasurably small deviations from Gaussianity. However any deviation from any of the above conditions, leaves its signature in the so-called primordial bispectrum (the Fourier-transform of the three-point correlation function). --Recall that a Gaussian field is completely specified by its power spectrum and that its bispectrum is zero--.
Thus constraining or detecting primordial non-Gaussianity opens a window in the mechanism of inflation and thus the physics of the early Universe.
Current data already constrain Gaussianity at the percent level, but forthcoming data will push such constraints in the "interesting" regime where many models predict a non-zero signal.
There are several approaches to go after small primordial non-Gaussianity . One option is to look at the early Universe: the earliest snapshot we can see is the Cosmic Microwave background (CMB). The advantage is that perturbations are still small, the Universe has not evolved much from the initial conditions (evolution is mostly linear) and observations should thus probe them more directly. The disadvantage is that the signal is also small, so the signal-to-noise is high.
The other approach is to look at the late-time Universe, perturbations are large (so the signal is large) but the Universe underwent non-linear evolution which in itself, introduces a spurious non-Gaussian signal. As usual one's man trash is another man's treasure: the gravitational non-Gaussian signal carries also a wealth of information. For example it can be used to find out how closely galaxies trace the dark matter distribution  or wether gravity behaves as predicted by General relativity  -- we have precise tests of gravity only up to solar-system scales, and the law of gravity is then extrapolated 13 orders of magnitude to horizon-size scales. We do not know of any other physical law that was extrapolated that far without a hitch ---.
The CMB non-Gaussianity is usually tested with the CMB bispectrum, which, however, is not free of contaminants, in fact the CMB photons we observe had to travel through the entire Universe, thus the CMB is distorted by the intervening structures. Beyond the effect of foregrounds, active galactic nuclei and galaxy clusters, I find that the linear and non-linear growth of foreground large-scale structures leaves a strong signal in the CMB bispectrum which is very hard to distinguish for a primordial signal.
Again, one man's trash is another man's treasure: such spurious signal carries a wealth of information about the evolution and content of the Universe; forthcoming experiments will detect it at high statistical significance so I am investigating if and how such information can be extracted and used.
On the late-Universe side, there are three ways to avoid the small primordial signal to be swamped by the gravitationally-induced non-Gaussianity. One can go to high redshift and measure the bispectrum there. Alternatively one can look at observables that depend in a simple way on the initial (or linearly evolved) initial conditions. These are the halo abundance (also called the halo mass function: the abundance of halos as a function of mass and redshift) and their clustering.
While dark matter halos are not directly observable, galaxies and galaxy clusters inhabit (and light up) dark matter halos, making them easily detectable.
The calculation of the so-called non-Gaussian mass function has recently received renewed interest and I have been interested in refining analytical predictions and calibrate them with N-body simulations.
Of particular interest is the effect of primordial non-Gaussianity on the clustering of halos. Briefly, for Gaussian initial conditions the clustering of halos depend only on the dark matter primordial power spectrum, but for non-Gaussian initial conditions it depends on all higher order correlations, that is it carries information on primordial non-Gaussianity. In the past two years I have been exploring such an approach which turns out to be robust, extremely promising and has even been hailed as a "discovery tool".
Forthcoming data will be able to access a wide region of the parameter space spanned by various models and thus is expected to rule out a wide range of them (and possibly therefore rule in some).
Isocurvature modes as a contaminant?
Current data allow a small level of isocurvature modes. I find that even for future data, a small amount of primordial isocurvature modes can bias cosmological parameters as estimated from the CMB, in particular it can bias significantly the estimate of the sound horizon at radiation drag: the "standard ruler" used by Baryon Oscillation survey to measure the expansion history of the Universe. A biased estimate of such ruler may led us to mis-estimate the Universe's expansion history and thus to conclude e.g., that dark energy is dynamical or that the expansion does not match the growth of structure and therefore gravity is not well described by general relativity or that there is a coupling between dark matter and dark energy.
Preliminary investigation indicate, however that by combining CMB and large-scale structure data and allowing the level of isocurvature modes to be an extra parameter in the model, it is possible to avoid biases and still not degrade the error significantly.