The simplest single-scalar-field inflation models predict that primordial perturbations have nearly Gaussian initial conditions. A small degree of non-Gaussianity generally arises from self-coupling of the inflaton field, but this is expected to be very tiny [Salopek et al. 1989, Salopek 1992, Falk et al. 1993, Gangui et al. 1994, Gangui 1994]. More complicated models of inflation, such as two-field [Bartolo et al. 2002], warm [Gupta et al. 2002], or curvaton [Lyth et al. 2002] models may have small deviations from perfectly Gaussian initial conditions, and higher-order calculations of perturbation production suggest that non-Gaussianity may be significant even in slow-roll models [Acquaviva et al. 2002]. Although it is difficult, if not impossible, to predict the exact amplitude and precise form of non-Gaussianity from inflation, it is certainly reasonable to search for it.
Perhaps the most intuitive place to look for primordial non-Gaussianity is in the CMB. Since the CMB temperature fluctuations probe directly primordial density perturbations, non-Gaussianity in the density field should lead to proportionate non-Gaussianity in the temperature maps. So, for example, if the primordial distribution of perturbations is skewed, then there should be a skewness in the temperature distribution. Alternatively, one can study the effects of primordial non-Gaussianity in the distribution of mass in the Universe today. This should have the advantage that density perturbations have undergone gravitational amplification and should thus have a larger amplitude than in the early Universe. However, we must keep in mind that the matter distribution today is expected to be non-Gaussian, even if primordial perturbations are Gaussian. This can be seen just by noting that gravitational infall can lead to regions - e.g., galaxies or clusters - with densities of order 200 times the mean density, while the smallest underdensity, a void, has a fractional underdensity of only -1. However, non-Gaussianity in the galaxy distribution from gravitational infall from Gaussian initial conditions can be calculated fairly reliably (for an excellent recent review, see, e.g., Bernardeau et al. (2002)), and so the distribution today can be checked for consistency with primordial Gaussianity. In verdeone, we studied the relative sensitivity of galaxy surveys and CMB experiments to primordial non-Gaussianity. We considered two classes of non-Gaussianity: in the first, the gravitational potential is written (x, t) = g(x, t) + g^{2}(x, t), where g(x) is a Gaussian random field (so becomes Gaussian as 0); such a form of non-Gaussianity arises in some inflation models. In the second, the fractional density perturbation is written (x, t) = g(x, t) + g^{2}(x, t); this approximates the form of non-Gaussianity expected from topological defects. We then determined what the smallest detectable would be for both cases for future galaxy surveys and for CMB experiments. We found that in both cases the CMB would provide a more sensitive probe of . Conversely, if the CMB turns out to be consistent with primordial Gaussianity, then for all practical purposes, the galaxy distribution can safely be assumed to arise from Gaussian initial conditions.
Experimentally, the bispectrum from 2dF [Verde et al. 2002] and the Sloan Digital Sky Survey [Szapudi et al. 2002] have now been studied and found to be consistent with Gaussian initial conditions. A tentative claim of non-Gaussianity in the the COBE-DMR maps [Ferreira et al. 1998] in great excess of slow-roll-inflationary expectations [Wang & Kamionkowski 2000, Gangui & Martin 2000] was later found to be due to a very unusual and subtle systematic effect in the data [Banday et al. 2000].
The abundances of clusters provide other avenues toward detecting primordial non-Gaussianity. Galaxy clusters, the most massive gravitationally bound objects in the Universe presumably form at the highest-density peaks in the primordial density field, as indicated schematically in Fig. 3. Now suppose that instead of a Gaussian primordial distribution, we had a distribution with positive skewness, as shown in Fig. 3. In this case, we would expect there to be more high-density peaks, even for a distribution with the same variance, and thus more clusters [Robinson et al. 2000]. Thus, the cluster abundance can be used to probe this type of primordial non-Gaussianity.
Just as clusters are rare objects in the Universe today, galaxies were rare at redshifts z 3. In [Verde et al. 2001a], we considered the use of abundances of high-redshift galaxies as probes of primordial non-Gaussianity. We also found an expression that relates the excess abundance of rare objects to the parameter in the models discussed above and were thus able to compare the sensitivities of cluster and high-redshift-galaxy counts and the CMB to non-Gaussianity in the models considered above. We found that although the CMB was expected to be superior in detecting non-Gaussianity in the gravitational potential, the high-redshift-galaxy abundances may do better with non-Gaussianity in the density field.
In addition to producing more clusters, such a skew-positive distribution might change the distribution of the properties of such objects. In ourclusters, we considered the size-temperature relation. If we model the formation of a cluster as a spherical top-hat collapse, then the virial radius and the virial temperature can be determined as a function of the halo mass and the collapse redshift. We then modeled the x-ray-emitting gas to relate its size and temperature to the virial radius and temperature of the halo in which it lives in order to obtain better estimates for the x-ray isophotal radii and x-ray temperatures that are measured. Fig. 4 shows resulting contours of constant mass and constant collapse redshift in the cluster size-temperature plane. As shown there, halos that collapse earlier should lead to hotter and smaller clusters, and more massive halos should be hotter and bigger. If the primordial distribution is skew-positive, then the halos that house clusters will collapse over a wider range of redshifts, and as indicated in Fig. 5, this will lead to a broader scatter than in the size-temperature relation than is observed. In this way, primordial distributions with a large positive skewness can be constrained.
Figure 5. (a) Size-temperature distribution for LCDM and _{8} = 0.99 and Gaussian initial conditions. Each dot represents a simulated cluster, while the diamonds are data from M00. The line shows the size-temperature relation expected for clusters today. Panel (b) shows the same except that here we use _{8} = 1.5. Panel (c) shows the same as in (a) but with the skew-positive distribution of Robinson et al. (2000). From Verde et al. (2001b). |