Non-Gaussian Density Perturbations in Inflationary Cosmologies

The primordial mass density fluctuations may have arisen from quantum fluctuations in a (massless) scalar field that occurred during an inflationary era. We show that it is possible for primordial mass density fluctuations, which arose in this way, to be highly non-Gaussian. We also show that the “bad” infrared properties of the propagator for a massless scalar field in de Sitter space can translate itself into a power spectrum, for the the two-point spatial correlation of objects that do not trace the mass, which behaves like k, at small wave numbers k. ∗ Work supported in part by the U.S. Department of Energy under Contracts DEAC-03-81-ER40050 and DE-FG03-84-ER40172. † Sloan Fellow. Limits on the anisotropy of the microwave background radiation restrict the primordial fluctuations in the mass density to be small at early times. However, the fluctuations do grow in the matter-dominated era because of gravitational instability. Presumably the primordial fluctuations in the mass density determined the large-scale structure of the universe that is observed today. Understanding the nature of these fluctuations is one of the important issues in physics. If the length scales associated with the physical processes that generated the primordial fluctuations are very small, compared to astrophysically relevant length scales, then they probably can be neglected in the correlations of the mass density. This naturalness requirement of scale invariance restricts the power spectrum for the two-point correlation of the mass density fluctuations to have the Harrison-Zeldovich form, [1] but allows for many types of connected higher point correlations. A gaussian probability distribution is the simplest scale invariant choice for the primordial mass density fluctuations. This choice also has considerable predictive power. Provided there is no new physics occurring at very late times, once the type of fluctuations (e.g., adiabatic) and the type of dark matter are specified, only the normalization of the two-point correlation of the mass density fluctuations is left undetermined (we assume an Ω = 1 universe with vanishing cosmological constant). It appears that this simplest choice is encountering difficulties explaining the observed large-scaled structure of the universe. For example, with cold dark matter and adiabatic fluctuations, it is hard to explain the significant correlations of rich clusters of galaxies that occur at large distances. [2] This problem has lead to speculation that the primordial mass density fluctuations are not gaussian. [3] Since primordial fluctuations in the mass density with wavelengths that are less than the horizon length (i.e., 1/H) today had wavelengths greater than the horizon length at early times, it is difficult to come up with reasonable ways to generate these fluctuations. In fact, at the present time, there are only two schemes. In one it is the presence of topological defects called cosmic strings that generate the fluctuations. [4] The other scheme has the fluctuations in the energy density arising from quantum fluctuations in a scalar field during an inflationary era. [5][6] The smallness of the primordial fluctuations restricts the field that is driving the inflation to be very weakly coupled so that its quantum fluctuations (and the corresponding fluctuations in the mass density) are approximately gaussian. [7] The purpose of this letter is to show that it is possible to have cosmologies where quantum fluctuations in a field during an inflationary era give rise to scale invariant non-gaussian primordial fluctuations in the mass density. We shall do this by constructing a model where quantum fluctuations