Non--Gaussian Primordial Fluctuations

We analyze the non–Gaussian primordial fluctuations which are inescapably contributed by scalar fields $\mathrm{\Phi }$ with vanishing expectation values, $⟨\mathrm{\Phi }⟩=0$, present during inflation in addition to the inflaton field. For simplicity we take $\mathrm{\Phi }$ to be non–interacting and minimally coupled to gravity. $\mathrm{\Phi }$ is a Gaussian variable, but the energy density fluctuations contributed by such a field are ${\chi }^2$–distributed. We compute the three–point function $\text{xxxT}$ for the configuration of an equilateral triangle (with side length $\mathrm{\ell }$) and the skewness $\text{dddRR}$, {\it i.e.} the third moment of the one–point probability distribution of the spatially smeared energy density contrast ${\text{de}}_R$, where $R$ is the smearing scale. The relative magnitudes of the non–Gaussian effects, $[{\xi }^{(N)}{]}^{1\mathrm{/}N}\mathrm{/}[{\xi }^{(2)}{]}^{1\mathrm{/}2}$, do not grow in time. They are given by numerical constants of order unity, independent of the scale $\mathrm{\ell }$. The “bi–skewness” $\text{dddRS}$ is positive. For smearing lengths $R\ll S$ this shows that in our model (in contrast to Gaussian models) voids are more quiet than high–density regions.