The three-point correlation function of the cosmic microwave background in inflationary models

Abstract

We analyze the temperature three-point correlation function and the skewness of the Cosmic Microwave Background (CMB), providing general relations in terms of multipole coefficients. We then focus on applications to large angular scale anisotropies, such as those measured by the COBE DMR, calculating the contribution to these quantities from primordial, inflation generated, scalar perturbations, via the Sachs-Wolfe effect. Using the techniques of stochastic inflation we are able to provide a universal expression for the ensemble averaged three-point function and for the corresponding skewness, which accounts for all primordial second-order effects. These general expressions would moreover apply to any situation where the bispectrum of the primordial gravitational potential has a hierarchical form. Our results are then specialized to a number of relevant models: power-law inflation driven by an exponential potential, chaotic inflation with a quartic and quadratic potential and a particular case of hybrid inflation. In all these cases non-Gaussian effects are small: as an example, the mean skewness is much smaller than the cosmic rms skewness implied by a Gaussian temperature fluctuation field.

Inflation and String Theory

“…Local non-Gaussianity.-The shape (1.66) arises from the following ansatz in real space [114,115]: 69) where R g (x) is a Gaussian random field. In momentum space, the signal peaks for squeezed triangles, e.g.…”

Gauge-invariant second-order perturbations and non-Gaussianity from inflation

“…is a symmetric transverse and trace-free tensor (∂ i χ (r) ij = 0, χ i(r) i = 0) and 3 . Here and in the following latin indices are raised and lowered using δ ij and δ ij , respectively.…”

“…So far, the problem of calculating the bispectrum of perturbations produced during inflation has been addressed by either looking at the effect of inflaton self-interactions (which necessarily generate non-linearities in its quantum fluctuations) in a fixed de Sitter background [2], or by using the so-called stochastic approach to inflation [3] 1 , where back-reaction effects of field fluctuations on the background metric are partially taken into account. An intriguing result of the stochastic approach is that the dominant source of non-Gaussianity actually comes from non-linear gravitational perturbations, rather than by inflaton self-interactions.…”

Non-Gaussianity in two-field inflation

“…Interestingly, for standard single-field inflation, the non-linear parameters representing the bispectrum [12,[20][21][22][23][24][25][26][27] and trispectrum [28][29][30] are all of order the slowroll parameters (i.e., at the percent level) and will not be accessible to CMB experiments. However, if inflation is described by some non-minimal modification, such as multiple fields or higher derivative operators in the inflationary Lagrangian, then non-Gaussianity might be observable in the near future.…”