Testing for Gaussianity Through the Three Point Temperature Correlation Function

One of the crucial aspects of density perturbations that are produced by the standard inflation scenario is that they are Gaussian where seeds produced by topological defects tend to be non-Gaussian. The three point correlation function of the temperature anisotropy of the cosmic microwave background radiation (CBR) provides a sensitive test of this aspect of the primordial density field. In this paper, this function is calculated in the general context of various allowed non-Gaussian models. It is shown that by COBE and the forthcoming South Pole and Balloon CBR anisotropy data may be able to test provide a crucial test of Gaussianity. PACS number: 98.80.Bp, 98.80.Dr Testing for the Gaussianity of the primordial fluctuation spectrum is of critical importance to many cosmological models. In particular, traditional cosmic inflation [1] specifically predicts a Gaussian density fluctuation spectrum. The scale invariant quantum fluctuations generated during the inflationary epoch are expected to serve as the primordial density perturbations which develop into the large scale structures we observe today[2]. Competing models for structure formation, including topological defects originating from cosmological phase transitions [3] and non-standard inflation models[4], will also generate a scale invariant (or nearly scale invariant) power spectrum for density perturbations similar to that of inflation. However, the statistics of these latter fluctuations are non-Gaussian. Thus, the Gaussianity of the fluctuations provides a unique handle in discriminating different structure formation scenarios. In this letter, we will discuss how to test this aspect of the primordial density field through the temperature anisotropy of the cosmic microwave background radiation (CBR). As we showed [5], in momentum space, the lowest order deviation from Gaussianity is described by the bispectrum of the gravitational potential φ, Pφ(k1, k2) =< φk1φk2φ−k1−k2 >. When the perturbation is adiabatic so that the temperature anisotropy is related to the gravitational potential φ at the last scattering surface through the Sachs-Wolfe [6] formula: δT T = φ 3 , (1) the three point temperature correlation function is related to the bispectrum through ξT (m̂, n̂, l̂) = 1 27 · ∫ Pφ(k1, k2, k3)e i(k̂1m̂+ik̂2n̂+ik̂3 0δ(~ k1+ ~ k2+ ~ k3) dk1d k2d k3 (2π)9 , (2) where η0 = 2H −1 0 is the distance to the last scattering surface, and m̂, n̂, l̂ are the beam directions. A non-vanishing three point function clearly indicates that the bispectrum is not zero. Note that for Gaussian primordial perturbations, the bispectrum is strictly zero in all cases. Thus, the three