The Microwave Background Bispectrum, Paper I: Basic Formalism

TL;DR

The paper develops a formalism to compute and measure the CMB bispectrum, the three-point statistic that captures non-Gaussianity from non-linear evolution. It defines the harmonic coefficients $a_{lm}$ and the angle-averaged bispectrum $B_{l_1 l_2 l_3}$, and outlines a practical measurement approach including noise and cosmic variance considerations. Applying the formalism to the second-order Rees-Sciama effect, it derives a reduced expression for the RS bispectrum $B^{(1)}_{l_1 l_2 l_3}$ and shows the signal is too small to be detected by MAP/Planck. The work highlights that, although RS is undetectable, other low-redshift lensing-induced bispectra (studied in companion work) may be detectable, underscoring the bispectrum’s potential to reveal non-Gaussian physics in the CMB with future analyses.

Abstract

In this paper, we discuss the potential importance of measuring the CMB anisotropy bispectrum. We develop a formalism for computing the bispectrum and for measuring it from microwave background maps. As an example, we compute the bispectrum resulting from the 2nd order Rees-Sciama effect, and find that is undetectable with current and upcoming missions.

Figures (3)

• Figure 1: The coefficients $b_{l_m}$ for various cosmologies, as defined in the text. Note that for low $\Omega$ models, the signal is significantly larger than $\Omega_m=1$ models at all l.
• Figure 2: The amount of information gained by increasing $l_3$, as given by $\chi^2=\sum (\langle B_{l_1 l_2 l_3}(\Omega_m)\rangle -\langle B_{l_1 l_2 l_3}(\Omega_m=0.3)\rangle )^2/\langle B_{l_1 l_2 l_3}^2\rangle$, where we have used the PLANCK detection sensitivity. Since there is very little contribution above $l_3=700$, the MAP and PLANCK missions will be equally sensitive to this effect. Our fiducial model here and throughout is $\Omega_m=0.3$, $\Omega_{\Lambda}=0.7$. The shape of this spectrum comes from several competing effects. At large $l$, more modes are included in the sum, providing a larger signal. However, the signal per mode becomes increasingly weak at a logarithmic rate for high l, as suggested by the plot of the coefficients in Figure 1. Finally, the features in the spectrum above are caused by variations in the expected noise. Troughs correspond to peaks in the CMB spectrum.
• Figure 3: The total value of $\chi^2(\Omega_m,\Omega_m=0.3)$ for variations in $\Omega_m$ for both PLANCK and MAP. All values of $\Omega_m$ produce signals which are consistent with $\Omega_m=0.3$.