The Microwave Background Bispectrum, Paper II: A Probe of the Low Redshift Universe

TL;DR

This paper develops a formalism for the CMB bispectrum $B_{l_1 l_2 l_3}$ to probe late-time, low-redshift physics by cross-correlating gravitational lensing with ISW and SZ effects. It derives the angular-averaged bispectrum from a three-term temperature decomposition and computes the ISW-lensing and SZ-lensing coefficients $b^{ISW}_l$ and $b^{SZ}_l$, including practical high-$l$ approximations. The results indicate SZ-lensing dominates at high multipoles and can constrain the mean density of ionized gas and gas physics, while ISW-lensing provides information on the evolution of gravitational potentials and the dark energy equation of state. Planck should distinguish among models with high significance, whereas MAP is expected to primarily detect SZ-lensing and enable complementary cross-correlations with large-scale structure and X-ray data, highlighting the bispectrum as a powerful tool for late-time cosmology.

Abstract

Gravitational fluctuations along the line-of-sight from the surface of last scatter to the observer distort the microwave background in several related ways: The fluctuations deflect the photon path (gravitational lensing), the decay of the gravitational potential produces additional fluctuations (ISW effect) and scattering off of hot gas in clusters produce additional fluctuations (Sunyaev-Zel'dovich effect). Even if the initial fluctuations generated at the surface of last scatter were Gaussian, the combination of these effects produce non-Gaussian features in the microwave sky. We discuss the microwave bispectrum as a tool for measuring a studying this signal. For MAP, we estimate that these measurements will enable us to determine the fraction of ionized gas and to probe the time evolution of the gravitational potential.

Figures (5)

• Figure 1: The $b_l$ coefficients as a function of $l$. The solid line shows the coefficient from the ISW effect, while the the dashed line shows the coefficient arising from the Sunyaev Zel'dovich effect. Each of the plots is for an $\Omega_m=0.3$, $\Omega_\Lambda=0.7$, $h=0.65$ cosmology. We can estimate the contribution that the different coefficients will have through dimensional analysis. Since $c_l\propto l^{-2}$, the Wigner 3-j symbols $\propto l^{-1/2}$ (equation \ref{['eq:bispec']}) and the signal per $l_3$ is expected to go as $B_{l_1 l_2 l_3}\times l^{-4}$ (see equation \ref{['eq:chi2']}) the $\chi^2/\Delta l_3$ will be roughly constant if $b_{l}\propto l^{-3}$. Thus we have normalized the kernels with an $l^3$ prefactor in order to estimate their importance.
• Figure 2: As in the previous figure, but the coefficients, $b_l$, are shown only for the SZ-lensing coupling. Here we have varied $\Omega_m$ to illustrate that the shape of the coefficients, as well as the normalization, vary with $\Omega_m$, and thus, $\Omega_m$ cannot be considered a degenerate parameter with the product, $y_0 b_{gas}$.
• Figure 3: The $\chi^2$ differences in the ISW-lensing bispectrum, the SZ-lensing bispectrum, and their sums between a fiducial model $\Omega_m=0.3$, $\Omega_\Lambda=0.7$ and a test model with a flat cosmology and varying $\Omega_m$. The solid line shows the $\chi^2$ for the PLANCK experiment, while the dashed line shows $\chi^2$ for MAP 4 year results and the the dotted line shows the MAP 1 year result. with 1 $\sigma$ uncertainty.
• Figure 4: The same as the previous figure, but with variations in the equation, $w$. For a cosmological constant, $w=-1$ (our fiducial model), while $w=-1/3$ is a curvature-like term. Both MAP and PLANCK will be able to distinguish between extreme cases; MAP would predict $w<-0.4$ and PLANCK would give $w<-0.8$.
• Figure 5: The $\chi^2$ per mode ($l_3$) as given by the bispectrum. The solid line shows the results from the ISW-lensing effect measured by PLANCK, while the dashed line shows the SZ-lensing effect as measured by MAP (4 yr. results). In general, the signal is at higher wavenumber (smaller scale) for the SZ-lensing effect than for the ISW-lensing effect. However, PLANCK's increased sensitivity at higher wavenumber distorts this relation in this plot.