The Bispectrum and the Trispectrum of the Ostriker and Vishniac Effect

TL;DR

The paper develops analytical expressions for the Fourier-space bispectrum and trispectrum of the Ostriker–Vishniac effect in the linear and mildly nonlinear regimes, leveraging the Limber approximation to identify the dominant vector-like contributions. It demonstrates that even moments dominate due to the OV's vector nature, making the trispectrum a more sensitive statistic than the bispectrum for this secondary CMB signal. Using a flat $\Lambda$CDM framework and two reionization histories, it provides concrete formulas for the dominant OV and kSZ terms, including nonlinear extensions via a Pea–Dodds-style prescription for the density field, and assesses detectability with MAP/Planck-like instruments. The results indicate the OV bispectrum is unlikely to be detected even with ideal data, but the OV trispectrum could be detectable by Planck or future arcminute-scale experiments, especially when nonlinearities are included, offering a potential window into reionization and nonlinear structure formation. The methodology, based on turbulence-inspired spectral tensor techniques and generalized Limber equations, is broadly applicable to other vector-like secondary anisotropies and enhances our ability to quantify and separate non-Gaussian signatures in the CMB.

Abstract

We present analytical expressions for the Fourier analog of the CMB three-point and four-point correlation functions, the spatial bispectrum and trispectrum, of the Ostriker and Vishniac effect in the linear and mildly nonlinear regime. Through this systematic study, we illustrate a technique to tackle the calculation of such statistics making use of the effects of its small-angle and vector-like properties through the Limber approximation. Finally we discuss its configuration dependence and detectability in the context of Gaussian theories for the currently favored flat Lambda CDM cosmology.

Figures (4)

• Figure 1: The linear (label L) and nonlinear (label NL) OV power spectrum for the fiducial $\Lambda$CDM model. The dot dash lines correspond to $z_r=8$ and the solid lines to $z_r=17$. In both cases we assume $\Delta z_r= 0.1(1+z_r)$ .
• Figure 2: Left panel--- Linear (label L) flat-sky bispectrum of the OV effect and its nonlinear extension (label NL). Right panel --- Contribution to $\chi^2$ per log interval in $\ell$ for the OV full-sky linear bispectrum with no instrumental noise ( top), Planck noise ( middle) and MAP noise ( bottom) included in the variance. We used the specifications in the table \ref{['table1']}. All the plots were calculated for the fiducial $\Lambda$CDM model. The dot dash lines correspond to $z_r=8$ and the solid lines to $z_r=17$. We assumed $\Delta z_r= 0.1(1+z_r)$. The total $S/N$ for the OV full-sky linear bispectrum for MAP, Planck and a perfect experiment respectively are: $5.0\times10^{-5}$, $1.2\times10^{-3}$ and $4.4\times10^{-3}$ assuming $z_r=8$, and $3.7\times10^{-5}$, $2.0\times10^{-3}$ and $1.4\times10^{-2}$ assuming $z_r=17$.
• Figure 3: Left panel--- Linear and Right panel --- nonlinear flat-sky trispectrum of the OV effect for geometrical configurations such that $-0.95 \leq \epsilon \leq 0.00$ in steps of $0.05$. The amplitude of the trispectrum decreases as $\epsilon$ decreases from $0.00$ to $-0.95$. Because the power is symmetric in $\epsilon$ around $0.00$ we only plotted the negative $\epsilon$s. All the plots were calculated for the fiducial $\Lambda$CDM model. The dot dash lines correspond to $z_r=8$ and the solid lines to $z_r=17$. We assumed $\Delta z_r= 0.1(1+z_r)$.
• Figure 4: Contribution to the $\chi^2$ per log interval in $\ell$ for $\epsilon = -0.95$ for the OV full-sky linear/nonlinear trispectrum with no instrumental noise ( top), Planck noise ( middle) and MAP noise ( bottom) included in the variance. Again we used the specifications in the table \ref{['table1']}. All the plots were calculated for the fiducial $\Lambda$CDM model. The dot dash lines correspond to $z_r=8$ and the solid lines to $z_r=17$. We assumed $\Delta z_r= 0.1(1+z_r)$. The higher amplitudes for each of the experiments correspond to the contributions from the full-sky nonlinear trispectrum. The horizontal line at $d\chi^2/dln(l)=1$ shows the minimum detection threshold. The total $S/N$ for the OV full-sky linear trispectrum for MAP, Planck and a perfect experiment respectively are: $1.4\times10^{-3}$, $1.1$ and $27.4$ assuming $z_r=8$, and $2.9\times10^{-3}$, $4.7$ and $149$ assuming $z_r=17$.