Skew-spectra: a generalization to spin-$s$

TL;DR

This work extends skew-spectra to arbitrary spin fields by developing a spin-weighted harmonic framework that yields cross-spectra between a field and its quadratic products, enabling efficient extraction of non-Gaussian information tied to the angular bispectrum. By expressing generalized skew-spectra in terms of the angular bispectrum with Wigner-$3j$ algebra, the authors formulate a versatile, probe-agnostic tool applicable to spin-$s$ fields and, in particular, to weak lensing where mass-mapping is avoided. The paper provides a comprehensive theory and practical prescriptions, including parity considerations, filtering, and domain choices, and demonstrates ΛCDM predictions for various skew-spectra using BiHalofit and Planck-like cosmology. The approach promises to harness the extensive power-spectrum infrastructure (e.g., Pseudo-$C_ell$ methods) for fast, robust non-Gaussian inference in large cosmological data sets and may be extended to CMB polarization analyses.

Abstract

Skew-spectra allow us to extract non-Gaussian information by taking the square of a map and finding the power spectrum of this new map with the original map. This allows us to use much of the infrastructure of power spectra and avoid the intricacies of estimating three point statistics. In this paper we present the first extension of skew-spectra to arbitrary spin-$s$ fields, as a means to extract non-Gaussian information efficiently from cosmological data sets like cosmic shear or CMB polarization. We apply the formalism to weak lensing in the context of large scale structure, and discuss different ways of combining fields to build skew-spectra, all while avoiding the problems associated with mass-mapping. We provide plots of these new statistics for $Λ$CDM and vary cosmological parameters.

Figures (4)

• Figure 1: The compression factors present in Eqs. (\ref{['eq:ClEEgammasquared_lensing']})-(\ref{['eq:ggg']}) (these are the factors multiplied by the angular bispectrum at a given $(\ell,\ell_1,\ell_2)$ configuration before summing over $\ell_1$ and $\ell_2$, where we only consider $\ell_1+\ell_2+\ell=\text{even}$ contributions) plotted for $\ell_1/\ell$ against $\ell_2/\ell$, all shown at a slice of $\ell=30$. The top left (right) plot corresponds to the compression factors related to $\gamma^*\gamma$ ($\gamma^2$) shown in Eqs. (\ref{['eq:ClEEgammastar_lensing']}) and (\ref{['eq:galaxy_gamma_gammastar']}) (Eqs. (\ref{['eq:ClEEgammasquared_lensing']}) and (\ref{['eq:galaxy_gamma_squared']})). The bottom left (right) plot shows the compression factors related to $\gamma g$ ($g^2$) shown in Eqs. (\ref{['eq:gamma_gamma_galaxy']}) and (\ref{['eq:galaxy_gamma_galaxy']}) (Eqs. (\ref{['eq:gamma_galaxy_galaxy']}) and (\ref{['eq:ggg']})). The squeezed ($\ell_1/\ell \ll \ell_2/\ell$) and folded ($\ell_1/\ell=\ell_2/\ell=1/2$) configurations are illustrated in the top left plot at $(0,1)$ and $(0.5,0.5)$ respectively. We show on the plots a choice of $\ell_1^{\rm{max}}=40$ and $\ell_2^{\rm{max}}=40$ to demonstrate the effect of filtering the domain of $\ell_1,\ell_2$, as the regions below and to the left of these lines contribute to the angular power spectrum at a given $\ell$.
• Figure 2: Theoretical predictions based on BiHalofit for the angular power spectra shown in Eqs.(\ref{['eq:ClEEgammasquared_lensing']})-(\ref{['eq:ggg']}) formed from the quadratic quantities: $\gamma^2$, $\gamma^*\gamma$, $\gamma g$ and $g^2$. The power spectra for Eqs.(\ref{['eq:ClEEgammasquared_lensing']})-(\ref{['eq:galaxy_gamma_gammastar']}), (${E^\gamma\times E^{\gamma^2}}$, ${E^\gamma\times E^{\gamma^*\gamma}}$, ${g\times E^{\gamma^2}}$, and ${g\times E^{\gamma^*\gamma}}$), are shown in the left-hand plot. The remaining statistics of Eqs.(\ref{['eq:gamma_galaxy_galaxy']})-(\ref{['eq:ggg']}) (${g\times g^2}$, ${E^\gamma\times g^2}$, ${\gamma \times E^{\gamma g}}$, and ${g\times E^{\gamma g}}$) are shown in the right-hand plot. All the statistics are shown for the fiducial Planck 2018 cosmology planckresults, i.e. $\{\sum m_\nu ,\Omega_b,\Omega_c,h,n_s,\sigma_8\}= \{0.06,0.0493,0.264,0.6766,0.9665,0.8102 \}$, with redshift bins for the lensing sample of $z_\gamma \sim \mathcal{N}(\bar{z}=0.5,\sigma_z=0.05)$ and for the galaxy sample of $z_g \sim \mathcal{N}(\bar{z}=0.25,\sigma_z=0.05)$. We neglect the contribution from $B$ modes and parity odd signals. We choose the domain of $\ell_1$ and $\ell_2$ to be and $[\ell_1^{\rm{min}}, \ell_1^{\rm{max}}]=[10,600]$ and $[\ell_2^{\rm{min}}, \ell_2^{\rm{max}}]=[10,600]$.
• Figure 3: Ratio of power spectra for $E^\gamma\times E^{\gamma^2}$ of Eq. (\ref{['eq:ClEEgammasquared_lensing']}) for different cosmologies relative to the fiducial Planck planckresults results, taking $\{\sum m_\nu ,\Omega_b,\Omega_c,h,n_s,\sigma_8\}= \{0.06,0.0493,0.264,0.6766,0.9665,0.8102 \}$, with the same redshift bin of $z_\gamma \sim \mathcal{N}(\bar{z}=0.5,\sigma_z=0.05)$ for each of the shear quantities. We use the domain of the sum in the statistics of $[\ell_1^{\rm{min}},\ell_1^{\rm{max}}]=[10,600]$ and $[\ell_2^{\rm{min}},\ell_2^{\rm{max}}]=[10,600]$.
• Figure 4: The same as Fig. \ref{['fig:gamma_squared_ratios_real_cosmologies']} but for the ratio of power spectra for $E^\gamma\times E^{\gamma^*\gamma}$ of Eq. (\ref{['eq:ClEEgammastar_lensing']}).