Constraints from CMB lensing tomography with projected bispectra

Abstract

We measure the angular power spectrum and bispectrum of the projected overdensity of photometric DESI luminous red galaxies, and its cross-correlation with maps of the Cosmic Microwave Background lensing convergence from \planck. This analysis is enabled by the use of the ``filtered-squared bispectrum'' approach, introduced in previous work, which we generalise here to the case of cross-correlations between multiple fields. The projected galaxy bispectrum is detected at very high significance (above $30σ$ in all redshift bins), and the galaxy-galaxy-convergence bispectrum is detected above $5σ$ in the three highest-redshift bins. We find that the bispectrum is reasonably well described over a broad range of scales by a tree-level prediction using the linear galaxy bias measured from the power spectrum. We carry out the first cosmological analysis combining projected power spectra and bispectra under a relatively simple model, and show that the galaxy bispectrum can be used in combination with the power spectrum to place a constraint on the amplitude of matter fluctuations, $σ_8$, an on the non-relativistic matter fraction $Ω_m$. We find that data combinations involving the galaxy bispectrum recover constraints on these parameters that are in good agreement with those found from the traditional ``2$\times$2-point'' combination of galaxy-galaxy and galaxy-convergence power spectra, across all redshift bins.

Figures (14)

• Figure 1: Filters used in this analysis. The colourful set represents the 4 filters used in Section \ref{['ssec:results.detection']} to detect the FSB on a wide range of scales; the hatched filter is the single filter used for deriving cosmological constraints in Section \ref{['ssec:results.constraints']}. The latter remains below the conservative $\ell_\mathrm{max}^B$ cut in all redshift bins, and the former are defined below the $2N_\mathrm{side}$ threshold, beyond spherical harmonic transforms may become unreliable. We also show the pixel window function (solid black line) corresponding to the map resolution.
• Figure 2: The $ggg$ FSBs measured in all four redshift bins. Each column corresponds to one redshift bin, and each row to one of the four colourful filters shown in Fig. \ref{['fig:filters']} -- here represented by the greyed-out band. The FSB measurements are shown in blue, and the black solid line shows the theoretical fit for published values of $b_1$whiteCosmologicalConstraintsTomographic2022 and for a fiducial cosmology (Table \ref{['tab:fiducialcosmo']}). These theory predictions are used as templates in Eq. \ref{['eq:snr2']} to report the $\text{SNR}_2$ shown on the colour bars on the right of each FSB. Beyond the conservative cuts used in our inference pipeline (see Section \ref{['ssec:results.constraints']}), we show the predictions as dashed lines to emphasise that the tree-level bispectrum is not necessarily a good theoretical description of the power at those scales. It is worth noting that the FSBs in all but the first filter (i.e. the first row in this plot) also lie in this small-scale regime. The blue shaded area represents the amount of noise (Eq. \ref{['eq:noiseggg']}) contributing to the overall theory prediction. Note that we plot $\ell \Phi^{ggg}_{LL \ell}$ instead of $\Phi^{ggg}_{LL \ell}$ to improve readability at higher multipoles.
• Figure 3: The $gg\kappa$ FSBs measured in all four redshift bins. Each column corresponds to one redshift bin, and each row to one of the four colourful filters shown in Fig. \ref{['fig:filters']} -- here represented by the greyed-out band. The FSB measurements are shown in blue, and the black solid line shows the theoretical fit for published values of $b_1$whiteCosmologicalConstraintsTomographic2022 and for a fiducial cosmology (Table \ref{['tab:fiducialcosmo']}). These theory predictions are used as templates in Eq. \ref{['eq:snr2']} to report the $\text{SNR}_2$ shown on the colour bars on the right of each FSB. Beyond the conservative cuts used in our inference pipeline (see Section \ref{['ssec:results.constraints']}), we show the predictions as dashed lines to emphasise that the tree-level bispectrum is not necessarily a good theoretical description of the power at those scales. The blue shaded area represents the amount of noise (Eq. \ref{['eq:noiseggk']}) contributing to the overall theory prediction. Note that we plot $\ell \Phi^{gg\kappa}_{LL \ell}$ instead of $\Phi^{gg\kappa}_{LL \ell}$ to improve readability at higher multipoles.
• Figure 4: A visual representation of the data vectors used to derive constraints on $\sigma_8$ and $\Omega_m$, with power spectra $C_\ell^{gg}$ (yellow data points) and $C_\ell^{g\kappa}$ (red, multiplied by a factor of $10$ for readability) measurements on the left, and FSBs $\Phi^{ggg}_{LL \ell}$ (blue) and $\Phi^{gg\kappa}_{LL \ell}$ (purple) on the right. Each row corresponds to a different redshift bin, starting from lowest redshift at the top. The multipole cuts evolve accordingly (the number of data points included in the analysis thus increases with redshift). The new filtered range (for a comparison with previous filters, see Fig. \ref{['fig:filters']}) is shown in grey, and fits within the most restrictive $\ell_\mathrm{max}^B$ cut (lowest redshift bin). The black lines show theory predictions for the best-fit values obtained from constraining the models with the full data vector, $\Phi^{ggg}_{LL \ell} + C_\ell^{gg} + C_\ell^{g\kappa}$. These best-fit values are reported in Table \ref{['tab:cosmo']}, and the corresponding predictions are used as templates to report $\text{SNR}_2$ (Eq. \ref{['eq:snr2']}) indicated next to each FSB.
• Figure 5: Constraints on $\sigma_8$ and $b_1$ from three data combinations: $\Phi^{ggg}_{LL \ell} + C_\ell^{gg}$ (blue contours in top panel), $\Phi^{ggg}_{LL \ell} + C_\ell^{g\kappa}$ (red contours, middle panel), and $C_\ell^{gg} + C_\ell^{g\kappa}$ (yellow contours, bottom panel). The black contours show the degeneracy directions of individual power spectrum or FSB measurements, with the filled contours showing the combined constraints. The constraints shown were obtained for redshift bin $z_3$.