Cosmology without window functions. II. Cubic estimators for the galaxy bispectrum

When analyzing the galaxy bispectrum measured from spectroscopic surveys, it is imperative to account for the effects of nonuniform survey geometry. Conventionally, this is done by convolving the theory model with the window function; however, the computational expense of this prohibits full exploration of the bispectrum likelihood. In this work, we provide a new class of estimators for the unwindowed bispectrum, a quantity that can be straightforwardly compared to theory. This builds upon the work of Philcox [Phys. Rev. D 103, 103504 (2021)] for the power spectrum and comprises two parts (both obtained from an Edgeworth expansion): a cubic estimator applied to the data and a Fisher matrix, which deconvolves the bispectrum components. In the limit of weak non-Gaussianity, the estimator is minimum variance; furthermore, we give an alternate form based on Feldman-Kaiser-Peacock weights that is close to optimal and easy to compute. As a demonstration, we measure the binned bispectrum monopole of a suite of simulations using both conventional estimators and our unwindowed equivalents. Computation times are comparable, except that the unwindowed approach requires a Fisher matrix, computable in an additional $\mathcal{O}(100)$ CPU hours. Our estimator may be straightforwardly extended to measure redshift-space distortions and the components of the bispectrum in arbitrary separable bases. The techniques of this work will allow the bispectrum to straightforwardly be included in the cosmological analysis of current and upcoming survey data.

Figure 1

Unwindowed bispectrum measurements from 999 Patchy simulations at $z=0.57$, obtained using the cubic estimators of this work. In the first panel, we show the bin-averaged triangle sides $\{k_1,k_2,k_3\}$ corresponding to each one-dimension wavenumber bin, with the second panel giving the associated binned bispectrum estimates (normalized by $k_1{k}_2{k}_3$), averaged across all mocks. The third panel displays the fractional error in the bispectrum relevant for a single mock dataset. We show triangle configurations corresponding to scalene ($k_1\ne k_2\ne k_3$), isosceles ($k_1=k_2<k_3$ or $k_1<k_2=k_3$) and equilateral ($k_1=k_2=k_3$) triangles in green, blue and red respectively; the error-bars for isosceles (equilateral) triangles are inflated by a factor $\sim 2$ ($\sim 6$) as expected. To compute these bispectra, we assume an FKP weighting (12), a Nyquist frequency of $k_{\text{Nyq}}=0.3h{\mathrm{Mpc}}^{-1}$, $N_{\mathrm{mc}}=50$ Monte Carlo simulations, and do not forward model pixelation effects (cf. Appendix pp1). We additionally show results from the windowed bispectrum estimator in purple; whilst these have smaller fractional errors, this is primarily due to increased bin-to-bin covariances, as demonstrated in Fig. 3. The figure shows results only for the bispectrum monopole (averaged over triangle rotations); the estimators of this work could be simply extended to include higher multipoles sourced by redshift-space distortions.

Figure 2

Fisher matrix for the unwindowed bispectrum estimates shown in Fig. 1. As discussed in the main text, this removes effects of the survey window function, practically acting as a deconvolution matrix. Here, we plot the matrix using the same binning scheme as Fig. 1, normalizing by the diagonal elements for clarity. The largest correlations are between bins $k_1^{\prime }=k_1\pm \mathrm{\Delta }k$, $k_2^{\prime }=k_2$, $k_3^{\prime }=k_3$ (or permutations); other correlations are found to be small.

Figure 3

Correlation matrices for the unwindowed and windowed bispectrum measurements shown in Fig. 1. These are computed from the sample covariance of 999 individual Patchy bispectra and follow the same bin ordering as Fig. 1. While the unwindowed spectra have a close-to-diagonal covariance, this is not true for the windowed measurements and highlights the smoothing effect of the window function.

Figure 4

Dependence of the unwindowed bispectra on the Fourier-space grid size. This compares two sets of bispectra: those computed with a Nyquist frequency $k_{\text{Nyq}}=0.3h{\mathrm{Mpc}}^{-1}$ (shown in Fig. 1) and those with $k_{\text{Nyq}}=0.2h{\mathrm{Mpc}}^{-1}$. The top panel shows the difference between the two estimates as a fraction of the statistical error of the fiducial estimates, while, in the bottom panel, we give the ratio of statistical errors. The bins are ordered in the same fashion as Fig. 1, and both datasets include 100 Patchy simulations. Here, increasing the number of grid points is found to have an insignificant effect, except on the smallest scales.

Figure 5

As in Fig. 4, but assessing the impact of the number of Monte Carlo simulations used to define the Fisher matrix. Here we compare the case of $N_{\mathrm{mc}}=100$ to the fiducial results with $N_{\mathrm{mc}}=50$. The error is expected to scale as $\sqrt{1+1/N_{\mathrm{mc}}}$; this matches that found in the figure.

Figure 6

As in Fig. 4, but testing the impact of forward modeling the pixelation effects, as described in Appendix pp1. This is found to have a very small effect in practice, since the survey size is much larger than the pixel width.

Figure 7

As in Fig. 4, but assessing the importance of the second term in ${\widehat{q}}_{\alpha }$ [Eq. (51)], which is ignored in conventional bispectrum estimators. Removal of this term is found to significantly increase the error on large scales; this is consistent with the expected behavior found in Appendix pp2 and highlights the importance of this contribution in $f_{\mathrm{NL}}$-based studies.

Figure 8

As in Fig. 4, but adopting maximum-likelihood weights ${\mathsf{C}}^{-1}$ [Eq. (a4)] rather than the short-scale FKP approximation [Eq. (12)]. As in Ref. [74], this is not expected to significantly improve constraints from a BOSS-like survey, since the window function contains mostly large-scale power and the shot noise is large. Here, we find an $\sim 5\%$ decrease in the error bars; however, this is associated with a nontrivial underprediction of the bispectrum. This may arise from the difficulty of inverting the survey mask (since it contains zeros, which do not quite match those of the dataset due to pixelation effects). In practice, we expect the difference to be absorbed in bispectrum bias parameters in any cosmological analysis; thus, this is unlikely to be a cause for concern.