Near optimal bispectrum estimators for large-scale structure

Clustering of large-scale structure provides significant cosmological information through the power spectrum of density perturbations. Additional information can be gained from higher-order statistics like the bispectrum, especially to break the degeneracy between the linear halo bias $b_1$ and the amplitude of fluctuations ${\sigma }_8$. We propose new simple, computationally inexpensive bispectrum statistics that are near optimal for the specific applications like bias determination. Corresponding to the Legendre decomposition of nonlinear halo bias and gravitational coupling at second order, these statistics are given by the cross-spectra of the density with three quadratic fields: the squared density, a tidal term, and a shift term. For halos and galaxies the first two have associated nonlinear bias terms $b_2$ and $b_{s^2}$, respectively, while the shift term has none in the absence of velocity bias (valid in the $k\to 0$ limit). Thus the linear bias $b_1$ is best determined by the shift cross-spectrum, while the squared density and tidal cross-spectra mostly tighten constraints on $b_2$ and $b_{s^2}$ once $b_1$ is known. Since the form of the cross-spectra is derived from optimal maximum-likelihood estimation, they contain the full bispectrum information on bias parameters. Perturbative analytical predictions for their expectation values and covariances agree with simulations on large scales, $k\lesssim 0.09h/\mathrm{Mpc}$ at $z=0.55$ with Gaussian $R=20h^{-1}\mathrm{Mpc}$ smoothing, for matter-matter-matter, and matter-matter-halo combinations. For halo-halo-halo cross-spectra the model also needs to include corrections to the Poisson stochasticity.

Figure 1

Theory contributions (67) to halo-halo-halo cross-spectra scaling like $b_1^3$ (dashed lines), $b_1^2{b}_2$ (dash-dotted lines) and $b_1^2{b}_{s^2}$ (dotted lines) for squared density ${\delta }_{\mathrm{h}}^2(\mathbf{x})$ (blue), shift term $-{\mathrm{\Psi }}_{\mathrm{h}}^i(\mathbf{x}){\partial }_i{\delta }_{\mathrm{h}}(\mathbf{x})$ (red) and tidal term $s_{\mathrm{h}}^2(\mathbf{x})$ (green), evaluated for fixed bias parameters $b_1=1$, $b_2=0.5$ and $b_{s^2}=2$, Gaussian smoothing with $R_G=20h^{-1}\mathrm{Mpc}$, at $z=0.55$, with linear matter power spectra in integrands. Thin gray lines show the large-scale (low $k$) limit given by Eq. (70). The cross-spectra are divided by the partially smoothed FrankenEmu emulator matter power spectrum $W_R^{3/2}{P}_{\mathrm{mm}}^{\text{emu}}$ [45, 46, 47, 48] for plotting convenience.

Figure 2

Matter-matter-matter cross-spectra measured from 10 realizations at $z=0.55$ (crosses with error bars), compared with leading order theory prediction of Eq. (61) (solid lines), neglecting shot noise. Upper panels show cross-spectra divided by the partially smoothed emulator matter power spectrum $W_R^{3/2}{P}_{\mathrm{mm}}^{\text{emu}}$, lower panels show the ratio of measured cross-spectra over their theory expectation (61). Gaussian smoothing is applied with $R_G=20h^{-1}\mathrm{Mpc}$ (left) and $R_G=10h^{-1}\mathrm{Mpc}$ (right). Different colors represent different cross-spectra (squared density in blue, shift term in red and tidal term in green).

Figure 3

Test of $b_2$ and $b_{s^2}$ contributions to matter-matter-halo cross-spectra. First, ${\widehat{b}}_1$ is obtained from ${\widehat{P}}_{\mathrm{hm}}/{\widehat{P}}_{\mathrm{mm}}$ at $k<0.04h/\mathrm{Mpc}$. Then $b_2$ and $b_{s^2}$ are obtained by fitting the model (66) to the measured excess cross-spectra ${\widehat{P}}_{D[{\delta }_{\mathrm{m}}^R]{\delta }_{\mathrm{h}}^R}-{\widehat{b}}_1{\widehat{P}}_{D[{\delta }_{\mathrm{m}}^R]{\delta }_{\mathrm{m}}^R}$ (crosses). The best-fit total model (solid lines) consists of the $b_2$ contribution (dash-dotted lines) and the $b_{s^2}$ contribution (dotted lines), while shot noise is neglected. Different plots show different mass bins (increasing from upper left to lower right). The upper subpanels show excess cross-spectra divided by the partially smoothed emulator matter power $W_R^{3/2}{P}_{\mathrm{mm}}^{\text{emu}}$, the lower subpanels show measured excess cross-spectra divided by their theory expectation. The fit is obtained from the grey shaded region, assuming estimated standard errors of the mean without any covariances. Best-fit bias parameters, reduced ${\chi }^2$ and halo mass range are reported at the top of each plot. Gaussian smoothing with $R_G=20h^{-1}\mathrm{Mpc}$ is applied to matter and halo densities.

Figure 4

Measured halo-halo-halo cross-spectra (crosses) compared against theory [thick solid, Eq. (67)] with bias parameters ${\widehat{b}}_1$ from ${\widehat{P}}_{\mathrm{hm}}/{\widehat{P}}_{\mathrm{mm}}$ and $b_2$ and $b_{s^2}$ from ${\widehat{P}}_{D[{\delta }_{\mathrm{h}}^R]{\delta }_{\mathrm{m}}^R}-{\widehat{b}}_1{\widehat{P}}_{D[{\delta }_{\mathrm{m}}^R]{\delta }_{\mathrm{m}}^R}$ for $R_G=20h^{-1}\mathrm{Mpc}$ smoothing. Upper panels also show theory contributions scaling like $b_1^3$ (dashed lines), $b_1^2{b}_2$ (dash-dotted lines) and $b_1^2{b}_{s^2}$ (dotted lines), as well as the halo-halo-halo shot noise contribution (thin solid lines), which is assumed to be Poissonian [i.e. ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=0$ in Eq. (77)]. The reduced ${\chi }^2$ on top of the plots quantifies the (dis)agreement between halo-halo-halo cross-spectra measurements and model for the fixed bias parameters. It is computed over the gray region, neglecting covariances. Note that the shot noise contribution fluctuates on very large scales because it is computed using the ensemble-averaged estimated halo-halo power spectrum.

Figure 5

Same as Fig. 4 if the shot noise correction ${\mathrm{\Delta }}_1$ in Eq. (77) is fitted to measured halo-halo-halo cross-spectra, fixing ${\mathrm{\Delta }}_2={\mathrm{\Delta }}_1/{\overline{n}}_{\mathrm{h}}$ (still keeping $b_1$, $b_2$ and $b_{s^2}$ fixed to the values obtained from matter-halo and matter-matter-halo measurements).

Figure 6

Same as Fig. 5 for two higher mass bins and treating both shot noise corrections ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_2$ in Eq. (77) as independent parameters.

Figure 7

Left: ratio of estimated over theoretical standard deviation of matter-matter-matter cross-spectra [using $D=E$ and $a=b=\mathrm{m}$ in Eq. (78)]. Right: correlations between matter-matter-matter cross-spectra at the same scale $k=k^{\prime }$ predicted by theory [Eq. (79), dashed lines] and estimated from simulations (solid lines). Both panels assume Gaussian smoothing with $R_G=20h^{-1}\mathrm{Mpc}$ and use the linear matter power spectrum in theory expressions. For smaller smoothing scale $R$, the zero crossing would move to higher $k$.

Figure 8

Same as Fig. 7 but for matter-matter-halo cross-spectra (top) and halo-halo-halo cross-spectra (bottom), using the halo mass bin with $b_1=1.98$. Theoretical standard deviations (78) are evaluated using the estimated ensemble-averaged halo-halo power spectrum including shot noise for $P_{\mathrm{hh}}$ and the linear matter power spectrum for $P_{\mathrm{mm}}$. Theory correlations of Eq. (79) are the same as for the matter-matter-matter case. Results for other mass bins are similar (not shown).

Figure 9

Results of fitting $b_1$, $b_2$, $b_{s^2}$ and $A_{\text{shot}}\equiv -{\overline{n}}_{\mathrm{h}}{\mathrm{\Delta }}_1$ (imposing ${\mathrm{\Delta }}_2={\mathrm{\Delta }}_1/{\overline{n}}_{\mathrm{h}}$) to halo-halo-halo cross-spectra involving the squared density (blue), shift term (red) or tidal term (green), or all three combined (black), choosing Gaussian $R=20h^{-1}\mathrm{Mpc}$ smoothing and $k_{\max }=0.09h/\mathrm{Mpc}$ for the halo mass bin with $b_1=1.98$. The 2D contours of the posterior show 68% and 95% confidence regions corresponding to the full volume of $V=26.3h^{-3}{\mathrm{Gpc}}^3$ (i.e. errors in a single realization would be larger by a factor of $\sqrt{10}$). Thin black lines show the maximum-likelihood points corresponding to the black contours. The joint likelihood for the three halo-halo-halo cross-spectra is assumed to be Gaussian in the cross-spectra, with nonzero covariance between cross-spectra at $k=k^{\prime }$ given by the theory expression of Eq. (78). The green contours are somewhat uncertain because it is not clear how well the Markov chain Monte Carlo chains sampled the elongated degeneracies.

Figure 10

Same as Fig. 1 but for $R_G=10h^{-1}\mathrm{Mpc}$.

Figure 11

Same as Fig. 3 but for $R_G=10h^{-1}\mathrm{Mpc}$ smoothing.