Evidence for quadratic tidal tensor bias from the halo bispectrum

The relation between the clustering properties of luminous matter in the form of galaxies and the underlying dark matter distribution is of fundamental importance for the interpretation of ongoing and upcoming galaxy surveys. The so-called local bias model, where galaxy density is a function of local matter density, is frequently discussed as a means to infer the matter power spectrum or correlation function from the measured galaxy correlation. However, gravitational evolution generates a term quadratic in the tidal tensor and thus nonlocal in the Eulerian density field, even if this term is absent in the initial conditions (Lagrangian space). Because the term is quadratic, it contributes as a loop correction to the power spectrum, so the standard linear bias picture still applies on very large scales; however, it contributes at leading order to the bispectrum for which it is significant on all scales. Such a term could also be present in Lagrangian space if halo formation were influenced by the tidal field. We measure the corresponding coupling strengths from the matter-matter-halo bispectrum in numerical simulations and find a nonvanishing coefficient for the tidal tensor term. We find no scale dependence of the inferred bias parameters up to $k\sim 0.1h{\mathrm{Mpc}}^{-1}$ and that the tidal effect is increasing with halo mass. While the local Lagrangian bias picture is a better description of our results than the local Eulerian bias picture, our results suggest that there might be a tidal tensor bias already in the initial conditions. We also find that the coefficients of the quadratic density term deviate quite strongly from the theoretical predictions based on the spherical collapse model and a universal mass function. Both quadratic density and tidal tensor bias terms must be included in the modeling of galaxy clustering of current and future surveys if one wants to achieve the high precision cosmology promise of these data sets.

Figure 1

Matter (black points) and halo-matter-matter (red points) bispectra as a function of triangle shape for configuration $k_1=0.052h{\mathrm{Mpc}}^{-1}$, $k_2=0.06h{\mathrm{Mpc}}^{-1}$. The black solid line shows the tree-level prediction for the matter bispectrum, the red solid line has $b_1$ only and dashed and dashed-dotted lines are adding $b_2$ and $b_{s^2}$. The list of bias parameters behind the theoretical cross bispectra in the legend indicates the parameters that were considered for the corresponding curve. The error bars are estimated from the box-to-box variance of the bispectrum measurement. Note that this is only a small fraction of the total bispectrum information that our simulations contain, and the log scale also deemphasizes what are actually significant differences between the fit with and without $s^2$ (these will be highlighted later).

Figure 2

Convergence of the measured $b_2$ (upper panels) and $b_{s^2}$ (lower panels) parameters with increasing maximum $k$ mode for the four mass bins. The horizontal red and blue lines show the constraints obtained for our fiducial $k_{\max }=0.07h{\mathrm{Mpc}}^{-1}$. The pivot data point is highlighted by the gray shaded region.

Figure 3

Residual shape dependence of the halo bispectrum for our reduced bispectrum defined in Eq. (35). The blue data points with error bars show the result of the combined reduced bispectrum defined in Eq. (37) including all the configurations up to $k_{\max }=0.07h{\mathrm{Mpc}}^{-1}$. The horizontal dashed line shows the model including $b_2$ only, and the solid blue line shows the model including both $b_2$ and $b_{s^2}$.

Figure 4

Left panel: Mass dependence of the bias parameters and theoretical predictions. The points with error bars are our best-fit parameters for ${\widehat{b}}_1$, $2{\widehat{b}}_2$ and ${\widehat{b}}_{s^2}$. The numerical values of the data points are given in Table . The curves show the corresponding theoretical bias functions as calculated using the relations in Sec. 2e. The measurements for ${\widehat{b}}_1$ are in good agreement with the theory; there is a clear deviation for the ${\widehat{b}}_{s^2}$ and ${\widehat{b}}_2$ measurement for the two central mass bins. Right panel: Ratio of the simulation halo matter and matter power spectra ${\widehat{P}}_{\mathrm{hm}}(k)/{\widehat{P}}_{\mathrm{mm}}(k)$ and first-order bias parameters inferred using the data points highlighted by the shaded region.