tokens: the Q 21,C ii ,C ii bispectrum should be identical to that of the Q 21, δ , δ bispectrum. That is, both of these cross-bispectra are expected to follow the form. Consequently, we study the statistics of Q 21, δ , δ here rather than Q 21,C ii ,C ii since the latter is more difficult to model. The main shortcoming of this approach is that it neglects the additional nonlinearities in the [C ii ] biasing relation, and these effects may reduce the agreement with the perturbative formula. In estimating the signal-to-noise ratio (S/N) at which may be measured in future surveys (Section 6 ), we assume a linear biasing model for [C ii ] emission. This allows us to predict the [C ii ] auto-spectrum and 21 cm-[C ii ] cross-spectrum (as required for the S/N calculation) from the statistics of the simulated density field. It may be possible to construct a subgrid model to include unresolved small halos with the correct statistical properties (e.g., McQuinn et al. 2007 ). However, the approach adopted in that work would need to be extended to capture the correct quadratic halo bias and is beyond the scope of this paper. 3.4. Estimating the Bispectrum In order to estimate the bispectrum from our 21cmFAST simulation outputs, we follow the method described in Smith et al. ( 2008 ). Throughout we consider two characterizations of the wavevector arguments. In both cases we calculate the average bispectrum in linearly spaced wavevector bins. Our first characterization is for illustration purposes only. Here we fix the magnitude of and while varying the angle θ 12 between the two wavevectors. The third side of the triangle, , is determined by these three parameters and the requirement of translation invariance (.) We consider spherical bins around each wavevector of thickness Δ k = k f , where k f = 2 π / L box = 0.00785 Mpc −1 is the fundamental mode of our simulation box. Our second characterization sets a minimum ( k min ) and a maximum ( k max ) magnitude for each wavevector argument. That is, we enforce for each wavevector , along with the usual constraint that . Owing to homogeneity and isotropy, all bispectra have three arguments: , , and . Our bins are cubes of side length 4 k f in Fourier space. This characterization captures nearly all the available modes. As discussed in Section 2 , we also enforce that k 1 ≥ k 2 ≥ k 3 . In estimating bispectra, we loop through all of the simulated Fourier modes that lie within this wavevector range. We determine B 21, δ , δ and P 21, δ from the same set of Fourier modes to reduce sample variance in our estimates of Q 21, δ , δ . In general, one would like to make k min and k max as small and large as possible, respectively. However, our estimates at small k are noisy, and the perturbative formula becomes less accurate at high k . We chose values of k min and k max throughout that give the best results, but we also show results for different values of k min and k max in Appendix A . 4. Results First, we consider
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