To describe the H i bispectrum in redshift space, we apply the standard redshift space kernels (see Heavens et al. 1998; Scoccimarro et al. 1999 for derivations) such that
(10)$$\begin{eqnarray*}
B_\rm {H\,\small {I}}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3}) &amp;=&amp; 2\, \overline{T}_\rm {H\,\small {I}}^2\left[Z_1(\boldsymbol{k}_{1}) Z_1(\boldsymbol{k}_{2}) Z_2(\boldsymbol{k}_{1}, \boldsymbol{k}_{2}) P_\text{lin}(k_{1}) P_\text{lin}(k_{2})\right.\nonumber\\
&amp;&amp;\left. +\, \text{cycl.}\right] D_\text{FoG}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3},\sigma _\text{B}),
\end{eqnarray*}$$where cycl. represents cyclic permutations which run over all possible pairs of $\boldsymbol{k}_{1},\boldsymbol{k}_{2}$ and $\boldsymbol{k}_{3}$. Plin represents the real-space, linear matter power spectrum for which we use the CLASS Boltzmann solver (Blas, Lesgourgues & Tram 2011; Lesgourgues 2011). Z1 is given in equation (7) and Z2 denotes the second-order kernel and is given by
(11)$$\begin{eqnarray*}
Z_2(\boldsymbol{k}_{i}, \boldsymbol{k}_{j}) &amp;=&amp; b_1 F_2(\boldsymbol{k}_{i}, \boldsymbol{k}_{j})+f \mu _{ij}^2 G_2(\boldsymbol{k}_{i}, \boldsymbol{k}_{j})+\frac{f \mu _{ij} k_{ij}}{2}\nonumber\\
&amp;&amp;\times \,\left[\frac{\mu _i}{k_i} Z_1(\boldsymbol{k}_{j})+\frac{\mu _j}{k_{j}} Z_1(\boldsymbol{k}_{i})\right] + \frac{b_2}{2},
\end{eqnarray*}$$where $\boldsymbol{k}_{ij} = \boldsymbol{k}_{i} + \boldsymbol{k}_{j}$ and $\mu _{ij} = \boldsymbol{k}_{ij}\cdot \hat{{\bf z}}/k_{ij}$. F2 and G2 denote the second-order kernels for the real-space density and velocity fields and are given by
(12)$$\begin{eqnarray*}
F_2(\boldsymbol{k}_{i}, \boldsymbol{k}_{j}) = \frac{5}{7} + \frac{m_{ij}}{2}\left(\frac{k_i}{k_{j}} + \frac{k_{j}}{k_i}\right) + \frac{2}{7} m_{ij}^2,
\end{eqnarray*}$$(13)$$\begin{eqnarray*}
G_2(\boldsymbol{k}_{i}, \boldsymbol{k}_{j}) = \frac{3}{7} + \frac{m_{ij}}{2}\left(\frac{k_i}{k_{j}} + \frac{k_{j}}{k_i}\right)+\frac{4}{7} m_{ij}^2 ,
\end{eqnarray*}$$where $m_{ij} = (\boldsymbol{k}_{i}\cdot \boldsymbol{k}_{j})/(k_i k_{j})$. The final term in equation (10), DFoG, is a phenomenological factor to address some non-linear RSD effects not sufficiently modelled by the redshift kernels alone. On smaller scales, internal motion inside virialized structures produces a radial smearing to the density field in redshift space, known as the fingers-of-god (FoG) effect (Jackson 1972). It is common to include a term which describes the FoG (Taruya, Nishimichi & Saito 2010), even when including higher order perturbation theory terms and should be seen as a phenomenological damping required to correct for non-linear effects (Verde et al. 1998; Gil-Marín et al. 2014). For our choice of model, this factor is given by (Gil-Marín et al. 2015)
(14)$$\begin{eqnarray*}
D_\text{FoG}(\boldsymbol{k}_{1},\boldsymbol{k}_{2},\boldsymbol{k}_{3},\sigma _\text{B}) = \left[1 + \frac{1}{2}\left(k_{1}^2 \mu _1^2 + k_{2}^2\mu _2^2 + k_{3}^2\mu _3^2\right)^2 \sigma _\text{B}^2\right]^{-2} . \nonumber\\
\end{eqnarray*}$$It is worth noting that it is necessary for galaxy surveys to also include modelling of shot noise caused by discreteness effects in their bispectra analyses. However, for H i IM, where unresolved signal is integrated over, this should not be a limiting factor (Spinelli et al. 2020). We therefore do not consider any treatment of shot noise in our analysis, making the assumption that this should be very low in H i IM observations.
Citation: Lesgourgues 2011
Citation function: Uses