 together with the HR2 measurements ({\it data points}) comparing CLPT ({\it thin solid}), TCLPT with $R=1{\,\mathrm{Mpc}/h}$ ({\it thick dashed}) and the Lagrangian scale $R=R_{\rm L}(M)$ ({\it thick dotted}) for 3 selected halo masses (low, intermediate, high). {\it upper panel} The mean pairwise velocity $v_{12}$ normalized by linear theory $v_{12}^{\text{lin}}$. {\it lower panel} The parallel part of the pairwise velocity dispersion  $\sigma_{\|}^2$.}
\label{fig:velstatcomp}
\end{figure}

\begin{figure}[t!]
\includegraphics[width=0.46\textwidth]{sigmashift}
\caption{The residual or `shift' between the HR2 $\hat \sigma^2_{\perp}$ and several theoretical predictions for $\sigma_{\perp}^2$. The curves for $ \sigma^2_{\rm shift, ||}$ and $ \sigma^2_{\rm shift, \perp}$ cannot be distinguished by eye.}
\label{fig:sigmashift}
\end{figure}

\begin{figure}[h!]
\includegraphics[width=0.47\textwidth]{z0quadrupolecompCLPTshift}
\caption{The effect of shifting CLPT predictions for $\v{\sigma}_{12}^2$ to agree with HR2 at $149{\,\mathrm{Mpc}/h}$ on the quadrupole of the redshift space halo correlation function predicted by the GSM \eqref{GSM}. All models use the same function $\xi_X$ and $v_{12}$ calculated with TCLPT at $R=1 {\,\mathrm{Mpc}/h}$, while $\v{\sigma}_{12}^2$ is either calculated with CLPT ($R=0$) or TCLPT with $R=R_{\rm L}(M)$ and either used directly (`$\v{\sigma}_{12}^2$') or shifted (`$\bar{\v{\sigma}}_{12}^2$') to match HR2 at larger scales.}
\label{fig:z0quadrupoleShift}
\end{figure}

In Fig.\,\ref{fig:velstatcomp}  the results for $\bar \sigma^2_\perp$ and $\bar \sigma^2_{||}$ and the data points are normalised by the mass-independent result of linear theory, see \cite{RW11},
\begin{align}
\label{sig12lin}
\notag  \sigma^2_{||,\rm lin}&=  2\sH^2 f^2 \left(R_{\rm NL}^2 -\frac{1}{2\pi^2} \int_0^\infty d k\, P_{\rm lin}(k) \left( j_0(k r)- \frac{2 j_1(k r)}{k r}\right) \right)\\
 \sigma^2_{\perp,\rm lin}& =  2\sH^2 f^2 \left(R_{\rm NL}^2 - \frac{1}{2\pi^2} \int_0^\infty d k\, P_{\rm lin}(k) \frac{ j_1(k r)}{k r}\right) \\
\notag R_{\rm NL}^2&= \frac{1}{6 \pi^2} \int_0^\infty dk\, P_{\rm lin}(k) \,.
\end{align}
 One can clearly see that the shape of $\v{\sigma}_{12}^2$, namely $\bar \sigma^2_\perp$ and $\bar \sigma^2_{||}$, matches best the $N$-body data for TCLPT with a Lagrangian smoothing scale $R_{\rm L}(M)$. 
 Thus we observe that within TCLPT both the shape and amplitude of $\v{\sigma}_{12}^2$ are optimised for $R=R_{\rm L}(M)$ compared to TCLPT with $R=1{\,\mathrm{Mpc}/h}$ or CLPT. 
 Hence, we conclude that the optimal smoothing scale is $R=1{\,\mathrm{Mpc}/h}$ for the mean pairwise velocity (just as for the real space correlation function) but $R_{\rm L}(M)$ for the pairwise velocity dispersion. 

Since the $\bar{\v{\sigma}}_{12}^2$ of CLPT is better than linear perturbation theory, at least for $r>30 {\,\mathrm{Mpc}/h}$, but $\sigma_{\rm shift}^2$ is very large, see Fig.\,\ref{fig:sigmashift}, one might be tempted to simply use $\bar{\v{\sigma}}^2_{12}$ in the GSM, as has been done in \cite{RW11,WRW14}. Although at linear order, any constant, isotropic velocity dispersion does not alter the redshift space correlation function as explained in \cite{RW11} it does affect the GSM.
 In Fig.\,\ref{fig:z0quadrupoleShift} we investigate the impact of using different versions of the pairwise velocity dispersion, while keeping all other ingredients of the GSM fixed, in particular $\xi_X$ and $v_{12}$ have been calculated with TCLPT and $R=1{\,\mathrm{Mpc}/h}$. 
We observe that for $R=R_{\rm L}(M)$ switching between the  shifted $\bar{\v{\sigma}}_{12}^2$ and the original $\v{\sigma}_{12}^2$ does have a smaller impact than switching between shifted and original prediction of CLPT ($R=0$) as is expected from the fact that $\bar{\v{\sigma}}_{12}^2$ and $\v{\sigma}_{12}^2$ are closer to each other in the former case and very different in the latter.
In to addition to not being well justified, using  the quantity $\bar{\v{\sigma}}_{12}^2$ can be dangerous since it can become easily negative for relevant scales and therefore requires some further ad-hoc fudging when used in the GSM.
In the rest of this paper and in particular for all results presented in Sec.\,\ref{sec:results} we therefore use the un-shifted $\v{\sigma}_{12}^2$.





 






\subsubsection{Comparison of TCLPT and TZA}

In \cite{W14} the `Zel'dovich Streaming Model' was proposed, which is identical to combining the GSM and CLPT from \cite{WRW14}, with the exception that only first order (or ZA) displacements $\v{\Psi}_{\rm Z}$ are considered, instead of $\v{\Psi}_{\rm Z,2,3}$ as done in CLPT. 
In \cite{W14} it was found that for the redshift space quadrupole the ZA exhibits a better agreement with the $N$-body data than CLPT when galaxy-sized halos are considered.
Note however, that in \cite{W14} the linear input power spectrum has been additionally smoothed on a scale $R=1{\,\mathrm{Mpc}/h}$ compared to \cite{WRW14}, such that in our nomenclature that model considered in \cite{W14} is TZA with $R=1{\,\mathrm{Mpc}/h}$.

We therefore investigate in more detail the relative performance of TZA and TCLPT.
From the $\chi^2$ values in Table \ref{biasfittab} it follows that TZA/ZA is slightly worse than CLPT/TCLPT for the real space halo correlation function on large scales. 
Now we demonstrate that TZA is also worse than TCLPT for the correlation function on small scales and the mean pairwise velocity.


\begin{figure}[t!]
\includegraphics[width=0.48\textwidth]{ZARealRatioLinSmallrSmallM}
\caption{Comparison of the real space correlation function between TCLPT ({\it thick dashed}) and TZA ({\it thin dotted}) for small scales and  the four lowest halo masses, both with $R=1 {\,\mathrm{Mpc}/h}$. The halo correlation function $\xi_X$ is normalized by linear theory $\xi_{X}^{\text{lin}}$.}
\label{fig:xiRealZeldo}
\end{figure}



In Figs.\,\ref{fig:xiRealZeldo} and \ref{fig:velstatZeldo} we compare the prediction computed from TCLPT with that from TZA, both with $R=1{\,\mathrm{Mpc}/h}$ for $\xi_X$ and $v_{12}$ respectively.\footnote{Note that $v_{12}^{\rm lin}$ does not coincide with the $v_{12}$ of ZA because the mean pairwise velocity from \eqref{GSMparam} is weighted with the density and not linearised in ZA. The same is true for $\v{\sigma}_{12}^2$.} 
We do not show the pairwise velocity dispersion $\v{\sigma}_{12}^2$ because it resembles the result from linear theory on large scales and hence gives a reasonable amplitude but a less accurate scale dependence than TCLPT.
We also studied the redshift space correlation function obtained when feeding those ingredients into the GSM. As expected TCLPT always performs better than TZA.
Thus the Zel'dovich approximation is clearly improved by the inclusion