\frac{\left( {\bm p}_j\cdot {\bm x}'\right)^2 {\bm x}' + \epsilon^2 \left( {\bm p}_j\cdot {\bm x}'\right) {\bm p}_j }{\left( x'^2 + \epsilon^2 \right)^{5/2}} \right. \nonumber \\
\left. -\frac{{\bm x}'}{\left( x'^2 + \epsilon^2 \right)^{3/2}} \right]
\label{realspacedipole}\end{aligned}$$ is the velocity field generated at ${\bm x}_{i}$ by a hydrodynamic dipole located at ${\bm x}_{j}$ with the orientation ${\bm p}_{j}$. Here, $\kappa = F l/\mu$ is the reduced dipolar strength, where $F$ is the magnitude of the forces applied to the fluid, $l$ is the dipolar length, and $\mu$ is the viscosity of the fluid; ${\bm x}' = {\bm x}_i -{\bm x}_j$, and $x'$ denotes the length of ${\bm x}'$. The dipole consists of two regularised Stokeslets, that were introduced by Cortez *et al.* [@Cortez2005], with $\epsilon$ being the regularisation length of the order of swimmer size. For pushers, $\kappa > 0$.

The main goal of our work is to calculate spatial and temporal correlations of the fluid velocity in microswimmer suspensions described by the model above. Both quantities can be succinctly expressed through a combined correlation function $$\begin{aligned}
&C(R,T) \nonumber \\
&\qquad = \lim_{t\rightarrow\infty} \frac{1}{V}\int d{\bm x} \,\overline{U^{\alpha}\left( {\bm x}, t \right) U^{\alpha}\left( {\bm x} + {\bm R}, t+T \right)},
\label{CRTgeneral}\end{aligned}$$ where $U^{\alpha}\left( {\bm x}, 