= 0,
\label{EqF1MF}\end{aligned}$$ that has been extensively studied before [@Saintillan2008; @Saintillan2008a; @Subramanian2009; @Hohenegger2010; @Koch2011; @Saintillan2013; @Krishnamurthy2015]. One of the solutions of this equation is given by a constant, which is fixed to $F_1^{\text{MF}}({\bm z}, t)=1/(4\pi V)$ by the normalisation condition Eq.. This solution, which is valid at any number density, corresponds to a homogeneous and isotropic suspension of microswimmers. For pushers $(\kappa>0)$, this state loses its stability [@Saintillan2008; @Saintillan2008a; @Subramanian2009; @Hohenegger2010; @Stenhammar2017] at the critical number density of microswimmers $n_{crit} = 5\lambda/(B\kappa)$, while for pullers $(\kappa<0)$, the homogeneous and isotropic state is always linearly stable within the mean-field approximation.

The homogeneous and isotropic mean-field solution implies that $N F_1^{\text{MF}}\sim n \sim O(1)$ is finite in the thermodynamic limit. This, in turn, implies that, to leading order, $G\sim O(N^{-2})$, $H\sim O(N^{-3})$, etc. A more comprehensive discussion of this statement, together with the required rescaling of the correlation functions, system parameters, and time is given elsewhere [@Stenhammar2017].

Building upon these results, here we *assume* that upon approaching the thermodynamic limit, $F_1$ is well-approximated by $F_1^{\text{MF}}$, since the r.h.s. of Eq. is $O(1/N)$ compared its l.h.s. In the homogeneous and isotropic state, the mean-field velocity vanishes $\mathcal{U}^\alpha_{\text{MF}}({\bm x}) = 0$, since the integral 