K_2(x),\end{aligned}$$ with $K_2(x)$ being the modified Bessel function of the second kind, is close to unity for $x<1$, and quickly approaches zero for $x>1$. It will serve as a regularisation of the integrals over $k$, suppressing contributions from lengthscales smaller than the size of individual microswimmers.

Performing the Fourier and Laplace transforms of Eq., we obtain after re-arranging $$\begin{aligned}
&\hat{h}({\bm k}, {\bm p}, s) = \frac{1}{\sigma({\bm k},{\bm p}, s)}
\Bigg[\hat{h}_0({\bm k}, {\bm p}) + \hat{\xi}({\bm k}, {\bm p},s) \nonumber \\
& \qquad + \frac{\lambda}{4\pi} I^{(0)}({\bm k},s)
+ \frac{15 \lambda}{4\pi} \Delta A(k\epsilon) \bigg\{ (\hat{\bm k}\cdot {\bm p}) I^{(1)}({\bm k},{\bm p}, s) \nonumber \\
& \qquad\qquad\qquad\qquad\qquad - (\hat{\bm k}\cdot {\bm p})^2 I^{(2)}({\bm k}, s)
\bigg\} 
\Bigg].
\label{hfullsolution}\end{aligned}$$ Here, $\hat{\xi}({\bm k}, {\bm p},s)$ is the Fourier-Laplace transform of the noise, $\sigma({\bm k},{\bm p}, s) = s + \lambda + i v_s ({\bm k}\cdot{\bm p})$, and we defined $$\begin{aligned}
I^{(0)}({\bm k},s) &= \int d{\bm p } \,\hat{h}({\bm k}, {\bm p}, s), \label{I0} \\
I^{(1)}({\bm k},{\bm p}, s) &= \int d{\bm p}'  (\hat{\bm k}\cdot {\bm p}')({\bm p}\cdot {\bm p}') \hat{h}({\bm k}, {\bm p}', s), \label{I1} \\
I^{(2)}({\bm k},s) &= \int d{\bm p }\, (\hat{\bm k}\cdot {\bm p})^2 \hat{h}({\bm k}, {\bm p}, s). \label{I2}\end{aligned}$$ In Eq., 