This, finally, leads to a direct relationship between the field $h$, which encodes the statistical properties of fluctuations in the suspension, and the combined correlation function $$\begin{aligned}
&C(R,T) = \lim_{t\rightarrow\infty} \frac{1}{V}\int d{\bm x} \int d{\bm z}'d{\bm z}''  \nonumber \\
&  \times u_d^\alpha({\bm x}; {\bm z}') u_d^\alpha({\bm x}+{\bm R}; {\bm z}'')  \langle h({\bm z}', t) h({\bm z}'', t+T) \rangle.
\label{CRTtmp}\end{aligned}$$

Dynamics of the auxiliary field $h$ {#subsection:h}
-----------------------------------

Here, we explicitly find the solution to Eq. together with Eqs. and . Since Eq. is linear in $h$, we introduce the Fourier $$\begin{aligned}
h({\bm z}, t) = \frac{1}{(2\pi)^3} \int d{\bm k} e^{i {\bm k}\cdot{\bm x}} \hat{h}({\bm k}, {\bm p}, t),
\label{FT}\end{aligned}$$ and the Laplace transforms $$\begin{aligned}
\hat{h}({\bm k}, {\bm p}, s) = \int_0^\infty dt e^{-s t} \hat{h}({\bm k}, {\bm p}, t).\end{aligned}$$ We will also require the Fourier transform of the regularised dipolar field, Eq., which is given by $$\begin{aligned}
& u_d^\nu({\bm x}; {\bm z}') = \frac{-i \kappa}{(2\pi)^3}  \int d{\bm k} \, e^{i {\bm k}\cdot ({\bm x}-{\bm x}')}  \nonumber \\
& \qquad\qquad\qquad \times  \frac{A(k \epsilon)}{k} (\hat{\bm k} \cdot {\bm p}') \left( \delta^{\nu\delta} - \hat{k}^\nu \hat{k}^\delta \right) p'^{\delta},
\label{udfourier}\end{aligned}$$ where $\hat{\bm k}={\bm k}/k$, and $k=|{\bm k}|$. The function $A$, defined as $$\begin{aligned}
A(x)=\frac{1}{2} x^2 