\dots~,$$ where the $\dots $ denote terms with negative powers of $t$, related to space-time curvature, which are subleading in the limit $t \to \infty$.

For the spatial part of (\[stress\]) we consider the OPE $a(t(z))^2 (\partial X^i(z))^2 {\cal D}^{(q)}(w)$ as $z \to w$. Again, for convenience we shall do the time and space contractions separately: $$\begin{aligned}
&~& t^{2p}(z)\cdot {\cal D}^{(q)}(w) = \int d\omega {\cal {\tilde D}}^{(q)}(\omega) 
t^{2p}(z)\cdot
e^{i\omega (t(w)-t_0)} = \nonumber \\
&~& \int _0^{\infty} \frac{d\nu}{\Gamma (-2p)} \nu^{-1-2p}
\int d\omega {\cal {\tilde D}}^{(q)}(\omega) e^{-\nu t(z)}\cdot
e^{i\omega (t(w)-t_0)}  \end{aligned}$$ Using the OPE $e^{-\nu t(z)} \cdot e^{-i\omega (t(w)-t_0)} \sim 
|z-w|^{i\nu\omega}e^{-\nu t(z) - i\omega (t(z)-t_0) + {\cal O}(z-w)}$ one obtains (as $z \sim w$): $$\begin{aligned}
&~& t(z)^{2p}\cdot{\cal D}^{(q)}(w)=
\int _0^\infty \frac{d\nu}{\Gamma (-2p)}\nu^{-1-2p}e^{-\nu t(z)}
{\cal D}^{(q)} (t-t_0 -\nu {\rm ln}|z-w|) = \nonumber \\
&~& t^{2p}~\int _0^\infty \frac{d\nu}{\Gamma (-2p)}\nu^{-1-2p}e^{-\nu}
{\cal D}^{(q)} (t-t_0 -\frac{\nu}{t} {\rm ln}|z-w|) = \nonumber \\ 
&~& t^{2p}\left[{\cal D}^{(q)}(t-t_0) -\frac{1}{t}{\rm ln}|z-w|
\frac{\Gamma (1-2p)}{\Gamma (-2p)}\frac{d}{dt}{\cal D}^{(q)}(t-t_0)
+ {\cal O}(t-t_0)^{q-2}\right]\end{aligned}$$ We now observe that $\tfrac{d{\cal D}^{(q)}}{dt}=q{\cal D}^{(q-1)} - 
\epsilon~{\cal D}^{(q)} $, where both terms have vacuum expectation values of the same order in $\epsilon$, as we shall see below, and hence both should be kept in our perturbative expansion.

Expanding the various terms around 