elements between [*on-shell*]{} physical states. In the context of $\sigma$-models, we are working with, the physical state condition implies the constraint of the vanishing of the world-sheet stress-energy tensor $2T=-(\partial t)^2 + a(t)^2(\partial X^i)^2 =0$. This condition allows $(\partial X^i)^2 $ to be expressed in terms of $(\partial t)^2$, which is consistent even at a correlation function level in the case of very target times $t \gg t_0$, since in that case, the correlator $\langle X^i t \rangle$ is subleading, as mentioned previously. Implementing this, it can be then seen that the OPE between the spatial parts of $T$ and ${\cal D}^{(q)}$ is: $$\begin{aligned}
&~& a^2(t)(\partial X^i)^2 \cdot {\cal D}^{(q)} = 
t^{-2p}(\partial t)^2 \cdot
\{ {\cal D}^{(2p+q)}(t-t_0)+ 
\left(2p~t_0 - 2p~\epsilon~{\rm ln}|z-w|\right)~{\cal D}^{(2p+q-1)}+ 
\nonumber \\
&~& + \left(\frac{t_0^{2}}{2}2p(2p-1)+[2pq + 
(2p-4p^2)\epsilon~t_0]~{\rm ln}|z-w|\right)
{\cal D}^{(2p+q-2)}(t-t_0)+ \nonumber \\
&~& +{\cal O}([t-t_0]^{2p+q-3})\}.\end{aligned}$$ Performing the appropriate contractions, and adding to this result the OPE of the temporal part of $T$ with ${\cal D}^{(q)}$, i.e. the quantity $-\frac{\epsilon^2}{2}{\cal D}^{(q)} + q\epsilon {\cal D}^{(q-1)} 
+ \frac{1}{2}q(q-1){\cal D}^{(q-2)}$, we obtain: $$\begin{aligned}
&~& T\cdot {\cal D}^{(q)}|_{\rm on-shell}=\left(-2p\epsilon
-pt_0~\epsilon^2 + p\epsilon^2~{\rm ln}(a/L)\right)~{\cal D}^{(q-1)}
+  \nonumber \\
&~& + \{t_0^2\epsilon^2~2p(2p+1) - 3\epsilon^2p(2p+q){\rm ln}(a/L)
- 2\epsilon^3(p+p^2)t_0{\rm ln}(a/L) - 2p^2\epsilon^4{\rm ln}^2(a/L)
+ \nonumber \\
