the coordinate times of the detection and emission and is determined by the integral $$\begin{aligned}
\label{eq:tof_T1}
\Delta \tau_{ED}(t) = {\rm t}(\lambda_{D}) - {\rm t}(\lambda_{E}) =
\int\limits_{\gamma_{ED}}\sqrt{{\rm g}_{ij}\left[\rm{t}(\lambda),{\rm \bf x(\lambda)}\right]\frac{d{\rm x}^{i}}{d\lambda}\frac{d{\rm x}^{j}}{d\lambda}}\,d\lambda,\end{aligned}$$ where $[{\rm t}(\lambda),\gamma_{ED}(\lambda)]=[{\rm t}(\lambda),{\rm \bf x}(\lambda)]$ parametrizes the null geodesic of the light ray from the emitter to the detector with the arrival time at the detector ${\rm t}(\lambda_{D})=t$. In the synchronous coordinates trajectories of photons in the perturbed background (\[eq:b:synch\]) differ in general from straight lines but as was shown by Finn [@Finn09] (see also [@AB2012],[@TA10]) the integral on the right hand side of Eq.(\[eq:tof\_T1\]) is unchanged when computed along the [*unperturbed*]{} trajectory $$\begin{aligned}
\label{eq:unpert_tr}
{\rm t}_{0}(\lambda) = \lambda, \qquad {\rm \bf x}_{0}(\lambda) = {\bf x}_{D} - {\bf n}_{ED}(t - \lambda).\end{aligned}$$ This gives to the leading order $$\begin{aligned}
\label{eq:tof_T2a}
\Delta \tau_{ED}(t) & = &
L + \frac{{\bf n}_{ED}\otimes{\bf n}_{ED}}{2}:
\int\limits_{t-L}^{t}{\rm \bf h}\left[\rm{t}_{0}(\lambda),{\rm \bf x}_{0}(\lambda)\right]\,d\lambda \\
\label{eq:tof_T2b}
& = &
L + \frac{{\bf n}_{ED}\otimes{\bf n}_{ED}:{{\boldsymbol \epsilon}}^{p}}{2}\int\limits_{t-L}^{t}h_{p}\left[v(\lambda)\right]\,d\lambda, \nonumber\end{aligned}$$ where in the last line of Eq.(\[eq:tof\_T2a\]) we have specified the $p$–polarized plane wave having the (spatial) polarization tensor ${{\boldsymbol \epsilon}}^{p}$ propagating along the vector ${{\boldsymbol \Omega}}$ with the velocity ${\rm v}$, ${\rm \bf h}(t,{\bf x})={{\boldsymbol \epsilon}}^{p}\,h_{p}(t-{{\boldsymbol \Omega}}\cdot{\bf x}/{\rm v})$, and we 