Soft synchronous gauge in the perturbative gravity

An attempt to directly use the synchronous gauge ($g_{0 \lambda} = - \delta_{0 \lambda}$) in perturbative gravity leads to a singularity at $p_0 = 0$ in the graviton propagator. This is similar to the singularity in the propagator for Yang-Mills fields $A^a_\lambda$ in the temporal gauge ($A^a_0 = 0$). There the singularity was softened, obtaining this gauge as the limit at $\varepsilon \to 0$ of the gauge $n^\lambda A^a_\lambda = 0$, $n^\lambda = (1, - \varepsilon (\partial^j \partial_j )^{- 1} \partial^k )$. Then the singularities at $p_0 = 0$ are replaced by negative powers of $p_0 \pm i \varepsilon$, and thus we bypass these poles in a certain way. Now consider a similar condition on $n^\lambda g_{\lambda \mu}$ in perturbative gravity, which becomes the synchronous gauge at $\varepsilon \to 0$. Unlike the Yang-Mills case, the contribution of the Faddeev-Popov ghosts to the effective action is nonzero, and we calculate it. In this calculation, an intermediate regularization is needed, and we assume the discrete structure of the theory at short distances for that. The effect of this contribution is to change the functional integral measure or, for example, to add non-pole terms to the propagator. This contribution vanishes at $\varepsilon \to 0$. Thus, we effectively have the synchronous gauge with the resolved singularities at $p_0 = 0$, where only the physical components $g_{j k}$ are active and there is no need to calculate the ghost contribution.