Nonpropagation of scalar in the deformed Hov{r}ava-Lifshitz gravity

We study the propagation of a scalar, the trace of $h_{ij}$ in the deformed Ho\v{r}ava-Lifshitz gravity with coupling constant $\lambda$. It turns out that this scalar is not a propagating mode in the Minkowski spacetime background. In this work, we do not choose a gauge-fixing to identify the physical degrees of freedom and instead, make it possible by substituting the constraints into the quadratic Lagrangian.