Weak field limit and gravitational waves in higher-order gravity

We derive the weak field limit for a gravitational Lagrangian density $L_{g}=(R+a_{0}R^{2}+\sum_{k=1}^{p} a_{k}R\Box^{k}R)\sqrt{-g}$ where higher-order derivative terms in the Ricci scalar $R$ are taken into account. The interest for this kind of effective theories comes out from the consideration of the infrared and ultraviolet behaviors of gravitational field and, in general, from the formulation of quantum field theory in curved spacetimes. Here, we obtain solutions in weak field regime both in vacuum and in the presence of matter and derive gravitational waves considering the contribution of $R\Box^{k}R$ terms. By using a suitable set of coefficients $a_{k}$, it is possible to find up to $(p+2)$ normal modes of oscillation with six polarization states with helicity 0 or 2. Here $p$ is the higher order term in the $\Box$ operator appearing in the gravitational Lagrangian. More specifically: the mode $\omega_{1}$, with $k^{2}=0$, has transverse polarizations $\epsilon_{\mu\nu}^{\left(+\right)}$ and $\epsilon_{\mu\nu}^{\left(\times\right)}$ with helicity 2; the $(p+1)$ modes $\omega_{m}$, with $k^{2}\neq0$, have transverse polarizations $\epsilon_{\mu\nu}^{\left(1\right)}$ and non-transverse ones $\epsilon_{\mu\nu}^{\left(\text{TT}\right)}$, $\epsilon_{\mu\nu}^{\left(\text{TS}\right)}$, $\epsilon_{\mu\nu}^{\left(L\right)}$ with helicity 0.