Gravitational radiation in higher order non-local gravity

In this paper we examine gravitational radiation in higher order non-local gravity described by the non-local gravitational Lagrangian density $\mathcal{L}_{g}=R+\sum_{h=1}^{n}a_{h}R\Box^{-h}R$. This non-local theory of gravitation always exhibits the tensor transverse gravitational radiation for $k_{1}^{2}=0$, corresponding to the angular frequency $\omega_{1}$, composed of two standard $(+)$ and $(\times)$ polarization modes, massless and of helicity 2. Furthermore, it shows, under suitable constraint and $n\geq 2$, an additional massive transverse scalar gravitational radiation with helicity 0. It is composed of $n-1$ modes associated to $n-1$ angular frequencies $\omega_{2},\ldots,\omega_{n}$, each of which of breathing polarization $(b)$ to lowest order in $\gamma$, a parameter that takes into account the difference in speed between the slightly massive wave and the massless one. Thanks to NP formalism, we find that the $E(2)$ class of non-local gravitational waves is $N_{3}$, according Petrov classification, where the presence or absence of all modes are observer independent. Also, the scalar radiation is forbidden for $n=1$ and $n=2$ cases, when some conditions are satisfied. Finally, in $\Box^{-1}$ gravity where $n=1$, a possible degenerate case with a continuous infinity of transverse massive scalar breathing modes appears under a particular constraint, which reproduces in two-dimensional spacetime the Polyakov effective action.