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 L-g = R + Sigma(n)(h=1) a(h)R square R-h. 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 (x) polarization modes, massless and of helicity 2. Furthermore, it shows, under suitable constraint and n >= 2, an additional massive transverse scalar gravitational radiation with helicity 0. It is composed of n-1 modes associated ton-1 angular frequencies omega(2), ..., omega(n), each of which is 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 to 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 square(-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.