Considerations on gravitational waves in higher-order local and non-local gravity

The detection of gravitational wave modes and polarizations could constitute an extremely important signature to discriminate among different theories of gravity. According to this statement, it is possible to prove that higher-order non-local gravity has formally the same gravitational spectrum of higher-order local gravity. In particular, we consider the cases of $f \left( R, \Box R, \Box^2 R, \cdots, \Box^n R \right) = R + \sum_{i=1}^n \alpha_i R \Box^i R$ gravity, linear with respect to both $R$ and $\Box^i R$ and $ f \left( R, \Box R \right) = R + \alpha \left(\Box R\right)^2 $ gravity, quadratic with respect to $\Box R$, where it is demonstrated the graviton amplitude changes if compared with General Relativity. We also obtain the gravitational spectrum of higher-order non-local gravity $ f \left( R, \Box^{-1} R, \Box^{-2} R, \cdots, \Box^{-n} R \right) = R + \sum_{i=1}^n \alpha_i R \Box^{-i} R$. In this case, we have three state of polarization and $n+3$ oscillation modes. More in detail, it is possible to derive two transverse tensor $(+)$ and $(\times)$ standard polarization modes of frequency $\omega_{1}$, massless and with 2-helicity; $n+1$ further scalar modes of frequency $\omega_{2},\dots,\omega_{n+2}$, massive and with 0-helicity, each of which has the same mixed polarization, partly longitudinal and partly transverse.