• Bardeen variables
• gauge-invariant
• general relativity
• gravitational waves
• higher-derivative gravity
• inspiral
• linearized gravity
• modified gravity
• quadratic gravity

We derived the 4th order Higher-Derivative (HD) modification to General Relativity (GR), treating GR as a low-energy approximation of a more fundamental, unknown theory. In the weak-field approximation, we linearized HD gravity, and using the 3+1 Helmholtz-York decomposition, we obtained Bardeen’s gauge-invariant variables. We found the equations of motion for these variables, which included 4th-order differential operators and coupled scalar potentials, and we solved them. Adopting a prescription chosen to avoid tachyons, we obtained conditions on the signs of the coupling constants, which agree with those found in the literature in the context of QFT. We observed that there are 6 massive propagating degrees of freedom and 2 massless ones, as also concluded by several authors using other methods. We noted that this solution is not easily tractable because the traditional multipole expansion cannot be applied, and it is unclear how to recover the GR limit. We calculated the stress-energy pseudo-tensor as a function of the Bardeen potentials in the vacuum region of propagation. Then, assuming small coupling constants, we perturbatively solved the equations for the Bardeen variables. By applying the simplifying vacuum relations, we concluded that the gravitational wave luminosity of HD gravity is the same as in GR, without any modification arising from propagation. Therefore, the only possible effect might be hidden in the HD modifications to the orbits of the source.