• BH ringdown
• quasinormal modes
• Regge-Wheeler equation
• Schwarzschild black hole

Gravitational perturbation theory on a black hole (BH) background predicts that the solution to the linearised Einstein's equations can be described via quasinormal modes (QNMs): solutions oscillating with characteristic frequencies. The modifier quasi- signals that these modes are damped, since energy is radiated to infinity by gravitational waves (GWs).
The leading order approximation is believed to break down in several interesting scenarios. The linearised theory however does not allow its limits of validity to be determined. For instance, after a BH binary merger, the starting time of the linear ringdown cannot be predicted. It has been claimed that QNMs succeed in fitting the GW signal up until its peak---which, if true, is very surprising.
We contribute to the debate by considering second-order perturbations on a Schwarzschild BH. Their dynamics follows the same equation as the first order, though with a source quadratic in the first order, which we characterise. We note that particular coincidences in QNM frequencies may lead to (quasi)secular terms; hence we study a method, based on the renormalisation group, to eliminate them. We then suggest that the same method may be extended to improve perturbation theory even in the absence of secular terms, although further research is needed.
Our work may be of interest in GW modelling, BH theory and in other contexts where gravitational perturbation theory is used, such as cosmology.