# Transition of EMRIs through resonance: corrections to higher order in the on-resonance flux modification

## TL;DR

This paper extends models of EMRI resonances by calculating higher-order corrections to the on-resonance flux, providing a more precise understanding of the gravitational self-force effects during resonance in extreme mass ratio inspirals.

## Contribution

It introduces higher-order corrections to existing resonance models of EMRIs and validates the instantaneous frequency approach for analyzing resonance effects.

## Key findings

- Extended the Gair, Bender, and Yunes model to third order in flux modification.
- Developed a perturbative solution algorithm for non-linear resonance equations.
- Discussed the mathematical properties and scope of the perturbative approach.

## Abstract

Extreme mass ratio in-spirals (EMRIs) are candidate events for gravitational wave detection in the millihertz range (by detectors like LISA and eLISA). These events involve a stellar-mass black hole, or a similar compact object, descending in the gravitational field of a supermassive black hole, eventually merging with it. Properties of the in-spiralling trajectory away from resonance are well known and have been studied extensively, however little is known about the behaviour of these binary systems at resonance, when the radial and lateral frequencies of the orbit become commensurate. We describe the two existing models, the instantaneous frequency approach used by Gair, Bender, and Yunes, and the standard two timescales approach implemented by Flanagan and Hinderer. In both cases, the exact treatment depends on the modelling of the gravitational self-force, which is currently not available. We extend the results in Gair, Bender and Yunes to higher order in the on-resonance flux modification, and argue that the instantaneous frequency approach is also a valid treatment of the resonance problem. The non-linear differential equations which arise in treating resonances are interesting from a mathematical view point. We present our algorithm for perturbative solutions and the results to third order in the infinitesimal parameter, and discuss the scope of this approach.