# Impact of multiple modes on the black-hole superradiant instability

## TL;DR

This paper investigates how multiple modes influence the superradiant instability of black holes caused by ultralight bosonic fields, revealing complex dynamics and conditions that affect the instability's development and observational constraints.

## Contribution

It introduces a multimode analysis of black-hole superradiance, showing the impact of mode interactions and seed energy on the instability's evolution and end-state.

## Key findings

- Multimode interactions can suppress or enhance superradiant instability.
- Seed energy levels significantly influence the instability's growth.
- Nonsuperradiant modes have negligible effects at very low seed energies.

## Abstract

Ultralight bosonic fields in the mass range $\sim (10^{-20}-10^{-11})\,{\rm eV}$ can trigger a superradiant instability that extracts energy and angular momentum from an astrophysical black hole with mass $M\sim(5,10^{10})M_\odot$, forming a nonspherical, rotating condensate around it. So far, most studies of the evolution and end-state of the instability have been limited to initial data containing only the fastest growing superradiant mode. By studying the evolution of multimode data in a quasi-adiabatic approximation, we show that the dynamics is much richer and depend strongly on the energy of the seed, on the relative amplitude between modes, and on the gravitational coupling. If the seed energy is a few percent of the black-hole mass, a black hole surrounded by a mixture of superradiant and nonsuperradiant modes with comparable amplitudes might not undergo a superradiant unstable phase, depending on the value of the boson mass. If the seed energy is smaller, as in the case of an instability triggered by quantum fluctuations, the effect of nonsuperradiant modes is negligible. We discuss the implications of these findings for current constraints on ultralight fields with electromagnetic and gravitational-wave observations.