Little is known about dark matter, despite the numerous searches for its constituents. Fortunately, everything falls in the same way, so possible imprints of dark matter can be found in gravitational fields. In particular, if ultralight bosons exist in nature, they would make spinning black holes unstable. How does such instability evolve in realistic scenarios? And what can it teach us about the existence of dark matter?

In our recent CQG paper, we take the first step to address these questions by studying how a light scalar field grows near a spinning (Kerr) black hole in a process known as “superradiant instability”.

Superradiance is a fascinating phenomenon involving dissipative systems, whereby energy is transferred from one medium to another, typically stimulated by wave scattering. Classically, the event horizon of a black hole is a perfect absorber, so it’s no surprise that superradiance can occur in black-hole geometries, as understood already in the 1970s by Zeldovich, Starobinsky, and others. Thus, a low-frequency wave scattered off a Kerr black hole can be amplified at the expense of the black-hole mass and angular momentum. Because a black hole is pure geometry, this process effectively extracts energy from the vacuum, even at the classical level!

If such a wave is trapped near the black hole, it can sustain and amplify itself in a process akin to a laser. In fact, *any* spinning black hole is unstable against small fluctuations of a massive particle with integer spin! Insiders like to call this process a “black-hole bomb”. So, how come we observe gigantic black holes essentially in all galactic centers? Shouldn’t they “explode” through this superradiant mechanism? This is not the case because the timescale of the superradiant instability is typically very large. For example, the instability timescale for a stellar-mass black hole associated with the Higgs boson is roughly 10^{22} years, a trillion times longer than the age of the Universe!

On the other hand some dark-matter candidates, like axion-like particles and so-called hidden photons, can be light enough as to make astrophysical black holes unstable on timescales shorter than a second. If such fast instability occurs in nature, it has to leave some imprint on the dynamics of massive black holes.

The picture above has emerged as the outcome of a 40-year long effort to understand black-hole superradiant instabilities. Despite a considerable amount of work on this subject, our understanding was limited to a linearized analysis, in which the fluctuations are assumed to be small and their backreaction on the metric neglected. This limitation is again related to timescales: even in the most optimistic scenarios the timescales (in units of the black-hole dynamical time) are just too long to follow the development of the instability in full General Relativity, even if one resorts to supercomputing facilities. Thus, the most natural question, namely,*what’s the final state of the superradiant instability?*is still waiting for an answer.

Last year, we were writing a book on superradiance and got stuck with this problem. How could we possibly say anything useful on superradiant instabilities if we didn’t even know how the latter evolve? It would be like buying the finest furnishings for a house with shaky foundations: it could be a complete waste of time. Therefore, we stopped writing this for a while and decided we wouldn’t continue until this issue was clarified. This was the reasoning for writing our CQG paper.

As it turned out, the key to solve the issue was again lurking in the timescales involved in the process. This is because it is so challenging to simulate the superradiant evolution on a computer. The system is suitable for a *quasi-adiabatic approximation*: over the dynamical timescale of the black hole, the scalar field can be considered almost stationary and its backreaction on the geometry can be neglected as long as the scalar energy is small compared to the black-hole mass. The quasi-adiabatic approximation is a standard tool in physics and, in fact, it is elevated to the status of a theorem (the adiabatic theorem) in the context of quantum mechanics. By treating the evolution quasi-adiabatically, not only could we follow the development of the instability, but we also managed to include several important effects such as the emission of gravitational waves and gas accretion.

Our results put the intuition borrowed from the linearized analysis on a firmer basis: as a result of the instability, a non-spherical bosonic condensate grows around a spinning black hole extracting energy and angular momentum, until superradiance stops and the “cloud” is slowly reabsorbed by the black hole and dissipated through gravitational waves at infinity. Therefore, a neat prediction of this scenario is that massive black holes should have a maximum spin lower than the Kerr bound and can be endowed with large bosonic condensates. Because such condensates can survive for cosmological times, they would be practically indistinguishable from a full-fledged black-hole hair. Thus, superradiant instabilities circumvent the celebrated black-hole *no-hair theorem* in a subtle way: namely, by making a black hole slightly non-stationary but only on extremely long timescales. However, our results show that the energy-density of this hair is typically tiny and can affect the black-hole geometry only marginally.

The gravitational waves emitted by these *de facto* hairy black holes leave a peculiar fingerprint, which is potentially detectable with upcoming interferometers. Furthermore, some striking predictions of the superradiant instability can already be tested: because highly-spinning black holes are unstable, we should only observe black holes that rotate sufficiently slow. Technically, this accounts for depleted regions in the black-hole mass-spin diagram as shown in the Figure (insiders refer to these regions as “holes” in the Regge plane). However, because astrophysicists do observe fast-spinning black holes, such measurements can be turned around to constrain the mass of light exotic particles.

This was done in the past to put stringent theoretical bounds on the mass of axions, hidden photons and massive gravitons, in a region of the parameter space which is practically inaccessible to experiments. Our paper shows that those constraints (derived within a linearized analysis) are solid and prone to exciting developments, such as the inclusion of bosonic self-interactions, generation of precise gravitational-wave templates, and a more systematic analysis of other dark-matter candidates.

*Read the full article in Classical and Quantum Gravity:
Black holes as particle detectors: evolution of superradiant instabilities
Paolo Pani, Richard Brito and Victor Cardoso 2015 Class. Quantum Grav. *

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