By Xinlu Huang and Masha Baryakhtar

**Why are there no "gravitational atoms"? **

Our world is made up of atoms: i.e. nuclei and electrons held together by electromagnetism. At the same time, though not all particles have electric charge, they all have a gravitational “charge” (i.e. their mass). And gravity is universal and attractive. So it is only natural to ask: why don’t we observe any "atoms" bound by gravity?

The answer is that gravity is weak: a small magnet can lift a nail against the gravitational pull of the entire earth. The weakness of gravity means that a gravitational “hydrogen atom” would have a radius larger than the size of the observable universe. Thus to make a gravitational atom, we need to look to places where gravity is strong—like around black holes!

Imagine a very light particle of mass μ with Compton wavelength λ = h / (μ c ) comparable to the size of the black hole event horizon (or just "horizon" for short). The particle can form a bound state with the black hole: in essence, this is a gravitational “atom” with a black hole “nucleus” and the light orbiting particle as the “electron.”

*Above: a schematic showing a "gravitational atom": a spinning black hole nucleus surrounded by a purple axion probability cloud.*

Comparing the Coulomb force law F = e^{2} / 4 π r^{2 } = α / r^{2 }, where α is the normal electromagnetic fine-structure constant, to Newton’s law of gravitation: F = G_{N}m_{1} m_{2} / r^{2 }, we see that by simply defining the gravitational “fine-structure constant” as α_{grav} = G_{N} m_{1} m_{2, }we can map everything we know about the hydrogen atom to the gravitational atom. (There are corrections to Newton’s law close to a black hole, but these are small in our approximations). For example, the gravitational atom has quantized energy levels with quantum numbers (n, l, m) with the energies of typical hydrogen atom energy levels.

Similar to the hydrogen atom, the size of the gravitational atom is given by the “Bohr radius” r = h / (μ c α_{grav}). We need a very light particle to form our gravitational atom: for heavier particles, the size of the atom would be inside the black hole horizon.

The horizon of a solar-mass black hole is on the order of 10 km, which is the Compton wavelength of a particle with roughly the mass 10^{-11}eV/c^{2}—10^{17} times lighter than the electron! However, there are good reasons for such ultralight particles to exist. The best example is the QCD axion, which was proposed nearly 40 years ago as a solution to the strong-CP problem in the Standard Model and is an excellent and still completely viable candidate for the dark matter of our universe. Another aspect of the QCD axion that makes these gravitational atoms interesting is that it is a boson, and so is not subject to the Pauli exclusion principle. This means that unlike the hydrogen atom, the gravitational atom can, in principle, have an unlimited number of axions populating each energy level.

**Black hole "superradiance" **

A light particle in a bound state with a black hole is a beautiful idea, but can it be observed? How would such a gravitational atom form?

Black hole superradiance is not something out of Marvel Comics, but rather the process where a particle (or light) wave passing near a rotating black hole exits the black hole environment with a larger amplitude than the one with which it came in, by extracting energy and momentum from the black hole. This process only happens if the particle satisfies the superradiance condition,

E < m Ω _{BH}

for a hydrogen-like bound state with total energy E and magnetic quantum number *m*, and where Ω _{BH} is the angular velocity of the black hole horizon. The energy and momentum gain happen if the particle’s angular velocity is less than the angular velocity of the black hole.

While it may sound mysterious, black hole superradiance is just one manifestation of a phenomenon that appears in a variety of systems. The most famous is “inertial motion superradiance”—otherwise known as Cherenkov radiation (also discussed in this previous KIPAC blogpost). In Cherenkov radiation, a non-accelerating charged particle spontaneously emits radiation if it is moving faster than the speed of light in the medium, and radiation that scatters inside a cone behind the particle is amplified. Similarly, superradiance of light waves occurs for an axisymmetric conductor (the Zel’dovich cylinder) rotating at a constant angular velocity. A light wave is amplified when the angular velocity of the object is faster than the angular phase velocity of the light. Black hole superradiance is the analogous effect, with the cylinder rotational velocity replaced by the black hole horizon angular velocity, and electromagnetic interaction replaced by gravity.

The superradiance process populates all levels that satisfy the superradiance condition, with the superradiating levels closest to the black hole growing the fastest. The process happens spontaneously. For bosons, the occupation number in an atomic level will grow exponentially starting from zero, forming a macroscopic “cloud” around the black hole, amounting very rapidly to a few percent of the black hole mass! A single level stops growing when the black hole spins down enough and no longer satisfies the superradiance condition—by this point, the level can be filled with more than 10^{75} particles!

The time needed for the number of particles in a level to double is typically 10^{7} times the light-crossing time of the black hole or longer; for a solar-mass black hole, this can be as short as 100 seconds. This is quite fast compared to timescales on which the black holes can change due to astrophysical processes like accretion, which takes on the order of 10^{8} years: if the superradiance condition is satisfied, superradiance can dominate the evolution of black holes.

The idea of black hole superradiance has been around for many decades, but only recently have people considered superradiance of solar mass black holes—this requires a new particle beyond the Standard Model but can have observable effects.

**How do we see the black hole “atom”?**

Astrophysical black holes make perfect detectors for the QCD axion, or any other ultralight bosons which interact almost as weakly as gravity and are very difficult to observe in a lab.

There may be as many as a billion solar-mass black holes in our galaxy alone. Since the black holes form from collapse or collisions of larger objects, they are often rapidly rotating: to conserve angular momentum the object must speed up its rotation as it collapses. Once a black hole forms, if it is spinning quickly enough and if any light bosonic particles with suitable masses exist, superradiance will automatically start to populate levels of the “atom” with these particles. The black hole spins down until the superradiance condition is no longer satisfied.

The fact that black holes can lose a significant fraction of their spin through this process already places constraints on very light particles.

In the plot above, the blue-shaded regions (each corresponding to different angular momentum levels) are affected by superradiance in the presence of a QCD axion with mass 10^{-11} eV. On the y-axis we plot the black hole spin, which ranges from zero (non-spinning) to 1 (extremal, where the horizon velocity approaches the speed of light), and on the x-axis we plot the black hole mass in terms of solar masses. If a black hole is created with spin and mass within the blue region, it will spin down rapidly in as short as a few years—this is very fast compared to the Eddington accretion time of 100 million years (the typical time it takes for a black hole to grow by a factor of two). The points are stellar black holes measurements with error bars, which tell us that is it unlikely that a particle with such mass exists (in which case there would not be any old black holes in the blue shaded regions today). This provides the first ever constraint on such light particles, relying only on their gravitational interaction with the black hole—that is, only their mass.

When there are a lot of particles in the “cloud”, there are two potentially observable processes that take place. Just like in an atom, particles can transition from higher energy states to lower energy states, but instead of charged electrons emitting a photon, these emit a graviton. And unlike electrons, if the particles are their own antiparticles, they can also pair annihilate into a graviton in the gravitational field of the black hole. Since the occupation numbers of the cloud are enormous, the number of gravitons emitted can be very large, and since the particles are all in one state, the gravitational radiation is coherent and monochromatic, which makes it a perfect candidate for detection with upcoming gravitational wave observatories such as the Laser Interferometer Gravitational Wave Observatory (LIGO)—which consists of L-shaped laser interferometers at each of its two sites, thus the "half-diamond" reference in the title). While the processes are short on astrophysical timescales—10 years for transitions, and thousands of years for annihilations—they are very long on human timescales. Advanced LIGO could observe the same “atom” for the whole time the experiment is running!

To see how many such atoms we might be able to observe, we estimate the number of black hole candidates in our universe that are surrounded by a particle cloud. For a given particle mass, there is a range of black hole masses and spins that give signals which are observable from black holes in our Milky Way. Transition signals are more rare since they require two levels to be populated at once and last for a shorter amount of time. Advanced LIGO could optimistically hope to see one such transition event.

Signals from annihilations are more promising—hundreds of thousands of events could be observed with target advanced LIGO sensitivity!

*Above: Estimates for number of events expected to be observed at aLIGO as a function of the particle mass are shown. Each event lasts thousands of years or longer. The vertical gray shaded region is disfavored by current black hole spin measurements. The width of the blue band corresponds to our estimates of astrophysical uncertainties, including the mass and spin distributions of black holes in the Milky Way. The energy of two annihilating axions is directly converted to the gravitational wave, so the frequency of the signal is approximately twice the axion mass (shown on the scale at the top). *

Once a signal is observed, it can be studied in more detail—for example, changes in signal power and small frequency drifts could help establish it is as coming from the gravitational atom.

Future detectors at lower frequencies are in the planning stages (AGIS, eLISA); these would be able to see the signatures of gravitational atoms for even lighter particles around supermassive black holes in the centers of galaxies.

Advanced LIGO is coming online within a year and may be the first experiment to observe gravitational waves. With the help of black hole superradiance, LIGO could also discover a new particle (perhaps the long-sought-after QCD axion) in the process!

And* that *would be quite a discovery, indeed.

----------------- Further Reading:

To learn more details, see our paper (also on the arXiv).

A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper,and J. March-Russell, Phys.Rev., D81, 123530 (2010), arXiv:0905.4720; A. Arvanitaki and S. Dubovsky, Phys. Rev. D 83, 044026 (2011).