How self-interactions change boson-cloud limits from spins and gravitational waves

This guide reads the arXiv version 2 of the paper, dated 16 February 2021, which was published as Phys. Rev. D 103, 095019 (2021). Baryakhtar, Galanis, Lasenby and Simon, Black hole superradiance of self-interacting scalar fields, arXiv:2011.11646v2 We read the full text of the sections and figures named below. We did not compare it line by line with the journal's typeset version, and we have not reproduced any of its figures. The only results this site recomputed are Dolan's Table I and one slice of an amplification table, described in solving for the bound cloud's growth rate reproducibly and recomputing a published amplification table.

Why a free-field calculation is not enough

The growth rates in the earlier chapters treat the boson as a free field. Brito, Cardoso and Pani, Superradiance, §5.6.2 A real axion-like particle has a potential whose quartic term gives an attractive self-interaction. The paper writes its strength as $\lambda=\mu^2/f^2$, where $\mu$ is the boson mass and $f$ the decay constant: a smaller $f$ means a stronger self-interaction.

Once a cloud is dense, self-interactions move bosons between levels and push some of them out to infinity or into the hole. When self-interactions are strong enough, the $211$ level, the $l=m=1$ ground state of the earlier chapters, stops growing at an occupation set by $f$, well below the size it would need to remove much of the hole's spin. The cloud then sits in a quasi-equilibrium, and the spin drops only slowly.

Figure 3: which level wins

Figure 3 maps the plane of boson mass $\mu$ and decay constant $f$ for a black hole of $10\,\mathrm M_\odot$ that starts at spin $a_\ast=0.9$ and lives $10^{10}$ years. Different shadings mark where the hole is spun down to the $211$ threshold, where self-interactions saturate the $211$ cloud first, and where the faster-growing $322$ level takes over, which happens for $\mu\gtrsim4\times10^{-12}\,\mathrm{eV}$. Gray hatching marks regions where other levels matter and the evolution was not fully analysed. A dashed blue line shows the QCD axion relation between mass and decay constant, and a band shows axion-like particles that could make up the dark matter.

What to take from it: the exclusion is a region in two parameters, not a range of masses. Read the hatched regions as open questions.

Figure 9: a spin history, not a growth rate

Figure 9 shows the spin of a $10\,\mathrm M_\odot$ hole against time for two masses, $\mu=8\times10^{-13}\,\mathrm{eV}$ with $\alpha\approx0.06$ and $\mu=2.5\times10^{-12}\,\mathrm{eV}$ with $\alpha\approx0.19$, Read it next to Sec. VI, where the authors state that large self-interactions suppress spin-down.

Our solver puts the free-field growth at the same couplings in context. At $a=0.9$ the $211$ amplitude grows by a factor $e$ in about $16$ years for $\alpha=0.06$ and in about $13$ hours for $\alpha=0.19$. Both are tiny compared with $10^{10}$ years, which is why free-field spin-down looks unavoidable and why the self-interaction, not the growth rate, decides the outcome in the paper's analysis. This comparison is our own addition to the paper.

Figure 10: limits from measured spins

Figure 10 applies the analysis to five X-ray binaries: Cyg X-1, GRS 1915+105, M33 X-7, GRO J1655-40 and LMC X-1. For each, solid lines enclose the parameters where spin-down would confidently have taken the hole below its measured spin within the error intervals; shaded regions show where it may have. The authors call the limits at $\alpha\gtrsim0.2$ and small $f$ estimates, because the evolution there involves several levels.

Three things limit what this figure can say:

1. The spins come from X-ray spectral modelling, which has its own systematic errors.
2. The age assumed for each system bounds how long the cloud had to act.
3. In Sec. VI the authors note that large self-interactions suppress spin-down, and that spins from gravitational-wave mergers still carry large uncertainties.

Figure 11 repeats the exercise for a supermassive hole of $10^7\,\mathrm M_\odot$ at $a_\ast=0.9$, with an Eddington time of $4\times10^8$ years as the age. The authors offer it as a guide only, since accretion and the hole's history are poorly known.

Figures 12 to 14: gravitational waves

Figure 12 gives the growth rates of levels with $l=m=3$ and $l=m=4$ at $a_\ast=0.9$ and $0.99$, from a continued-fraction calculation the paper cites. These enter the multi-level evolution.

Figure 13 shows the peak strain of the waves from pairs of $211$ bosons annihilating into gravitons, for a $10\,\mathrm M_\odot$ hole $1\,\mathrm{kpc}$ away, together with the signal duration and the Advanced LIGO design sensitivity. The signal is nearly monochromatic at about twice the boson frequency. Boundaries labelled $f_{AB}$ and $f_{BC}$ separate regimes. Above $f_{AB}$, where self-interactions are weak, the cloud behaves as in the purely gravitational case.

Two equations set the time dependence. While the cloud annihilates freely,

and at late times in the self-interacting regime the strain falls as $h\propto\sqrt{\tau_{\mathrm{scalar}}/t}$, with $\tau_{\mathrm{scalar}}\propto(f/M_{\mathrm{pl}})^4$. These are the paper's Eqs. (97) and (98). A stronger self-interaction gives a weaker signal that lasts longer.

Figure 14 projects how many such signals could be observed. For weak self-coupling the count reaches up to thousands; the shortest signals likely to be observed last about $10^4$ years. These numbers depend on assumed black-hole populations and are not observations.

What remains open

1. Regions marked as not fully analysed in Fig. 3, and the estimated limits at $\alpha\gtrsim0.2$ and small $f$ in Fig. 10.
2. Spin systematics in X-ray binaries and the large errors on merger spins.
3. Ages, accretion and environment for supermassive holes.
4. No boson cloud has been detected. Every gravitational-wave number in the paper is a projection.

Lab tasks

1. Pick the preset A slow, light cloud and read the e-fold time for a $10\,\mathrm M_\odot$ hole. Compare it with the $10^{10}$ years assumed in Fig. 3.
2. Raise the mass in the advanced group to $10^7\,\mathrm M_\odot$ and read the same quantity. Compare it with the $4\times10^8$ year age used in Fig. 11.
3. The Lab computes only free-field linear growth. Write down which panel of the paper corresponds most closely to what the Lab shows.

Exercises with explained answers

1. Why can a short e-fold time not by itself exclude a boson mass?
2. In Fig. 13, what happens to the signal as $f$ decreases?
3. Which results in the paper would change if a black hole's measured spin were revised downward?

The Lab's free-field growth corresponds to the weak self-interaction limit, $f$ above $f_{AB}$, and only to the early, linear phase. Try it in the black hole superradiance Lab.