Bosenovae and Nonlinear Cloud Collapse

An attractive self-interaction can make a superradiant boson cloud collapse before smooth spin-down reaches synchronization. The resulting “bosenova” is a model-specific nonlinear competition among gravitational binding, gradient pressure, self-attraction, horizon absorption, and outward radiation. Establishing it computationally requires more than observing a sharp amplitude spike: conserved budgets, resolution convergence, and sensitivity to the potential and approximations must all be reported.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr horizons; Black-Hole Perturbation Theory supplies bound modes; Formulations of General Relativity supplies nonlinear evolution; Relativistic Matter and Compact Objects supplies axion-like potentials and self-gravity; Computational Relativity supplies convergence methodology.

Cloud energy and attractive self-interaction

For the field and occupation-number estimates in this section, also set $\hbar=1$, so the boson mass $\mu$ is an inverse length and $N$ is dimensionless. For an axion-like scalar,

$$V(\phi)=\mu^2f_a^2 \left[1-\cos\left(\frac{\phi}{f_a}\right)\right] =\frac12\mu^2\phi^2 -\frac{1}{24}\frac{\mu^2}{f_a^2}\phi^4+\cdots .$$

The negative quartic term is attractive. The quartic expansion is useful only while $|\phi|/f_a\ll1$; unlike the truncated potential, the full cosine potential is bounded below. In a nonrelativistic cloud with approximate occupation $N$, characteristic size $R$, and black-hole mass $M$, the energy above the rest mass $\mu N$ has the schematic form

$$E(R)\sim \frac{N}{2\mu R^2} -\frac{GM\mu N}{R} -C_{\rm self}\frac{N^2}{f_a^2R^3} +E_{\rm selfgrav}.$$

Here $C_{\rm self}\gt0$ depends on the chosen trial profile; the coefficients of the gradient and gravitational terms are likewise only scale estimates. A real scalar has no exact particle-number symmetry, so $N$ is an approximate nonrelativistic occupation, not an exactly conserved Noether charge. Gradient energy resists compression; black-hole gravity binds the cloud; the attractive quartic becomes increasingly important at small $R$. Relativistic gradients, higher terms in the periodic potential, absorption, and ejection must be included once contraction leaves this regime.

The loss of a metastable minimum can be seen without a simulation. Ignore self-gravity and write the trial energy as $E=A N/R^2-BN/R-CN^2/R^3$, with positive constants $A,B,C$. Its stationary radii satisfy

$$B R^2-2AR+3CN=0, \qquad R_\pm=\frac{A\pm\sqrt{A^2-3BCN}}{B}.$$

The larger root is a local minimum and the smaller a barrier. They merge when $N=A^2/(3BC)$; above that value the trial energy has no metastable minimum. This demonstrates a mechanism, not a quantitative bosenova threshold: profile changes, angular structure, self-gravity, and breakdown of the quartic approximation alter the result.

Yoshino and Kodama simulate this process for a sine-Gordon field on a fixed Kerr background and identify collapse, strong mode mixing, black-hole absorption, and outward emission Yoshino and Kodama 2012, §§3–4, pp. 8–23, PDF. Because the metric is not fully evolved, the calculation controls scalar nonlinearity but only approximates gravitational backreaction.

Continuum problem and diagnostics

A computational claim should specify:

• Einstein–scalar or fixed-Kerr Klein–Gordon equations and the exact potential;
• $GM\mu$, $f_a$, black-hole spin, initial cloud modes and amplitudes;
• horizon-regular inner boundary or excision, outer boundary, angular representation, and time integrator;
• invariant or asymptotically defined cloud energy, Noether-like occupation proxy where applicable, horizon flux, and radiated flux;
• maximum local field amplitude and multipolar energy distribution.

On a fixed stationary Kerr background, the Killing-energy budget is

$$E_{\rm initial} =E_{\rm cloud}(t)+\Delta E_H(t)+E_\infty(t)+\epsilon_E(t),$$

with positive $\Delta E_H$ defined as net energy entering the horizon and $E_\infty$ as outward radiation. Superradiant intervals can make $\Delta E_H$ negative. There is a corresponding angular-momentum budget. In fully dynamical gravity one must instead use the appropriate total asymptotic and horizon charges, including gravitational radiation; a matter-only energy integral is not an exact conserved total. The residual $\epsilon_E$ must decrease under refinement. A sudden drop in coordinate cloud energy without matching flux is not evidence for physical collapse.

Benchmark and convergence

Before the nonlinear threshold, the code should reproduce linear bound-state frequency and growth or decay rate at the same $GM\mu$ and spin. In the weak-field limit it should recover the hydrogenic spectrum. A collapse threshold should then be bracketed with a sequence of initial amplitudes and shown stable under at least three effective spatial resolutions or an equivalent spectral/tolerance sequence.

Report norms such as

$$\|\mathcal C\|_2, \qquad \left|\epsilon_E\right|/E_{\rm initial}, \qquad \left|\epsilon_J\right|/J_{\rm initial},$$

and convergence of the collapse time, peak field, horizon absorption, and ejected fraction. Here $\mathcal C$ denotes Einstein constraint residuals only when the metric is evolved; fixed-Kerr runs instead test their field equation and flux balances. Threshold times can be especially resolution-sensitive because a small phase or amplitude error is exponentially amplified.

No simulation was rerun for this page. The cited computations are evaluated through their published equations, tests, parameter choices, and limitations.

Failure modes and interpretation

A principal failure mode is an outer-boundary reflection mistaken for a recurrent burst. Others include angular truncation that suppresses mode mixing, insufficient radial resolution during contraction, coordinate amplitudes compared at different points, and energy definitions that omit horizon flux. On a fixed background, neglect of the cloud’s stress-energy becomes inconsistent once its energy is no longer small relative to $M$.

A bosenova does not necessarily destroy the cloud. Part may fall through the horizon, part may escape, and a remnant can resume superradiant growth, producing cycles. Whether cycles persist after full gravitational backreaction and in generic environments is not settled.

Check the numerical claim

A run shows a sharp central amplitude peak whose time moves earlier by 12%, 7%, and 4% across successive grid refinements, while the energy residual remains 8% and does not converge. Is the bosenova time established?

Solution

No. The event is qualitatively suggestive, but its time is not converged and the nonclosing energy budget could account for the apparent collapse or ejection. The calculation needs another resolved sequence, decreasing constraint and budget residuals, stable horizon and infinity fluxes, and a benchmark linear mode at the same parameters before quoting a physical threshold or time.

References

• Yoshino, Hirotaka, and Hideo Kodama, “Bosenova Collapse of Axion Cloud around a Rotating Black Hole”, Progress of Theoretical and Experimental Physics 2012, 043E02, 2012. Open PDF.

Further reading

• Brito, Richard, Vitor Cardoso, and Paolo Pani, Superradiance, Lecture Notes in Physics 971, second edition, Springer, 2020. Open PDF.
• Yoshino, Hirotaka, and Hideo Kodama, “Gravitational Radiation from an Axion Cloud around a Black Hole: Superradiant Phase”, Progress of Theoretical and Experimental Physics 2014, 043E02, 2014. Open PDF.