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6395f35 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | # Kerr Superradiant $2p$ Scalar Cloud with Self-Gravity Shift, Boson Number & Annihilation GW Signal
A stellar-mass Kerr black hole at cosmological redshift $z_{\text{cos}} = 0.065$ with line-of-sight peculiar velocity $v_{\text{pec}} = -850.0\text{ km s}^{-1}$ (approaching the observer relative to the local Hubble flow) is born with initial gravitating mass $M_0 = 12.5\;M_\odot$ and initial dimensionless spin parameter $a_{*,0} = 0.94$.
An ultralight real scalar boson field of mass $\mu_b$ with initial gravitational fine-structure parameter $\alpha_{b,0} \equiv \frac{G M_0 \mu_b}{\hbar c} = 0.220$ undergoes superradiant growth exclusively in the $m = 1, \ell = 1, n = 2$ ($2p$) hydrogenic bound state (where $n = 2$ is the principal quantum number). Including both the Newtonian hydrogenic Bohr binding energy of the $n = 2$ state in the gravitational potential of the black hole of instantaneous mass $M$ ($\alpha_b(M) = \alpha_{b,0}\,M/M_0$) and the attractive self-gravity binding shift of the condensate of mass $M_{\text{cloud}} = M_0 - M$ parameterized by $\eta_{\text{sg}} = 0.450$, each boson in the cloud has eigenfrequency
$$\frac{\omega_b(M)}{\mu_b c^2 / \hbar} = 1 - \frac{\alpha_b(M)^2}{2n^2} - \eta_{\text{sg}}\left(\frac{M_0 - M}{M_0}\right)\alpha_b(M)^2,$$
and carries eigenenergy $\hbar\omega_b(M)$ and azimuthal angular momentum $m\hbar$. The black hole extracted mass and angular momentum go entirely into the $2p$ scalar cloud until superradiance saturates at the threshold $\omega_b(M_{\text{sat}}) = m\,\Omega_H(M_{\text{sat}}, a_{*,\text{sat}})$, after which pairs of bound scalar bosons in the saturated cloud annihilate into gravitons ($2\text{ bosons} \to 1\text{ graviton}$).
Adopted CGS constants:
- $G = 6.67430 \times 10^{-8}\text{ cm}^3\text{ g}^{-1}\text{ s}^{-2}$, $c = 2.99792458 \times 10^{10}\text{ cm s}^{-1} = 299792.458\text{ km s}^{-1}$
- $M_\odot = 1.98847 \times 10^{33}\text{ g}$, $h = 6.62607015 \times 10^{-27}\text{ erg s}$ ($\hbar = \frac{h}{2\pi}$)
Report:
- `initial_horizon_angular_velocity_rad_s`: initial horizon angular velocity $\Omega_H(M_0, a_{*,0})$ in $\text{rad s}^{-1}$, to 5 significant figures.
- `saturated_black_hole_spin_a_sat`: final dimensionless black hole spin $a_{*,\text{sat}}$ at superradiant saturation, to 6 significant figures.
- `saturated_cloud_boson_number`: total number of scalar bosons $N_b$ occupying the $2p$ cloud at superradiant saturation, to 5 significant figures.
- `observed_annihilation_gw_frequency_hz`: observed monochromatic gravitational-wave cyclic frequency $f_{\text{GW,obs}}$ in Hz arriving at Earth from $2\text{-boson} \to 1\text{-graviton}$ annihilation in the saturated cloud, to 6 significant figures.
Write `candidate.json` in the working directory as a flat JSON object:
```json
{
"problem_id": "ASTRO2",
"initial_horizon_angular_velocity_rad_s": <number>,
"saturated_black_hole_spin_a_sat": <number>,
"saturated_cloud_boson_number": <number>,
"observed_annihilation_gw_frequency_hz": <number>
}
```
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