Mergers, Accretion, and Spin Evolution

A black hole remembers growth through its mass and angular-momentum vector, but that memory is incomplete. A long prograde disk, many small randomly oriented episodes, and binary mergers drive different spin distributions only after alignment, radiative capture, recoil, host retention, and observational selection are specified. The central lesson is therefore conditional: spin is a demographic tracer of growth, not a unique label for a formation channel.

Required background. Probability and inference for gravity supplies hierarchical population inference and selection effects.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr spin and orbital structure; Relativistic Matter and Compact Objects supplies compact-remnant formation; Black-Hole Astrophysics supplies accretion and spin observables; Relativistic Cosmology supplies merger histories and cosmic-time weighting.

Spin changes as mass and angular momentum are added

In the site conventions,

$$a_*=\frac{J_\bullet}{G M_\bullet^2}, \qquad -1\leq a_*\leq1.$$

The sign is meaningful only relative to a declared orbital or disk axis. For equatorial thin-disk accretion, a rest-mass element $dm_0$ crossing from the innermost stable circular orbit gives, before radiation captured by the hole is added,

$$dM_\bullet=E_{\rm ISCO}\,dm_0, \qquad dJ_\bullet=L_{\rm ISCO}\,dm_0.$$

Differentiating the dimensionless spin gives

$$\frac{da_*}{d\ln M_\bullet} = \frac{L_{\rm ISCO}}{G M_\bullet E_{\rm ISCO}}-2a_*.$$

The first term adds specific angular momentum; the second reflects the $M_\bullet^2$ in the denominator. Coherent prograde inflow normally spins the hole up, whereas retrograde inflow has a larger ISCO lever arm and can spin it down efficiently. In the ideal radiatively efficient, geometrically thin, equatorial model, capture of disk photons balances further spin-up near $a_*\simeq0.998$ Thorne 1974, §§ II–III, pp. 507–511. That value is a result for a particular radiation field and inner boundary condition, not a universal maximum for thick, magnetized, stressed, or jet-producing flows.

The same accretion history also changes the radiative efficiency,

$$\eta_{\rm NT}=1-E_{\rm ISCO}(a_*),$$

so spin and mass growth feed back on each other. A rapidly spinning prograde hole radiates more energy per unit supplied rest mass in this model and therefore gains less mass at fixed emitted luminosity. A demographic calculation using this thin-disk closure must update the luminosity-to-growth conversion as spin changes; a phenomenological model with fixed efficiency is a different approximation.

Coherence, alignment, and chaotic episodes

For a finite accretion episode, the accounting is vectorial:

$$\boldsymbol J_{\rm after} = \boldsymbol J_{\rm before} +\Delta\boldsymbol J_{\rm acc} +\Delta\boldsymbol J_{\rm EM},$$

Here $\Delta\boldsymbol J_{\rm EM}$ is the signed change due to electromagnetic torque, including extraction. A warped disk exchanges angular momentum with the hole before most of its mass reaches the horizon. Whether the inner flow aligns, counteraligns, tears, or remains time dependent depends on the disk angular momentum, thickness, viscosity or MHD stress, warp propagation, episode duration, and outer feeding direction.

The chaotic-accretion proposal is not merely “random directions.” It requires episodes small enough that their angular momenta do not force every disk into the same prograde configuration. In the analytic limit $J_d\lesssim J_\bullet$, some initially retrograde disks can counteralign, so repeated independent episodes alternate efficient spin-down with spin-up and tend to avoid persistently extremal spin King and Pringle 2006, pp. L90–L92. If episodes are massive, their directions correlate with a galactic gas reservoir, or alignment completes before accretion, the same nominally random supply can instead produce coherent spin-up.

Thus “coherent” and “chaotic” name parameter regimes, not two exhaustive source classes. A defensible model declares the episode-mass distribution, angular correlation time, disk angular momentum, alignment prescription, and stopping rule. Its null test is to randomize only the orientations while preserving episode masses and ask whether the predicted spin distribution truly changes.

Mergers, remnants, and recoil

For an isolated binary, use the initial center-of-momentum frame and let $E_i$ and $\boldsymbol J_i$ be its total initial energy and angular momentum, including orbital contributions. After gravitational waves escape and the remnant settles,

$$E_f=E_i-E_{\rm GW}, \qquad \boldsymbol P_f=-\boldsymbol P_{\rm GW}, \qquad M_f^2=E_f^2-|\boldsymbol P_f|^2,$$ $$\boldsymbol J_f^{\rm total} = \boldsymbol J_i-\boldsymbol J_{\rm GW}, \qquad \boldsymbol v_{\rm kick} =-\frac{\boldsymbol P_{\rm GW}}{E_f}.$$

The remnant spin is its intrinsic angular momentum in its rest frame; total angular momentum about the initial origin can also include the recoiling remnant’s orbital contribution. In the usual small-recoil approximation $E_f\simeq M_f$, giving $\boldsymbol v_{\rm kick}\simeq-\boldsymbol P_{\rm GW}/M_f$. Likewise $M_f\simeq M_1+M_2-E_{\rm GW}$ assumes an initially widely separated, slowly moving binary and counts radiation from that same starting state. At finite separation, binding and kinetic energy belong in $E_i$; subtracting only the subsequent radiation from $M_1+M_2$ does not close the energy budget.

Numerical-relativity fits interpolate these radiated quantities across mass ratio and spin vectors. An equal-mass, nonspinning, quasicircular binary leaves a remnant with $a_f$ near $0.69$, illustrating that orbital angular momentum alone produces substantial spin; generic initial spins broaden and tilt the remnant distribution Rezzolla et al. 2008, §§ II–IV. Unequal masses and asymmetric spin configurations radiate net linear momentum. Large in-plane spin components can produce strong recoils, while alignment can suppress them Lousto and Zlochower 2011, pp. 1–4.

The astrophysical outcome depends on the host escape speed, not on the kick alone. A retained remnant can become a second-generation binary; a marginal kick can displace it and interrupt accretion; an escaping remnant reduces the occupation fraction. Gas that aligns spins before merger simultaneously changes the remnant spin and retention probability. Merger-driven growth therefore cannot be represented by adding masses and drawing a spin from a universal remnant distribution.

From intrinsic histories to observed spin samples

Electromagnetic and gravitational-wave measurements see different populations. Continuum fitting and reflection spectroscopy require luminous disks and use an assumed inner-flow model. At fixed supplied mass rate, high prograde spin can raise radiative efficiency and the volume over which a flux-limited source is detected. Method validity, inclination, absorption, disk state, and publication selection further reshape the observed distribution Reynolds 2021, §§ 3–5.

Gravitational waves from stellar-mass binaries constrain combinations such as

$$\chi_{\rm eff} = \frac{m_1\boldsymbol\chi_1+m_2\boldsymbol\chi_2} {m_1+m_2}\boldsymbol{\cdot}\hat{\boldsymbol L},$$

more directly than both component spin vectors. The GWTC-5.0 population release reports a dominant low-spin binary-black-hole population together with evidence, under several hypermodels, for a smaller rapidly spinning component; it also emphasizes model dependence in joint mass, mass-ratio, spin, and redshift structure LVK 2026, § IV.4.3 and § VIII. This is evidence about merging stellar-mass binaries selected by the detector network, not a measurement of active galactic nuclei or of all stellar remnants.

A population comparison must therefore forward-select each proposed intrinsic history through the applicable electromagnetic or gravitational-wave likelihood. Naively juxtaposing an X-ray spin histogram with a merger value of $\chi_{\rm eff}$ mixes mass scales, source states, spin summaries, and selection functions.

Results and open questions

The evidence cutoff is 2026-08-18. The inverse problem remains unresolved: several histories overlap after alignment, recoil, and method selection.

Growth position	Evidence or result	Counterexample or independent check	Scope, limitation, or open status
Prolonged coherent prograde accretion	Thin-disk angular-momentum addition drives high positive $a_*$ and higher radiative efficiency	Integrate the mass–spin equations with photon capture and recover the thin-disk equilibrium	Thick disks, magnetic stress, winds, and changing orientation alter the boundary condition
Many small chaotic episodes	Alternating prograde and retrograde episodes can keep spins moderate while adding mass	Preserve episode masses but randomize orientations; vary $J_d/J_\bullet$ and alignment time	Large episodes or coherent host gas can erase the proposed behavior
Merger-dominated growth	Orbital and component angular momenta create a broad remnant-spin and recoil distribution	Reproduce numerical-relativity remnant fits and close mass, spin, and momentum budgets	Mass ratio, spin tilts, eccentricity, and host retention remain required inputs
Gas-assisted alignment before merger	Aligned spins reduce some recoils and can preserve rapid rotation	Compare the alignment time with inflow, binary-hardening, and merger times	Warp transport and multiphase feeding remain open questions
Electromagnetic spin demographics	Luminous accretors can show strong apparent support for high spins	Forward-select a mock intrinsic population through efficiency and method validity	Flux, state, inclination, modeling, and publication selection favor particular sources
GWTC-5.0 binary population	Hierarchical inference supports mostly modest spins plus model-dependent substructure	Refit common injections with alternative spin–mass hypermodels and sensitivity estimates	Measures merging stellar binaries; compressed spin parameters are not full vectors

The discriminating calculation evolves the same seeded population under explicit episode masses and orientations, alignment physics, binary mass ratios and spins, numerical-relativity remnant fits, recoil, and host escape speeds. It then applies each survey’s likelihood and selection, predicts mass, spin, alignment, occupation, recoil, radiative-efficiency, and merger-rate observables together, and performs held-out posterior-predictive tests. Fitting only one spin histogram does not distinguish the channels.

The interpretation could change after a revised spin catalog with a shared-systematics population model, a new merger-remnant release, or a direct alignment constraint; a radiation-MHD result that changes the accreted angular-momentum or equilibrium-spin prescription could also change the conclusion.

Interpretation and scope

Spin retains information about growth only statistically. Coherent accretion, episode-scale disorder, mergers, and recoil leave correlated signatures in mass, alignment, occupation, efficiency, and host environment; those correlations are more discriminating than spin alone.

Generic host demographics without black-hole inference are outside this page’s scope. The duty-cycle analysis supplies accretion histories, while cross-method mass and spin consistency explains how shared nuisances must enter an observational comparison.

Common pitfalls

Calling every low spin chaotic. Natal spin, inefficient angular-momentum transport, mergers, and measurement selection can also produce a low inferred spin.

Applying the thin-disk limit universally. The Thorne value depends on a thin radiative disk and photon capture; magnetized and geometrically thick flows have different fluxes and torques.

Ignoring retention. A remnant spin distribution without recoil and host escape speeds is not a growth model for an occupied galaxy population.

Update a spin vector

A toy black hole begins with mass $M$ and spin $a_*=0.60$ along $+\hat z$, so $J_i=0.60GM^2$. An episode increases the gravitational mass to $1.10M$ and adds angular momentum $\Delta J=+0.30GM^2$ along $+\hat z$. Compute the new spin. Repeat for the same magnitude added along $-\hat z$. What does this calculation omit?

Solution

For the prograde episode,

$$a_{*,f} = \frac{J_i+\Delta J}{G(1.10M)^2} = \frac{0.60+0.30}{1.21} \simeq0.744.$$

For the retrograde episode,

$$a_{*,f} = \frac{0.60-0.30}{1.21} \simeq0.248.$$

The denominator’s growth matters: adding positive angular momentum does not mean adding the same amount to $a_*$. The toy inputs already specify the mass and angular momentum that reach the hole. A physical calculation must obtain them from $E_{\rm ISCO}$ and $L_{\rm ISCO}$ or an MHD flux, allow vector misalignment and alignment torques, subtract radiated or extracted energy and angular momentum, and verify $|a_*|\leq1$.

References

• King, Andrew R., and James E. Pringle. “Growing Supermassive Black Holes by Chaotic Accretion.” Monthly Notices of the Royal Astronomical Society: Letters 373 (2006): L90–L92. doi:10.1111/j.1745-3933.2006.00249.x.
• LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration. “GWTC-5.0: Population Properties of Merging Compact Binaries.” Preprint, 2026. arXiv:2605.27226; GWTC-5.0 data release.
• Lousto, Carlos O., and Yosef Zlochower. “Hangup Kicks: Still Larger Recoils by Partial Spin–Orbit Alignment of Black-Hole Binaries.” Physical Review Letters 107 (2011): 231102. doi:10.1103/PhysRevLett.107.231102.
• Reynolds, Christopher S. “Observational Constraints on Black Hole Spin.” Annual Review of Astronomy and Astrophysics 59 (2021): 117–154. doi:10.1146/annurev-astro-112420-035022.
• Rezzolla, Luciano, et al. “On the Final Spin from the Coalescence of Two Black Holes.” Physical Review D 78 (2008): 044002. doi:10.1103/PhysRevD.78.044002.
• Thorne, Kip S. “Disk-Accretion onto a Black Hole. II. Evolution of the Hole.” The Astrophysical Journal 191 (1974): 507–519. doi:10.1086/152991.