Post-Recombination BHL Accretion

• The paper demonstrates that Bondi–Hoyle–Lyttleton accretion can concentrate over 10^3 solar masses of baryons by z ~400, enabling direct collapse into IMBH seeds.
• It utilizes quantitative BHL formulas with defined parameters, showing an accretion rate of ~10^-3 M☉/yr over a 2×10^8 yr period, critical for early mass buildup.
• The study highlights that disk instability and suppressed fragmentation in UDMH promote rapid inward mass flow, offering insights into early supermassive black hole formation.

Post recombination Bondi–Hoyle–Lyttleton (BHL) accretion describes a scenario in which gas accretes onto ultra-dense dark matter halos (UDMH) of mass $M \sim 10^5\,M_\odot$ that have formed around the cosmological recombination epoch. This process allows the concentration of $\gtrsim 10^3\,M_\odot$ of baryons at very early epochs (redshifts $z \sim 400$), providing conditions for the collapse of intermediate-mass black hole (IMBH) seeds. This formation channel exploits the rare occurrence of strong small-scale curvature fluctuations that give rise to UDMH but remain compatible with current Cosmic Microwave Background (CMB) spectral distortion constraints, and proceeds with characteristic thermal, dynamical, and stability properties suppressing fragmentation and promoting direct collapse (Subramanian et al., 5 Jan 2026).

1. Accretion Physics: Bondi–Hoyle–Lyttleton Framework

The BHL accretion rate onto a compact massive object moving through a uniform medium is given by

$$\dot{M}_{\text{BHL}} = 4\pi \frac{G^2 M_{\rm halo}^2 \rho_b(z)}{[c_s^2(z) + v_\infty^2(z)]^{3/2}}$$

with $M_{\rm halo}$ denoting the mass of the UDMH (here, $\sim 10^5\,M_\odot$), $\rho_b(z) = \rho_{b,0} (1+z)^3$ the mean baryon density ($\rho_{b,0} \simeq 6 \times 10^{-31}$g cm$^{-3}$), $c_s(z)$ the baryonic sound speed, and $\gtrsim 10^3\,M_\odot$0 the relative streaming velocity between baryons and dark matter. For $\gtrsim 10^3\,M_\odot$1, the baryonic gas remains Compton coupled to the CMB with temperature $\gtrsim 10^3\,M_\odot$2K, yielding $\gtrsim 10^3\,M_\odot$3km s$\gtrsim 10^3\,M_\odot$4. The streaming velocity rms $\gtrsim 10^3\,M_\odot$5km s$\gtrsim 10^3\,M_\odot$6 at $\gtrsim 10^3\,M_\odot$7 evolves as $\gtrsim 10^3\,M_\odot$8.

At redshifts $\gtrsim 10^3\,M_\odot$9–$z \sim 400$0, both sound speed and streaming velocities are comparable, justifying the use of an effective velocity $z \sim 400$1. For $z \sim 400$2, $z \sim 400$3g cm$z \sim 400$4, and $z \sim 400$5cm s$z \sim 400$6, the BHL accretion rate evaluates to $z \sim 400$7/yr (Subramanian et al., 5 Jan 2026).

2. Quantitative Accretion and Mass Growth

Integrating this rate over the period between $z \sim 400$8 and $z \sim 400$9 (corresponding to a time interval $$\dot{M}_{\text{BHL}} = 4\pi \frac{G^2 M_{\rm halo}^2 \rho_b(z)}{[c_s^2(z) + v_\infty^2(z)]^{3/2}}$$0yr), the total baryonic inflow is $$\dot{M}_{\text{BHL}} = 4\pi \frac{G^2 M_{\rm halo}^2 \rho_b(z)}{[c_s^2(z) + v_\infty^2(z)]^{3/2}}$$1. The calculation can be refined:

• For constant halo mass: $$\dot{M}_{\text{BHL}} = 4\pi \frac{G^2 M_{\rm halo}^2 \rho_b(z)}{[c_s^2(z) + v_\infty^2(z)]^{3/2}}$$2
• Allowing for $$\dot{M}_{\text{BHL}} = 4\pi \frac{G^2 M_{\rm halo}^2 \rho_b(z)}{[c_s^2(z) + v_\infty^2(z)]^{3/2}}$$3 growth: $$\dot{M}_{\text{BHL}} = 4\pi \frac{G^2 M_{\rm halo}^2 \rho_b(z)}{[c_s^2(z) + v_\infty^2(z)]^{3/2}}$$4 Here, $$\dot{M}_{\text{BHL}} = 4\pi \frac{G^2 M_{\rm halo}^2 \rho_b(z)}{[c_s^2(z) + v_\infty^2(z)]^{3/2}}$$5 is an efficiency factor determined by physical processes such as Compton drag and Hubble expansion, which are negligible for $$\dot{M}_{\text{BHL}} = 4\pi \frac{G^2 M_{\rm halo}^2 \rho_b(z)}{[c_s^2(z) + v_\infty^2(z)]^{3/2}}$$6 ($$\dot{M}_{\text{BHL}} = 4\pi \frac{G^2 M_{\rm halo}^2 \rho_b(z)}{[c_s^2(z) + v_\infty^2(z)]^{3/2}}$$7) (Subramanian et al., 5 Jan 2026).

3. Thermal Evolution and Cooling Constraints

The gravitating gas is shock-heated to the virial temperature,

$$\dot{M}_{\text{BHL}} = 4\pi \frac{G^2 M_{\rm halo}^2 \rho_b(z)}{[c_s^2(z) + v_\infty^2(z)]^{3/2}}$$8

thus $$\dot{M}_{\text{BHL}} = 4\pi \frac{G^2 M_{\rm halo}^2 \rho_b(z)}{[c_s^2(z) + v_\infty^2(z)]^{3/2}}$$9K at $M_{\rm halo}$0 and $M_{\rm halo}$1K at $M_{\rm halo}$2. Atomic cooling (principally collisional excitation of H I) is efficient for $M_{\rm halo}$3K and rapidly cools gas to this floor ($M_{\rm halo}$4yr), but below this temperature, the cooling rate sharply declines. Critically, molecular hydrogen (H$M_{\rm halo}$5) cooling and further fragmentation are suppressed at these epochs due to CMB photons dissociating H$M_{\rm halo}$6 formation intermediaries (H$M_{\rm halo}$7 for $M_{\rm halo}$8, H$M_{\rm halo}$9 for $\sim 10^5\,M_\odot$0), and maintaining H$\sim 10^5\,M_\odot$1 rotational levels in thermal equilibrium for $\sim 10^5\,M_\odot$2K ($\sim 10^5\,M_\odot$3). This inhibition prevents catastrophic fragmentation, enabling most of the gas to collapse coherently (Subramanian et al., 5 Jan 2026).

4. Collapse Dynamics and Disk Formation

The free-fall time for accretion of the accumulated gas is

$\sim 10^5\,M_\odot$4

for $\sim 10^5\,M_\odot$5, with virial radius $\sim 10^5\,M_\odot$6pc. Even modest angular momentum (dimensionless spin $\sim 10^5\,M_\odot$7–$\sim 10^5\,M_\odot$8) halts spherically symmetric collapse at a scale $\sim 10^5\,M_\odot$9–$\rho_b(z) = \rho_{b,0} (1+z)^3$0pc, producing a centrifugally supported disk with vertical scale height $\rho_b(z) = \rho_{b,0} (1+z)^3$1 (Subramanian et al., 5 Jan 2026). This suggests that the process transitions from spherical infall to disk-dominated collapse at sub-parsec scales.

5. Disk Instability and Rapid Mass Inflow

The compact, massive disk is generically gravitationally unstable (Toomre $\rho_b(z) = \rho_{b,0} (1+z)^3$2),

$\rho_b(z) = \rho_{b,0} (1+z)^3$3

where $\rho_b(z) = \rho_{b,0} (1+z)^3$4, $\rho_b(z) = \rho_{b,0} (1+z)^3$5, $\rho_b(z) = \rho_{b,0} (1+z)^3$6, $\rho_b(z) = \rho_{b,0} (1+z)^3$7km s$\rho_b(z) = \rho_{b,0} (1+z)^3$8, $\rho_b(z) = \rho_{b,0} (1+z)^3$9km s$\rho_{b,0} \simeq 6 \times 10^{-31}$0, $\rho_{b,0} \simeq 6 \times 10^{-31}$1–$\rho_{b,0} \simeq 6 \times 10^{-31}$2pc. Disk instability rapidly redistributes mass inwards via spiral arms, torques, and effective gravitational viscosity ($\rho_{b,0} \simeq 6 \times 10^{-31}$3–$\rho_{b,0} \simeq 6 \times 10^{-31}$4), resulting in inflow timescales $\rho_{b,0} \simeq 6 \times 10^{-31}$5–$\rho_{b,0} \simeq 6 \times 10^{-31}$6yr. This culminates in the formation of central supermassive stars and/or direct-collapse black holes of $\rho_{b,0} \simeq 6 \times 10^{-31}$7 at $\rho_{b,0} \simeq 6 \times 10^{-31}$8 a few hundred (Subramanian et al., 5 Jan 2026).

6. Seed Abundance and Cosmological Implications

The comoving number density of such UDMH can be estimated for curvature perturbations of $\rho_{b,0} \simeq 6 \times 10^{-31}$9–$^{-3}$0 at wavenumber $^{-3}$1 Mpc$^{-3}$2 ($^{-3}$3) using Press–Schechter theory, yielding $^{-3}$4–$^{-3}$5 Mpc$^{-3}$6. These densities are comparable to the present-day galaxy abundance and are sufficient to explain the appearance of $^{-3}$7–$^{-3}$8 supermassive black holes observed at $^{-3}$9 by the James Webb Space Telescope, assuming subsequent Eddington-limited growth (Subramanian et al., 5 Jan 2026). A plausible implication is that early UDMH-driven BHL accretion could resolve the origin of luminous quasars at the highest redshifts.

7. Summary Table

Physical Parameter	$c_s(z)$0 Value	$c_s(z)$1 Value
$c_s(z)$2 (g/cm$c_s(z)$3)	$c_s(z)$4	$c_s(z)$5
$c_s(z)$6 (km/s)	$c_s(z)$7	$c_s(z)$8
$c_s(z)$9 (km/s)	$\gtrsim 10^3\,M_\odot$00	$\gtrsim 10^3\,M_\odot$01
$\gtrsim 10^3\,M_\odot$02 ($\gtrsim 10^3\,M_\odot$03/yr)	$\gtrsim 10^3\,M_\odot$04	$\gtrsim 10^3\,M_\odot$05
$\gtrsim 10^3\,M_\odot$06 (yr)	$\gtrsim 10^3\,M_\odot$07	$\gtrsim 10^3\,M_\odot$08

These parameter values underpin the scenario's feasibility for rapid, efficient seed black hole production by post-recombination BHL accretion onto rare, ultra-dense minihalos (Subramanian et al., 5 Jan 2026).