Impact of Ly$\alpha$ radiation force on super-Eddington accretion onto a massive black hole

The viability of super-Eddington accretion remains a topic of intense debate, crucial for understanding the formation of supermassive black holes in the early universe. However, the impact of Ly$\alpha$ radiation force on this issue remains poorly understood. We investigate the propagation of the Ly$\alpha$ photons and evaluate the Ly$\alpha$ radiation force within a spherically symmetric accreting HI gas onto the central black hole. We solve the radiation transfer equation, incorporating the destruction processes of Ly$\alpha$ photons through two-photon decay and collisional de-excitation. We find that the Ly$\alpha$ photons, originating in the HII region around black holes, suffer from multiple resonance scattering before being destroyed via two-photon decay and collisional de-excitation. Hence, the Ly$\alpha$ radiation force undergoes a significant amplification, surpassing gravity at the innermost section of the HI region. This amplification, quantified as the force multiplier, reaches approximately 130 and remains nearly constant, regardless of the optical depth at the line center, provided the optical thickness of the flow is within the range of $10^{10-14}$. The requisite lower limit of the product of gas density and black hole mass to realize the super-Eddington accretion is found to be in the range $(2-{40}) \times 10^9 M_\odot\,{\rm cm}^{-3}$, which is a few to tens of times larger than the minimum value obtained without accounting for the Ly$\alpha$ radiation force. The pronounced amplification of the Ly$\alpha$ radiation force poses a substantial challenge to the feasibility of super-Eddington accretion.