Lyman-$\alpha$ radiation pressure regulates star formation efficiency

Order-unity star formation efficiencies (SFE) in early galaxies may explain the overabundance of bright galaxies observed by JWST at high redshift. Here we show that Lyman-$\alpha$ (Ly$\alpha$) radiation pressure limits the gas mass converted into stars, particularly in primordial environments. We develop a shell model including Ly$\alpha$ feedback, and validate it with one-dimensional hydrodynamical simulations. To account for Ly$\alpha$ resonant scattering, we adopt the most recent force multiplier fits, including the effect of Ly$\alpha$ photon destruction by dust grains. We find that, independently of their gas surface density $\Sigma_g$, clouds are disrupted on a timescale shorter than a free-fall time, and even before supernova explosions if $\Sigma_g \gtrsim 10^3\,M_{\odot}\ \rm pc^{-2}$. At $\log(Z/Z_{\odot}) = -2$, relevant for high-redshift galaxies, the SFE is $0.01 \lesssim \hat{\epsilon}_{*} \lesssim 0.66$ for $10^3 \lesssim\Sigma_g [M_{\odot}\ \rm pc^{-2}] \lesssim 10^5$. The SFE is even lower for decreasing metallicity. Near-unity SFEs are possible only for extreme surface densities, $\Sigma_{g} \gtrsim 10^5\;M_{\odot}\ \rm pc^{-2}$, and near-solar metallicities. We conclude that Ly$\alpha$ radiation pressure severely limits a possible extremely efficient, feedback-free phase of star formation in dense, metal-poor clouds.