Constraints on the early Universe star formation efficiency from galaxy clustering and halo modeling of H$\alpha$ and [O III] emitters

We develop a theoretical framework to provide observational constraints on the early Universe galaxy-halo connection by combining measurements of the UV luminosity function (UVLF) and galaxy clustering via the 2-point correlation function (2PCF). We implemented this framework in the FRESCO and CONGRESS JWST NIRCam/grism surveys by measuring the 2PCF of spectroscopically selected samples of H$\alpha$ and [OIII] emitters at $3.8<z<9$ in 124 arcmin$^2$ in GOODS-N and GOODS-S. By fitting the 2PCF and UVLF at $3.8<z<9$ we inferred that the H$\alpha$ and [OIII] samples at $\langle z \rangle \sim4.3, 5.4$ and $7.3$ reside in halos of masses of log$(M_{\rm h}/$M$_{\odot}) = 11.5$, $11.2$, $11.0$ respectively, while their galaxy bias increases with redshift with values of $b_{\rm g} = 4.0$, $5.0$, $7.6$. These halos do not represent extreme overdense environments at these epochs. We constrain the instantaneous star formation efficiency (SFE), defined as the ratio of the star formation rate over the baryonic accretion rate as a function of halo mass. The SFE rises with halo mass, peaks at $\sim20\%$ at $M_{\rm h} \sim 3 \times 10^{11}\, M_{\odot}$, and declines at higher halo masses. The SFE-$M_{\rm h}$ shows only a mild evolution with redshift with tentative indications that low mass halos decrease but the high mass halos increase in efficiency with redshift. The scatter in the $M_{\rm UV}-M_{\rm h}$ relation, quantified by $\sigma_{\rm UV}$, implies stochasticity in the UV luminosities of $\sim 0.7$ mag, relatively constant with z. Extrapolating our model to $z>9$ shows that a constant SFE-$M_{\rm h}$ fixed at $z=8$ cannot reproduce the observed UVLF and neither high maximum SFE nor high stochasticity alone can explain the high abundances of luminous galaxies seen by JWST. Extending the analysis of the UVLF and 2PCF to $z>9$ measured from wider surveys will be crucial in breaking degeneracies.