The role of galaxies and AGN in reionizing the IGM - II: metal-tracing the faint sources of reionization at $5\lesssim z\lesssim 6$

We present a new method to study the contribution of faint sources to the UV background using the 1D correlation of metal absorbers with the intergalactic medium (IGM) transmission in a Quasi Stellar Object (QSO) sightline. We take advantage of a sample of $25$ high signal-to-noise ratio QSO spectra to retrieve $150$ triply-ionised carbon (\cfour) absorbers at $4.5\lesssim z\lesssim 6.2$,ofwhich$37$ systems whose expected H{~\small I} absorption lie in the Lyman-$\alpha$ forest. We derive improved constraints on the cosmic density of \cfour ,at $4.3<z<6.2$ and infer from abundance-matching that \cfour ,absorbers trace $M_{\text{UV}}\lesssim -16$ galaxies. Correlation with the Lyman-$\alpha$ forest of the QSOs indicates that these objects are surrounded by a highly opaque region at $r\lesssim 5$ cMpc/h followed by an excess of transmission at $r\gtrsim 10$ cMpc/h detected at $2.7\sigma$. This is in contrast to equivalent measurements at lower redshifts where only the opaque trough is detected. We interpret this excess as a statistical enhancement of the local photoionisation rate due to clustered faint galaxies around the \cfour ,absorbers. Using the analytical framework described in Paper I of this series, we derive a constraint on the average product of the escape fraction and the Lyman continuum photon production efficiency of the galaxy population clustered around the \cfour ,absorbers, $\log \langle f_{$\text{esc}$}\xi_{$\text{ion}$}\rangle /[{\rm erg^{-1}~Hz}] = 25.01^{+0.30}_{-0.19}$. This implies that faint galaxies beyond the reach of current facilities may have harder radiation fields and/or larger escape fractions than currently detected objects at the end of the reionisation epoch.