The ALPINE-ALMA [CII] Survey: Modelling ALMA and JWST lines to constrain the interstellar medium of $z\sim 5$ galaxies

In this work, we devise a model for estimating UV and optical line emission (i.e., CIII] $1909$A, H$β$, [OIII] $5007$A, H$α$, [NII] $6583$A) tracing HII regions in the interstellar medium (ISM) of galaxies at $z\sim4-6$ from the ALMA Large Programme ALPINE. The aim is to investigate the impact of binary stars in the stellar population along with an abrupt quenching in the Star Formation History (SFH) on line emission. This is crucial for understanding the ISM's properties in early galaxies and identifying new star formation tracers in high-$z$ galaxies. The model simulates HII+Photodissociation Region (PDR) complexes through radiative transfer in 1D slabs, characterized by gas density ($n$), ionisation parameter ($U$), and metallicity ($Z$). It considers: (a) heating from star formation (SF), simulated with Starburst99 and BPASS to quantify binary stars impact; (b) constant, exponentially declining, and quenched SFH scenarios. For each galaxy, we select theoretical ratios from CLOUDY models between [CII] line emission, tracing PDRs, and nebular lines from HII regions, using these to derive expected optical/UV lines from observed [CII]. We find binary stars strongly impact line emission post-quenching, keeping UV photon flux higher for longer, maintaining free electron temperature and ionised column density in HII regions up to 5 Myr after quenching. We constrain ISM properties of our subsample, finding a low ionisation parameter $\log U{\approx}-3.8\pm 0.2$ and moderate/high densities $\log(n/\rm cm^{-3}){\approx}2.9\pm 0.6$. Finally, we derive UV/optical line luminosities-SFR relations for different burstiness parameters ($k_s$). In the fiducial BPASS model, relations show negligible SFH dependence but depend on $k_s$, while in the SB99 case, dependence is on SFH. We propose their use for characterising the burstiness of high-$z$ galaxies.