Constraining the [CII] luminosity function from the power spectrum of line-intensity maps at redshift 3.6

Forthcoming measurements of the line-intensity mapping (LIM) power spectrum (PS) are expected to provide valuable constraints on astrophysical and cosmological quantities. We focus on the [CII] luminosity function (LF) at high redshift, which remains poorly constrained, especially at the faint end. We present forecasts for the Deep Spectroscopic Survey (DSS) that is to be conducted with the Fred Young Submillimeter Telescope (FYST) at $z\simeq3.6$. We also make predictions for surveys with a ten times larger sky coverage and/or a $\sqrt{10}$ times higher sensitivity, accounting for the Lorentzian spectral profile of Fabry-P\'erot interferometers and the impact of their resolving power $R$. Motivated by the halo-occupation properties of [CII] emitters in the MARIGOLD simulations, we derived a luminosity-mass relation by abundance matching two ALPINE LFs to the halo mass function. This relation was then used in a halo-model framework to predict the PS and its uncertainty. Bayesian inference on mock PS data provided forecasts for the first two LF moments and Schechter parameters. Depending on the true LF, the DSS is expected to be able to detect clustering and shot-noise components with signal-to-noise ratios of $\gtrsim2$. At $R=100$, spectral smoothing masks redshift-space distortions, rendering the damping scale $\sigma$ unmeasurable. For $R\gtrsim500$, $\sigma$ is distinguishable from instrumental effects, though degeneracies with amplitude parameters increase. Joint fits to the PS and LF yield precise constraints on the Schechter normalisation and cutoff luminosity, while the faint-end slope remains uncertain (unless the true value approaches $-2$). An increased survey sensitivity offers greater gains than a wider area. A higher spectral resolution improves the access to physical parameters, but intensifies degeneracies. This highlights key design trade-offs in LIM surveys.