Constraining bursty star formation histories with galaxy UV and H$\alpha$ luminosity functions and clustering

The observed prevalence of galaxies exhibiting bursty star formation histories (SFHs) at $z\gtrsim6$ has created new challenges and opportunities for understanding their formation pathways. The degenerate effects of the efficiency and burstiness of star formation on the observed UV luminosity function are separable by galaxy clustering. However, quantifying the timescales of burstiness requires more than just the continuum UV measurements. Here we develop a flexible semi-analytic framework for modeling both the amplitude of star formation rate (SFR) variations and their temporal correlation, from which the luminosity function and clustering can be derived for SFR indicators tracing different characteristic timescales (e.g., UV continuum and H$\alpha$ luminosities). Based on this framework, we study the prospect of using galaxy summary statistics to distinguish models where SFR fluctuations are prescribed by different power spectral density (PSD) forms. Using the Fisher matrix approach, we forecast the constraints on parameters in our PSD-based model that can be extracted from mock JWST observations of the UV and H$\alpha$ luminosity functions and clustering bias factors at $z\sim6$. These constraints demonstrate the feasibility of constraining the burstiness of high-$z$ galaxies solely from their one-point and two-point statistics and underscore the importance of combining tracers of both long-term and short-term SFR variations. Our flexible framework can be readily extended to characterize the SFH of high-redshift galaxies with a wider range of observational diagnostics.