Stochastic modeling of star-formation histories I: the scatter of the star-forming main sequence

We present a framework for modelling the star-formation histories of galaxies as a stochastic process. We define this stochastic process through a power spectrum density with a functional form of a broken power-law. Star-formation histories are correlated on short timescales, the strength of this correlation described by a power-law slope, $\alpha$, and they decorrelate to resemble white noise over a timescale that is proportional to the timescale of the break in the power spectrum density, $\tau_{\rm break}$. We use this framework to explore the properties of the stochastic process that, we assume, gives rise to the log-normal scatter about the relationship between star-formation rate and stellar mass, the so-called galaxy star-forming main sequence. Specifically, we show how the measurements of the normalisation and width ($\sigma_{\rm MS}$) of the main sequence, measured in several passbands that probe different timescales, give a constraint on the parameters of the underlying power spectrum density. We first derive these results analytically for a simplified case where we model observations by averaging over the recent star-formation history. We then run numerical simulations to find results for more realistic observational cases. As a proof of concept, we use observational estimates of the main sequence scatter at $z\sim0$ and $M_{\star}\approx10^{10}~M_{\odot}$ measured in H$\alpha$, UV+IR and the u-band, and show that combination of these point to $\tau_{\rm break}=178^{+104}_{-66}$ Myr, when assuming $\alpha=2$. This implies that star-formation histories of galaxies lose "memory" of their previous activity on a timescale of $\sim200$ Myr, highlighting the importance of baryonic effects that act over the dynamical timescales of galaxies.