Stochastic modelling 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 time-scales, the strength of this correlation described by a power-law slope, alpha, and they decorrelate to resemble white noise over a time-scale that is proportional to the time-scale of the break in the power spectrum density, tau(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 normalization and width (sigma(MS)) of the main sequence, measured in several passbands that probe different time-scales, 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 similar to 0 and M-star approximate to 10(10) M-circle dot measured in H alpha, UV+IR, and the u-band. The result is degenerate in the tau(break)-alpha space, but if we assume alpha = 2, we measure tau(break) = 170(-85)(+169) Myr. This implies that star-formation histories of galaxies lose 'memory' of their previous activity on a time-scale of similar to 200 Myr.