Decoding the Star-Forming Main Sequence or: How I Learned to Stop Worrying and Love the Central Limit Theorem

Star-formation rates (SFR) of disk galaxies strongly correlate with stellar mass, with a small dispersion in SSFR at fixed mass, sigma~0.3 dex. With such small scatter this star-formation main sequence (SFMS) has been interpreted as deterministic and fundamental. Here we demonstrate that it is a simple consequence of the central limit theorem. Our derivation begins by approximating in situ stellar mass growth as a stochastic process, much like a random walk (where the expectation of SFR at any time is equal to the SFR at the previous time). We then derive expectation values for median SSFR of star-forming disks and their scatter over time. We generalize the results for stochastic changes in SFR that are not independent of each other but are correlated over time. For unbiased samples of (disk) galaxies, we derive an expectation that <SSFR> should be independent of mass, decline as 1/T, and have a relative scatter that is independent of mass and time. The derived SFMS and its evolution matches published data to z=10 with sufficient accuracy to constrain cosmological parameters. The framework reproduces several important observables, including: the scatter in SSFR at fixed mass; the SFHs of nearby dwarf galaxies and the Milky Way; and the scatter in the Tully-Fisher relation. The evolution of the mass function is less well reproduced and we discuss ways to generalize the framework to include other sources of stellar mass such as mergers. The predicted dispersion in SSFR has consequences for the classification of quiescent galaxies, as such galaxies have heterogeneous formation histories, and many may only be temporarily diminished in their star-formation activity. The implied dispersion in SFHs, and the SFMS's insensitivity to timescales of stochasticity, thus substantially limits the ability to connect massive galaxies to their progenitors over long cosmic baselines. [TRUNC.]