Modelling the M*-SFR relation at high redshift: untangling factors driving biases in the intrinsic scatter measurement.

We present a method to self-consistently propagate M$_{*}$ and SFR ($\Psi$) uncertainties onto intercept, slope and intrinsic scatter estimates for a simple model of the main sequence of star forming galaxies where $\Psi = \alpha + \beta$M$_{*} + \mathcal{N}(0,\sigma)$. From simple idealised models set up with broad-band photometry from NIRCam filters at $z\sim5$, we test the method and compare to methods in the literature. Simplifying the $\Psi$ estimate by basing it on dust-corrected MUV can help to reduce the impact of template set degeneracies on slope and intercept estimates, but act to bias the intrinsic scatter estimate. Our test scenarios employ standard star-formation histories used in SED-fitting in the literature. We find that priors in age for constant SFHs, and age and $\tau_{SFR}$ for delayed histories, act to corral constraints in the M$_{*}-\Psi$ plane. This means that one might infer better $\Psi$ constraints than are actually possible because of the underlying priors. These priors are also incompatible with searching for possible mass-dependence of intrinsic scatter in the main sequence, even for the M$_{*}$ estimates being used with independent $\Psi$ estimates. $\Psi$ estimates obtained from H$\alpha$ with dust attenuation correction using H$\beta$ are subject to uncertain stellar H$\beta$ absorption if SFHs are stochastic. Strong constraints on rest-frame UV slope are therefore required in addition to H$\alpha$ and H$\beta$ fluxes when investigating an increase in scatter to low stellar masses. These can be obtained with NIRSpec R100 spectra. We show that, for simple exposure time calculations assuming point sources without dust, we should be able to probe to log(m$_{*}$/M$_{\odot})\sim8.5$ with JWST at $z\sim5$.