Scatter in Sunyaev--Zel'dovich effect scaling relations explained by inter-cluster variance in mass accretion histories

X-ray and microwave cluster scaling relations are immensely valuable for cosmological analysis. However, their power is limited by astrophysical systematics that bias mass estimates and introduce additional scatter. Turbulence injected into the intracluster medium via mass assembly contributes substantially to cluster non-thermal pressure support, a significant source of such uncertainties. We use an analytical model to compute the assembly-driven non-thermal pressure profiles of haloes based on Monte Carlo-generated accretion histories. We introduce a fitting function for the average non-thermal pressure fraction profile, which exhibits minimal dependence on redshift at fixed peak height. Using the model, we predict deviations from self-similarity and the intrinsic scatter in the Sunyaev--Zel'dovich effect observable-mass scaling relation ($Y_\mathrm{SZ}-M$) due solely to inter-cluster variation in mass accretion histories. We study the dependence of $Y_\mathrm{SZ}-M$ on aperture radius, cosmology, redshift, and mass limit. The model predicts $5-9\%$ scatter in $Y_\mathrm{SZ}-M$ at $z=0$, increasing as the aperture used to compute $Y_\mathrm{SZ}$ increases from $R_\mathrm{500c}$ to $5R_\mathrm{500c}$. The predicted scatter lies slightly below that of studies based on non-radiative hydro-simulations, illustrating that assembly history variance is likely responsible for a substantial fraction of scatter in $Y_\mathrm{SZ}-M$. This should be regarded as a lower bound, which will likely increase with the use of an updated gas density model that incorporates a more realistic response to halo assembly. As redshift increases, $Y_\mathrm{SZ}-M$ deviates more from self-similarity and scatter increases. We show that the $Y_\mathrm{SZ}-M$ residuals correlate strongly with the recent halo mass accretion rate, potentially providing an opportunity to infer the latter.