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

The X-ray and microwave observable-mass scaling relations of galaxy clusters are immensely valuable tools for cosmological analysis, but their power is currently limited by astrophysical systematic uncertainties that bias cluster mass estimates and introduce additional scatter. Turbulent energy injected into the intracluster medium via mass assembly contributes substantially to the cluster's non-thermal pressure support, which is a significant source of such uncertainties. We use a simple analytical model to compute the mass assembly-driven non-thermal pressure profiles of individual haloes based on Monte Carlo-generated halo assembly histories from extended Press--Schechter theory. After combining this with the Komatsu--Seljak model for the total pressure to obtain thermal pressure profiles, 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 variations in halo mass assembly histories. We study the dependence of the slope, normalization, and scatter of $Y_\mathrm{SZ}-M$ on aperture radius, cosmology, redshift, and halo 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 hydrodynamical simulations, illustrating that assembly history variance alone is likely to be responsible for a substantial fraction of the scatter in $Y_\mathrm{SZ}-M$. As redshift increases, $Y_\mathrm{SZ}-M$ deviates more from self-similarity and scatter increases. Lastly, we show that the residuals of $Y_\mathrm{SZ}-M$ correlate strongly with the recent halo mass accretion rate, potentially providing an opportunity to infer the latter.