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X-ray and microwave cluster scaling relations are immensely valuable forcosmological analysis. However, their power is limited by astrophysicalsystematics that bias mass estimates and introduce additional scatter.Turbulence injected into the intracluster medium via mass assembly contributessubstantially to cluster non-thermal pressure support, a significant source ofsuch uncertainties. We use an analytical model to compute the assembly-drivennon-thermal pressure profiles of haloes based on Monte Carlo-generatedaccretion histories. We introduce a fitting function for the averagenon-thermal pressure fraction profile, which exhibits minimal dependence onredshift at fixed peak height. Using the model, we predict deviations fromself-similarity and the intrinsic scatter in the Sunyaev--Zel'dovich effectobservable-mass scaling relation ($Y_\mathrm{SZ}-M$) due solely tointer-cluster variation in mass accretion histories. We study the dependence of$Y_\mathrm{SZ}-M$ on aperture radius, cosmology, redshift, and mass limit. Themodel predicts $5-9\%$ scatter in $Y_\mathrm{SZ}-M$ at $z=0$, increasing as theaperture used to compute $Y_\mathrm{SZ}$ increases from $R_\mathrm{500c}$ to$5R_\mathrm{500c}$. The predicted scatter lies slightly below that of studiesbased on non-radiative hydro-simulations, illustrating that assembly historyvariance is likely responsible for a substantial fraction of scatter in$Y_\mathrm{SZ}-M$. This should be regarded as a lower bound, which will likelyincrease with the use of an updated gas density model that incorporates a morerealistic 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 massaccretion rate, potentially providing an opportunity to infer the latter.

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