A detailed background model for the CUPID-Mo $0 \nu \beta \beta$ experiment

CUPID-Mo, located in the Laboratoire Souterrain de Modane (France), is a demonstrator for CUPID, a next generation search for $0 \nu \beta \beta$ in $^{100}$Mo. CUPID-Mo consists of 20 $\sim 200g$ Li$_2^{100}$MoO$_4$ scintillating bolometers with 20 Ge light detectors. It has demonstrated excellent crystal radiopurity ($^{238}$U $ ^{232}$Th chains 0.3 - 1 $\mu Bq/kg$ for relevant isotopes), $\alpha$, $\beta/\gamma$ particle discrimination ( $>99.9\%$) and energy resolution ($\sim 7 keV$ FWHM at $2615 keV$). CUPID-Mo has placed the leading limit on the half life of $0 \nu \beta \beta$ in $^{100}$Mo of $T_{1/2}^{0 \nu}>1.4 \cdot 10^{24}yr$ with $90\%$ c.i. In this talk, we present a detailed model to disentangle the various background sources using a Geant4 Monte-Carlo simulation. This enables a precision analysis of the $2\nu \beta \beta$ decay spectrum in $^{100}$Mo. This decay has a short half-life of $\sim 7\cdot 10^{18} yrs$, and CUPID-Mo will provide statistics competitive to much larger experiments but with a much better signal to noise ratio. We then apply these results to the CUPID background model which with a $\sim 100 \times$ increase in mass and significant reduction of passive material will have an unprecedented sensitivity to $0\nu \beta \beta$ in $^{100}$Mo.