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The CUPID-Mo experiment, located at Laboratoire Souterrain de Modane(France), was a demonstrator experiment for CUPID. It consisted of an array of20 Li$_2^{100}$MoO$_4$ (LMO) calorimeters each equipped with a Ge lightdetector (LD) for particle identification. In this work, we present the resultof a search for two-neutrino and neutrinoless double beta decays of $^{100}$Moto the first 0$^+$ and $2^+$ excited states of $^{100}$Ru using the fullCUPID-Mo exposure (2.71 kg$\times$yr of LMO). We measure the half-life of$2\nu\beta\beta$ decay to the $0^{+}_1$ state as $T_{1/2}^{2\nu \rightarrow0_1^+}=7.5\pm 0.8 \ \text{(stat.)} \ ^{+ 0.4}_{-0.3} \ \text{(syst.)} )\times10^{20} \ \mathrm{yr}$. The bolometric technique enables measurement of theelectron energies as well as the gamma rays from nuclear de-excitation and thisallows us to set new limits on the two-neutrino decay to the $2_1^+$ state of$T^{2\nu \rightarrow 2_1^+}_{1/2}>4.4\times 10^{21} \ \mathrm{yr} \ \text{(90 %c.i.)}$ and on the neutrinoless modes of $T_{1/2}^{0\nu\rightarrow2_1^+}>2.1\times10^{23} \ \mathrm{yr}\ \text{(90 % c.i.)}$,$T_{1/2}^{0\nu\rightarrow 0_1^+}>1.2\times10^{23} \ \mathrm{yr}\ \text{(90 %c.i.)}$. Information on the electrons spectral shape is obtained which allowsus to make the first comparison of the single state (SSD) and higher state(HSD) $2\nu\beta\beta$ decay models for the $0_1^+$ excited state of$^{100}$Ru.

New measurement of double beta decays of $^{100}$Mo to excited states of $^{100}$Ru with the CUPID-Mo experiment