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In this note we compute the 1-loop partition function of spin-$s$ fields onEuclidean de Sitter space $S^{2n+1}$ using the quasinormal mode method. Insteadof computing the quasinormal mode frequencies from scratch, we use the analyticcontinuation prescription $L_{\text{AdS}}\to iL_{\text{dS}}$, appearing in thedS/CFT correspondence, and Wick rotate the normal mode frequencies of fields onthermal $\text{AdS}_{2n+1}$ into the quasinormal mode frequencies of fields onde Sitter space. We compare the quasinormal mode and heat kernel methods ofcalculating 1-loop determinants, finding exact agreement, and furthermoreexplicitly relate these methods via a sum over the conformal dimension. Wediscuss how the Wick rotation of normal modes on thermal $\text{AdS}_{2n+1}$can be generalized to calculating 1-loop partition functions on the thermalspherical quotients $S^{2n+1}/\mathbb{Z}_{p}$. We further show that thequasinormal mode frequencies encode the group theoretic structure of thespherical spacetimes in question, analogous to the recent analysis made forthermal AdS in (1910.07607) and (1910.11913).

Higher spin partition functions via the quasinormal mode method in de Sitter quantum gravity