Accurate Simultaneous Constraints on the Dust Mass, Temperature and Emissivity Index of a Galaxy at Redshift 7.31

We present new multi-frequency ALMA continuum observations of the massive [$\log_{10}(M_\star/M_\odot) = 10.3_{-0.2}^{+0.1}$], UV-luminous [$M_\mathrm{UV} = -21.7 \pm 0.2$] $z=7.31$ galaxy REBELS-25 in Bands 3, 4, 5, and 9. Combining the new observations with previously-taken data in Bands 6 and 8, we cover the dust continuum emission of the galaxy in six distinct bands -- spanning rest-frame $50-350\,\mu$m -- enabling simultaneous constraints on its dust mass ($M_\mathrm{dust}$), temperature ($T_\mathrm{dust}$) and emissivity index ($\beta_\mathrm{IR}$) via modified blackbody fitting. Given a fiducial model of optically thin emission, we infer a cold dust temperature of $T_\mathrm{dust} = 32_{-6}^{+9}\,$K and a high dust mass of $\log_{10}(M_\mathrm{dust}/M_\odot) = 8.2_{-0.4}^{+0.6}$, and moderately optically thick dust does not significantly alter these estimates. If we assume dust production is solely through supernovae (SNe), the inferred dust yield would be high, $y = 0.7_{-0.4}^{+2.3}\,M_\odot$ per SN. Consequently, we argue grain growth in the interstellar medium of REBELS-25 also contributes to its dust build-up. This is supported by the steep dust emissivity index $\beta_\mathrm{IR} = 2.5 \pm 0.4$ we measure for REBELS-25, as well as by its high stellar mass, dense interstellar medium, and metal-rich nature. Our results suggest that constraining the dust emissivity indices of high-redshift galaxies is important not only to mitigate systematic uncertainties in their dust masses and obscured star formation rates, but also to assess if dust properties evolve across cosmic time. We present an efficient observing setup to do so with ALMA, combining observations of the peak and Rayleigh-Jeans tail of the dust emission.