Measuring the Hubble constant from the cooling of the CMB monopole

The cosmic microwave background (CMB) monopole temperature evolves with the inverse of the cosmological scale factor, independent of many cosmological assumptions. With sufficient sensitivity, real-time cosmological observations could thus be used to measure the local expansion rate of the Universe using the cooling of the CMB. We forecast how well a CMB spectrometer could determine the Hubble constant via this method. The primary challenge of such a mission lies in the separation of Galactic and extra-Galactic foreground signals from the CMB at extremely high precision. However, overcoming these obstacles could potentially provide an independent, highly robust method to shed light on the current low-/high-$z$ Hubble tension. An experiment with 3000 linearly spaced bins between 5 GHz and 3 THz with a sensitivity of 1 $\mathrm{mJy\sqrt{yr}~sr^{-1}}$ per bin, could measure $H_0$ to 3% over a 10 year mission, given current foreground complexity. This sensitivity would also enable high-precision measurements of the expected $\Lambda$CDM spectral distortions, but remains futuristic at this stage.