Carbonate-Silicate Cycle Predictions of Earth-like Planetary Climates and Testing the Habitable Zone Concept

In the conventional habitable zone (HZ) concept, a CO$_{2}$-H$_2$O greenhouse maintains surface liquid water. Through the water-mediated carbonate-silicate weathering cycle, atmospheric CO$_{2}$ partial pressure (pCO$_{2}$) responds to changes in surface temperature, stabilizing the climate over geologic timescales. We show that this weathering feedback ought to produce a log-linear relationship between pCO$_{2}$ and incident flux on Earth-like planets in the HZ. However, this trend has scatter because geophysical and physicochemical parameters can vary, such as land area for weathering and CO$_2$ outgassing fluxes. Using a coupled climate and carbonate-silicate weathering model, we quantify the likely scatter in pCO$_2$ with orbital distance throughout the HZ. From this dispersion, we predict a two-dimensional relationship between incident flux and pCO$_2$ in the HZ and show that it could be detected from at least 83 ($2{\sigma}$) Earth-like exoplanet observations. If fewer Earth-like exoplanets are observed, testing the HZ hypothesis from this relationship could be difficult.