Probing the Capability of Future Direct-imaging Missions to Spectrally Constrain the Frequency of Earth-like Planets

A critical question in astrobiology is whether exo-Earth candidates (EECs) are Earth-like, in that they originate life that progressively oxygenates their atmospheres similarly to Earth. We propose answering this question statistically by searching for O₂ and O₃ on EECs with missions such as HabEx or LUVOIR. We explore the ability of these missions to constrain the fraction, f _E, of EECs that are Earth-like in the event of a null detection of O₂ or O₃ on all observed EECs. We use the Planetary Spectrum Generator to simulate observations of EECs with O₂ and O₃ levels based on Earth's history. We consider four instrument designs—LUVOIR-A (15 m), LUVOIR-B (8 m), HabEx with a starshade (4 m, "HabEx/SS"), and HabEx without a starshade (4 m, "HabEx/no-SS")—as well as three estimates of the occurrence rate of EECs (η _earth): 24%, 5%, and 0.5%. In the case of a null detection, we find that for η _earth = 24%, LUVOIR-A, LUVOIR-B, and HabEx/SS would constrain f _E to ≤0.094, ≤0.18, and ≤0.56, respectively. This also indicates that if f _E is greater than these upper limits, we are likely to detect O₃ on at least one EEC. Conversely, we find that HabEx/no-SS cannot constrain f _E, due to the lack of a coronagraph ultraviolet channel. For η _earth = 5%, only LUVOIR-A and LUVOIR-B would be able to constrain f _E, to ≤0.45 and ≤0.85, respectively. For η _earth = 0.5%, none of the missions would allow us to constrain f _E, due to the low number of detectable EECs. We conclude that the ability to constrain f _E is more robust to uncertainties in η _earth for missions with larger aperture mirrors. However, all missions are susceptible to an inconclusive null detection if η _earth is sufficiently low.