How informative are summaries of the cosmic 21-cm signal?

The cosmic 21-cm signal will bring data-driven advances to studies of the Cosmic Dawn (CD) and Epoch of Reionization (EoR). Radio telescopes such as the SKA will eventually map the HI fluctuations over the first billion years - the majority of our observable Universe. With such large data volumes, it becomes increasingly important to develop "optimal" summary statistics, allowing us to learn as much as possible about the CD and EoR. In this work we compare the constraining power of several 21-cm summary statistics, using the determinant of the Fisher information matrix, $\det F$. Since we do not have an established "fiducial" model for the astrophysics of the first galaxies, we compute the distribution of $\det F$ across the prior volume. Using a large database of cosmic 21-cm lightcones that include realizations of telescope noise, we compare the following summaries: (i) the spherically-averaged power spectrum (1DPS), (ii) the cylindrically-averaged power spectrum (2DPS), (iii) the 2D Wavelet scattering transform (WST), (iv) a recurrent neural network (RNN), (v) an information-maximizing neural network (IMNN), and (vi) the combination of 2DPS and IMNN. Our best performing individual summary is the 2DPS, having relatively high Fisher information throughout parameter space. Although capable of achieving the highest Fisher information for some parameter choices, the IMNN does not generalize well, resulting in a broad distribution. Our best results are achieved with the concatenation of the 2DPS and IMNN. The combination of only these two complimentary summaries reduces the recovered parameter variances on average by factors of $\sim$6.5 - 9.5, compared with using each summary independently. Finally, we point out that that the common assumption of a constant covariance matrix when doing Fisher forecasts using 21-cm summaries can significantly underestimate parameter constraints.