Global 21 cm signal experiments: A designer’s guide

The global (i.e., spatially averaged) spectrum of the redshifted 21 cm line has generated much experimental interest lately, thanks to its potential to be a direct probe of the epoch of reionization and the dark ages, during which the first luminous objects formed. Since the cosmological signal in question has a purely spectral signature, most experiments that have been built, designed, or proposed have essentially no angular sensitivity. This can be problematic because with only spectral information, the expected global 21 cm signal can be difficult to distinguish from foreground contaminants such as galactic synchrotron radiation, since both are spectrally smooth and the latter is many orders of magnitude brighter. In this paper, we establish a systematic mathematical framework for global signal data analysis. The framework removes foregrounds in an optimal manner, complementing spectra with angular information. We use our formalism to explore various experimental design trade-offs, and find that (1) with spectral-only methods, it is mathematically impossible to mitigate errors that arise from uncertainties in one's foreground model; (2) foreground contamination can be significantly reduced for experiments with fine angular resolution; (3) most of the statistical significance in a positive detection during the dark ages comes from a characteristic high-redshift trough in the 21 cm brightness temperature; (4) measurement errors decrease more rapidly with integration time for instruments with fine angular resolution; and (5) better foreground models can help reduce errors, but once a modeling accuracy of a few percent is reached, significant improvements in accuracy will be required to further improve the measurements. We show that if observations and data analysis algorithms are optimized based on these findings, an instrument with a 5\ifmmode^\circ\else\textdegree\fi{} wide beam can achieve highly significant detections (greater than $5\ensuremath{\sigma}$) of even extended (high $\ensuremath{\Delta}z$) reionization scenarios after integrating for 500 h. This is in strong contrast to instruments without angular resolution, which cannot detect gradual reionization. Ionization histories that are more abrupt can be detected with our fiducial instrument at the level of tens to hundreds of $\ensuremath{\sigma}$. The expected errors are similarly low during the dark ages, and can yield a $25\ensuremath{\sigma}$ detection of the expected cosmological signal after only 100 h of integration.