Detection of Cosmic Structures using the Bispectrum Phase. I.
  Mathematical Foundations
  Many low-frequency radio interferometers are aiming to detect very faint
spectral signatures from structures at cosmological redshifts, particularly of
neutral Hydrogen using its characteristic 21 cm spectral line. Due to the very
high dynamic range needed to isolate these faint spectral fluctuations from the
very bright foregrounds, spectral systematics from the instrument or the
analysis, rather than thermal noise, are currently limiting their sensitivity.
Failure to achieve a spectral calibration with fractional inaccuracy $\lesssim
10^{-5}$ will make the detection of the critical cosmic signal unlikely. The
bispectrum phase from interferometric measurements is largely immune to this
calibration issue. We present a basis to explore the nature of bispectrum phase
in the limit of small spectral fluctuations. We establish that they measure the
intrinsic dissimilarity in the transverse structure of the cosmic signal
relative to the foregrounds, expressed as rotations in the underlying phase
angle. Their magnitude is related to the strength of the cosmic signal relative
to the foregrounds. Using a range of sky models, we detail the behavior of
bispectrum phase fluctuations using standard Fourier-domain techniques and find
it comparable to existing approaches, with a few key differences. Mode-mixed
foreground contamination is more pronounced than in existing approaches because
the bispectrum phase is a product of three individual interferometric phases.
The multiplicative coupling of foregrounds in the bispectrum phase fluctuations
results in the mixing of foreground signatures with that of the cosmic signal.
We briefly outline a variation of this approach to avoid extensive mode-mixing.
Despite its limitations, the interpretation of results using bispectrum phase
is possible with forward-modeling. Importantly, it is an independent and a
viable alternative to existing approaches.
