Invariants in Co-polar Interferometry: an Abelian Gauge Theory

Abstract

An $N$-element interferometer measures correlations among pairs of array elements. Closure invariants associated with closed loops among array elements are immune to multiplicative, element-based ("local") corruptions that occur in these measurements. Till recently, it has been unclear how a complete set of independent invariants can be analytically determined. We view the local, element-based corruptions in co-polar correlations as gauge tranformations belonging to the gauge group $\text{GL}(1,\mathbb{C})$. Closure quantities are then naturally gauge invariant. We use this to provide a simple and effective formalism, and identify the complete set of independent closure invariants from co-polar interferometric correlations using only quantities defined on $(N-1)(N-2)/2$ elementary and independent triangular loops. The $(N-1)(N-2)/2$ closure phases and $N(N-3)/2$ closure amplitudes (totaling $N^2-3N+1$ real invariants), familiar in astronomical interferometry, naturally emerge from this formalism, which unifies what has required separate treatments until now. We do not require auto-correlations, but can easily include them if reliably measured. This unified view clarifies issues relating to noise and inference of object model parameters. It also allows us to extend the rule of parallel transport associated with Pancharatnam phase in optics to apply to amplitudes as well. The framework presented here extends to $\text{GL}(2,\mathbb{C}$) for full polarimetric interferometry as presented in a companion paper, which generalizes and clarifies earlier work. Our findings are relevant to state of the art co-polar and full polarimetric very long baseline interferometry measurements to determine features very near the event horizons of blackholes at the centers of M87, Centaurus~A, and the Milky Way.

13 pages (including references), 0 figures, 5 appendices, accepted (in press) in Physical Review D. See also companion paper arXiv:2108.11400

References in corpus (13)

• First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole
• First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole
• First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole
• First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring
• First M87 Event Horizon Telescope Results. II. Array and Instrumentation
• First M87 Event Horizon Telescope Results. III. Data Processing and Calibration
• First M87 Event Horizon Telescope Results. VII. Polarization of the Ring
• Imaging the Surface of Altair
• Few Skewed Disks Found in First Closure-Phase Survey of Herbig Ae/Be stars
• Event Horizon Telescope observations of the jet launching and collimation in Centaurus A
• Detection of Cosmic Structures using the Bispectrum Phase. II. First Results from Application to Cosmic Reionization Using the Hydrogen Epoch of Reionization Array
• Invariants in Polarimetric Interferometry: a non-Abelian Gauge Theory
• Detection of Cosmic Structures using the Bispectrum Phase. I. Mathematical Foundations

Cited by in corpus (9)

• Bayesian Black Hole Photogrammetry
• Invariants in Polarimetric Interferometry: a non-Abelian Gauge Theory
• Imaging a ring-like structure and the extended jet of M87 at 86 GHz
• Search for the Epoch of Reionisation with HERA: Upper Limits on the Closure Phase Delay Power Spectrum
• Prospects of using closure traces directly for imaging in Very Long Baseline Interferometry
• Interferometric Image Reconstruction using Closure Invariants and Machine Learning
• Two-dimensional Light Beam Shape Characterization using Interferometric Closure Amplitudes
• Very-Long Baseline Interferometry Imaging with Closure Invariants using Conditional Image Diffusion
• Establishing a relationship between the cosmological 21 cm power spectrum and interferometric closure phases