Index of Common Formulas

This formula index lists where each relation first appears in a settled form. When a later chapter needs the same set of quantities, return here and to the corresponding place in the main text first to check symbols, units, orders of magnitude and conditions of validity.

Event Tables, Counts and Likelihoods

Topic	Location	Check when using
Photon event-table fields	Chapter A Guided Tour: What Does a Telescope Actually Record, and Where Is This Book Taking You?; instrument-level extensions in Chapter Detectors, Clocks, and Event Tables	Time reference, detector or telescope index, frequency channel, polarization, quality flags, spatial pixel and weight.
Time calibration	Chapter Detectors, Clocks, and Event Tables	Raw TDC time, clock drift, optical-path delay, barycentric correction and absolute synchronization.
Poisson counts	Chapter A Guided Tour: What Does a Telescope Actually Record, and Where Is This Book Taking You?	Whether the expected count includes source variability, background, exposure, selection function and dead time.
Unbinned point-process likelihood	Chapter Correlators and Event-Table Data Analysis	Whether event times are preserved; whether the rate model varies with time, energy or quality window.
Binned Poisson likelihood	Chapter Correlators and Event-Table Data Analysis; pedagogical projection in Chapter Teaching Experiments and Computational Experiments	Whether the bin width is appropriate relative to the physical time scale, the instrument response and the statistical count requirement.
Delay and pair count	Chapter Correlators and Event-Table Data Analysis	Delay window, time-shift background, accidental coincidences, duplicate events and quality selection.
\(g^{(2)}\) estimator	Chapter Correlators and Event-Table Data Analysis	Normalization, background window, width of the zero-delay peak and uncorrelated-channel checks.
Covariance and posterior	Chapter Correlators and Event-Table Data Analysis	Whether the data points are correlated; whether the prior and likelihood are written on the same data vector.

Coherence, Visibility and Intensity Interferometry

Topic	Location	Check when using
Complex visibility	Chapter Spatial Coherence, van Cittert–Zernike, and Intensity Interferometry	Angular-coordinate units, projected baseline, wavelength, narrow-field approximation, bandwidth averaging and source model.
Uniform disk	Chapter Spatial Coherence, van Cittert–Zernike, and Intensity Interferometry	Whether the angular diameter is a radius or a diameter; whether limb darkening is needed; whether the baseline reaches near the first null.
Siegert relation	Chapter The Coherence Functions g^{(1)}, g^{(2)} and the Siegert Relation; spatial HBT in Chapter Spatial Coherence, van Cittert–Zernike, and Intensity Interferometry	Whether the thermal-light, chaotic-light or Gaussian-field approximation holds; whether polarization, spectrum and number of modes change the contrast.
Intensity-interferometry observation model	Chapter Spatial Coherence, van Cittert–Zernike, and Intensity Interferometry; calibration model in Chapter Common Misconceptions	Whether zero-baseline contrast, peak shape, instrumental crosstalk, selection function and statistical noise are kept separate.
Coherence time	Chapter The Coherence Functions g^{(1)}, g^{(2)} and the Siegert Relation; instrument-channel form in Chapter Detectors, Clocks, and Event Tables	Filter bandwidth, line width, central wavelength and frequency units.
Contrast dilution	Chapter The Coherence Functions g^{(1)}, g^{(2)} and the Siegert Relation	Electronic response, correlation bin, effective number of modes, polarization averaging and spectral averaging.
Background dilution	Chapter Detectors, Clocks, and Event Tables	Target flux fraction, night-sky brightness, companion stars, line-to-continuum ratio, dark counts and calibrator stars.
Intensity-interferometry SNR	Chapter Spatial Coherence, van Cittert–Zernike, and Intensity Interferometry; observational feasibility in Chapter Observation Design, Error Budget, and Feasibility	Photon rate, telescope area, integration time, equivalent bandwidth, $
Number of baselines	Chapter Spatial Coherence, van Cittert–Zernike, and Intensity Interferometry	Adding telescopes increases the number of correlation tasks quadratically; data rate and calibration complexity rise in step.

Instrument Response and Data Rates

Topic	Location	Check when using
Data rate	Chapter Detectors, Clocks, and Event Tables	Sampling time, number of bits, number of spectral channels, number of telescopes and real-time correlator capacity.
Response convolution	Chapter Detectors, Clocks, and Event Tables	The response kernel formed jointly by the detector pulse, cabling, amplifier, digitizer and software correlation window.
Jitter budget	Chapter Detectors, Clocks, and Event Tables	Detector jitter, clock synchronization, fiber delay, trigger jitter and barycentric correction.
Event-table quality selection	Chapters Correlators and Event-Table Data Analysis and Teaching Experiments and Computational Experiments	Clouds, saturation, events near dead time, anomalous background, bad channels and target altitude.
Decomposition of calibration terms	Chapter Common Misconceptions	Astrophysical terms, instrumental terms, selection-function terms and random errors must not be mixed into a single free constant.

Estimation Theory and Mode Measurement

Topic	Location	Check when using
Fisher information	Chapter Correlators and Event-Table Data Analysis; spatial-mode example in Chapter Quantum Estimation, the Rayleigh Limit, and SPADE Sub-Rayleigh Resolution	Whether the parameters, data vector, covariance, derivatives and prior match the actual observation.
Cramér–Rao lower bound	Chapter Quantum Estimation, the Rayleigh Limit, and SPADE Sub-Rayleigh Resolution	The bound is a result under local, unbiased or asymptotic conditions; systematic errors are not automatically included.
Quantum Fisher information for a Gaussian source	Chapter Quantum Estimation, the Rayleigh Limit, and SPADE Sub-Rayleigh Resolution	Whether the source is a weak, equally bright, incoherent two-point source; whether the PSF is known.
SPADE mode probabilities	Chapter Quantum Estimation, the Rayleigh Limit, and SPADE Sub-Rayleigh Resolution	Mode basis, centroid localization, crosstalk matrix, background and finite photon number.
SII Fisher matrix	Chapter Quantum Estimation, the Rayleigh Limit, and SPADE Sub-Rayleigh Resolution	$
Mixed statistics	Chapter Common Misconceptions	The flux fractions of several independent components dilute the correlated excess quadratically.
Fano factor	Chapter Common Misconceptions	Slowly varying sources, dead time and afterpulsing can all produce non-Poisson counts.
Multiple testing	Chapter Common Misconceptions	Time bins, frequency channels, baselines, number of targets and a posteriori windows all enter the trial count.

Astrophysical Source Models

Topic	Location	Check when using
Brightness temperature	Chapter The Quantum Language of Astrophysical Radiation Mechanisms	Whether the Rayleigh–Jeans approximation applies; whether angular scale, distance and flux density are independently constrained.
Thermal radiation and occupation number	Chapter The Quantum Language of Astrophysical Radiation Mechanisms	Frequency, temperature and mode occupation number determine whether thermal-light statistics are appreciable.
Synchrotron radiation and polarization	Chapter The Quantum Language of Astrophysical Radiation Mechanisms	Electron energy spectrum, magnetic field, viewing angle and Faraday effect.
Maser gain	Chapter The Quantum Language of Astrophysical Radiation Mechanisms	Population inversion, velocity-coherence length, degree of saturation and pump fluctuations.
Stellar effective temperature	Chapter Stars as Quantum Light Sources; case-study version in Chapter First-Generation Quantum-Astronomy Science Cases	Angular diameter, bolometric flux, extinction and calibrator stars.
Binary-star visibility	Chapter Stars as Quantum Light Sources; science case in Chapter First-Generation Quantum-Astronomy Science Cases	Flux ratio, angular separation, position angle, multi-epoch orbit and mirror degeneracy.
Line/continuum separation	Chapter Stars as Quantum Light Sources; case-study version in Chapter First-Generation Quantum-Astronomy Science Cases	Line flux fraction, continuum angular scale and filter leakage.
Transient angular expansion	Chapter Bursts, Transients, and Multi-Messenger Quantum Astronomy; case-study version in Chapter First-Generation Quantum-Astronomy Science Cases	Trigger time, velocity model, departures from spherical symmetry and epoch-to-epoch evolution.
Type Ia distance toy model	Chapter Teaching Experiments and Computational Experiments	Photospheric velocity, explosion time, angular-radius error and radiative-transfer model.

Propagation, Cosmology and Quantum Networks

Topic	Location	Check when using
Dispersion delay	Chapter Propagation Effects: Plasma, Dust, and Gravitational Lensing	Frequency units, DM decomposition, intra-channel broadening and plasma approximation.
Faraday rotation	Chapter Propagation Effects: Plasma, Dust, and Gravitational Lensing	Polarization-angle convention, frequency coverage, intrinsic angle and foreground subtraction.
Scattering convolution	Chapter Propagation Effects: Plasma, Dust, and Gravitational Lensing	Multipath propagation, pulse broadening, deconvolution and selection effects.
Gravitational lensing	Chapter Propagation Effects: Plasma, Dust, and Gravitational Lensing	Mass model, source position, time delay and microlensing.
Axion polarization rotation	Chapter Dark Matter, Axions, and the Polarization Quantum Channel	Frequency dependence, time modulation, polarization calibration and the ordinary Faraday term.
CMB likelihood	Chapter Quantum Problems in Cosmology	\(C_\ell\), covariance, foregrounds and cosmic variance.
Quantum-network resources	Chapter Quantum-Network Telescopes	Link loss, storage time, frequency conversion, fidelity and the astronomical photon rate actually available.
Observational error budget	Chapter Observation Design, Error Budget, and Feasibility	Whether statistical, calibration, background, model and selection-function errors are all included.
Proposal milestones	Chapter From White Paper to Research Plan	Verifiable observables, target precision, null tests, deliverable data and failure criteria.