Index of Frequently Used Formulas

This index points to the first stable appearance of the relations used throughout the book. When a later chapter reuses the same quantities, return here and to the corresponding main text for the symbols, units, scales, and assumptions.

Event Tables, Counts, and Likelihoods

Topic	Location	Check before use
Photon event-table fields	Chapter Mathematical and Physical Foundations, Eq. (1); instrument-level extension in Chapter Detectors, clocks, and event tables, Eq. (100)	Time standard, detector or telescope identifier, frequency channel, polarization, quality flag, spatial pixel, and weight.
Timing calibration	Chapter Detectors, clocks, and event tables, Eq. (101)	Raw TDC time, clock drift, optical-path delay, barycentric correction, and absolute synchronization.
Poisson counts	Chapter Mathematical and Physical Foundations, Eq. (9)	Whether the expected count includes source variability, background, exposure, the selection function, and dead time.
Unbinned point-process likelihood	Chapter Data analysis for event tables, Eq. (117)	Whether event times are retained; whether the rate model varies with time, energy, or quality window.
Binned Poisson likelihood	Chapter Data analysis for event tables, Eq. (118); pedagogical projection in Chapter Teaching experiments and computational experiments, Eq. (353)	Whether the bin width is compatible with the physical time scale, instrument response, and required counts per bin.
Delays and pair counts	Chapter Data analysis for event tables, Eq. (120)	Delay window, time-shift background, accidental coincidences, repeated events, and quality cuts.
\(g^{(2)}\) estimator	Chapter Data analysis for event tables, Eq. (122)	Normalization, background window, zero-delay peak width, and checks with uncorrelated channels.
Covariance and posterior	Chapter Data analysis for event tables, Eqs. (127) and (131)	Whether data points are correlated; whether the prior and likelihood are written for the same data vector.

Coherence, Visibility, and Intensity Interferometry

Topic	Location	Check before use
Complex visibility	Chapter Mathematical and Physical Foundations, Eq. (5); VCZ form in Chapter Spatial coherence and intensity interferometry, Eq. (92)	Angular-coordinate units, projected baseline, wavelength, narrow-field approximation, bandwidth averaging, and source model.
Uniform disk	Chapter Mathematical and Physical Foundations, Eq. (6); first null in Chapter Spatial coherence and intensity interferometry, Eq. (94)	Whether the angular size is a radius or a diameter; whether limb darkening is needed; whether the baselines sample the first null.
Siegert relation	Chapter Mathematical and Physical Foundations, Eq. (16); spatial HBT form in Chapter Spatial coherence and intensity interferometry, Eq. (95)	Whether the thermal, chaotic, or Gaussian-field approximation is valid; whether polarization, spectrum, and mode number change the contrast.
Intensity-interferometry observing model	Chapter Spatial coherence and intensity interferometry, Eq. (96); calibration model in Chapter Common pitfalls, Eq. (357)	Whether zero-baseline contrast, peak shape, instrumental crosstalk, selection function, and statistical noise are separated.
Coherence time	Chapter Mathematical and Physical Foundations, Eq. (17)	Filter bandwidth, line width, central wavelength, and frequency units.
Contrast dilution	Chapter Mathematical and Physical Foundations, Eq. (18); instrumental form in Chapter Photon statistics and coherence functions, Eq. (83)	Electronic response, correlation bin, effective number of modes, polarization averaging, and spectral averaging.
Background dilution	Chapter Mathematical and Physical Foundations, Eq. (19); instrumental background in Chapter Detectors, clocks, and event tables, Eq. (114)	Target flux fraction, night sky, companion stars, line continuum, dark counts, and calibrator stars.
SII signal-to-noise ratio	Chapter Spatial coherence and intensity interferometry, Eq. (98); observing feasibility in Chapter Observing design, error budgets, and feasibility calculations	Photon rate, telescope area, integration time, equivalent bandwidth, $
Number of baselines	Chapter Spatial coherence and intensity interferometry, Eq. (99)	Adding telescopes increases the correlation workload quadratically; data rate and calibration complexity rise with it.

Instrument Response and Data Rate

Topic	Location	Check before use
Data rate	Chapter Detectors, clocks, and event tables, Eq. (102)	Sampling time, number of bits, spectral channels, telescope count, and real-time correlator capacity.
Response convolution	Chapter Detectors, clocks, and event tables, Eq. (107)	The effective response kernel includes the detector pulse, cables, amplifier, digitizer, and software correlation window.
Jitter budget	Chapter Detectors, clocks, and event tables, Eq. (108)	Detector jitter, clock synchronization, fiber delay, trigger jitter, and barycentric correction.
Event-table quality selection	Chapters Data analysis for event tables and Teaching experiments and computational experiments	Clouds, saturation, events near dead time, anomalous background, bad channels, and target elevation.
Calibration decomposition	Chapter Common pitfalls, Eq. (357)	Astrophysical terms, instrumental terms, selection-function terms, and random errors should not be hidden inside one free constant.

Estimation Theory and Mode Measurements

Topic	Location	Check before use
Fisher information	Chapter Mathematical and Physical Foundations, Eq. (36); event-table model in Chapter Data analysis for event tables, Eq. (130)	Parameters, data vector, covariance, derivatives, and priors must match the actual observation.
Cramer–Rao lower bound	Chapter Quantum estimation, the Rayleigh limit, and sub-resolution information, Eq. (138)	The bound is local and usually assumes unbiased or asymptotic estimators; systematic errors are not included automatically.
Gaussian-source quantum Fisher information	Chapter Quantum estimation, the Rayleigh limit, and sub-resolution information, Eq. (141)	Whether the source is weak, equal-brightness, and mutually incoherent; whether the PSF is known.
SPADE mode probabilities	Chapter Quantum estimation, the Rayleigh limit, and sub-resolution information, Eq. (142)	Mode basis, centroid registration, crosstalk matrix, background, and finite photon number.
SII Fisher matrix	Chapter Quantum estimation, the Rayleigh limit, and sub-resolution information, Eq. (147)	$
Mixture statistics	Chapter Common pitfalls, Eq. (358)	For independent components, flux fractions dilute the correlation excess quadratically.
Fano factor	Chapter Common pitfalls, Eq. (359)	Slow source variability, dead time, and afterpulsing can all produce non-Poisson counts.
Multiple testing	Chapter Common pitfalls, Eq. (360)	Time bins, frequency channels, baselines, number of targets, and post-selected windows all contribute to the number of trials.

Astrophysical Source Models

Topic	Location	Check before use
Brightness temperature	Chapter The quantum language of astrophysical radiation mechanisms, Eqs. (149) and (150)	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, Eq. (154)	Frequency, temperature, and mode occupation determine whether thermal photon statistics are observable.
Synchrotron radiation and polarization	Chapter The quantum language of astrophysical radiation mechanisms, Eqs. (158) and (160)	Electron spectrum, magnetic field, viewing angle, and Faraday effects.
Maser gain	Chapter The quantum language of astrophysical radiation mechanisms, Eq. (163)	Population inversion, velocity-coherent length, saturation, and pump fluctuations.
Stellar effective temperature	Chapter Stars as quantum light sources, Eq. (169); case-study version in Chapter First-generation quantum-astronomy science cases, Eq. (346)	Angular diameter, bolometric flux, extinction, and calibrator stars.
Binary visibility	Chapter Stars as quantum light sources, Eq. (175); science-case version in Chapter First-generation quantum-astronomy science cases, Eq. (348)	Flux ratio, angular separation, position angle, multi-epoch orbit, and mirror degeneracy.
Line/continuum separation	Chapter Stars as quantum light sources, Eq. (177); case-study version in Chapter First-generation quantum-astronomy science cases, Eq. (349)	Line flux fraction, continuum angular scale, and filter leakage.
Transient angular expansion	Chapter Explosions, transients, and multi-messenger quantum astronomy, Eq. (232); case-study version in Chapter First-generation quantum-astronomy science cases, Eq. (350)	Trigger time, velocity model, asymmetry, and epoch-to-epoch evolution.
Type Ia distance toy model	Chapter Teaching experiments and computational experiments, Eqs. (355) and (356)	Photospheric velocity, explosion time, angular-radius uncertainty, and radiative-transfer model.

Propagation, Cosmology, and Quantum Networks

Topic	Location	Check before use
Dispersion delay	Chapter Propagation effects: plasma, dust, and gravitational lensing, Eq. (264)	Frequency units, DM decomposition, intra-channel smearing, and plasma approximation.
Faraday rotation	Chapter Propagation effects: plasma, dust, and gravitational lensing, Eqs. (268) and (269)	Polarization-angle convention, frequency coverage, intrinsic angle, and foreground subtraction.
Scattering convolution	Chapter Propagation effects: plasma, dust, and gravitational lensing, Eq. (271)	Multipath propagation, pulse broadening, deconvolution, and selection effects.
Gravitational lensing	Chapter Propagation effects: plasma, dust, and gravitational lensing, Eqs. (279) and (283)	Mass model, source position, time delay, and microlensing.
Axion polarization rotation	Chapter Dark matter, axions, and polarization quantum channels, Eqs. (294) and (300)	Frequency dependence, time modulation, polarization calibration, and ordinary Faraday terms.
CMB likelihood	Chapter Quantum questions in cosmology, Eq. (326)	\(C_\ell\), covariance, foregrounds, and cosmic variance.
Quantum-network resources	Chapter Quantum network telescopes, Eqs. (334), (335), and (336)	Link loss, storage time, frequency conversion, fidelity, and usable astronomical photon rate.
Observing error budget	Chapter Observing design, error budgets, and feasibility calculations, Eq. (344)	Statistical, calibration, background, model, and selection-function errors must all be included.
Proposal milestones	Chapter From white paper to research plan, Eqs. (362), (363), and (364)	Acceptable observables, target precision, null tests, data products, and failure criteria.