(app:a)=
# 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.

(appsec:a-observables)=
## Event Tables, Counts, and Likelihoods
| Topic | Location | Check before use |
|:---------------|:----------------------|:--------------------------|
| Photon event-table fields | Chapter {ref}`chap:01`, Eq. {eq}`eq:event-row`; instrument-level extension in Chapter {ref}`chap:06`, Eq. {eq}`eq:ch06-raw-event` | Time standard, detector or telescope identifier, frequency channel, polarization, quality flag, spatial pixel, and weight. |
| Timing calibration | Chapter {ref}`chap:06`, Eq. {eq}`eq:ch06-calibrated-time` | Raw TDC time, clock drift, optical-path delay, barycentric correction, and absolute synchronization. |
| Poisson counts | Chapter {ref}`chap:01`, Eq. {eq}`eq:poisson` | Whether the expected count includes source variability, background, exposure, the selection function, and dead time. |
| Unbinned point-process likelihood | Chapter {ref}`chap:07`, Eq. {eq}`eq:ch07-point-process-likelihood` | Whether event times are retained; whether the rate model varies with time, energy, or quality window. |
| Binned Poisson likelihood | Chapter {ref}`chap:07`, Eq. {eq}`eq:ch07-binned-poisson`; pedagogical projection in Chapter {ref}`chap:20`, Eq. {eq}`eq:ch20-binned-count` | Whether the bin width is compatible with the physical time scale, instrument response, and required counts per bin. |
| Delays and pair counts | Chapter {ref}`chap:07`, Eq. {eq}`eq:ch07-pair-count` | Delay window, time-shift background, accidental coincidences, repeated events, and quality cuts. |
| $g^{(2)}$ estimator | Chapter {ref}`chap:07`, Eq. {eq}`eq:ch07-g2-estimator` | Normalization, background window, zero-delay peak width, and checks with uncorrelated channels. |
| Covariance and posterior | Chapter {ref}`chap:07`, Eqs. {eq}`eq:ch07-covariance` and {eq}`eq:ch07-posterior` | Whether data points are correlated; whether the prior and likelihood are written for the same data vector. |

(appsec:a-coherence)=
## Coherence, Visibility, and Intensity Interferometry
| Topic | Location | Check before use |
|:---------------|:----------------------|:--------------------------|
| Complex visibility | Chapter {ref}`chap:01`, Eq. {eq}`eq:visibility`; VCZ form in Chapter {ref}`chap:05`, Eq. {eq}`eq:ch05-vcz` | Angular-coordinate units, projected baseline, wavelength, narrow-field approximation, bandwidth averaging, and source model. |
| Uniform disk | Chapter {ref}`chap:01`, Eq. {eq}`eq:uniform-disk`; first null in Chapter {ref}`chap:05`, Eq. {eq}`eq:ch05-first-null` | 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 {ref}`chap:01`, Eq. {eq}`eq:siegert-hbt`; spatial HBT form in Chapter {ref}`chap:05`, Eq. {eq}`eq:ch05-hbt-spatial` | Whether the thermal, chaotic, or Gaussian-field approximation is valid; whether polarization, spectrum, and mode number change the contrast. |
| Intensity-interferometry observing model | Chapter {ref}`chap:05`, Eq. {eq}`eq:ch05-correlation-model`; calibration model in Chapter {ref}`chap:21`, Eq. {eq}`eq:ch21-calibration-model` | Whether zero-baseline contrast, peak shape, instrumental crosstalk, selection function, and statistical noise are separated. |
| Coherence time | Chapter {ref}`chap:01`, Eq. {eq}`eq:coherence-time` | Filter bandwidth, line width, central wavelength, and frequency units. |
| Contrast dilution | Chapter {ref}`chap:01`, Eq. {eq}`eq:g2-dilution`; instrumental form in Chapter {ref}`chap:04`, Eq. {eq}`eq:ch04-contrast-dilution` | Electronic response, correlation bin, effective number of modes, polarization averaging, and spectral averaging. |
| Background dilution | Chapter {ref}`chap:01`, Eq. {eq}`eq:background-dilution`; instrumental background in Chapter {ref}`chap:06`, Eq. {eq}`eq:ch06-background-dilution` | Target flux fraction, night sky, companion stars, line continuum, dark counts, and calibrator stars. |
| SII signal-to-noise ratio | Chapter {ref}`chap:05`, Eq. {eq}`eq:ch05-snr`; observing feasibility in Chapter {ref}`chap:18` | Photon rate, telescope area, integration time, equivalent bandwidth, $|V|^2$, and systematic floor. |
| Number of baselines | Chapter {ref}`chap:05`, Eq. {eq}`eq:ch05-baseline-number` | Adding telescopes increases the correlation workload quadratically; data rate and calibration complexity rise with it. |

(appsec:a-instrument)=
## Instrument Response and Data Rate
| Topic | Location | Check before use |
|:---------------|:----------------------|:--------------------------|
| Data rate | Chapter {ref}`chap:06`, Eq. {eq}`eq:ch06-data-rate` | Sampling time, number of bits, spectral channels, telescope count, and real-time correlator capacity. |
| Response convolution | Chapter {ref}`chap:06`, Eq. {eq}`eq:ch06-response-convolution` | The effective response kernel includes the detector pulse, cables, amplifier, digitizer, and software correlation window. |
| Jitter budget | Chapter {ref}`chap:06`, Eq. {eq}`eq:ch06-jitter-budget` | Detector jitter, clock synchronization, fiber delay, trigger jitter, and barycentric correction. |
| Event-table quality selection | Chapters {ref}`chap:07` and {ref}`chap:20` | Clouds, saturation, events near dead time, anomalous background, bad channels, and target elevation. |
| Calibration decomposition | Chapter {ref}`chap:21`, Eq. {eq}`eq:ch21-calibration-model` | Astrophysical terms, instrumental terms, selection-function terms, and random errors should not be hidden inside one free constant. |

(appsec:a-estimation)=
## Estimation Theory and Mode Measurements
| Topic | Location | Check before use |
|:---------------|:----------------------|:--------------------------|
| Fisher information | Chapter {ref}`chap:01`, Eq. {eq}`eq:fisher`; event-table model in Chapter {ref}`chap:07`, Eq. {eq}`eq:ch07-fisher` | Parameters, data vector, covariance, derivatives, and priors must match the actual observation. |
| Cramer--Rao lower bound | Chapter {ref}`chap:08`, Eq. {eq}`eq:ch08-crb` | The bound is local and usually assumes unbiased or asymptotic estimators; systematic errors are not included automatically. |
| Gaussian-source quantum Fisher information | Chapter {ref}`chap:08`, Eq. {eq}`eq:ch08-qfi-gaussian` | Whether the source is weak, equal-brightness, and mutually incoherent; whether the PSF is known. |
| SPADE mode probabilities | Chapter {ref}`chap:08`, Eq. {eq}`eq:ch08-spade-probability` | Mode basis, centroid registration, crosstalk matrix, background, and finite photon number. |
| SII Fisher matrix | Chapter {ref}`chap:08`, Eq. {eq}`eq:ch08-sii-fisher` | $|V|^2$ errors, baseline coverage, parameter degeneracies, and model assumptions. |
| Mixture statistics | Chapter {ref}`chap:21`, Eq. {eq}`eq:ch21-mixture` | For independent components, flux fractions dilute the correlation excess quadratically. |
| Fano factor | Chapter {ref}`chap:21`, Eq. {eq}`eq:ch21-fano` | Slow source variability, dead time, and afterpulsing can all produce non-Poisson counts. |
| Multiple testing | Chapter {ref}`chap:21`, Eq. {eq}`eq:ch21-trials` | Time bins, frequency channels, baselines, number of targets, and post-selected windows all contribute to the number of trials. |

(appsec:a-sources)=
## Astrophysical Source Models
| Topic | Location | Check before use |
|:---------------|:----------------------|:--------------------------|
| Brightness temperature | Chapter {ref}`chap:09`, Eqs. {eq}`eq:ch09-brightness-temperature` and {eq}`eq:ch09-frb-tb` | Whether the Rayleigh--Jeans approximation applies; whether angular scale, distance, and flux density are independently constrained. |
| Thermal radiation and occupation number | Chapter {ref}`chap:09`, Eq. {eq}`eq:ch09-planck-occupation` | Frequency, temperature, and mode occupation determine whether thermal photon statistics are observable. |
| Synchrotron radiation and polarization | Chapter {ref}`chap:09`, Eqs. {eq}`eq:ch09-synch-critical` and {eq}`eq:ch09-synch-polarization` | Electron spectrum, magnetic field, viewing angle, and Faraday effects. |
| Maser gain | Chapter {ref}`chap:09`, Eq. {eq}`eq:ch09-maser-gain` | Population inversion, velocity-coherent length, saturation, and pump fluctuations. |
| Stellar effective temperature | Chapter {ref}`chap:10`, Eq. {eq}`eq:ch10-teff`; case-study version in Chapter {ref}`chap:19`, Eq. {eq}`eq:ch19-teff` | Angular diameter, bolometric flux, extinction, and calibrator stars. |
| Binary visibility | Chapter {ref}`chap:10`, Eq. {eq}`eq:ch10-binary`; science-case version in Chapter {ref}`chap:19`, Eq. {eq}`eq:ch19-binary-visibility` | Flux ratio, angular separation, position angle, multi-epoch orbit, and mirror degeneracy. |
| Line/continuum separation | Chapter {ref}`chap:10`, Eq. {eq}`eq:ch10-line-visibility`; case-study version in Chapter {ref}`chap:19`, Eq. {eq}`eq:ch19-line-continuum` | Line flux fraction, continuum angular scale, and filter leakage. |
| Transient angular expansion | Chapter {ref}`chap:13`, Eq. {eq}`eq:ch13-angular-expansion`; case-study version in Chapter {ref}`chap:19`, Eq. {eq}`eq:ch19-expansion` | Trigger time, velocity model, asymmetry, and epoch-to-epoch evolution. |
| Type Ia distance toy model | Chapter {ref}`chap:20`, Eqs. {eq}`eq:ch20-typeia-theta` and {eq}`eq:ch20-typeia-likelihood` | Photospheric velocity, explosion time, angular-radius uncertainty, and radiative-transfer model. |

(appsec:a-propagation-network)=
## Propagation, Cosmology, and Quantum Networks
| Topic | Location | Check before use |
|:---------------|:----------------------|:--------------------------|
| Dispersion delay | Chapter {ref}`chap:14`, Eq. {eq}`eq:ch14-dispersion-delay` | Frequency units, DM decomposition, intra-channel smearing, and plasma approximation. |
| Faraday rotation | Chapter {ref}`chap:14`, Eqs. {eq}`eq:ch14-rm-angle` and {eq}`eq:ch14-rm-definition` | Polarization-angle convention, frequency coverage, intrinsic angle, and foreground subtraction. |
| Scattering convolution | Chapter {ref}`chap:14`, Eq. {eq}`eq:ch14-scattering-convolution` | Multipath propagation, pulse broadening, deconvolution, and selection effects. |
| Gravitational lensing | Chapter {ref}`chap:14`, Eqs. {eq}`eq:ch14-lens-equation` and {eq}`eq:ch14-time-delay-distance` | Mass model, source position, time delay, and microlensing. |
| Axion polarization rotation | Chapter {ref}`chap:15`, Eqs. {eq}`eq:ch15-axion-rotation` and {eq}`eq:ch15-event-likelihood` | Frequency dependence, time modulation, polarization calibration, and ordinary Faraday terms. |
| CMB likelihood | Chapter {ref}`chap:16`, Eq. {eq}`eq:ch16-cmb-likelihood` | $C_\ell$, covariance, foregrounds, and cosmic variance. |
| Quantum-network resources | Chapter {ref}`chap:17`, Eqs. {eq}`eq:ch17-entanglement-rate`, {eq}`eq:ch17-memory-time`, and {eq}`eq:ch17-effective-fidelity` | Link loss, storage time, frequency conversion, fidelity, and usable astronomical photon rate. |
| Observing error budget | Chapter {ref}`chap:18`, Eq. {eq}`eq:ch18-error-budget` | Statistical, calibration, background, model, and selection-function errors must all be included. |
| Proposal milestones | Chapter {ref}`chap:22`, Eqs. {eq}`eq:ch22-readiness-vector`, {eq}`eq:ch22-data-vector`, and {eq}`eq:ch22-milestone` | Acceptable observables, target precision, null tests, data products, and failure criteria. |