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Data Analysis for Precision 21 cm Cosmology

Adrian Liu, J. Richard Shaw

arXiv:1907.08211v2astro-ph.IMastro-ph.CO

TL;DR

21 cm cosmology can probe poorly surveyed cosmic epochs and vast ranges of scale, but extracting its signal requires careful analysis of severe foreground and instrumental systematics. This review presents analysis methods spanning measurement, foreground mitigation, statistical characterization, and pipeline design, emphasizing that successful experiments are software telescopes as well as hardware.

  • Problem

    21 cm observations could fill gaps in cosmic history and probe fundamental cosmology across a vast range of scales, but measurements remain entangled with polarized foregrounds and other contaminants.

  • Method

    The review provides a pedagogical synthesis of 21 cm analysis methods, including measurement equations, interference mitigation, foreground treatment, and statistical summaries.

  • Results

    The review establishes that 21 cm experiments require hardware-aware analysis pipelines, with m-mode analysis providing an exact mapping from polarized sky multipoles to measured visibility Fourier coefficients for transit telescopes.

  • Takeaways & Limitations

    Successful 21 cm cosmology depends on software algorithms and analysis pipelines alongside telescope hardware, because foregrounds and instrumental effects constrain usable cosmological information.

  • Takeaways & Limitations

    Foreground subtraction can over-subtract cosmological signal, while calibration errors and cable reflections can leak contamination beyond the foreground wedge; hybrid mitigation remains an open question.

Abstract

from arXiv · show

The redshifted 21 cm line is an emerging tool in cosmology, in principle permitting three-dimensional surveys of our Universe that reach unprecedentedly large volumes, previously inaccessible length scales, and hitherto unexplored epochs of our cosmic timeline. Large radio telescopes have been constructed for this purpose, and in recent years there has been considerable progress in transforming 21 cm cosmology from a field of considerable theoretical promise to one of observational reality. Increasingly, practitioners in the field are coming to the realization that the success of observational 21cm cosmology will hinge on software algorithms and analysis pipelines just as much as it does on careful hardware design and telescope construction. This review provides a pedagogical introduction to state-of-the-art ideas in 21 cm data analysis, covering a wide variety of steps in a typical analysis pipeline, from calibration to foreground subtraction to mapmaking to power spectrum estimation to parameter estimation.

1. INTRODUCTION

21 cm cosmology can survey an observationally incomplete cosmic timeline in three dimensions using neutral hydrogen, but realizing this promise requires sophisticated data analysis alongside specialized instruments.

  • 1. INTRODUCTION: Existing observations leave a substantial gap between roughly 400,000 and 1.5 billion years after the Big Bang, motivating complementary probes such as the redshifted 21 cm line.
  • 1. INTRODUCTION: 21 cm observations can map neutral hydrogen in three dimensions, using redshift for line-of-sight distance and reaching large survey volumes and early epochs.Neutral hydrogen also traces matter, ionization state, and temperature.
  • 1. INTRODUCTION: Recent progress includes cross-correlation detections at z < 1, a tentative sky-averaged signal near z ∼17, and upper limits from spatial-fluctuation experiments.
  • 1. INTRODUCTION: The review introduces analysis ideas and problems across 21 cm cosmology, treating theory and instrumentation mainly as background for calibration, foreground subtraction, mapmaking, and statistical inference.

2. SCIENCE WITH THE 21 cm LINE

The 21 cm line encodes cosmological and astrophysical information through hydrogen’s spin temperature, brightness temperature, and spatial fluctuations across cosmic epochs. Its signals can probe the Dark Ages, Cosmic Dawn, reionization, and broader cosmological parameters.

  • 2. SCIENCE WITH THE 21 cm LINE: The 21 cm brightness temperature contrasts the spin temperature with the CMB and depends on neutral fraction, density, velocity gradients, and thermal state.The optical depth is typically ≲4%, motivating a Taylor expansion of the brightness-temperature expression.
  • 2. SCIENCE WITH THE 21 cm LINE: 21 cm observations target either the sky-averaged global signal or spatial fluctuations reconstructed as maps or summarized statistically.
  • 2. SCIENCE WITH THE 21 cm LINE: During the Dark Ages, linear matter fluctuations provide a clean high-redshift probe with many Fourier modes and potential sensitivity to primordial-spectrum features, running, non-Gaussianity, and relic gravitational waves.
  • 2. SCIENCE WITH THE 21 cm LINE: During Cosmic Dawn, Lyα coupling produces an absorption signal carrying information about the density field and Lyα flux, while reionization generates clustered ionized regions around overdense galaxy-forming regions.
  • 2. SCIENCE WITH THE 21 cm LINE: Combining intensity mapping with BAO, power-spectrum shape, and future CMB data could constrain the sum of neutrino masses to ≲20 meV.

3. OBSERVATIONAL 21CM COSMOLOGY

21 cm observations require instruments designed for faint, broadband, three-dimensional measurements and analysis methods that connect interferometric sampling to maps and power spectra. Instrument geometry, observing strategy, and computational cost jointly determine accessible modes and systematics.

  • 3. OBSERVATIONAL 21CM COSMOLOGY: 21 cm instruments generally require high sensitivity, broad frequency coverage, access to relevant spatial scales, and temporal stability because the signal is faint and contaminants are strong.Designs may require 1000 hrs or more of integration time.
  • 3. OBSERVATIONAL 21CM COSMOLOGY: Interferometers correlate antenna pairs to measure visibilities that approximately sample sky Fourier modes, which can be transformed into images or statistical measurements.The primary beam smears each baseline’s nominal Fourier-mode response.
  • 3. OBSERVATIONAL 21CM COSMOLOGY: Earth rotation moves baselines through the uv plane, while conjugate points carry no independent information because the sky temperature is real-valued.
  • 3. OBSERVATIONAL 21CM COSMOLOGY: Power spectra capture important reionization physics: as ionized bubbles grow, power shifts from high k to low k before declining when the neutral fraction becomes sufficiently small.
  • 3. OBSERVATIONAL 21CM COSMOLOGY: Regular arrays can reduce visibility-processing cost from the quadratic baseline count toward O(Nant log Nant), while full-sky approximations may require more rigorous spherical treatments.

4. CURRENT STATUS OF OBSERVATIONS

21 cm observations have progressed from theoretical promise toward increasingly stringent upper limits and cross-correlation detections, using instruments with distinct designs and analysis pipelines.

  • Current results: Recent observations include cross-correlation detections at z < 1 and a tentative sky-averaged signal near z ∼17.
  • Instruments: Instrument designs range from general-purpose interferometers and single dishes to arrays optimized for EoR power-spectrum sensitivity.GMRT, MWA, PAPER, LOFAR, GBT, HERA, CHIME, HIRAX, and OVRO-LWA exemplify differing observing strategies and capabilities.
  • Analysis approaches: MWA analyses use multiple pipelines for validation, while PAPER bypasses mapmaking and estimates power spectra directly from visibilities with the delay-spectrum approach.
  • Analysis approaches: OVRO-LWA results combine m-mode mapmaking, Karhunen–Loève foreground projection, and quadratic-estimator power-spectrum estimation.
  • Current results: No 21 cm auto-power-spectrum detection has yet been made, although published upper limits have become increasingly stringent.Figure 7 summarizes limits evaluated at each experiment’s most competitive k bin, so direct comparison is intrinsically imperfect.

5. FUNDAMENTALS OF INTERFEROMETRY

Interferometry models how sky emission, antenna response, polarization, and propagation combine into measured voltages and visibilities. Its fundamental measurement equation incorporates curved-sky and polarization effects exactly.

  • Measurement fundamentals: An antenna measures a direction-weighted sum of incoming electromagnetic plane waves, with its complex vector beam encoding sensitivity, polarization orientation, and relative delay.
  • Instrument response: The antenna beam is difficult to measure or simulate accurately, making beam characterization a major challenge for 21 cm cosmology.
  • Sky statistics: The sky electric field is modeled statistically as a Gaussian random signal whose two-point statistics encode its emission properties.Spectral and spatial incoherence simplifies the relationship between the measured field and its statistical description.
  • Polarization: The coherency matrix and four Stokes parameters describe total intensity and polarization across sky position and frequency.Stokes V is generally assumed negligible for typical astrophysical emission, while the statistical properties can remain correlated across frequencies and angles.
  • Measurement fundamentals: The fundamental interferometric equation maps sky emission through antenna responses into measured data while including curved-sky and polarization effects.

6. INSTRUMENTAL SYSTEMATICS

Instrumental systematics alter the measured signal through gains, beams, polarization mixing, and cross talk, demanding calibration precision far beyond ordinary radio measurements.

  • Systematic requirements: Systematics must often be understood better than the foreground-to-signal ratio, which can be ≲10^-4, to prevent foreground leakage into 21 cm data.
  • Gain variations: Unknown and time-varying gains limit combining baselines and frequencies for unbiased mapmaking and foreground removal.Thermal environments produce partially independent, difficult-to-predict variations across signal chains.
  • Gain variations: Cross talk requires gains to be represented as complex matrices rather than scalars, complicating calibration between polarizations or nearby signal paths.
  • Primary beams: Frequency-dependent, spatially complex beams and feed-to-feed differences can generate non-redundancy and bias redundant-calibration solutions.Different analysis methods impose different requirements on beam knowledge, sometimes approaching the foreground-to-signal precision level.
  • Polarization leakage: Polarization leakage can transfer spectrally structured polarized emission into total intensity, hindering foreground removal.Leakage of feed-oriented Q into I arises from beam-amplitude mismatch, while U leakage arises from non-orthogonal beam responses.

7. FOREGROUNDS

Foregrounds are far brighter than the cosmological 21 cm signal and must be separated using their approximate spectral smoothness, but instrumental effects can proliferate foreground modes and remain a central unresolved obstacle.

  • Foreground brightness: Foregrounds dominate the sky relative to the 21 cm signal, with Galactic synchrotron emission reaching hundreds of kelvin near 408 MHz and increasing toward lower frequencies.
  • Inference challenge: Unlike CMB measurements, 21 cm observations have unique cosmological information in every frequency channel, creating too many cosmological and foreground variables for the available measurements.
  • Spectral separation: Most foreground-mitigation methods rely on foreground spectral smoothness to discard foreground modes while retaining cosmological modes.This assumption is supported by empirical and physical models but may fail if non-smooth components matter at required precision.
  • Polarization foregrounds: Polarized synchrotron emission undergoes Faraday rotation, producing rapid spectral variation that can leak into total intensity and limit foreground removal.At frequencies around 300 MHz, typical rotation measures can rotate the polarization vector fully within roughly 10 MHz.
  • Instrumental coupling: Instrumental effects can cause foreground mode proliferation, leaking contamination into spectral modes where it is not expected.The interaction between foregrounds and instrumental systematics is described as the field’s chief obstacle, with no definitive solution yet.

8. INTERLUDE: FROM OUR UNIVERSE TO VISIBILITIES AND BACK

The paper traces how cosmological radiation becomes interferometric visibilities corrupted by foregrounds and instrumental systematics, then outlines analysis steps that recover scientific information.

  • Interferometers transform radiation carrying cosmological and astrophysical information into visibilities affected by foreground contamination and instrumental systematics.
  • Calibration, mapmaking, power-spectrum estimation, and foreground mitigation are presented as successive analysis tools for extracting 21 cm information.
  • Power-spectrum estimation can be performed with or without first constructing a sky map.

9. CALIBRATION

Calibration seeks to recover true visibilities from measurements containing unknown antenna gains and noise. Because the unrestricted problem is underdetermined, practical methods impose assumptions about the sky, instrument, or both.

  • 9. CALIBRATION: Calibration solves for unknown complex antenna gains to recover the true visibility from measured visibility corrupted by gains and noise.The measured visibility is associated with a baseline between antennas i and j.
  • 9. CALIBRATION: With Nant antennas, the general calibration problem has more unknown true visibilities and gain factors than measured baselines, making it underdetermined.
  • 9.1. Calibration by fitting to a sky model: A single bright point-source model reduces calibration to Nant(Nant −1) measurements and 2Nant + 1 unknown quantities, making the problem solvable.
  • 9.1. Calibration by fitting to a sky model: Sky-model calibration iteratively alternates between fitting complex gains and updating model visibilities until convergence.The approach can use increasingly realistic sky models, including multiple point sources.
  • 9.1. Calibration by fitting to a sky model: Diffuse low-frequency emission complicates point-source calibration, motivating long-baseline calibration while leaving diffuse-model parametrization an open question.Long baselines are used because diffuse emission primarily affects short baselines, whereas localized sources contribute across baseline lengths.
  • 9.2. Calibration using self consistency: Redundant calibration reduces sky-emission assumptions but leaves absolute amplitude, phase, and linear phase-gradient degeneracies unresolved.Hybrid covariance-based calibration can relax perfect redundancy and model partial correlations between nominally redundant baselines.

10. MAP-MAKING

Map-making converts calibrated visibility data into sky maps through statistical inversion, but direct visibility analysis can be preferable for power-spectrum estimation. Linear estimators face noise and unmeasured-mode problems, while m-mode formalism enables computationally efficient, essentially exact polarized imaging for drift-scan telescopes.

  • Linear Map-makers: Map-making is essential for producing interpretable sky products but is not required for further 21 cm analysis, especially power-spectrum estimation.Analysis from maps can sometimes be more problematic than proceeding directly from visibilities.
  • Linear Map-makers: Map-making statistically inverts the visibility-to-sky forward model, accounting for Gaussian noise and a generally non-invertible projection matrix.The discretized data are visibility samples, while the sky is represented by pixels and related through a forward-projection matrix.
  • Linear Map-makers: The maximum-likelihood map can severely upweight noise-dominated, low-sensitivity modes, adding large amounts of noise below the telescope’s resolution limit.Regularization is therefore required to control the inversion of low-sensitivity modes.
  • Map-making with m-modes: For drift-scan telescopes, m-mode formalism is essentially exact, wide-field, fully polarized, and often makes otherwise impractical matrix operations tractable.Its generality is limited because the formalism applies only to drift-scan telescopes.
  • Map-making with m-modes: m-mode operations save O(mmax) for matrix-vector products and O(mmax^2) for matrix-matrix operations; with mmax ≳10^3, computation can be reduced by ≳10^6.The savings apply when the relevant approximations allow operations to be performed independently for each m.

11. POWER SPECTRUM ESTIMATION

Power-spectrum estimation provides a central but incomplete summary of 21 cm fluctuations, whose non-Gaussian structure motivates complementary statistics. The paper develops quadratic estimators and specialized delay-spectrum and m-mode methods, emphasizing trade-offs among variance, window functions, correlations, computational cost, and foreground sensitivity.

  • Scope and limitations: Power spectra do not capture all 21 cm information because reionization produces strongly non-Gaussian brightness-temperature fluctuations beyond two-point statistics.A brightness-temperature histogram shows a zero-temperature spike and skewness that are absent from a power-spectrum description.
  • Quadratic estimators: Quadratic estimators use the data covariance to estimate power spectra and can generalize to angular, cross-angular, and spherical Fourier-Bessel statistics.Their flexibility follows from treating covariance responses to bandpowers in a common formalism.
  • Estimator trade-offs: The M = F−1 estimator is unbiased and reaches the Cramer–Rao error bound, but its narrow window functions can produce larger errors than variance-minimizing alternatives.Narrow k bins contain less information, whereas broader windows average over more independent modes.
  • Estimator trade-offs: The minimum-variance quadratic estimator achieves the smallest possible error bars, while normalization choices trade error size against window widths and error correlations.The framework requires inverse-covariance weighting, but different invertible normalization matrices preserve information while changing estimator properties.
  • Scope and limitations: Gaussian assumptions can make simplified bandpower covariance expressions inaccurate during the Epoch of Reionization, where ionized bubbles generate non-Gaussian fluctuations.The full covariance expression does not require Gaussianity, but its simpler reduction does.
  • Specialized approaches: Delay-spectrum estimation remains close to baseline visibilities, relaxing some calibration and late-stage data-cut requirements, while m-mode analysis preserves computational savings for power-spectrum estimation.m-mode independence enables separate estimates for each m and subsequent averaging, but relies on statistical isotropy that foregrounds violate.

12. FOREGROUND MITIGATION

Foreground mitigation in 21 cm analyses combines spectral modeling, mode projection, wedge avoidance or decorrelation, cross-correlation, and RFI flagging, each with distinct trade-offs. Instrument chromaticity produces the foreground wedge, while calibration and intrinsic foreground structure can extend contamination beyond ideal predictions.

  • 12.1.1. Parametric fits: Smooth-function subtraction is motivated by spectrally smooth foregrounds but performs poorly on fine angular scales and can be difficult to apply in log-log space because interferometers discard the zero mode.The discarded angular mean means spectra are not guaranteed to remain positive.
  • 12.1.3. Mode projection: Blind data-driven projection captures unknown instrument and sky characteristics but risks over-subtracting cosmological signal; KL modes reduce this risk by ordering modes by signal-to-contaminant ratio.The number of modes to project remains uncertain, and projecting more modes inevitably begins to remove signal.
  • 12.1.5. Avoiding foregrounds in the foreground wedge: Foreground mode mixing produces the characteristic wedge: contamination decreases toward higher k∥, with constant-contamination lines satisfying k⊥∝k∥.The wedge’s detailed profile can become more complicated with realistic beams and foregrounds, but its overall shape remains robust.
  • 12.1.5. Avoiding foregrounds in the foreground wedge: Intrinsic foreground spectral structure can move the wedge to higher k∥, while calibration errors and cable reflections can leak contamination beyond it.These effects reduce the Fourier space available for cosmological measurements and motivate hybrid subtraction-plus-avoidance strategies.
  • 12.1.6. Decorrelating the foreground wedge: M-matrix wedge decorrelation has mixed success: it can reduce high-k foreground contamination but suffers numerical instability and relies on stationarity violated by frequency-dependent foreground amplitudes.Its stability depends on small differences in long window-function tails, while the underlying inversion assumption fails for nonstationary foregrounds.
  • 12.4. Radio Frequency Interference: Machine-learning RFI flaggers are generally competitive with state-of-the-art algorithms and can be orders of magnitude faster after training, but often miss single-pixel time-frequency blips.Their speed can provide substantial computational savings for large next-generation telescopes with enormous data volumes.

13. POWER SPECTRA TO PARAMETERS/SCIENCE

Power spectra compress 21 cm fluctuations for comparison with theoretical models and parameter inference, but evolving light-cone fields and expensive predictions require extensions to standard analyses.

  • Power spectra and inference: Power spectra provide a useful but potentially lossy compression of non-Gaussian 21 cm fields for constraining theoretical models.Compression is lossless for Gaussian fluctuations but can discard information otherwise.
  • Parameter estimation: Bayesian inference compares measured power-spectrum bandpowers with model predictions to sample posterior distributions for physical parameters.The data are power-spectrum bandpowers, while the inferred parameters belong to the physical model; MCMC sampling produces the posterior.
  • Light-cone effects: Light-cone evolution breaks translation invariance, so the power spectrum no longer captures all field information.The survey spans changing redshifts, producing off-diagonal Fourier-space covariance.
  • Light-cone effects: Frequency-pair angular spectra, wavelet bases, or restricted frequency ranges offer alternatives for handling evolving fields, with different completeness and modelling trade-offs.Cℓ(ν,ν′) captures frequency correlations, wavelets seek a more diagonal covariance basis, and ignoring evolution may be acceptable over sufficiently small ranges.
  • Computationally expensive model predictions: Emulators replace expensive exact model evaluations with interpolation models, enabling MCMC forecasts and analyses across astrophysical and cosmological parameter spaces.They are built from exact evaluations at selected parameter-grid points and have been demonstrated for 21 cm forecasts.
  • Model comparison: Bayesian evidence can distinguish reionization model classes and help select foreground parameterizations, including in applications to global-signal data.The cited uses include forecasts for power-spectrum measurements and evaluation of different foreground models.

14. POWER SPECTRUM ALTERNATIVES AND VARIANTS

Beyond the conventional 21 cm power spectrum, cross-correlations, higher-order statistics, image-based methods, and voxel distributions can recover complementary information, though practical limitations remain.

  • 14.1. Cross correlations: Cross power spectra extend 21 cm analysis by combining the line with other structure tracers, including galaxies, CO, [CII], and emission-line maps.The review discusses candidate tracers across redshift, including galaxy surveys near z ∼0.8 and several intensity-mapping lines at higher redshift.
  • 14.1. Cross correlations: Cross-correlation signal-to-noise requires overlap in both sky/redshift coverage and Fourier-space scales, so foreground cleaning can remove modes needed for the measurement.Suppressing smooth line-of-sight modes can destroy low-k∥ modes and weaken cross-correlations with projected fields.
  • 14.2. Higher-order statistics: Higher-order statistics exploit non-Gaussian mode coupling from nonlinear evolution or lensing to access information absent from two-point power spectra.These approaches include direct higher-point measurements and tidal reconstruction, while lensing provides another source of mode correlations.
  • 14.2. Higher-order statistics: Bispectrum measurements may offer favorable sensitivity scaling, but their feasibility remains uncertain because studies often assume highly precise foreground removal and omit calibration errors.The cited sensitivity scaling is t^-3/2 for the bispectrum versus t^-1 for the power spectrum.
  • 14.3. Bispectrum phases: Closure-phase measurements are independent of per-antenna gain calibration factors, and delay-spectrum processing may help separate foregrounds from cosmological signals.The phase of the angular bispectrum is identified with the closure phase in radio astronomy.
  • 14.2. Higher-order statistics: Voxel PDFs provide an alternative summary of high-redshift 21 cm non-Gaussianity without requiring an infinite hierarchy of higher-order correlation functions.The review presents the voxel probability distribution as an alternative to computing the trispectrum and higher moments.
  • 14.4. Imaging: 21 cm images can provide probabilistic context for high-redshift galaxy environments even when resolution prevents definitive bubble-membership assignments.Maps can predict the probability that a pixel is ionized or neutral despite angular-scale mismatches.
  • 14.4. Imaging: CNNs have recovered astrophysical parameters and performed model selection in simulations, but foregrounds and systematics must be addressed before robust real-data applications.Examples include ionizing efficiency, reionization duration, and distinguishing galaxy-driven from AGN-driven scenarios.

15. GLOBAL SIGNAL MEASUREMENTS

The global 21 cm signal provides information about ionization, heating, and Lyα coupling, but extracting it is difficult because smooth foregrounds can resemble the cosmological spectrum. Current experiments constrain reionization and test unusual Cosmic Dawn scenarios while systematic uncertainties remain important.

  • Global-signal measurements average the 21 cm brightness over all sky directions, yielding the monopole mode with information independent of spatial fluctuations.
  • A multiplicative degeneracy between fσ8 and T_b^2 can be broken using mildly nonlinear terms, with forecasts suggesting competitive fσ8 constraints.
  • The global signal traces ionization history during reionization and X-ray heating and Lyα coupling at higher redshifts.
  • Foreground mitigation is especially difficult because the smooth cosmological signal can resemble spectrally smooth foregrounds, potentially limiting reionization experiments to ruling out very rapid histories.
  • EDGES data ruled out reionization durations Δz ≲1 near z_mid ∼8.5 and Δz ≲0.4 for 6.5 < z_mid < 14.8, while combined analyses favored T_vir min between 10^4.5 and 10^5.7.
  • The claimed EDGES absorption feature at 78 MHz would imply an unusually cold intergalactic medium, but instrumental and analysis-systematic concerns remain under discussion.

16. CONCLUSION

21 cm cosmology has moved toward experimental reality, but its scientific return depends on controlling extreme sensitivity, foregrounds, systematics, and interconnected analysis choices. The review emphasizes that hardware-aware software pipelines remain essential and that no community consensus yet exists for the optimal analysis path.

  • 21 cm experiments increasingly have sufficient sensitivity for detection and characterization, while higher signal-to-noise also helps diagnose and mitigate systematics.
  • The central challenges are extreme sensitivity, foreground mitigation, and systematic control, whose interactions link calibration, map-making, power spectra, foreground removal, and parameter estimation.
  • 21 cm experiments are “software telescopes” because successful analysis pipelines must remain closely linked to the hardware for both spatial-fluctuation and global-signal measurements.
  • There is still no community consensus on the optimal route from raw radio-telescope data to the science enabled by redshifted 21 cm measurements.

A. DRIFT SCAN OBSERVATIONS

Drift-scan and tracking telescopes access Fourier modes differently as Earth rotates. Drift scans keep a fixed overall uv-plane footprint while time variation resolves modes within the primary-beam footprint, whereas tracking observations can expand mode coverage through rotation synthesis.

  • The flat-sky treatment is appropriate only when the primary beam samples a limited declination range, and the final discrete transform is exact only for particular integer mode conditions.
  • For drift scans, the measured modes are fixed and only north-south modes are convolved, so observed m modes map directly to sky m modes.
  • The small-angle limitation can be removed azimuthally with a Bessel-function treatment, but not in the polar direction because it is not strictly periodic.
  • Drift-scan observations measure a fixed set of Fourier modes, while tracking observations trace an ellipse through the uv-plane and fill gaps as Earth rotates.
  • Tracking and drift-scan analyses use different time behavior: the tracking effective Fourier vector varies with φ, whereas the drift-scan transform exploits a simple translational shift.
  • Extended drift-scan observing does not enlarge the overall uv-plane area, so instantaneous coverage must already be sufficient; a full sidereal day instead accesses individual azimuthal m modes.

B.2. Measuring P(k⊥, k∥) or P(k)

The quadratic-estimator framework supports power-spectrum measurements in multiple Fourier and angular bases, including P(k⊥, k∥), P(k), angular spectra, cross-spectra, and direct visibility estimators. Its simplest Fourier-space interpretation relies on large-survey approximations, while the full matrix formulation handles finite volumes.

  • Power spectra can be binned as P(k⊥, k∥) for systematic diagnosis or as scalar P(k) for final isotropic cosmological results.
  • For Fourier-space data, power-spectrum estimation reduces to inverse-covariance weighting followed by squaring the component at the wavevector of interest.
  • The simple Fourier-space interpretation assumes a large survey volume, whereas the full matrix quadratic estimator accommodates finite-volume effects.
  • The same quadratic-estimator formalism can estimate angular Cℓ spectra by forming an inverse-covariance-weighted map, transforming it into spherical harmonics, and summing over m.
  • Cross-power spectra between frequencies use cross-multiplication of spherical-harmonic coefficients from different frequencies rather than squaring one set.
  • Applying the estimator directly to visibilities yields a power-spectrum estimator, with frequency-dependent phase terms incorporating foreground-wedge phenomenology.
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