Angular Redshift Fluctuations in Cosmology

• Angular Redshift Fluctuations (ARF) are a cosmological observable that measure deviations in the mean redshift of matter tracers using only angular and redshift data.
• ARF is analyzed using spherical harmonics to extract angular power spectra that provide complementary insights into velocity fields, gravitational effects, and relativistic corrections.
• ARF enables tighter cosmological parameter constraints by reducing degeneracies when combined with angular density fields and CMB lensing, improving dark energy and non-Gaussianity measurements.

Angular Redshift Fluctuations (ARF) are a cosmological observable encoding the angular variations in the mean redshift of matter tracers across the sky, typically galaxies or quasars, within defined redshift shells. Unlike traditional three-dimensional clustering observables, ARF are constructed purely from angles and redshifts without requiring a fiducial cosmology to convert these into comoving positions. ARF are specifically sensitive to radial gradients—peculiar velocities, gravitational redshifts, and lensing magnification—that shift the redshift distribution within shells. As such, their two-point (and higher-order) angular statistics carry complementary information to standard angular density fields, particularly on large scales relevant to constraining primordial non-Gaussianity, cosmic growth, and tests of gravity and dark energy (Bermejo-Climent et al., 23 Jan 2026, Hernández-Monteagudo et al., 2020, Legrand et al., 2020).

1. Definition and Physical Interpretation

The ARF field, $\delta z(\hat n)$, measures, at each angular direction $\hat n$, the deviation of the local mean redshift from the global shell mean: $$\delta z(\hat n) = \frac{ \sum_{j \in \hat n} w_j (z_j - \bar z) }{ \sum_{j \in \hat n} w_j }$$ where $z_j$ are tracer redshifts in pixel $\hat n$, $w_j$ is a weight (unity or Gaussian in $z_j$), and $\bar z$ is the shell-wide mean. The observable can be generalized to photometric surveys, where Gaussian/redshift-dependent weights $W_j$ reflect the shell selection (Hernández-Monteagudo et al., 2024, Hernandez-Monteagudo et al., 2019).

Physically, ARF captures fluctuations in the line-of-sight distribution of tracers, including:

• Peculiar velocities, contributing $(1+z_H) v_{\parallel}/c$ to observed redshift.
• Gravitational redshifts and lensing magnification (significant for wide shells).
• Mean density perturbations, leading to density-weighted redshift deviations.

These effects make ARF a robust probe of cosmic velocity fields, the nature of gravity (through $f\sigma_8$ and $H(z)$), and primordial non-Gaussianity (Bermejo-Climent et al., 23 Jan 2026, Lima-Hernández et al., 2022).

2. Mathematical Formalism and Harmonic Statistic

ARF is analyzed in spherical harmonics: $$\delta z(\hat n) = \sum_{\ell m} a^{\rm ARF}_{\ell m} Y_{\ell m}(\hat n)$$ with angular power spectrum: $$C^{\rm ARF}_\ell = \langle |a^{\rm ARF}_{\ell m}|^2 \rangle$$

In linear theory, ARF cross- and auto-spectra relate to the 3D matter power spectrum $P(k)$ via: $$C^{XY}_\ell = 4\pi \int_0^\infty \frac{dk}{k} \mathcal{P}(k) I^X_\ell(k) I^Y_\ell(k)$$ where $\mathcal{P}(k) = k^3 P(k) / (2\pi^2)$, and $I^X_\ell(k)$ is the line-of-sight kernel for field $X$.

The ARF kernel, incorporating density and velocity terms, is: $$I^z_\ell(k) = \int d\chi \frac{dN}{d\chi} (z-\bar z) b_g(k, \chi) D(\chi) j_\ell(k\chi)$$ where $b_g$ is bias, $D$ is linear growth, and $j_\ell$ is the spherical Bessel function (Bermejo-Climent et al., 23 Jan 2026, Ferreira et al., 22 Apr 2025, Legrand et al., 2020).

For photometric surveys with uncertainty $\sigma_{\rm Err}$ the effective shell width is $$\sigma_{\rm tot}^2 = \sigma_z^2 + \sigma_{\rm Err}^2$$, which damps ARF power, especially at high redshift or large $\ell$ (Hernández-Monteagudo et al., 2024).

3. Velocity Sensitivity, Systematics, and Complementarity

ARF's unique weighting $(z-\bar z)$ makes it acutely sensitive to line-of-sight peculiar velocities:

$$\delta z(\hat n) \approx (1+z) \frac{v_\parallel(\hat n)}{c}$$

The velocity term dominates for narrow shells ($\sigma_z \lesssim 0.02$), and ARF exhibits strong ($>60\%$) correlation with projected velocity maps, but is almost uncorrelated with standard density maps under identical shell selection (Hernandez-Monteagudo et al., 2019, Hernández-Monteagudo et al., 2020, Ferreira et al., 22 Apr 2025).

Systematic errors in angular selection (stellar contamination, variable depth) are largely suppressed in ARF because redshift-independent biases contribute only to the shell monopole, which is subtracted by construction. This immunity holds provided systematics do not have redshift-dependent structure within the shell width (Hernandez-Monteagudo et al., 2019, Hernández-Monteagudo et al., 2024, Hernández-Monteagudo et al., 2020).

4. Applications in Cosmological Parameter Constraints

ARF carries independent and complementary information to the angular density field (ADF) and CMB lensing:

• Tomographic ARF maps measure $E(z) f(z) \sigma_8(z)$, providing constraints on cosmic growth and gravity index $\gamma$ (Hernández-Monteagudo et al., 2020).
• Joint analysis with ADF substantially reduces degeneracies, notably between $\sigma_8$ and galaxy bias $b_g$ (Legrand et al., 2020, Ferreira et al., 22 Apr 2025).
• For dark energy, combining ARF and ADF yields over an order-of-magnitude improvement in the Chevallier-Polarski-Linder Figure of Merit ($w_0$–$w_a$) compared to clustering alone (Legrand et al., 2020, Ferreira et al., 22 Apr 2025).
• The ARF+ADF Fisher determinant increases by $>10\times$ relative to ADF alone with optimal shell and binning schemes (Ferreira et al., 22 Apr 2025).
• In the context of primordial non-Gaussianity, ARF's dependence on (scale-dependent) bias enables $f_{\rm NL}$ constraints that improve on traditional density analyses; recent work combines ARF, ADF, and CMB lensing to derive $f_{\rm NL}=-3\pm 14$ at 68% CL, representing a 25% improvement over previous results and the tightest 2D two-point constraint after DESI DR1 (Bermejo-Climent et al., 23 Jan 2026).

5. Measurement, Surveys, and Methodology

ARF analysis proceeds by:

• Slicing a galaxy/quasar survey into tomographic redshift shells (Gaussian or top-hat selection).
• For each shell and sky pixel, computing the weighted mean redshift and constructing the fluctuation map.
• Projecting into spherical harmonics and extracting $C_\ell^{\rm ARF}$ using pseudo-$C_\ell$ estimators (e.g., NaMaster/MASTER), correcting for masks, pixelization, and shot noise (from random catalog realizations).
• Modeling covariance via correlated Gaussian simulations, empirical low-$\ell$ excess fitting, and constructing the full likelihood (MCMC/posterior sampling) to constrain cosmological parameters and nuisance terms, fixing baseline cosmology (e.g., Planck 2018) as necessary (Bermejo-Climent et al., 23 Jan 2026, Hernández-Monteagudo et al., 2024).

The Quaia quasar sample, built from Gaia DR3 and unWISE, is a benchmark for ARF analyses due to its sky coverage ($>60\%$), redshift range ($0.97$–$2.10$), and photometric redshift uncertainty ($\sigma_z \approx 0.06(1+z)$) (Bermejo-Climent et al., 23 Jan 2026). J-PLUS and other spectro-photometric surveys demonstrate ARF measurement feasibility at low $z$, setting bounds on $\sigma_{\rm Err}$ and bias evolution (Hernández-Monteagudo et al., 2024).

For BAO, ARF enables detection of the sound horizon feature in the joint $(\theta, \Delta z)$ plane, with maximal sensitivity for narrow shells and joint probe analysis (ADF+ARF), preserving information associated with peculiar velocities (Ferreira et al., 22 Apr 2025).

6. Relativistic Corrections and Cross-Correlations

ARF incorporates all linear-order relativistic corrections:

• Density, velocity (RSD, Doppler), lensing magnification, gravitational potential, and integrated Sachs-Wolfe (ISW) effects (Lima-Hernández et al., 2022).
• Velocity contributions dominate large angular scales ($\ell \lesssim 10$); lensing becomes significant for wider shells or higher $\ell$.
• ARF exhibits strong ($60$–$100\%$) cross-correlation with CMB lensing, and a robust ($S/N\sim4$–$5$) anti-correlation with ISW signal at high $z$ ($\sim2$), complementing standard clustering$\times$ISW which peaks at $z\lesssim1$ (Lima-Hernández et al., 2022).
• Combined ARF+ADF analysis enhances $\chi^2$ statistics for null ISW testing by up to $150\%$ for wide-shell analyses.

7. Implications for Future Large-Scale Structure Surveys

Angular Redshift Fluctuations are recommended as a standard observable for future wide-field surveys (DESI, Euclid, LSST, WFIRST, SKA), photometric and spectroscopic, owing to:

• Superior tomographic leverage—sub-percent redshift precision and narrow shells ($\sigma_z\lesssim0.01$) maximize velocity sensitivity.
• Robustness to angular systematics.
• Substantial gains in cosmological parameter constraints when jointly analyzed with ADF and CMB lensing.
• Ability to constrain growth rate, dark energy equation of state ($w_0, w_a$), gravity modifications ($\gamma$), and primordial non-Gaussianity ($f_{\rm NL}$).

ARF’s velocity dominance at low $\sigma_z$, complementarity with angular clustering, and straightforward harmonic analysis make it uniquely advantageous in the cosmological inference pipeline. Its application substantially tightens constraints on fundamental parameters and breaks degeneracies inherent in single-probe analyses (Bermejo-Climent et al., 23 Jan 2026, Legrand et al., 2020, Ferreira et al., 22 Apr 2025).