CosmicFlows-4 TF W1 Subsample

• The paper presents a robust calibration of the Tully-Fisher relation using mid-IR WISE W1 photometry and HI 21 cm spectroscopy, achieving a scatter of ~0.52 mag.
• It employs advanced photometric, kinematic, and selection methodologies with comprehensive corrections for inclination, extinction, and systematic biases.
• The resulting distances underpin analyses of peculiar velocities, Hubble constant estimations, and large-scale structure mapping in the local universe.

The CosmicFlows-4 Tully-Fisher W1 Subsample refers to the mid-infrared (3.4 μm) component of the fourth CosmicFlows project, which provides precise distances to spiral galaxies via the Tully-Fisher (TF) relation using photometry from the WISE W1 band. This subsample is a cornerstone for mapping peculiar velocities and large-scale structure in the local universe due to its robust calibration, large sample size, and minimized systematics achieved through consistent photometry and comprehensive corrections. The W1 TF subsample underpins modern cosmological analyses of the Hubble constant, the growth rate of structure, and bulk flows, and serves as a foundational data set for velocity field reconstructions at low redshift.

1. Photometric, Kinematic, and Selection Methodology

The CosmicFlows-4 TF W1 calibration utilizes WISE W1 (3.4 μm) imaging for mid-infrared luminosity coupled to rotational velocity estimates from HI 21 cm spectroscopy. The selection pipeline identifies spiral galaxies with robust HI line widths and secure W1 photometry, imposing an inclination cut (typically $i>45^\circ$) to ensure reliable deprojection of the line-of-sight velocity. The HI profile widths are measured and homogenized using a reference algorithm (e.g., Wmean50, as in the All Digital HI catalog (Courtois et al., 2014, Dupuy et al., 2021)), and all velocities are corrected for instrumental effects, spectral resolution, redshift, and turbulence, then deprojected as $W_i = W/\sin i$.

WISE W1 magnitudes are carefully corrected for foreground extinction, internal attenuation, k-correction, and aperture effects (see $M^{b,i,k,a}_{W1}$ in (Neill et al., 2014)). The final "adjusted" magnitude used in the calibration is

$$m^*_{W1} = m_{\rm total, W1} - A^{W1}_b - A^{W1}_k - A^{W1}_a - A^{W1}_i,$$

where each term corresponds to a specific correction detailed in the CosmicFlows-4 methodology (Kourkchi et al., 2020).

The sample is HI flux limited and includes thousands of galaxies with robust cross-matching to WISE, allowing uniform coverage across the sky (including special provisions for low-latitude and Zone of Avoidance fields).

2. Tully-Fisher Relation Calibration and Form

The calibration of the TF relation in W1 proceeds via simultaneous or iterative forward-modelling of the galaxy population. The fiducial (mid-infrared) TF relation has the canonical form:

$$M_{W1} = a_0 + a_1(\log W_i - 2.5) + \begin{cases} 0, & \text{if linear regime}\ a_2 (\log W_i - 2.5)^2, & \text{if curvature is significant} \end{cases}$$

Quadratic ("curved") fits are preferred for the W1 band, as evidence supports curvature especially at the high mass/luminosity end (Neill et al., 2014, Bell et al., 2022). Recent CosmicFlows-4 analyses yield representative parameters such as

• $a_0 \approx -20.48$, $a_1 \approx -8.36$, $a_2 \approx 3.60$ (curved fit, (Neill et al., 2014))
• Slope values of $-9.56$ to $-10.08$ are typical for linear or slightly curved parametrizations (Neill et al., 2014, Kourkchi et al., 2020, Bell et al., 2022) with a total scatter of $\sigma_{W1} \sim 0.52$–$0.68$ mag dominated by intrinsic and observational uncertainties.

All fits account for statistical biases (such as cluster incompleteness/Malmquist bias), and a morphological type correction is applied since early-type spirals are systematically offset from the fiducial Sc/Sd relation (Bell et al., 2022).

3. Systematic Corrections and Empirical Refinements

A suite of empirical refinements further optimizes the TF relation:

• Color corrections: A tight correlation exists between TF residuals and optical–infrared colors (e.g., $I - W1$), allowing definition of "pseudo-magnitudes" with reduced scatter (Neill et al., 2014, Kourkchi et al., 2020).
• Surface brightness and HI content: Residuals can also be minimized using corrections based on effective surface brightness and gas-to-light ("pseudo-color," such as $m_{21} - W1$) (Kourkchi et al., 2020, Kourkchi et al., 2020).
• Dust attenuation modelling: Where infrared photometry is unavailable, the internal extinction correction is predicted using a machine-learning random forest approach trained on galaxies with both optical and WISE data (Kourkchi et al., 2020).
• Calibration zero-point anchoring: The absolute scale of the TF relation is set using external calibrators with independent precise distances (e.g., Cepheids, TRGB stars, SNIa hosts) (Kourkchi et al., 2020, Boubel et al., 7 Aug 2024).
• Baryonic TF: In the baryonic Tully-Fisher formalism, stellar and gas mass estimates (using W1 luminosities and gas scaling from HI flux) yield a relation of the form $\log M_{\rm bary} = a\, \log W + b$ (Kourkchi et al., 2022, Duey et al., 2 Apr 2024).

Scatter in the W1 TF relation post-correction is typically $\sim$0.46–0.54 mag, and color/surface-brightness corrections further reduce this towards $\sim$0.45 mag (Neill et al., 2014, Kourkchi et al., 2020).

4. Cosmological Applications: Distances, Peculiar Velocities, and the Hubble Constant

The CosmicFlows-4 W1 TF subsample yields high-fidelity distance moduli for $\sim$10,000 galaxies to $\sim$200 Mpc. These distances provide the foundation for:

• Derivation of radial peculiar velocities: $v_{\rm pec} = v_{\rm obs} - H_0 d$.
• Mapping large-scale velocity and density fields: Wiener filtering and Hamiltonian Monte Carlo techniques reconstruct the underlying gravitational field and bulk flow structure (Courtois et al., 2022, Duangchan et al., 29 Jul 2025).
• Measurement of cosmological parameters: Combining distances with CMB-frame velocities yields robust local Hubble constant values:
  • $H_0 = 74.4 \pm 1.4\,(\rm stat) \pm 2.4\,(\rm sys)$ km s$^{-1}$ Mpc$^{-1}$ (clusters, (Neill et al., 2014))
  • $H_0 = 75.1 \pm 0.2$ (stat) km s$^{-1}$ Mpc$^{-1}$ (CosmicFlows-4 W1 main sample, (Kourkchi et al., 2020))
  • $H_0$ from contemporary Bayesian forward-modelling ranges from 73.3 to 75.8 km s$^{-1}$ Mpc$^{-1}$, with statistical–systematic uncertainties 0.2–3.5 km s$^{-1}$ Mpc$^{-1}$, depending on calibration choices (Boubel et al., 7 Aug 2024, Duangchan et al., 29 Jul 2025).
• Quantification of the growth rate of cosmic structure: The product $f\sigma_8$ measured from pairwise velocity statistics is consistent with Planck CMB values within errors, providing further constraints on cosmology (Courtois et al., 2022, Boubel et al., 2023).

5. Statistical Uncertainties and Systematic Limitations

The total statistical uncertainty per distance is set primarily by the intrinsic TF scatter (typically $\sim0.45$–$0.55$ mag in W1), HI line width errors, and photometric calibration. Propagation into the Hubble constant yields statistical errors as low as 0.2–1.2 km s$^{-1}$ Mpc$^{-1}$ for the full sample.

Systematic uncertainties remain dominated by:

• Zero-point calibration: Differences in fundamental calibrators (Cepheid vs. TRGB vs. SNIa) produce Hubble constant shifts of several percent (Boubel et al., 7 Aug 2024).
• Photometric methodology: Differences in extraction (isophotal vs. asymptotic magnitudes, masking, sky subtraction) introduce biases at the level of a few percent (Duey et al., 2 Apr 2024).
• Sample selection and Malmquist bias: Residual incompleteness or selection-dependent biases can differentially affect the low-luminosity end.
• Inclination corrections and HI profile modeling: Imperfect deprojection and differences in line width estimators (e.g., difference between W50 on HI vs. CO, gaussian double-peak fitting for CO as in (Tiley et al., 2016)) can induce systematic offsets.

Despite these, global internal consistency checks and cross-calibration across different photometric bands, survey methods, and calibrator subsamples bolster the reliability of the W1 distance scale.

6. Empirical Performance and Comparisons

Comparisons with other TF samples reveal:

• WISE W1, Spitzer 3.6 μm, and near-infrared K-band TF relations exhibit near-identical slopes and dispersions, indicating robustness and minimal sensitivity to stellar population variations (Lagattuta et al., 2013, Neill et al., 2014).
• Zero-point offsets between W1 and 3.6 μm are typically $0.1$–$0.2$ mag, well-characterized and correctable, and the median $(K-W1)$ color is $\sim$0, indicating both bands trace the old stellar content (Duey et al., 2 Apr 2024).
• The TF distances from the WALLABY pre-pilot survey, when cross-calibrated with WISE W1, achieve 5–10% distance precision, confirming that new HI-based samples are fully compatible with the W1 scale (Courtois et al., 2022).

7. Large-Scale Velocity Field Reconstruction and Isotropy

Application of the CF4 TF W1 distances to peculiar velocity reconstructions has established:

• The reconstructed bulk flow and growth rate are compatible with $\Lambda$CDM, with some component-level tensions (notably along supergalactic X) at the $3$–$4 \sigma$ level for certain distances; interpretation remains under debate (Duangchan et al., 29 Jul 2025).
• No statistically significant evidence is found for a dipolar (anisotropic) Hubble constant in the W1 subsample when selection effects, peculiar velocities, and Bayesian model comparison (e.g., Bayes factors) are fully accounted for. Any inferred zero-point dipole is consistent with local flow features or systematics, not with a fundamental anisotropy in $H_0$ (Stiskalek et al., 18 Sep 2025).

Summary Table: Key Formulae in the W1 Tully-Fisher Calibration

Relation Type	Formula (W1 band)	Typical Scatter
Linear TFR	$M_{W1} = a_0 + a_1(\log W_i - 2.5)$	$0.54$ mag
Curved TFR	$$M_{W1} = a_0 + a_1(\log W_i - 2.5) + a_2(\log W_i - 2.5)^2$$	$0.52$ mag
Baryonic TFR	$\log M_{\rm bary} = a \cdot \log W + b$	$0.17$ dex
Adjusted Apparent Magnitude	$$m^*_{W1} = m_{\rm total, W1} - A^{W1}_b - A^{W1}_k - A^{W1}_a - A^{W1}_i$$	---
Dust Correction (empirical, machine learning)	$A^{W1}_i = \gamma_{W1} \cdot \mathcal{F}_{W1}(i)$	---

Conclusion

The CosmicFlows-4 Tully-Fisher W1 subsample undergirds the most extensive and robust measurement of spiral galaxy distances at low redshift. Its precision and comprehensive correction methodology enable stringent tests of cosmological isotropy, the calibration of the local Hubble constant, and the reconstruction of velocity and density fields in the local universe. The W1 band, owing to its minimal dust sensitivity and stable zero point, serves as the preferred wavelength for TF distance work in the CosmicFlows framework and future large HI surveys (e.g., WALLABY) are expected to increase these data sets several-fold, further enhancing the power and reach of TF-based cosmography.