Cosmicflows-4 Compilation

• Cosmicflows-4 is the largest catalog of galaxy distances and peculiar velocities, unifying eight measurement methods through robust Bayesian calibration.
• It synthesizes photometric, spectroscopic, and geometric indicators to construct a high-fidelity 3D map of the local universe’s velocity and density fields.
• The compilation enables precise cosmological tests, yielding accurate measurements of the Hubble constant, bulk flows, and structure formation.

The Cosmicflows-4 (CF4) compilation is the largest and most comprehensive catalog of galaxy distances and inferred peculiar velocities assembled to date, comprising approximately 56,000 galaxies grouped into 38,000 systems. It synthesizes photometric, spectroscopic, and geometric distance indicators to construct a high-fidelity three-dimensional map of the local universe's velocity and density fields, enabling precise cosmological tests and measurements of the Hubble constant and bulk flows. CF4 is the result of systematic data acquisition from multiple international radio and optical surveys, advanced calibration methods, and robust Bayesian statistical merging.

1. Dataset Structure and Composition

CF4 compiles individual distances for 55,877–56,000 galaxies (depending on the precise cut and grouping protocol) merged into 38,000–38,065 groups, using eight primary methodologies. The catalog brings together three principal subsamples:

• “Others” (CF2-like):  ~23,000 groups at $cz/100\lesssim120$, distributed nearly isotropically outside the Zone of Avoidance; includes classical Tully–Fisher (TF), Fundamental Plane (FP), and smaller surveys.
• 6dFGS:  ~7,500 groups at $60\lesssim cz/100\lesssim160$; dominant in the Southern Galactic hemisphere.
• SDSS:  ~7,000 groups at $60\lesssim cz/100\lesssim300$; confined to the Northern Galactic hemisphere.

The catalog relies on the following primary tracers and distance methodologies:

Method	Number of Galaxies	Typical Distance Uncertainty
TF/BTFR	12,412 / 9,967	0.3–0.4 mag (15–20%)
Fundamental Plane (FP)	42,223	0.3 mag (15%)
SN Ia	1,008	0.1–0.15 mag (5–7%)
SBF	480	0.05 mag (SBF method)
SN II	94	0.3 mag (15%)
TRGB	489	—
Cepheid	76	—
Maser	6	—

The sample extends out to $cz\sim24,000$ km s$^{-1}$ ($z\lesssim0.08$), with the SN Ia subsample reaching $z\sim0.05$. Typical individual distance errors for galaxies are $\sim20\%$, corresponding to $\sim2000$ km s$^{-1}$ for $cz/100=100$.

2. Distance Indicators and Calibration Protocols

The absolute zero-point for all distance measurements is anchored by local geometric methods—chiefly Cepheid period–luminosity relations, Tip of the Red Giant Branch (TRGB) in the color-corrected $I$-band, and water maser distances to NGC 4258. Cross-calibration across methodologies leverages group overlaps using Bayesian Markov Chain Monte Carlo (MCMC) techniques, marginalizing over intrinsic zero-point offsets.

Tully–Fisher and Baryonic Tully–Fisher

For spirals, the TF relation in a given band $\lambda$ is parameterized as

$$M_\lambda = a_\lambda \log_{10} W_{mx}^i + b_\lambda,$$

where $W_{mx}^i$ is the inclination-corrected HI profile width, and $a_\lambda$, $b_\lambda$ are empirically fit. Extensions to the Baryonic TF Relation (BTFR) incorporate the sum of stellar and gas mass:

$$\log_{10}\left(M_b/M_\odot\right) = a + b [\log_{10}(W_{mx}^{i} / \mathrm{km\,s}^{-1}) - 2.5].$$

Calibration employs a two-step process: (1) cluster regression for the TF/BTFR slope and (2) zero-point alignment using 64 galaxies with independent Cepheid and/or TRGB distances, referenced to LMC and NGC 4258 (Kourkchi et al., 2022, Kourkchi et al., 2020, Kourkchi et al., 2020).

Fundamental Plane (FP)

For early-type galaxies, the FP in SDSS bands is formulated as

$$\log R_e = a \log \sigma + b \langle \mu_e \rangle + c,$$

with $R_e$ the effective radius, $\sigma$ the velocity dispersion, and $\langle \mu_e \rangle$ the mean surface brightness. Uncertainties are controlled via strict photometric and kinematic cuts (Tully et al., 2022).

Other Methods

• SN Ia: Standardized using light-curve shape and color corrections, yielding $\sim5\%$ uncertainties.
• Supernova II: Standardized through expansion velocity and color.
• SBF: I-band and IR HST calibrations yield $\sim0.05$ mag precision.

All methodologies are merged onto a common scale by maximizing the joint likelihood over group/distance correlations, with zero-point priors sampled in the MCMC merging (Tully et al., 2022).

3. HI Data Acquisition, Linewidths, and Sample Selection

The backbone of TF distances is the All-Digital HI Catalog, aggregating 21 cm width measurements from Parkes, Green Bank Telescope (GBT), and Arecibo (ALFALFA), uniformized through a common reduction pipeline.

Key HI width parameter:

• $W_{m50}$: Width at 50% of mean flux within the 90% flux window.
• Standardization through instrumental, redshift, and turbulence corrections yields $W_{mx}$, corrected for inclination as $W_{mx}^i = W_{mx} / \sin i$.

Inclinations are determined via the Galaxy Inclination Zoo (GIZ), a crowdsourced visual classification system, minimizing systematic bias and reducing error floors to 1°–5°, depending on $i$ (Kourkchi et al., 2020, Dupuy et al., 2021, Courtois et al., 2014).

Selection criteria for TF inclusion: S/N $>$ 10, $e_W<20$ km s$^{-1}$ in $W_{mx}$, $i>45^\circ$, morphologically classified as Sa or later, and well-determined photometric parameters. ALFALFA widths are harmonized as $W_{mx}=W_{alf}-6$ km s$^{-1}$. Internal extinction corrections exploit both parametric and machine learning (random forest) models, incorporating colors and surface-brightness.

4. Grouping Algorithms and Bayesian Merging

Galaxies are assigned to groups or clusters via friends-of-friends and virial scaling (using $R_{2t}\propto M^{1/3}$), providing a robust statistical basis for cross-method calibration and error suppression.

Bayesian inference treats each methodology's zero-point $\Delta\mu_s$ as a free parameter; the likelihood for group $n$ having distances $DM_n^{(s)}$ from method $s$ is

$$L_n(\{\Delta\mu_s\}) = \prod_s \frac{1}{\sqrt{2\pi\sigma_{n,s}^2}} \exp\left[-\frac{1}{2}\left(\frac{DM_n^{(s)}+\Delta\mu_s-\langle DM \rangle_n}{\sigma_{n,s}}\right)^2\right],$$

with the posterior sampled by the \texttt{emcee} implementation of affine-invariant MCMC (Tully et al., 2022).

Group merges are essential for suppressing non-linear virial motions, reducing peculiar-velocity noise, and leveraging overlaps in hybrid clusters (e.g., Coma: $\sim209$ FP, $\sim50$ TF, 7 SN Ia).

5. Velocity Field Reconstruction and Statistics

Peculiar velocities are extracted for groups via

$$V_{pec}^{ds} = \frac{fV_{cmb} - H_0 d}{1 + H_0 d / c},$$

with $f=1 + 0.5(1-q_0)z-\ldots$ the relativistic correction, or with alternative logarithmic formulations at large $z$.

Reconstruction of the 3D density ($\delta_L$) and velocity ($v$) fields utilizes Bias Gaussianization correction (BGc) to address the lognormal error distribution of distances. After Gaussianizing, the Wiener Filter (WF) and Constrained Realizations (CRs) deliver minimum-variance estimates:

$$s_{WF}(r) = \langle s(r) d^T \rangle \langle dd^T \rangle^{-1} d,$$

with the prior drawn from the linear $\Lambda$CDM power spectrum.

Key statistics:

• Mean overdensity: $$\Delta_L(R) = (3/4\pi R^3)\int_{|r|<R}\delta_L(r)\,d^3r$$
• Bulk velocity: $V_{bulk}(R) = (3/4\pi R^3)\int_{|r|<R}v(r)\,d^3r$

CF4 finds $V_{bulk}(R)$ and $\Delta_L(R)$ are within $1\sigma$ of cosmic variance using “Others” alone. Inclusion of the 6dFGS introduces a $3.4\sigma$ bulk flow excess at $R\sim250\,h^{-1}$ Mpc, aligning mostly with the Supergalactic X axis (Shapley Concentration), and a $–1.9\sigma$ underdensity at $R\simeq190\,h^{-1}$ Mpc (Hoffman et al., 2023).

6. Cosmological Results and Implications

The joint dataset yields the following key cosmological measures:

• Hubble constant ($H_0$): $74.6\pm0.8$ (stat) $\pm\sim3$ (sys) km s$^{-1}$ Mpc$^{-1}$ (global CF4), with internal consistency among TF, BTFR, and SN Ia—e.g., TF: $75.1\pm0.2$ (stat) km s$^{-1}$ Mpc$^{-1}$; BTFR: $75.5\pm2.5$ (Kourkchi et al., 2022, Kourkchi et al., 2020, Tully et al., 2022).
• Large-scale bulk flow: At $R=300\,h_{100}^{-1}$ Mpc, $|V_{b}|=230\pm136$ km s$^{-1}$, directed toward the Sloan Great Wall; tidal analysis finds $V_{bulk}^{tidal}\simeq129$ km s$^{-1}$, indicating $\sim20\%$ of CMB-frame motion arises from external structures (Hoffman et al., 2023, Courtois et al., 2022).
• Structure growth ($f\sigma_8$): $0.36\pm0.05$ (grouped), $0.38\pm0.04$ (ungrouped); SN Ia: $0.30\pm0.06$ (Courtois et al., 2022).
• The inferred velocity field exhibits moderate $(2$–$3)\sigma$ excess for bulk flows on $150$–$300\,h^{-1}$ Mpc scales, but within plausible cosmic-variance fluctuations of $\Lambda$CDM.

7. Significance and Applications

CF4 provides the densest grid of peculiar velocities and distances to date, enabling rigorous mapping and analysis of large-scale flows, bulk motions, density monopoles, and comparisons against cosmological simulations. The systematic merging of eight distance indicators, anchored by geometric calibrations and robust group assembly, establishes a low-bias, high-precision foundation for cosmic velocity field studies and presents vital empirical constraints on $H_0$ and structure formation.

CF4 data products underpin analyses of gravitational basins (e.g., Laniakea), statistical isotropy, and tests for tension with $\Lambda$CDM, facilitating critical investigations into both local and global cosmological parameters (Hoffman et al., 2023, Courtois et al., 2022, Tully et al., 2022).