Cosmicflows-4 & Pantheon+ Samples

• The paper demonstrates the integration of eight independent distance measurement techniques in Cosmicflows-4 and a refined SNe Ia recalibration in Pantheon+ to robustly constrain the Hubble constant and cosmic expansion.
• Cosmicflows-4 and Pantheon+ are comprehensive datasets offering heterogeneous and homogeneous extragalactic distance measurements that support rigorous tests of cosmic isotropy, structure growth, and dark energy dynamics.
• Advanced statistical methodologies—such as Bayesian MCMC, flux statistics, and covariance analysis—are applied to achieve superior error control and cross-validation in modern cosmological studies.

The Cosmicflows-4 and Pantheon+ samples are two of the most influential, high-precision compilations for extragalactic distance measurements and cosmic expansion studies to date. Cosmicflows-4 delivers a heterogeneous set of over 55,000 galaxy distances using eight independent techniques—spanning Cepheid variables, surface brightness fluctuations, megamasers, the tip of the red giant branch, fundamental plane, Tully–Fisher, Type Ia and core-collapse supernovae—providing complete and overlapping coverage in the local Universe. Pantheon+ is the most extensive homogeneous sample of Type Ia supernovae (SNe Ia), recalibrated and reanalyzed with sophisticated systematics control, extending out to $z\sim2.26$, and supporting distance ladder measurements that underpin modern constraints on $H_0$, dark energy, and the cosmic expansion history. Synergistic analysis of these datasets has enabled robust tests of cosmic isotropy and homogeneity, multipolar expansion, the growth rate of structure, improved statistical methodologies, and a sharpening of the persistent $H_0$ tension.

1. Composition, Methodology, and Calibration

Cosmicflows-4 assembles 55,877 galaxy distance moduli grouped into 38,065 halos with eight distinct methodologies: Tully–Fisher (TF), Fundamental Plane (FP), SNe Ia, surface brightness fluctuations (SBF), core-collapse SNe II, Cepheid period-luminosity relations (CPLR), tip of the red giant branch (TRGB), and maser distances. Zero-point calibration is achieved by anchoring to Cepheids, TRGB, and geometric maser measurements, with all distance scales merged using Bayesian Markov Chain Monte Carlo to solve for inter-method offsets across overlapping groups (Tully et al., 2022). This approach resolves the absolute distance scale and places all methodologies on a common system, providing robust internal cross-checks and minimizing random errors via statistical averaging over groups.

The Pantheon+ sample comprises 1701 well-calibrated SNe Ia light curves from 1550 SNe, with coverage from $z\sim0.001$ to $z\sim2.26$ (Brout et al., 2021). The “SuperCal–Fragilistic” recalibration improves photometric system cross-calibration (now covering 25 systems and 105 filters) using Pan-STARRS as a reference, simultaneously solving zeropoints and propagating the covariance into light-curve model retraining (SALT2). This yields a well-quantified systematic error contribution on the equation-of-state parameter, $\sigma_w=0.013$, and systematic error on $H_0$ under $0.2$ km s$^{-1}$ Mpc$^{-1}$—both subdominant to statistical errors and thus unable to explain the Hubble tension (Brout et al., 2021).

Dataset	Number of objects	Distance methods	Redshift range	Calibration anchor
Cosmicflows-4	55,877	8 (TF, FP, SN…)	$z\lesssim0.1$	Cepheid, TRGB, maser
Pantheon+	1550/1701 (SNe)	SNe Ia (homogeneous)	$0.001\leq z \leq2.2613$	Pan-STARRS/SH0ES

2. Statistical Methodologies and Error Control

Multiple statistical techniques are developed for SNe Ia cosmology:

• Magnitude statistics (MS): Traditional approach comparing observed and theoretical distance moduli using a standard $\chi^2$, sensitive to systematic uncertainties from calibration, color, and host galaxy effects (Wang et al., 2019).
• Flux statistics (FS): Implements flux-averaging in redshift bins, reducing sensitivity to weak-lensing and correlated systematics at the cost of larger statistical errors.
• Improved flux statistics (IFS): Combines MS at $z<z_\mathrm{cut}$ with FS at $z\geq z_\mathrm{cut}$, with $(z_\mathrm{cut},\Delta z)$ scanned to optimize the Figure of Merit (FoM).

The Pantheon dataset enables error bars on $w$ to be reduced by 38% (MS), 47% (FS), and 53% (IFS) relative to the JLA sample, and FoM improvements up to 373% (Wang et al., 2019). The Pantheon+ recalibration further propagates full correlated systematics via a covariance matrix affecting both the calibration and the SALT2 retraining, yielding a robust assessment of total uncertainties on $w$ and $H_0$ (Brout et al., 2021).

For cosmic flow measurements, CF4 utilizes Bayesian frameworks and forward modeling (e.g., Hamiltonian Monte Carlo) to reconstruct 3D density and velocity fields, or applies Wiener filtering and constrained realizations for linear fields (Courtois et al., 2022, Hoffman et al., 2023). Covariance and error estimation rely on mock catalogs that precisely reproduce survey selection and geometry (Qin et al., 2021).

3. Key Cosmological Results: Expansion, Structure, and Isotropy

Cosmicflows-4:

• The assembled distance scale supports $H_0=74.6\pm0.8~\mathrm{(stat)}\pm3~\mathrm{(sys)}$ km s$^{-1}$ Mpc$^{-1}$ anchored by Cepheids, TRGB, and masers (Tully et al., 2022).
• Peculiar velocities and bulk flows are measured out to $300$ $h^{-1}$ Mpc. The reconstructed bulk flow is $230\pm136$ km s$^{-1}$ at this scale, consistent with $\Lambda$CDM within cosmic variance and dominated by the direction of the Shapley Concentration (Courtois et al., 2022, Hoffman et al., 2023).
• Growth-rate measurements yield $f\sigma_8\sim 0.36\pm0.06$ for galaxies, $0.30\pm0.06$ for SNe Ia, agreeing with standard cosmology (Courtois et al., 2022).

Pantheon+/Pantheon:

• The equation-of-state parameter $w$ is tightly constrained; the methodology choice (MS, FS, IFS) affects both precision and cosmological interpretation, with MS favoring a “big rip” scenario ($w<-1$) and FS/IFS consistent with eternal expansion (Wang et al., 2019).
• Combined Pantheon+ analyses show the significance of binning (tomography): the low-redshift bin ($z<0.22$) dominates the constraining power on $\Omega_m$ and $H_0$ (Wang, 2022).
• There is no statistically significant evidence for evolution in $H_0$ or $\Omega_m$ across the sampled redshift range at the $2\,\sigma$ level.
• The inclusion of SH0ES Cepheid host calibration shifts $H_0$ from $\sim68$ km s$^{-1}$ Mpc$^{-1}$ (Pantheon+ only) to $73.4\pm1.1$ km s$^{-1}$ Mpc$^{-1}$ and increases the tension with Planck to $\sim2\,\sigma$ (Wang, 2022).

Isotropy and Multipolar Expansion:

• Both CF4 and Pantheon+ reveal small but significant anisotropies in the local expansion field. A multipolar decomposition shows the dipole at $2.2\times10^{-2}$ (fractional) at low redshift, with decreasing amplitude at higher $z$, and a significant quadrupole (about half the dipole) and detectable octupole—all aligned with axial symmetry at $(l,b)\sim(295^\circ,5^\circ)$ (Kalbouneh et al., 2 Oct 2025).
• Covariant cosmographic analysis demonstrates that these can be modeled non-perturbatively, with the quadrupole of the covariant Hubble parameter and dipole/octupole of the deceleration parameter reconstructing the luminosity distance with high precision out to $z\sim0.1$ (Kalbouneh et al., 2 Oct 2025).
• Tests with hemisphere comparison and Padé cosmography in Pantheon+ yield a preferred Hubble dipole direction and indicate potential anisotropy at $4-5\,\sigma$ significance for $H_0$, suggesting the need for further scrutiny (Hu et al., 21 Jun 2024).

4. Systematics, Sample Variance, and Statistical Method Developments

Pantheon+ and CF4 studies systematically address statistical and astrophysical systematics:

• Sample variance in SNe Ia $H_0$ is quantified: in Pantheon, the uncertainty is consistent with a top-hat sphere of $R\sim220 h^{-1}$ Mpc, with sample variance errors $\sim0.4$ km s$^{-1}$ Mpc$^{-1}$—far too small to account for the $H_0$ tension without violating other constraints (Zhai et al., 2023).
• Statistical analyses show Pantheon+ SNe Ia residuals are better modeled with a Student's t-distribution rather than a Gaussian, reducing uncertainties on $H_0$ and $\Omega_M$ by up to $40\%$ and sharpening cosmological tensions (Dainotti et al., 2023).
• Alternative likelihoods assessing SN Ia absolute magnitude homogeneity favor a two-parameter model (with a transition at $d_\mathrm{crit}\sim20$ Mpc). This model fits the Pantheon+ sample substantially better, with Monte Carlo tests indicating that the observed $M_<-M_>$ tension is unlikely to occur by chance (Perivolaropoulos et al., 2023).
• Covariance tests using Gaussian Processes set an upper bound on additional correlated SN magnitude error ($\sigma<0.031$ mag at $95\%$ CL), finding no evidence for missing covariance that could resolve the Hubble tension (Bidenko et al., 2023).

5. Implications for Cosmology: $H_0$ Tension, Bulk Flows, and Dark Energy

The synergy of Cosmicflows-4 and Pantheon+ enables cross-validation of the local distance scale, local velocity field, and constraints on cosmological dynamics:

• The CF4 measurement ($H_0 = 74.6 \pm 0.8~\mathrm{km~s^{-1}~Mpc^{-1}}$) and Pantheon+ ($H_0 \sim 73.4$ km s$^{-1}$ Mpc$^{-1}$, SH0ES-calibrated) remain in $4-6\,\sigma$ tension with Planck CMB ($H_0 \sim 67$ km s$^{-1}$ Mpc$^{-1}$), even after full calibration systematics and sample variance are accounted for (Tully et al., 2022, Brout et al., 2021, Zhai et al., 2023).
• Multipolar decompositions and anisotropy signatures identified in both SNe Ia and peculiar velocity catalogues reveal a coherent, axially symmetric expansion rate fluctuation field with detectable dipole, quadrupole, and octupole out to $~300~h^{-1}$ Mpc (Kalbouneh et al., 2 Oct 2025).
• Covariant cosmographic frameworks explain these patterns model-independently, requiring only a few multipolar parameters of the Hubble and deceleration fields to accurately describe luminosity distance data out to $z\sim0.1$ (Kalbouneh et al., 2 Oct 2025).
• Isotropy analyses and likelihood tests suggest local inhomogeneities or transitions (e.g., at 20–40 Mpc in the SNe Ia sample) could affect the calibration of the distance ladder and, by extension, the derived $H_0$; these features are rare but observed (Perivolaropoulos, 2023, Perivolaropoulos et al., 2023).
• New statistical diagnostics (e.g., effective running Hubble constant, power-law fits to $H_0(z)$) and tomographic binning reveal small but robust trends hinting at nontrivial local expansion and possible deviations from FLRW homogeneity, while reasserting the need for model-independent analyses (Fazzari et al., 4 Jun 2025, Simone et al., 8 Nov 2024).

6. Future Directions: Integrated Cosmography and Ongoing Tensions

Future exploitation of CF4 and Pantheon+ will involve:

• Further mapping expansion rate multipoles to higher $z$ and disentangling cosmological from astrophysical origins of observed anisotropy.
• Integrating new SNe Ia samples and next-generation galaxy distances (e.g., from ZTF and DESI) for even greater sky coverage and statistical power.
• Cross-comparison of effective running Hubble constant diagnostics across independent samples (peculiar velocity, SNe Ia, and BAO) to distinguish quintessence/phantom-like dark energy evolution.
• Continued tomographic, covariant cosmographic, and anisotropy testing for robust, model-independent assessments of the cosmological principle.
• Ongoing reconciliation—or possible accommodation—of observed local-global discrepancies in $H_0$ and the structure of the cosmic velocity field, where all current evidence indicates neither calibration systematics nor sample variance are sufficient remedies.

These datasets, methodologies, and cross-validations form the empirical backbone for present and future cosmological model testing, refinement of the cosmic distance ladder, and the search for new gravitational or astrophysical phenomena underlying the still unresolved $H_0$ tension, cosmic isotropy, and local expansion rate anomalies.