Expansion Rate Fluctuation Field Analysis

• Expansion Rate Fluctuation Field is a model-independent construct that quantifies directional variations in local cosmic expansion without assuming FLRW geometry.
• It leverages redshift–independent distance measurements and spherical harmonic decomposition to isolate key multipolar anisotropies such as the dipole, quadrupole, and octupole.
• The nonperturbative cosmographic framework enables precise reconstruction of local expansion dynamics and links these variations to global cosmological parameters.

The expansion rate fluctuation field is an observable and theoretical construct that quantifies anisotropies and inhomogeneities in the cosmic expansion rate, especially in the local universe, without presupposing an underlying Friedmann–Lemaître–Robertson–Walker (FLRW) geometry or a perturbative decomposition into peculiar velocities. It encapsulates deviations in the redshift–distance relation attributable to directional and spatial variations in the local kinematics of the cosmic fluid, as captured directly from redshift‐independent distance measurements. This framework enables a model‐independent and nonperturbative cosmographic analysis of the cosmic expansion, offering new insights into local dynamical features and their connection to global cosmological parameters and symmetries (Kalbouneh et al., 2 Oct 2025).

1. Construction and Measurement of the Expansion Rate Fluctuation Field

The expansion rate fluctuation field, denoted by η(z, n), is constructed from observed redshift and luminosity distance data by removing the overall (monopole) expansion rate on each spherical shell centered on the observer: $$\eta(z, n) = \log\left(\frac{z}{d_L(z, n)}\right) - \frac{1}{4\pi} \int_S \log\left(\frac{z}{d_L(z, n)}\right) d\Omega$$ where $d_L(z, n)$ is the measured luminosity distance in direction $n$ at redshift $z$, and the integral ensures monopole removal. This definition ensures that η measures only the anisotropic fluctuations relative to the mean expansion rate (Kalbouneh et al., 2 Oct 2025).

Implementation utilizes large compilations of redshift–independent distances, such as the Cosmicflows-4 (CF4) sample (which combines Tully–Fisher, Fundamental Plane, and SNIa distances for over 55,000 galaxies) and the Pantheon+ Type Ia supernova sample (∼700 SNe Ia), covering $0.01 < z < 0.1$ (∼30–300 h⁻¹ Mpc) (Kalbouneh et al., 2 Oct 2025). The analysis is typically performed in discrete redshift bins ('shells'), balancing statistical precision against angular resolution.

2. Multipolar Decomposition and Anisotropy Structure

The field η(z, n) is decomposed into spherical harmonic (SH) components: $$\eta(z, n) = \sum_{\ell=1}^{\infty} \sum_{m=-\ell}^{\ell} \eta_{\ell m}(z) \, Y_{\ell m}(n)$$ Empirical results demonstrate that at low redshifts (z ≈ 0.01–0.03), the dipole ($\ell=1$) dominates, with an amplitude $\sim 2.2 \times 10^{-2}$, translating to ∼2% fluctuation relative to the monopole. The quadrupole ($\ell=2$) is significant at about half the dipole amplitude and remains robust as redshift increases, even as the dipole amplitude falls by a factor of two across the probed range ($z \lesssim 0.1$). The octupole ($\ell=3$) is detected at lower redshift but becomes less significant at higher z due to increased noise (Kalbouneh et al., 2 Oct 2025).

Directional analyses reveal strong axial symmetry: the dipole, quadrupole, and octupole axes are aligned around (l ≈ 299°, b ≈ 5°) in Galactic coordinates.

Multipole	Low-z Amplitude	Alignment	Evolution with z
Dipole	~0.022	Aligned (l ≈ 299°, b ≈ 5°)	Decreases, ~halved at high z
Quadrupole	~0.011 (∼½ dipole)	Same axis as dipole	Persists, no clear trend
Octupole	Nonzero at low z	Same axis as dipole	Unconstrained at high z

The constancy and alignment of the quadrupole with the dipole throughout 30–300 h⁻¹ Mpc suggest persistent, large-scale anisotropic features in the local expansion field.

3. Covariant Cosmographic Interpretation

The deviations observed in η are interpreted using covariant cosmography (CC)—a framework in which the redshift–distance relation is expanded as a Taylor series in z, with direction-dependent cosmographic parameters: $$d_L(z, n) = \frac{z}{\mathbb{H}(n)} \left[1 + \frac{1-\mathbb{Q}(n)}{2}z - \frac{1-\mathbb{Q}(n)-\mathbb{J}(n)+\mathbb{R}(n)+3\mathbb{Q}^2(n)}{6}z^2 + \dots \right]$$ Here, $\mathbb{H}(n)$ is the covariant Hubble parameter, $\mathbb{Q}(n)$ is the deceleration parameter, $\mathbb{J}(n)$ the jerk, and $\mathbb{R}(n)$ the curvature, all potentially possessing nontrivial angular structure. The field η, upon expansion, reveals that its dipole is associated primarily with the dipole of the deceleration parameter, while the quadrupole is sourced predominantly by the quadrupole of the covariant Hubble parameter: $$\eta_1(z) \simeq \frac{\mathbb{Q}_1}{2\ln 10} z, \quad \eta_2(z) \propto \frac{\mathbb{H}_2}{\mathbb{H}_0} + \mathcal{O}(z)$$ The empirical results show that the dominant multipoles in η are accounted for by a strong quadrupole in $\mathbb{H}$ and both dipole and octupole terms in $\mathbb{Q}$ (Kalbouneh et al., 2 Oct 2025). This compact parameterization suffices to reconstruct the local luminosity distance field out to $z \sim 0.1$ with high accuracy and without recourse to the FLRW background or peculiar velocity corrections.

4. Model Independence and Methodological Robustness

The measurement of the expansion rate fluctuation field is explicitly model-independent and nonperturbative. No assumption is made regarding the metric, underlying global geometry, or the use of Einstein’s equations. The construction depends solely on direct observables—redshifts and (logarithmic) distances—while removing the monopole to avoid dependence on absolute calibration or selection function uncertainties (Kalbouneh et al., 2 Oct 2025). The invariant nature of the CC expansion further removes the need for peculiar velocity modeling, sidestepping systematics such as the Malmquist bias and uncertainties in local flow reconstructions.

Robustness is checked by cross-analyzing multiple distance tracers (e.g., early- and late-type galaxies, SNIa), confirming that the observed anisotropies and their multipolar architecture do not hinge on any specific method or population.

5. Redshift Evolution and Cosmographic Implications

The evolution of the multipoles with redshift provides insight into the spatial extension and physical origin of the anisotropies:

• The declining dipole amplitude with increasing z indicates that bulk flow–like motions dominate on smaller (∼30–60 h⁻¹ Mpc) scales and taper off at larger radii.
• The persistence and alignment of the quadrupole suggest a large-scale tidal shear or gravitational stretching, not simply explained by random flows or local inhomogeneities.

These results favor an interpretation in terms of a coherent quadrupolar geometry in the local cosmic expansion, aligned with large-scale structure (e.g., due to massive attractors or underdense regions) and encoded in the low-rank multipoles of the CC expansion.

6. Mathematical Formalism

Key relationships from the analysis include: $$\eta(z, n) = \log\left(\frac{z}{d_L(z, n)}\right) - \frac{1}{4\pi} \int_S \log\left(\frac{z}{d_L(z, n)}\right) d\Omega$$

$$\eta(z, n) = -\mathcal{M}(z) + \log\mathbb{H}_0(n) - \frac{1-\mathbb{Q}_0(n)}{2\ln10}z + \ldots$$

with the SH decomposition

$$\eta(z, n) = \sum_{\ell=1}^{\infty} \sum_{m=-\ell}^{\ell} \eta_{\ell m}(z) Y_{\ell m}(n)$$

and parameter inference via direct fitting of the η_{ℓm} as a function of z.

7. Significance and Future Directions

The expansion rate fluctuation field provides a systematic tool for probing anisotropies in local cosmic expansion, with direct cosmographic interpretation and resilience to model-dependent biases. The robust detection of dipole, quadrupole, and octupole features—and their alignment—offers constraints on local cosmic kinematics, the matter distribution, and possible deviations from the isotropic cosmological model on scales up to several hundred Mpc.

The methodology is immediately applicable to next-generation surveys with improved distance precision and sky coverage, such as the Zwicky Transient Facility and anticipated SNIa samples, promising sub-km s⁻¹ Mpc⁻¹ constraints on the covariant Hubble constant and high-precision mapping of higher-order cosmographic multipoles. These results will facilitate rigorous tests of the Cosmological Principle, cosmic isotropy, and the dynamics of local expansion, with possible implications for the origin of the Hubble tension and emerging large-scale cosmic anomalies (Kalbouneh et al., 2 Oct 2025).