Covariant Cosmography: A Geometric Framework

• Covariant cosmography is a geometric, model-independent method that reconstructs the universe’s expansion history via series expansion of observable quantities.
• It systematically expands the luminosity distance in redshift using covariant parameters like Hubble, deceleration, jerk, curvature, and snap to capture anisotropies and inhomogeneities.
• The framework also corrects for observer motion and local structural biases, enabling high-precision cosmological measurements from modern surveys.

Covariant cosmography is the model-independent, fully geometric approach for reconstructing the expansion history and local structure of the universe by systematically expanding observable quantities—primarily the luminosity distance–redshift relation—in series controlled by covariantly-defined kinematic parameters. Unlike traditional FLRW cosmography, which presupposes large-scale homogeneity and isotropy, the covariant formulation allows for arbitrary inhomogeneities and anisotropies by encoding the information directly in observer-dependent, direction-resolved multipoles of parameters such as the Hubble, deceleration, jerk, curvature, and snap, all constructed from the matter expansion tensor and its derivatives. This approach is critical for contemporary cosmology, where local expansion anisotropies, nonlinear structures, and systematic effects from observer motion render the assumptions of the cosmological principle insufficient for high-precision inference.

1. Definition of Covariant Cosmography and Theoretical Framework

Covariant cosmography generalizes kinematic cosmography by constructing all observable cosmological parameters and series expansions directly from locally measurable, fully covariant geometric quantities, avoiding reliance on background solutions or perturbative frameworks. The key object is the expansion tensor of the cosmic matter flow, $\Theta^{\mu\nu}$, and null geodesic tangent vectors $k^\mu$ (for photons). Direction-dependent observables, such as the luminosity distance $d_L(z, n)$ along a line-of-sight $n$, are expanded in redshift: $$d_L(z, n) = d_L^{(1)}(n) z + d_L^{(2)}(n) z^2 + d_L^{(3)}(n) z^3 + d_L^{(4)}(n) z^4 + \mathcal{O}(z^5)$$ where each coefficient encodes a covariant cosmographic parameter:

• $d_L^{(1)}(n) = 1/\mathbb{H}_0(n)$: Hubble parameter (monopole, dipole, quadrupole, etc.)
• $$d_L^{(2)}(n) = [1 - \mathbb{Q}_0(n)]/(2 \mathbb{H}_0(n))$$: deceleration parameter and its anisotropies
• $d_L^{(3)}(n)$: combinations involving jerk $\mathbb{J}_0(n)$ and curvature $\mathbb{R}_0(n)$
• $d_L^{(4)}(n)$: incorporates snap $\mathbb{S}_0(n)$ and higher terms

Key covariant definitions include: $$\mathbb{H} \equiv k_\mu k_\nu \Theta^{\mu\nu};\qquad \mathbb{Q} \equiv -3 + \frac{k_\mu k_\nu k_\alpha \nabla^\alpha \Theta^{\mu\nu}}{\mathbb{H}^2};\qquad \mathbb{J},\,\mathbb{R},\,\mathbb{S}~\text{and higher derived from further derivatives}$$ These are direction-dependent (indexed by $n$), naturally forming a spherical (or Legendre, for special symmetries) multipole structure that characterizes anisotropy and inhomogeneity (Kalbouneh et al., 22 Jan 2024, Kalbouneh et al., 8 Aug 2024, Sarma et al., 3 Oct 2025).

2. Multipolar Structure and Degrees of Freedom

Covariant cosmographic parameters carry a rich multipole structure:

• The Hubble, deceleration, jerk, curvature, and snap parameters have monopole (ℓ = 0), dipole (ℓ = 1), quadrupole (ℓ = 2), ... moments up to ℓ = 5 for the (fourth-order) snap (dotriacontapole).
• The full multipole expansion (no symmetry imposed) for the snap includes 36 d.o.f. (degrees of freedom) (Kalbouneh et al., 8 Aug 2024).
• The general expansion for $d_L(z, n)$ up to fourth-order requires a total of 86 d.o.f. for an unconstrained description of local geometry, derived from the sum of (2ℓ+1) for each allowed multipole order and parameter.
• Observational indications of axial symmetry in the local Hubble diagram (evident for $z \lesssim 0.1$) allow reduction to Legendre multipoles with just one d.o.f. per ℓ. In this case, the dominant set relevant for unbiased estimation is 12 (Kalbouneh et al., 8 Aug 2024).

This multipole decomposition encodes both intrinsic geometry and observer-induced anisotropy, allowing a direct and model-independent link to the underlying cosmic metric.

3. Impact of Local Inhomogeneity, Anisotropy, and Observer Motion

The covariant approach explicitly separates:

• Intrinsic geometric/kinematic anisotropies: multipoles of cosmographic parameters arising from local mass/velocity structure, e.g., a spherically symmetric LTB inhomogeneity seen off-center gives non-zero dipole/quadrupole components (Sarma et al., 3 Oct 2025).
• Observer motion effects: boosts relative to the matter frame introduce a dipole in measured expansion rates (Doppler and aberration), and, if not properly corrected, a spurious octupole in the Hubble parameter (Maartens et al., 2023).
• Biases in parameter estimation: incomplete expansions or neglect of higher multipoles such as snap lead to systematic errors in low-order parameters (e.g., deceleration dipole estimates biased by unmodeled $\ell=3$ snap multipoles) (Kalbouneh et al., 8 Aug 2024).

A key result is a "dictionary" relating multipoles of the covariant parameters in general spacetimes (whether inhomogeneous/exact or perturbative) to those arising in linear perturbation theory or exact LTB solutions (Sarma et al., 3 Oct 2025). For instance, the expansion coefficients in the luminosity distance are matched to the covariantly defined Hubble, deceleration, jerk, and curvature parameters along the line-of-sight.

4. Statistical and Observational Implementation

Extraction of covariant cosmographic multipoles from data uses:

• High-resolution redshift--distance datasets (e.g., supernovae), decomposed into angular multipoles via spherical harmonic or Legendre expansions, depending on the assumed symmetry (Kalbouneh et al., 22 Jan 2024, Kalbouneh et al., 8 Aug 2024).
• Third-order expansions are typically sufficient for percent-level accuracy in $z\lesssim0.1$; inclusion of the fourth-order snap is necessary to fully de-bias multipole estimators and accurately infer curvature at low redshift (Kalbouneh et al., 8 Aug 2024).
• Direct inversion is possible: the fluctuation field

$$\eta(z, n) = \log\left(\frac{z}{d_L(z, n)}\right) - \mathcal{M}(z)$$

is decomposed and linked to the angular structure of $\mathbb{H}_0(n)$, $\mathbb{Q}_0(n)$, $\mathbb{J}_0(n)$, $\mathbb{R}_0(n)$, $\mathbb{S}_0(n)$ (Kalbouneh et al., 22 Jan 2024).

• Mock catalogs (e.g., ZTF supernovae) demonstrate that forthcoming surveys will constrain not only the monopole (mean expansion rate) but also dipole, quadrupole, and higher moments of all parameters with high precision (Kalbouneh et al., 8 Aug 2024).

Accurate modeling of the observer's motion and the transformation laws for observed distances (e.g., the correct use of boost-corrected luminosity distance $d_{L*} = d_L/(1+z)^2$) is essential to avoid spurious inferred anisotropies (Maartens et al., 2023).

5. Reliability and Limitation Regimes

Comparison with exact calculations in LTB models and linear perturbation theory (LPT) demonstrates:

• The covariant cosmographic expansion, up to third or fourth order, reproduces the exact Sachs-equation-based distances to within ~10% for low to moderate density contrasts ($\delta_c \lesssim 2.5$ near the observer, $\lesssim 1.5$ at large radii) (Sarma et al., 3 Oct 2025).
• LPT and the CC expansion agree in the small inhomogeneity limit; in regimes of strong inhomogeneity or non-linearity, only the fully covariant (or exact) approach remains reliable.
• Truncation at too low order or omitting higher multipoles biases the estimates, especially for anisotropy-sensitive quantities. The inclusion of the snap parameter and, if necessary, the next higher "pop" ($\ell=6$) should be considered as data quality increases (Kalbouneh et al., 8 Aug 2024).
• An explicit mapping (the "dictionary") between CC parameters and LPT quantities enables cross-validation and improved model testing (Sarma et al., 3 Oct 2025).

6. Applications and Future Prospects

The fully non-perturbative, covariant cosmographic formalism is directly applicable to:

• Testing the Cosmological Principle via measurement of local anisotropies in Hubble, deceleration, jerk, curvature, and snap (Kalbouneh et al., 22 Jan 2024, Kalbouneh et al., 8 Aug 2024).
• Interpreting the effect of local structures and observer location within realistic, inhomogeneous universes, beyond any FLRW assumption (Sarma et al., 3 Oct 2025).
• Constraining observer's velocity with respect to the matter frame by comparing the dipole of measured expansion to higher multipole moments (Maartens et al., 2023, Kalbouneh et al., 22 Jan 2024).
• Rigorous inference from upcoming large transient and supernova surveys (e.g., ZTF, LSST, Roman, Euclid) and cosmic drift measurements (Gaia, ELT, SKA) (Kalbouneh et al., 8 Aug 2024, Heinesen et al., 10 Jun 2024).
• Clarifying discrepancies in expansion rate measurements (Hubble tension) by providing the framework to correct for boost and geometric effects (Maartens et al., 2023).

This formalism is rapidly evolving, with the inclusion of higher-order multipoles (snap and beyond) and the development of anisotropy-debiasing methodologies expected to enhance the precision and depth of local cosmographic constraints. The approach will be pivotal for the next decade of high-precision cosmology, providing the tools to probe cosmic isotropy, understand structure formation dynamics, and rigorously test extensions to the standard cosmological model.