Directional Luminosity Distances

• Directional luminosity distances are a non-perturbative, direction-dependent generalization of the standard luminosity distance, capturing the effects of inhomogeneities and anisotropies in spacetime.
• They quantify path-dependent mappings between source luminosity and observed flux by incorporating redshift evolution, gravitational lensing, and peculiar velocity influences.
• This framework facilitates precise cosmological inference through numerical geodesic integration and two-point statistical analyses, essential for modern structure mapping.

Directional luminosity distances generalize the cosmological concept of luminosity distance by allowing for a fully non-perturbative, direction-dependent definition in arbitrary spacetimes, including inhomogeneous and anisotropic geometries. This quantity, often denoted $D_L(\hat n)$, encodes the path-dependent mapping between source luminosity and observed flux as photons traverse the real universe, subject to local expansion, lensing, peculiar velocities, and curvature. In both theory and application, the directional luminosity distance is fundamental to interpreting cosmological observations—such as those from standard candles or peculiar-velocity surveys—beyond the limits of homogeneous FLRW backgrounds and small-perturbation regimes.

1. Non-Perturbative Definition and Geometric Formalism

The directional luminosity distance is defined by considering a spray of null geodesics emanating from a source $S$ and converging at an observer $O$, with each geodesic specified by its initial direction $\hat n$ in the tangent space at $S$. The affine parameter, $\lambda$, along each geodesic is normalized so that $(g_{ab}\ell^a W^b)_S = -1$ for a chosen fiducial 4-velocity $W^a$ at $S$, where $\ell^a = dx^a/d\lambda$ is tangent to the null geodesic (Ivanov et al., 2018).

Given this normalization, the cross-sectional area per unit solid angle, $S$0, encapsulates both the expansion and focusing/defocusing properties of the null congruence. The van Vleck determinant, $S$1, defined by

$S$2

provides a robust, analytic measure of the deviation from inverse-square-law propagation due to spacetime curvature. The fundamental, fully non-perturbative expression for the directional luminosity distance is

$S$3

where $S$4 is the total redshift along the ray, $S$5 is the affine parameter interval between source and observer, and $S$6 encodes all curvature-induced focusing and caustics (Ivanov et al., 2018).

Directional dependence arises from the path-specific evolution of redshift, affine distance, and focusing, all of which depend on metric inhomogeneities and the local kinematic state of source and observer.

2. Linear Theory: Physical Decomposition and Fluctuation Statistics

In a perturbed FLRW universe, directional luminosity distances can be expanded around the background isotropic value to quantify fluctuations, which are essential probes of structure formation and gravitational effects. The fractional fluctuation

$S$7

admits a decomposition in the Newtonian gauge, with:

• Doppler terms: arising from source and observer peculiar velocities along the line of sight
• Local and integrated gravitational potentials: including both Sachs–Wolfe (SW) and Integrated SW contributions
• Gravitational lensing: encoded by the projected convergence integral
• Shapiro time delays and other projection effects

Quantitatively, in linear theory, these contributions (with explicit dependence on the Bardeen potentials $S$8, $S$9 and their time derivatives, as well as velocity terms) are provided in full in (Pantiri et al., 2024). The associated two-point statistics—correlation function $O$0 and its angular/power spectrum multipoles ($O$1, $O$2)—can be systematically constructed, capturing the directional statistics of $O$3 as a probe of both peculiar velocities and lensing by large-scale structure (Pantiri et al., 2024, Biern et al., 2017).

3. Exact Results in Inhomogeneous Cosmologies

In inhomogeneous spacetimes, particularly those with no global symmetries, $O$4 serves as a key observable for mapping the real geometry. For instance, in the Szekeres models—which include no rotational or translational symmetries—the full set of null geodesic equations (in $O$5 coordinates) are numerically integrated, with affine-parameter ODEs and accompanying transport equations for Sachs area propagation. The directional dependence is then explicit, with the observer's sky sampled by the initial conditions for geodesics and beam cross-sections.

The beam area transport (and hence $O$6) is calculated via

$O$7

where $O$8, $O$9 are constructed from the metric coefficients, and the index $\hat n$0 ranges over spacetime coordinates (Nwankwo et al., 2010). The observer can construct the full $\hat n$1 surface by numerically integrating the system for each sampled sky direction.

4. Directional Inversion Methods and Spherically Symmetric Examples

The directional luminosity distance can also be employed as an inversion tool to reconstruct inhomogeneous cosmological metrics from data. In the LTB class,

$\hat n$2

with $\hat n$3 for a central observer, one constructs ODEs for $\hat n$4 (and for off-center observers, extends to include the null geodesic equations for $\hat n$5 and/or $\hat n$6).

The directional $\hat n$7 for off-center observers is

$\hat n$8

enabling the comparison of directional distance data against the freely specifiable curvature functions $\hat n$9 (Romano et al., 2013). Matched Taylor expansions at the apparent horizon ensure numerical stability across caustics.

This approach allows for analytic setting of initial conditions, stable numerical crossing of the apparent horizon, and the possibility of exhaustive exploration of all curvature profiles compatible with a given $S$0.

5. Two-Point Statistics: Correlation Functions and Power Spectra

The covariance structure of $S$1 over the sky is central for cosmological inference. The two-point angular correlation function

$S$2

encodes, in linear theory:

• Dominance of peculiar velocity contributions at low redshift and large angles
• Lensing convergence becoming significant at higher redshift ($S$3) and small angular separation ($S$4)
• Subdominance of pure gravitational potential (geometric) terms if gauge invariance is enforced

The $S$5 multipoles are constructed as integrals over transfer kernels for each physical contribution and the matter power spectrum, with the lensing and velocity components each providing cosmological parameter sensitivity: $S$6, $S$7 (Biern et al., 2017).

In real surveys, this directional correlation formalism is essential for extracting constraints on the matter density, the growth function, and for discriminating between primary cosmological signals and biases from peculiar velocities and lensing.

6. Regimes of Applicability, Projection Effects, and Practical Considerations

Directional luminosity distances are sensitive to wide-angle and projection effects, especially in large-sky, low-redshift surveys. Corrections due to the changing line-of-sight orientation, window functions, and non-negligible relativistic projections must be incorporated. Quantitative results:

• Wide-angle and GR corrections enter the measured power spectrum multipoles at the few–15% level, with certain multipoles (e.g., dipole) absent in the plane-parallel limit but present when wide-angle corrections are included
• For typical peculiar-velocity survey parameters, wide-angle effects are detected at high significance (>10σ); GR corrections become detectable with high per-object distance accuracy and dense sampling (Pantiri et al., 2024)
• The non-perturbative formula for $S$8 remains valid up to the occurrence of caustics (where $S$9), which must be handled with care

Dedicated pipelines, such as GaPSE, implement the full (curved-sky, wide-angle, GR-corrected) statistics for cosmological analysis.

7. Significance and Applications in Modern Cosmology

Directional luminosity distances provide the theoretical and computational foundation for several high-precision cosmological analyses:

• Mapping and interpreting Type Ia supernovae Hubble diagrams with full account for local and line-of-sight structure
• Extracting 3D peculiar velocity fields and their power spectra from galaxy redshift–distance surveys while treating all relativistic and projection effects
• Direct inference of large-scale structure geometry, including the inversion for underlying metric properties in LTB, Szekeres, and related models
• Constraining the matter density parameter $\lambda$0 and the growth function $\lambda$1 via the redshift and angular dependence of the $\lambda$2 two-point function (Biern et al., 2017, Pantiri et al., 2024, Romano et al., 2013)

The separation of $\lambda$3 into redshift, affine, and focusing terms enables precise interpretation of observational signatures and systematic errors across all directions and redshifts. This robust, direction-dependent framework generalizes the isotropic FLRW result $\lambda$4 and remains valid in all spacetimes up to the emergence of caustics. It is thus indispensable for the next generation of cosmological surveys and for confronting the complexity of the actual, lumpy universe (Ivanov et al., 2018, Nwankwo et al., 2010).