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# Covariant cosmography: the observer-dependence of the Hubble parameter

## Details

**Authors** Roy Maartens, Jessica Santiago, Chris Clarkson, Basheer Kalbouneh, Christian Marinoni

**Year** 2024

**Publisher** Journal of Cosmology and Astroparticle Physics

**Kind of work** article

**Discipline** Cosmology

**Secondary disciplines** Astronomy, Physics

**Applied** false

[Read it at the publisher](https://doi.org/10.1088/1475-7516/2024/09/070) 
10.1088/1475-7516/2024/09/070

## In authors' words

### Abstract

The disagreement between low- and high-redshift measurements of the Hubble parameter is emerging as a serious challenge to the standard model of cosmology. We develop a covariant cosmographic analysis of the Hubble parameter in a general spacetime, which is fully model-independent and can thus be used as part of a robust assessment of the tension. Here our focus is not on the tension but on understanding the relation between the physical expansion rate and its measurement by observers -- which is critical for model-independent measurements and tests. We define the physical Hubble parameter and its multipoles in a general spacetime and derive for the first time the covariant boost transformation of the multipoles measured by a heliocentric observer. The analysis is extended to the covariant deceleration parameter. Current cosmographic measurements of the expansion anisotropy contain discrepancies and disagreements, some of which may arise because the correct transformations for a moving observer are not applied. A heliocentric observer will detect a dipole, generated not only by a Doppler effect, but also by an aberration effect due to shear. In principle, the observer can measure both the intrinsic shear anisotropy and the velocity of the observer relative to the matter -- without any knowledge of peculiar velocities, which are gauge dependent and do not arise in a covariant approach. The practical implementation of these results is investigated in a follow-up paper. We further show that the standard cosmographic relation between the Hubble parameter, the redshift and the luminosity distance (or magnitude) is not invariant under boosts and holds only in the matter frame. A moving observer who applies the standard cosmographic relation should correct the luminosity distance by a redshift factor -- otherwise an incorrect dipole and a spurious octupole are predicted.

### What they set out to do (purpose)

To derive, in a fully model-independent covariant framework, how the Hubble parameter and its higher multipoles depend on the motion of the observer measuring them.

### Who or what was studied (sample)

Not an empirical sample; a theoretical/mathematical cosmographic derivation applied to interpretation of current expansion-anisotropy measurements.

### How they did it (methods)

Covariant general-relativistic derivation of boost transformations for the Hubble parameter and its multipoles between the matter (comoving) frame and a moving (heliocentric) observer's frame, compared to standard cosmographic relations.

### What they found (results)

A heliocentric (moving) observer necessarily detects a dipole in the Hubble parameter generated by both a Doppler effect and a shear-induced aberration effect, and the standard cosmographic distance-redshift relation is not boost-invariant, so applying it without a frame correction produces a spurious dipole and octupole.

## Commentary

### In short

The finding demonstrates the Perspectives pattern because the measured value and multipole structure of the Hubble parameter change depending on the observer's reference frame, and a Relationships component because the paper derives the coupling, via Doppler and shear-induced aberration, linking observer motion to the measured dipole and higher multipoles.

**Patterns it shows** R, P

**Added** 2026-09-28

**How to cite this** Roy Maartens, Jessica Santiago, Chris Clarkson, Basheer Kalbouneh, Christian Marinoni (2024). Covariant cosmography: the observer-dependence of the Hubble parameter. Journal of Cosmology and Astroparticle Physics.