The Optical Expansion Scalar in Cosmological Perturbation Theory
Abstract
Within the geometric optics approximation, the optical expansion scalar describes the local rate of change in the amplitude of a wave or the area of a light bundle propagating through the Universe. As such, it is fundamental for describing almost all the observations we make in cosmology. It was recently pointed out that this quantity can, in principle, be reconstructed from measurements of the angular diameter distance DA and the effective observed expansion rate H. Motivated by this possibility, we introduce the observable ϑ ≡ θ/ωo, where θ is Sachs' optical expansion scalar, and ωo the observed photon frequency. We then derive the linear perturbative expression and angular power spectrum for ϑ in a flat FLRW universe with scalar perturbations. Comparing the power spectra with those of the fluctuations in DA and with power spectra from relativistic N-body simulations reveals no immediate striking advantage of studying the fluctuations of ϑ to those of DA. However, the anisotropies of ϑ do exhibit several unique qualities such as an enhanced sensitivity to the peculiar velocity and relativistic potentials. We suggest that our results may hint at ϑ being useful for developing consistency tests of cosmological observations and possibly probing departures from FLRW cosmology and modified gravity theories. The formalism developed and presented here provides the foundation for future studies into these possibilities.
Related
- Bridging perturbation theory and simulations: initial conditions and fast integrators for cosmological simulations
- Cosmological-perturbation equations from the total-angular-momentum formalism
- Entropy, Expansion, and χ‑Spectral Structuring: Operator‑Geometric Resolution of the Cosmological Order‑from‑Chaos Paradox
- The alternative scalar field dark sector of the Universe: Cosmological inflation, quintessence and ekpyrotic model in the framework of scalar-tensor cosmology
- Novel geometrical test of cosmological expansion from photometric data
BibTeX
@misc{oestreicher2026optical,
title = {{The Optical Expansion Scalar in Cosmological Perturbation Theory}},
author = {Alexander Oestreicher and Sofie Marie Koksbang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30108},
eprint = {2609.30108},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30108},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Optical Expansion Scalar in Cosmological Perturbation Theory
AU - Alexander Oestreicher
AU - Sofie Marie Koksbang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30108
UR - https://arxiv.org/abs/2609.30108
AB - Within the geometric optics approximation, the optical expansion scalar describes the local rate of change in the amplitude of a wave or the area of a light bundle propagating through the Universe. As such, it is fundamental for describing almost all the observations we make in cosmology. It was recently pointed out that this quantity can, in principle, be reconstructed from measurements of the angular diameter distance D_A and the effective observed expansion rate H. Motivated by this possibility, we introduce the observable ϑ ≡ θ/ω_o, where θ is Sachs' optical expansion scalar, and ω_o the observed photon frequency. We then derive the linear perturbative expression and angular power spectrum for ϑ in a flat FLRW universe with scalar perturbations. Comparing the power spectra with those of the fluctuations in D_A and with power spectra from relativistic N-body simulations reveals no immediate striking advantage of studying the fluctuations of ϑ to those of D_A. However, the anisotropies of ϑ do exhibit several unique qualities such as an enhanced sensitivity to the peculiar velocity and relativistic potentials. We suggest that our results may hint at ϑ being useful for developing consistency tests of cosmological observations and possibly probing departures from FLRW cosmology and modified gravity theories. The formalism developed and presented here provides the foundation for future studies into these possibilities.
ER -