Cosmic variance and ergodicity in finite systems with correlations
Abstract
We consider the difference between ensemble and volume average in cosmology. It is known that for sufficiently weak long-range correlations the root mean square of the difference, which we call ergodicity bias, decays like R−3/2 in the limit of large volume R3. We calculate the condition this imposes on the power spectrum of a Gaussian random field. We quantify the bias for finite R, and show that the R→∞ limit is of little relevance for cosmological observations when the measured scales and correlations extend to the size of the observable universe. We consider curvature, density, and velocity perturbations. On large scales the bias is important in all three cases. For the density perturbations, which are observationally the most relevant, the relative bias first exceeds 100% at the separation r=177 Mpc, and is larger than 100% for all r>560 Mpc. It should be taken into account when comparing ensemble and volume averages for large-scale structure. The bias is also large for the cosmic microwave background temperature perturbations on large angular scales, but this is not relevant for observations, as their analysis does not involve volume averaging.
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BibTeX
@misc{mukherjee2026cosmic,
title = {{Cosmic variance and ergodicity in finite systems with correlations}},
author = {Dipayan Mukherjee and Syksy Rasanen},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30838},
eprint = {2608.30838},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30838},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Cosmic variance and ergodicity in finite systems with correlations
AU - Dipayan Mukherjee
AU - Syksy Rasanen
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30838
UR - https://arxiv.org/abs/2608.30838
AB - We consider the difference between ensemble and volume average in cosmology. It is known that for sufficiently weak long-range correlations the root mean square of the difference, which we call ergodicity bias, decays like R^−3/2 in the limit of large volume R^3. We calculate the condition this imposes on the power spectrum of a Gaussian random field. We quantify the bias for finite R, and show that the R→∞ limit is of little relevance for cosmological observations when the measured scales and correlations extend to the size of the observable universe. We consider curvature, density, and velocity perturbations. On large scales the bias is important in all three cases. For the density perturbations, which are observationally the most relevant, the relative bias first exceeds 100% at the separation r=177 Mpc, and is larger than 100% for all r>560 Mpc. It should be taken into account when comparing ensemble and volume averages for large-scale structure. The bias is also large for the cosmic microwave background temperature perturbations on large angular scales, but this is not relevant for observations, as their analysis does not involve volume averaging.
ER -