The Cosmological Correlation Dimension Beyond the Linear Regime: Analytical Approximation and Parameter Sensitivity
Abstract
The correlation dimension (D2) provides a scale-dependent characterization of the transition towards homogeneity and a complementary perspective on the clustering of large-scale structure. In this work, we study its full dependence on scales and redshift connecting D2 directly to the matter power spectrum of a given cosmological model, and derive an approximate analytical relation based on an effective cut-off representation of the filtering kernel. We further perform two complementary local (derivative-based) and global (variance-based) sensitivity analyses within the w0waCDM model. In general, nonlinear evolution lowers D2 relative to the linear prediction at small scales, with the effect becoming more pronounced at low redshift. Our cut-off approximation, with kcut=α/r and α≃ 2.39, reproduces the exact D2 to better than 1\% for the representative cases considered. The sensitivity analysis identifies Ωm and As as the dominant parameters for the linear correlation dimension and its nonlinear-to-linear ratio, respectively, while w0 and wa induce scale- and redshift-dependent changes in the nonlinear contribution. These results provide a framework for characterizing the cosmological scale- and redshift-dependent features of D2 and its nonlinear correction.
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BibTeX
@misc{resco2026cosmological,
title = {{The Cosmological Correlation Dimension Beyond the Linear Regime: Analytical Approximation and Parameter Sensitivity}},
author = {Miguel Aparicio Resco and Florencia A. Teppa Pannia},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39867},
eprint = {2609.39867},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39867},
}
RIS
TY - UNPB
TI - The Cosmological Correlation Dimension Beyond the Linear Regime: Analytical Approximation and Parameter Sensitivity
AU - Miguel Aparicio Resco
AU - Florencia A. Teppa Pannia
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39867
UR - https://arxiv.org/abs/2609.39867
AB - The correlation dimension (D_2) provides a scale-dependent characterization of the transition towards homogeneity and a complementary perspective on the clustering of large-scale structure. In this work, we study its full dependence on scales and redshift connecting D_2 directly to the matter power spectrum of a given cosmological model, and derive an approximate analytical relation based on an effective cut-off representation of the filtering kernel. We further perform two complementary local (derivative-based) and global (variance-based) sensitivity analyses within the w_0w_aCDM model. In general, nonlinear evolution lowers D_2 relative to the linear prediction at small scales, with the effect becoming more pronounced at low redshift. Our cut-off approximation, with k_cut=α/r and α≃ 2.39, reproduces the exact D_2 to better than 1\% for the representative cases considered. The sensitivity analysis identifies Ω_m and A_s as the dominant parameters for the linear correlation dimension and its nonlinear-to-linear ratio, respectively, while w_0 and w_a induce scale- and redshift-dependent changes in the nonlinear contribution. These results provide a framework for characterizing the cosmological scale- and redshift-dependent features of D_2 and its nonlinear correction.
ER -