Evolution of clustering in cosmological models with time-varying dark energy

Observations favor cosmological models with a time-varying dark energy component. But how does dynamical dark energy (DDE) influence the growth of structure in an expanding Universe? We investigate this question using high-resolution $N$-body simulations based on a DDE cosmology constrained by first-year DESI data (DESIY1$+$DDE), characterized by a 4% lower Hubble constant ($H_0$) and 10% higher matter density ($\Omega_0$) than the Planck-2018 $\Lambda$CDM model. We examine the impact on the matter power spectrum, halo abundances, clustering, and Baryonic Acoustic Oscillations (BAO). We find that DESIY1$+$DDE exhibits a 10% excess in power at small scales and a 15% suppression at large scales, driven primarily by its higher $\Omega_0$. This trend is reflected in the halo mass function: DESIY1$+$DDE predicts up to 70% more massive halos at $z = 2$ and a 40% excess at $z = 0.3$. Clustering analysis reveals a 3.71% shift of the BAO peak towards smaller scales in DESIY1$+$DDE, consistent with its reduced sound horizon compared to Planck18 Measurements of the BAO dilation parameter $\alpha$, using halo samples with DESI-like tracer number densities across $0 < z < 1.5$, agree with the expected DESIY1$+$DDE-to-Planck18 sound horizon ratio. After accounting for cosmology-dependent distances, the simulation-based observational dilation parameter closely matches DESI Y1 data. We find that the impact of DDE is severely limited by current observational constraints, which strongly favor cosmological models -- whether including DDE or not -- with a tightly constrained parameter $\Omega_0h^2\approx 0.143$, within 1-2% uncertainty. Indeed, our results demonstrate that variations in cosmological parameters, particularly $\Omega_0$, have a greater influence on structure formation than the DDE component alone.