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 ($\mathrm{DESIY}1+\mathrm{DDE}$), characterized by a 4% lower Hubble constant ($H_0$) and 10% higher matter density (${\mathrm{\Omega }}_0$) than the Planck-2018 $\mathrm{\Lambda }$ cold dark matter model. We examine the impact on the matter power spectrum, halo abundances, clustering, and baryonic acoustic oscillations (BAOs). We find that $\mathrm{DESIY}1+\mathrm{DDE}$ exhibits a 10% excess in power at small scales and a 15% suppression at large scales, driven primarily by its higher ${\mathrm{\Omega }}_0$. This trend is reflected in the halo mass function: $\mathrm{DESIY}1+\mathrm{DDE}$ 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 toward smaller scales in $\mathrm{DESIY}1+\mathrm{DDE}$, 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 $\mathrm{DESIY}1+\mathrm{DDE}\text{−}\mathrm{to}\text{−}\mathrm{Planck}18$ 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 ${\mathrm{\Omega }}_0{h}^2\approx 0.143$, within 1%–2% uncertainty. Indeed, our results demonstrate that variations in cosmological parameters, particularly ${\mathrm{\Omega }}_0$, have a greater influence on structure formation than the DDE component alone.

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• Baryon acoustic oscillations
• Cosmological parameters
• Dark energy
• Large scale structure of the Universe
• Astrophysical & cosmological simulations