Signatures of Extended Dark Energy Parametrisations in Structure Formation under Background Constraints

We study structure formation in alternative cosmological models constrained by background observations, including $\Lambda$CDM, wCDM, the Chevallier-Polarski-Linder parametrisation and a flexible Chebyshev expansion of the dark energy equation of state. The models are constrained using baryon acoustic oscillations, cosmic microwave background, cosmic chronometers and strong lensing measurements. Using the best-fitting parameters, we generate cosmology-dependent initial conditions and perform N-body simulations to analyse the matter power spectrum, halo mass function and halo density profiles. Although all models remain broadly consistent with $\Lambda$CDM at the background level, differences in the physical matter density $\Omega_{0m}h^2$ and in the expansion history $H(z)$ lead to distinct growth histories that are amplified by non-linear evolution. We find a clear hierarchy in the power spectrum amplitude and in $\sigma_8$, with the Chebyshev and CPL models exhibiting enhanced small-scale power, earlier halo formation at $z\gtrsim2$ and a migration of excess toward higher masses at late times. The wCDM model displays milder and partially compensating effects driven by its different expansion history. When expressed in terms of the scaled radius $r/R_{200c}$, halo density profiles show a high degree of universality across cosmologies, indicating that internal halo structure is largely governed by the same gravitational dynamics. These results demonstrate that even modest background-level variations in $w(z)$ can translate into coherent non-linear signatures, highlighting the constraining power of large-scale structure observables in extended dark energy models.