Probing Time-Varying Dark Energy with DESI: The Crucial Role of Precision Matter Density ($\Omega_{m0}$) Measurements

Accurate measurements of fundamental cosmological parameters, especially the Hubble constant ($H_0$) and present-day matter density ($\Omega_{m0}$), are crucial for constraining dark energy (DE) evolution. We analyze the sensitivities of cosmological observables ($H(z)$, $D_L(z)$, $\EG$) to $\Omega_{m0}$, $\omega_0$, and $\omega_a$ under different parametrizations. Our results show observables are far more sensitive to $\Omega_{m0}$ than to DE equation of state parameters (e.g., at $z \sim 0.5$, $H(z)$'s $\Omega_{m0}$ sensitivity is $\sim 0.7$ vs. $\omega_a$'s $\sim 0.04$). This hierarchy mandates high-precision $\Omega_{m0}$ measurements to accurately constrain time-varying DE. We also find DE parameter sensitivity highly depends on parametrization; the standard CPL form shows low sensitivity to $\omega_a$, but $\omega(z) = \omega_0 + \omega_a \ln(1+z)$ significantly enhances it. Our analysis of DESI DR1/DR2 data confirms these theoretical limits: standalone DESI data primarily provides only upper limits for $\omega_a$, underscoring insufficient constraining power for a definitive time-varying DE detection. While combined datasets offer tighter constraints, interpretation requires caution due to parametrization influence. In conclusion, improving $\Omega_{m0}$ precision and adopting optimized parametrizations are imperative for future surveys like DESI to fully probe dark energy's nature.