Probing time-varying dark energy with DESI: the crucial role of precision matter density (Ω _m 0) measurements

Accurate measurements of fundamental cosmological parameters, especially the Hubble constant ( H₀ H 0 ) and present-day matter density ( Ω _m 0 Ω m 0 ), are crucial for constraining dark energy (DE) evolution. We analyze the sensitivities of cosmological observables ( H ( z ), DL(z) D L ( z ) , EG E G ) to Ω _m 0 Ω m 0 , ω ₀ ω 0 , and ω ₐ ω a under different parametrizations. Our results show observables are far more sensitive to Ω _m 0 Ω m 0 than to DE equation of state parameters (e.g., at z ~ 0.5 z ∼ 0.5 , H ( z )’s Ω _m 0 Ω m 0 sensitivity is ~ 0.7 ∼ 0.7 vs. ω ₐ ω a ’s ~ 0.04 ∼ 0.04 ). This hierarchy mandates high-precision Ω _m 0 Ω m 0 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 ω ₐ