Structural Limitations on Constraining the Time Evolution of Dark Energy

Cosmological constraints on a time-varying dark energy equation of state are fundamentally limited by the integral structure through which the equation of state enters cosmological observables. We rigorously derive the linear response kernel that maps perturbations in the equation of state \omega(z) to comoving distance fluctuations \delta D(z). By adopting a Fourier mode expansion \delta \omega(z) = \sin(kz), we obtain the exact analytic form of the distance response in terms of Sine and Cosine integrals. We show that this mapping involves a double integration over redshift, which acts as an intrinsic low-pass filter with a characteristic \sim k^{-2} scaling in redshift space. This structural limitation is visualized in a schematic diagram and confirmed by observational verification using the full covariance matrix of the Pantheon+ supernova dataset. Our analysis reveals a steep hierarchy in Fisher eigenvalues where the information content drops by an order of magnitude already at the second eigenmode. Consequently, distance-based probes effectively constrain only a single dominant mode of \omega(z). This implies that the difficulty in constraining dynamical parameters such as w_a is not due to data precision, but is a necessary consequence of the observable's integral nature, which renders it structurally blind to the instantaneous rate of change d\omega/da.