Complementary Roles of Distance and Growth Probes in Testing Time-Varying Dark Energy

TL;DR

This work analyzes how well time variation in the dark energy equation of state can be constrained by distance, expansion-rate, and growth observables using the eigenvalue spectrum of the Fisher information matrix. It shows that distance-based probes yield a strongly hierarchical spectrum, effectively constraining only a single mode due to the cumulative nature of their kernels, while growth observables, through their differential growth equation, can activate additional independent information directions. In realistic Euclid-like surveys, a second independent Fisher eigenmode only becomes observable once growth precision reaches percent-level or better, providing a concrete, model-independent criterion for testing dynamical dark energy. The results clarify the complementary roles of distance and growth probes and guide the design of future surveys to robustly probe time-dependent cosmic acceleration.

Abstract

Distance measurements have long provided the primary observational constraints on the expansion history of the Universe and the properties of dark energy. However, because such observables depend on cumulative line-of-sight integrals over the Hubble rate, their sensitivity to time-dependent features of the dark energy equation of state is intrinsically limited. In this work, we examine this limitation from an information-based perspective using the eigenvalue structure of the Fisher information matrix constructed from distance, expansion rate, and growth observables. We show that distance and expansion-rate data generically produce a strongly hierarchical Fisher spectrum dominated by a single information mode, reflecting an irreducible loss of sensitivity to temporal variations in dark energy. This behavior can be traced directly to the integrated kernel structure of geometric observables. Growth measurements, by contrast, respond through differential dynamics and can introduce additional independent information directions. Using both controlled mock data and survey-like configurations representative of next-generation experiments, we find that the impact of growth information depends not only on its nominal precision but also on the structure of the data covariance. In simplified mock setups, growth measurements can partially activate a second information direction even at moderate precision. In Euclid-like configurations, however, the information remains effectively one-dimensional until growth precision reaches the percent level, below which a second mode emerges rapidly. These results clarify the complementary roles of distance and growth probes and provide a model-independent criterion for assessing the physical content of cosmological constraints on dynamical dark energy.

Figures (1)

• Figure 1: Evolution of the Fisher eigenvalue hierarchy as a function of the fractional uncertainty of growth measurements, $\sigma_{f\sigma_8}$, with all other observational assumptions held fixed. The left panel shows results from controlled mock data experiments, while the right panel corresponds to Euclid-like survey specifications. In both cases, distance-only observables produce a strongly hierarchical Fisher spectrum dominated by a single eigenmode. When growth information is added, subdominant eigenvalues are lifted, indicating the emergence of additional information directions. In the controlled mock configuration (left), the second Fisher eigenmode becomes partially active already at moderate growth precision, leading to a gradual increase in the effective information dimensionality. In contrast, the Euclid-like configuration (right) exhibits a pronounced plateau, with the information remaining effectively one-dimensional down to the few-percent level. Only once the growth uncertainty reaches the percent or sub-percent regime does a rapid transition occur, signaling the activation of a second, genuinely independent Fisher eigenmode.