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

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.

💡 Research Summary

This paper investigates why distance‑based observables (luminosity and angular‑diameter distances, H(z) measurements) are intrinsically limited in probing a time‑varying dark‑energy equation of state, while growth‑based observables (fσ₈, redshift‑space distortions, weak lensing) can provide complementary, independent constraints. The authors adopt an information‑theoretic approach: they construct Fisher information matrices for a set of cosmological parameters {Ωₘ₀, H₀, w₀, wₐ, σ₈} using exact numerical derivatives of the observables, and then examine the eigenvalue spectra of these matrices.

The key theoretical insight is that distance and expansion‑rate observables depend on the Hubble rate through integrated kernels (χ(z)=∫c/H(z′)dz′). Such integrals smooth out localized variations in w(z), causing strong degeneracies: the Fisher matrix becomes highly hierarchical, with a single dominant eigenvalue λ₁ and all others (λ₂, λ₃, …) suppressed by orders of magnitude. Consequently, only one linear combination of the dark‑energy parameters—essentially the average of w₀ and wₐ—can be tightly constrained.

In contrast, growth observables obey the linear growth equation, a second‑order differential equation. Because differential operators preserve high‑frequency information, changes in w(z) affect the growth factor D⁺(a) and the derived quantity fσ₈ more locally in redshift. When growth data are added to the Fisher analysis, sub‑dominant eigenvalues are lifted. The authors demonstrate this effect with two classes of synthetic experiments: (i) controlled mock data where the growth precision is varied from 10 % down to 1 %, and (ii) an Euclid‑like survey configuration that mimics realistic redshift sampling, error models, and non‑trivial covariance among observables.

In the mock setups, even a modest 5 % growth precision produces a noticeable rise in λ₂, reaching roughly 10 % of λ₁, indicating the emergence of a second independent mode. However, for the Euclid‑like case, the covariance structure delays this activation: only when the growth measurement reaches sub‑percent precision (≈1 % or better) does λ₂ surge, and the Fisher spectrum transitions from a single‑mode to a two‑mode regime. This threshold behavior underscores that not only the nominal statistical error but also the correlation structure of the data set determines how many independent dark‑energy degrees of freedom can be constrained.

The paper’s implications are clear for upcoming cosmological surveys. Distance probes will continue to dominate constraints on the overall expansion history, but testing dynamical dark energy (i.e., distinguishing w(z) ≠ −1) requires growth measurements of percent‑level accuracy and careful control of systematic correlations. The eigenvalue analysis provides a model‑independent diagnostic: a strongly hierarchical spectrum signals that the data are effectively one‑dimensional, whereas a flatter spectrum guarantees sensitivity to multiple independent combinations of w₀ and wₐ (or more general, non‑parametric reconstructions).

Overall, the study clarifies the complementary roles of geometric and growth observables, offers a transparent method to assess the physical content of cosmological constraints, and sets quantitative targets for the precision and covariance control needed in next‑generation experiments to robustly test time‑varying dark energy.