Implications of a high growth index on the variation of $G$

A recent determination of the growth index indicates a value significantly higher than the $Λ$CDM prediction, suggesting that alternative scenarios to $Λ$CDM may be required. In this work, we investigate whether a time-varying Newton’s constant, $G_N$, can account for such a high growth index, $γ=0.063\pm0.025$. Adopting a phenomenological approach, we study two parameterizations of the effective gravitational coupling, $G_{\rm eff}$, one based on a Taylor expansion and another linked to the energy density parameter of Dark Energy. We constrain the models with Cosmic Chronometers (CC), Dark Energy Spectroscopic Instrument baryon acoustic oscillations (data release 2), CMB priors, and a gaussian likelihood for the growth index. We show that the constant $γ$ approximation is accurate for the parametrization linked to the energy density parameter of dark energy, but presents a non-negligible error for the other case, which we treat as a systematic error in the analysis. We find a $2.4σ-3.4σ$ tension level with constant $G_{\rm eff}$, depending on the parametrization. The results indicate that $G_{\rm eff}<G_N$ around the period of accelerated expansion, corresponding to a weaker effective gravitational interaction on cosmological scales, which leads to a suppression of the growth of cosmological structures.

💡 Research Summary

The paper addresses the recent measurement of the linear growth index γ = 0.633 ± 0.025, which is significantly higher than the ΛCDM prediction γ ≈ 0.55. Since a larger γ indicates a suppression of structure growth relative to the standard model, the authors explore whether a time‑varying Newtonian gravitational constant, G_N, could naturally produce such a high value. They adopt a phenomenological approach and consider two distinct parameterizations of the effective gravitational coupling G_eff(a).

The first parameterization is a second‑order Taylor expansion around the present epoch:
 G_eff/G_N = 1 + ½ g₂ (a − 1)²,
with g₁ set to zero to satisfy Solar‑System constraints and the requirement that G_eff ≈ G_N at Big‑Bang Nucleosynthesis. In this form, a positive g₂ leads to a decreasing G_eff at low redshift, which would suppress the growth of matter perturbations and raise γ.

The second parameterization ties the variation of G_eff directly to the dark‑energy density fraction:
 G_eff/G_N = 1 + μ₀ Ω_Λ(a)/Ω_Λ,
where μ₀ is a free dimensionless constant. A negative μ₀ similarly yields G_eff < G_N during the recent accelerated expansion, again leading to a higher γ.

Both models recover the standard Newtonian constant at high redshift, ensuring consistency with early‑Universe observables. The authors solve the linear perturbation equation for the matter density contrast,
 δ’’_m + (3/2)Ω_m(a) δ’_m − (3/2)