The Amplitude-Growth Degeneracy and Implied $A_s$ Diagnostic for Background-Inert Modified Gravity

Abstract

We prove that any background-inert perturbative coupling $ λ$ in coincident $ f(Q) $ gravity exhibits a degeneracy with the clustering amplitude $ σ_{80} $, when using compressed CMB distance priors. This degeneracy is, in fact, a direct materialization of a deeper $ A_s-D_0(λ) $ degeneracy between the primordial amplitude $ A_s $ and the present day growth factor $ D_0(λ) $. We outline a consistency check scheme, whose logic extends to any model in which the coupling is background inert, by computing $ A_s $ per posterior sample, needed to reproduce the $ σ_{80} $ preferred by the sampler. We perform our analysis with two dataset pipelines, based on the coupled/decoupled $ fσ_8(z) $ data. To ensure theoretical diversity, we include $ Λ$CDM and the Hybrid model in the $ f(Q) $ framework. Our results illustrate that adding the $ λ_0\sqrt{QQ_0} $ correction to the models inflates $ σ_{80} $ by $ 5\%-8\% $ as compared to its vanilla variant, while the Bayesian evidence disfavors every alternative considered - $Δ\log\mathcal{Z} = -0.5$ to $-2.6$ against $Λ$CDM on the same pipeline. Propagating this inflated $ σ_{80} $ through a per-sample computation of the implied primordial amplitude accounting for both the transfer function and the modified growth factor yields $\ln(10^{10}A_s)$ in $1.5σ-2.6σ$ tension with Planck 2018. Imposing the $ \ln(10^{10}A_s) $ constraint from Planck 2018 as an additional prior removes this inflation, pulling $ σ_{80} $ to the Planck value and $ λ_0 $ to values consistent with $ 0 $, with the implied amplitude recovering to within $ 0.2σ$ of Planck in every case. We find no model-dataset combination preferred over $ Λ$CDM.