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 more 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, applicable to models even outside the $ f(Q) $ class, by computing $ A_s $ needed to reproduce the $ σ_{80} $ predicted 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} $ to unphysical values, while showing moderate evidence in favor of the said models. However, this results in an increase of $ 20\%-30\% $ in $ A_s $ in $ 1.7σ-2.2σ$ tension with Planck values. We utilize the $ 1σ$ $ \ln(A_s) $ constraints from Planck as priors in order to fix the artificial increase in $ σ_{80} $ and find that all the constrained parameters return to their baseline values. Each model is penalized by around $ 2 $ units per extra parameter. Interestingly, the $ Λ$CDM$ +λ_0+\ln(A_s) $ + SDSS DR16 combination shows a weak preference over the vanilla $ Λ$CDM model, validated by the values of $ \log\mathcal{Z},\;AIC,\;DIC, $ and BIC.