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

We prove that any background-inert perturbative coupling $\lambda$ in coincident $f(Q)$ gravity exhibits a degeneracy with the clustering amplitude $\sigma_{80}$, when using compressed CMB distance priors. This degeneracy is, in fact, a direct materialization of a more deeper $A_s-D_0(\lambda)$ degeneracy between the primordial amplitude $A_s$ and the present day growth factor $D_0(\lambda)$. We outline a consistency check scheme, applicable to models even outside the $f(Q)$ class, by computing $A_s$ needed to reproduce the $\sigma_{80}$ predicted by the sampler. We perform our analysis with two dataset pipelines, based on the coupled/decoupled $f\sigma_8(z)$ data. To ensure theoretical diversity, we include $\Lambda$CDM and the Hybrid model in the $f(Q)$ framework. Our results illustrate that adding the $\lambda_0\sqrt{QQ_0}$ correction to the models inflates $\sigma_{80}$ 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 $1.7\sigma-2.2\sigma$ tension with Planck values. We utilize the $1\sigma$ $\ln(A_s)$ constraints from Planck as priors in order to fix the artificial increase in $\sigma_{80}$ 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 $\Lambda$CDM$+\lambda_0+\ln(A_s)$ + SDSS DR16 combination shows a weak preference over the vanilla $\Lambda$CDM model, validated by the values of $\log\mathcal{Z},\;AIC,\;DIC,$ and BIC.