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

Abstract

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 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, whose logic extends to any model in which the coupling is background inert, by computing $A_s$ per posterior sample, needed to reproduce the ${\sigma }_{80}$ preferred 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 ${\lambda }_0\sqrt{Q{Q}_0}$ correction to the models inflates ${\sigma }_{80}$ by $5\%-8\%$ as compared to its vanilla variant, while the Bayesian evidence disfavors every alternative considered - $\Delta \log \mathcal{Z}=-0.5$ to $-2.6$ against $\Lambda$CDM on the same pipeline. Propagating this inflated ${\sigma }_{80}$ 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 $1.5\sigma -2.6\sigma$ tension with Planck 2018. Imposing the $\ln (10^{10}{A}_s)$ constraint from Planck 2018 as an additional prior removes this inflation, pulling ${\sigma }_{80}$ to the Planck value and ${\lambda }_0$ to values consistent with $0$, with the implied amplitude recovering to within $0.2\sigma$ of Planck in every case. We find no model-dataset combination preferred over $\Lambda$CDM.

Comments: 16 pages, 14 figures, 3 tables