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

TL;DR

This paper investigates degeneracies in background-inert $f(Q)$ gravity models, revealing how they affect the clustering amplitude $\sigma_{80}$ and primordial amplitude $A_s$, and assesses model viability with observational data.

Contribution

It identifies a degeneracy between $A_s$ and the growth factor $D_0(\lambda)$ in $f(Q)$ gravity and proposes a consistency check scheme applicable beyond $f(Q)$ models.

Findings

Adding $\lambda_0\sqrt{QQ_0}$ inflates $\sigma_{80}$ to unphysical values.

Models show 20-30% increase in $A_s$, causing 1.7-2.2 sigma tension with Planck.

Including priors on $A_s$ restores parameters to baseline values, with models penalized by around 2 units per extra parameter.

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 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.