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

This paper demonstrates that background-inert perturbative couplings in coincident $f(Q)$ gravity create a degeneracy between the primordial amplitude $A_s$ and the growth factor, leading to unphysically high $\sigma_8$ values that can be resolved by imposing Planck $A_s$ priors, a constraint that ultimately penalizes the extended models with information criteria despite a weak statistical preference for the $\Lambda$CDM+$\lambda_0$+$\ln(A_s)$ variant.

The Gist

Imagine the universe as a giant, expanding balloon. For decades, scientists have used a standard recipe called ΛCDM (Lambda-Cold Dark Matter) to describe how this balloon inflates and how the "dust" on it (galaxies) clumps together. This recipe works incredibly well for the early universe, but when we look at the universe today, something feels slightly off. The dust seems to be clumping together a bit less than the recipe predicts. This is known as the $S_8$ tension.

To fix this, some scientists proposed a new ingredient for the recipe: a modification to gravity called $f(Q)$ gravity. Specifically, they added a tiny, invisible "spice" (a mathematical term called $\lambda_0$) that doesn't change how the balloon expands (the background) but does change how the dust clumps together (the growth).

Here is the story of what this paper discovered about that spice, explained simply:

1. The "Magic Trick" of the Data

The researchers ran a computer simulation to see if this new spice could fix the clumping problem. They used a specific set of rules (data) that includes measurements of the Cosmic Microwave Background (the "baby picture" of the universe) and how galaxies are moving today.

The Problem: The computer simulation found a "loophole." It discovered that if you add this spice ($\lambda_0$), the model can make the galaxies clump together more to match the data. However, to do this, the computer had to secretly cheat on a fundamental number called the Primordial Amplitude ($A_s$).

Think of it like a chef trying to make a soup taste saltier. Instead of adding salt, the chef secretly doubles the amount of water, which makes the soup taste different, but then claims the original recipe was just "under-salted." The computer found a way to make the model fit the data perfectly by inflating the "clumping" number, but it did so by forcing the "early universe" number to be physically impossible (about 20–30% higher than what we know from the baby picture of the universe).

2. The "Implied $A_s$" Detective Tool

The authors realized this was a statistical trick, not a real discovery. The computer was exploiting a "degeneracy"—a situation where two different knobs (the spice $\lambda_0$ and the clumping strength $\sigma_8$) can be turned together to get the same result, hiding the fact that the "early universe" knob is broken.

To catch this trick, they invented a new diagnostic tool called the "Implied $A_s$" check.

• How it works: Instead of just asking, "Does this model fit the data?" they asked, "If this model fits the data, what would the early universe have to look like?"
• The Result: When they applied this check, the models with the spice failed. They required an early universe that contradicts our best observations (Planck data). It was like finding out the "saltier soup" required a secret ingredient that doesn't exist in nature.

3. The "Firewall"

The paper shows that if you put a "firewall" on the simulation—forcing the computer to respect the known value of the early universe (using a Planck prior)—the magic trick stops.

• The computer can no longer cheat by inflating the clumping.
• The "spice" ($\lambda_0$) gets pushed back to zero or very small values.
• The models with the spice suddenly look worse than the standard recipe (ΛCDM) because they are now paying a penalty for having an extra ingredient that doesn't actually help.

4. The One Weird Exception

There was one tiny, strange case where the model with the spice still looked slightly better than the standard recipe, even after the firewall was put up. The authors call this a "weak preference." They are very careful to say this isn't a confirmed discovery; it's a tiny anomaly that needs more investigation with better data. It's like a coin that lands on its edge once in a million flips—you don't bet on it yet, but you don't ignore it either.

The Bottom Line

This paper is a warning label for future cosmology.

• The Trap: If you use simplified data (compressed CMB) to test new gravity theories that only affect how things clump (not how the universe expands), you might get a "false positive." The computer will tell you the new theory is great, but it's only great because it's cheating on the early universe numbers.
• The Solution: Always check the "Implied $A_s$." If a new theory requires the early universe to be totally different from what we already know, it's likely a statistical illusion, not a real discovery.

In short: The paper proves that a popular new gravity theory looks promising only because the math is playing a trick on us. Once we stop the trick, the theory goes back to being just a slightly more complicated version of the old, standard model, with no real advantage.

Technical Summary

Technical Summary: The Amplitude-Growth Degeneracy and Implied $A_s$ as Diagnostic for Background-Inert Modified Gravity

Problem Statement
Modern cosmology faces significant tensions, most notably the $S_8$ (or $\sigma_8$) tension, where late-time probes (Weak Lensing, Redshift-Space Distortions) suggest a lower clustering amplitude than extrapolated from Cosmic Microwave Background (CMB) data within the standard $\Lambda$CDM model. Modified gravity theories, such as $f(Q)$ gravity (Symmetric Teleparallel Gravity), are often invoked to resolve these discrepancies. However, a critical methodological vulnerability exists when analyzing these theories using "compressed" CMB distance priors (shift parameters $R$ and $l_a$) rather than full CMB likelihoods.

The paper identifies that when a modified gravity theory converges to standard $\Lambda$CDM before recombination but introduces a perturbative coupling ($\lambda$) that is inert to the background expansion (affecting only the growth of structure), a statistical degeneracy arises. Specifically, the sampler exploits a degeneracy between the primordial amplitude ($A_s$) and the present-day growth factor ($D_0$). Because compressed CMB priors implicitly fix $A_s$ to Planck values while leaving the background geometry constrained, the sampler artificially inflates the clustering amplitude $\sigma_8$ to accommodate the modified growth history, leading to statistically preferred but physically unphysical results.

Methodology
The authors analyze two specific models within the coincident $f(Q)$ framework:

1. $\Lambda$CDM: The standard reference model.
2. Hybrid $f(Q)$: A model containing a boundary term ($\alpha^2$) that is indistinguishable from $\Lambda$CDM in the background expansion but modifies perturbations.

Both models are tested with and without a specific perturbative correction: the $\lambda_0 \sqrt{Q}Q_0$ term. This term is "background-inert," meaning it does not alter the Hubble parameter $H(z)$ but modifies the effective gravitational constant $G_{eff}$ in the growth equation.

The analysis employs two dataset pipelines:

• BD2R: Compressed CMB + $E_g$ statistic + Pantheon+ SNe + DESI DR2 BAO + isolated RSD data.
• BSDp: Compressed CMB + $E_g$ statistic + Pantheon+ SNe + SDSS DR16 BAO (combined with RSD).

To diagnose the degeneracy, the authors propose a model-independent consistency check called the "Implied $A_s$" diagnostic. This metric calculates the primordial amplitude required to reproduce the $\sigma_8$ value predicted by the sampler given the modified growth factor, using the relation $A_s^{Implied} = A_s^{Planck} (\sigma_8 / \sigma_8^{Planck})^2$. If the implied $A_s$ deviates significantly from the Planck value, it indicates the sampler is exploiting the $A_s - D_0(\lambda)$ degeneracy.

Key Contributions

1. Identification of the $A_s - D_0(\lambda)$ Degeneracy: The paper proves that any background-inert perturbative coupling in $f(Q)$ gravity creates a degeneracy with $\sigma_8$ when using compressed CMB priors. The sampler "slides" along a $\sigma_8 - \lambda_0$ valley to find a better fit, inflating $\sigma_8$ without a corresponding adjustment in $A_s$ (which is fixed by the priors).
2. The Implied $A_s$ Diagnostic: The authors introduce a simple, post-processing consistency test. By mapping the best-fit $\sigma_8$ back to the required $A_s$, researchers can detect if a model's statistical preference is driven by an unphysical inflation of the clustering amplitude.
3. Quantification of the "Amplitude Compensation" Mechanism: The study quantifies the extent of this inflation, showing that the $\lambda_0$ correction inflates $\sigma_8$ to unphysical values (e.g., $0.90$ vs. $0.79$), requiring a $20\%-30\%$ increase in $A_s$ to be physically consistent, creating a $1.7\sigma - 2.2\sigma$ tension with Planck data.

Results

• Unconstrained Analysis: When $\lambda_0$ is allowed to vary freely, both the $\Lambda$CDM and Hybrid $f(Q)$ models show moderate statistical evidence (via $\Delta \log Z$, AIC, DIC) in favor of the $\lambda_0$ variants. However, this preference is driven by the artificial inflation of $\sigma_8$.
• Implied $A_s$ Tension: The "Implied $A_s$" values for the $\lambda_0$ models deviate significantly from Planck 2018 values. For the BD2R pipeline, the tension reaches $2.2\sigma$; for BSDp, it is $1.7\sigma - 1.8\sigma$.
• Effect of Priors: When a strict Gaussian prior on $\ln(A_s)$ (from Planck) is imposed, the degeneracy is broken. The $\sigma_8$ values return to baseline, and the statistical preference for the $\lambda_0$ models vanishes (or becomes weakly disfavored), with Information Criteria (AIC, BIC) penalizing the extra parameters as expected.
• Exception: A weak residual preference ($\Delta \log Z = +1.09$) for the $\Lambda$CDM + $\lambda_0$ + $\ln(A_s)$ model remains in the BSDp combination even after the degeneracy is closed. The authors flag this as a potential signal requiring further investigation with full-shape CMB likelihoods, noting it borders the "inconclusive-weak" regime.

Significance and Claims
The paper claims that the use of compressed CMB priors is insufficient for theories where perturbative degrees of freedom are decoupled from the background expansion. In such cases, the primordial amplitude is left unconstrained, allowing the sampler to generate unphysical fits that appear statistically superior.

The authors argue that the "Implied $A_s$" diagnostic serves as a necessary "firewall" or consistency check. It allows researchers to identify and reject models that rely on amplitude inflation to fit data, ensuring that statistical improvements are not mistaken for physical validity. The study concludes that future analyses of this class of modified gravity theories must either utilize full CMB likelihoods or enforce strict priors on $\ln(A_s)$ to prevent misinference, particularly as next-generation surveys provide sub-percent growth measurements.