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Inverse non-metricity in $f(Q)$ gravity: cosmology and observational constraints

This paper investigates a minimal $f(Q)$ gravity model featuring an inverse non-metricity term, finding that while it can alleviate the $H_0$ tension by predicting a higher expansion rate, it faces tight observational constraints from late-time data that largely negate its advantages over the standard $\Lambda$ CDM model.

Original authors: Luís Atayde, Simão Marques Nunes, Noemi Frusciante

Published 2026-03-31

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CC BY 4.0

Original authors: Luís Atayde, Simão Marques Nunes, Noemi Frusciante

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: Fixing the Universe's "Glitch"

Imagine the standard model of the universe (called $\Lambda$ CDM) as a perfectly tuned car engine. It runs smoothly, predicts where planets will be, and explains the cosmic background radiation. But, like any old car, it has a few annoying rattle noises:

The $H_0$ Tension: If you measure how fast the universe is expanding using the "old" way (looking at the baby universe/CMB), you get one speed. If you measure it using the "new" way (looking at nearby exploding stars), you get a faster speed. The engine seems to be running at two different RPMs at once.
The $S_8$ Tension: The universe seems to be clumping together (forming galaxies and clusters) a bit too much compared to what the standard model predicts.

The authors of this paper asked: What if the engine isn't broken, but the fuel is slightly different? They proposed a new type of "fuel" based on a theory called $f(Q)$ gravity.

The New Theory: Gravity as a "Rubber Band"

In standard physics (General Relativity), gravity is like the curvature of a trampoline. If you put a heavy bowling ball (Earth) on it, the fabric bends, and marbles roll toward it.

In this new paper, the authors look at a different version of gravity called Symmetric Teleparallel Gravity. Here, instead of bending the fabric, gravity is described by how the fabric stretches or shrinks (called non-metricity).

They proposed a specific recipe for this gravity, which they call the "Inverse Non-Metricity" model.

The Analogy: Imagine the universe is a giant rubber band. In the standard model, the rubber band stretches at a steady, predictable rate. In this new model, the rubber band has a weird property: as it gets older and stretches more, it gets stiffer in a specific way that changes how hard it pulls things together.

What Does This New Gravity Do?

The authors ran simulations to see how this "stiffer rubber band" changes the universe. Here are the three main things they found:

It Pulls Harder: The new gravity makes the "effective gravitational coupling" stronger. Think of it like turning up the volume on a speaker. The music (gravity) is louder.
- Result: Matter clumps together faster. Galaxies form more easily.
It Expands Faster: Because of this extra "pull," the universe's expansion history changes. It suggests the universe is expanding faster today than the standard model thinks.
- Result: This helps fix the $H_0$ tension. The new model predicts a faster expansion rate that matches the "local" measurements better.
It Changes the Cosmic Echo: The early universe was like a drum being hit, creating sound waves (which we see as the Cosmic Microwave Background). Because the expansion history changed, the "echo" of these waves shifts slightly.

The Catch: The "Neutrino" Compromise

Here is where the story gets tricky.

Because this new gravity pulls harder, it makes the universe clump together too much. This creates a new problem: the universe looks too clumpy compared to what we actually see in galaxy surveys (the $S_8$ tension gets worse).

The Solution? The authors introduced Massive Neutrinos.

The Analogy: Imagine the universe is a crowded dance floor.
- New Gravity: Everyone is dancing very energetically and grouping together tightly (clumping).
- Massive Neutrinos: Imagine a bunch of ghosts (neutrinos) running through the crowd. Because they are "ghosts," they don't stop to dance; they zip through the crowd, knocking people apart and preventing them from forming tight groups.

The paper found that if you combine this "stronger gravity" with "heavy ghosts" (neutrinos with significant mass), the ghosts cancel out the extra clumping.

The Trade-off: You fix the expansion speed ( $H_0$ ) and you fix the clumping ( $S_8$ ), BUT only if you assume neutrinos are much heavier than we currently think they are.

The Verdict: A "Minimal" but "Constrained" Alternative

The authors tested their model against a mountain of real data:

CMB (The Baby Picture): Planck satellite data.
BAO/RSD (The Ruler): Measurements of galaxy spacing.
SNIa (The Distance Markers): Exploding stars.
DES (The Lens): How gravity bends light from distant galaxies.

The Results:

CMB Only: The model looks okay. It's slightly better than the standard model at explaining the expansion speed.
Adding Real-World Data: When they added data about how galaxies are actually arranged and moving, the model started to struggle. The "stronger gravity" made the universe look too clumpy.
The Neutrino Fix: If they allowed the neutrino mass to be a free variable (not fixed), the model could fit the data again. But this requires neutrinos to be surprisingly heavy.

The Conclusion:
The paper concludes that this "Inverse Non-Metricity" model is a minimal solution (it doesn't add many new parameters) and it can solve the expansion speed problem. However, it is heavily constrained. It only works if we accept that neutrinos are much heavier than standard physics suggests.

Without heavy neutrinos, the model is ruled out by the data. With heavy neutrinos, it works, but it's not necessarily "better" than the standard model; it's just a different way to get the same result, with a very specific cost.

Summary in One Sentence

This paper explores a new theory where gravity gets "stiffer" over time, which speeds up the universe's expansion (fixing one problem) but makes galaxies clump too much (creating another problem), a problem that can only be solved if we assume invisible particles called neutrinos are much heavier than we thought.

1. Problem Statement

The standard $\Lambda$ CDM cosmological model, while successful, faces persistent theoretical and observational challenges:

Theoretical Issues: The cosmological constant ( $\Lambda$ ) problem and the coincidence problem.
Observational Tensions:
- $H_0$ Tension: A significant discrepancy ( $\sim 5\sigma$ ) between early-universe inferences (CMB) and late-universe local measurements (distance ladder) of the Hubble constant.
- $S_8$ Tension: A discrepancy in the amplitude of matter fluctuations ( $\sigma_8$ ) between CMB data and weak lensing/large-scale structure (LSS) surveys, where LSS typically prefers lower clustering.

The authors investigate Modified Gravity (MG) as a geometric alternative to dark energy. Specifically, they focus on Symmetric Teleparallel Gravity (STG), where gravity is attributed to non-metricity ( $Q$ ) rather than curvature or torsion. They analyze a specific model where the Lagrangian function $f(Q)$ includes an inverse non-metricity term:
$f(Q) = Q + \frac{M^4}{Q}$
This model is the non-metricity analogue of inverse-curvature $f(R)$ gravity. The primary goal is to determine if this minimal model (introducing no free parameters beyond $\Lambda$ CDM) can resolve cosmological tensions while remaining consistent with observational data.

2. Methodology

The study employs a comprehensive theoretical and observational framework:

Theoretical Framework:
- Field Equations: Derived from the action of $f(Q)$ gravity in the coincident gauge ( $\Gamma^\alpha_{\mu\nu}=0$ ).
- Background Dynamics: Solved for the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. The model introduces an effective "dark Q-fluid" with density $\rho_Q \propto M^4/H^2$ .
- Linear Perturbations: Analyzed using the quasi-static approximation (valid on sub-horizon scales). The key modification is an effective gravitational coupling $\mu(a)$ :
  $\mu(a) = \frac{1}{f_Q} = \left[ 1 - \frac{\Omega_Q^0}{3} \left(\frac{H_0}{H(a)}\right)^4 \right]^{-1}$
  Since $\mu > 1$ , gravity is effectively stronger than in General Relativity (GR) at late times, enhancing structure growth.
- Neutrino Mass: The impact of massive neutrinos ( $\Sigma m_\nu$ ) was investigated in two scenarios: fixed minimal mass ($0.06$ eV) and varying mass ( $\Sigma m_\nu \in [0.01, 2]$ eV). Neutrino free-streaming suppresses structure growth, potentially counteracting the MG enhancement.
Numerical Implementation:
- The model was implemented in the Einstein-Boltzmann solver MGCAMB to compute CMB temperature/lensing spectra and matter power spectra self-consistently.
Observational Data & Analysis:
- Datasets:
  - PLK18: Planck 2018 CMB data (TT, TE, EE).
  - PBRS: PLK18 + BAO (SDSS DR7, 6dF, DR12) + RSD + SNIa (JLA).
  - PBRSD: PBRS + DES Year-1 (galaxy clustering, cosmic shear, galaxy-galaxy lensing).
- Statistical Tools: Markov Chain Monte Carlo (MCMC) via MGCosmoMC and parameter constraints derived using GetDist.
- Model Comparison: Used $\chi^2$ and Deviance Information Criterion (DIC) to compare $f(Q)$ against $\Lambda$ CDM.

3. Key Contributions

Parameter Minality: Demonstrated that the inverse non-metricity model ( $f(Q) = Q + M^4/Q$ ) introduces no additional free parameters relative to $\Lambda$ CDM. The parameter $M$ is fully determined by the present-day matter density parameter $\Omega_m^0$ .
Signatures of Inverse $f(Q)$ : Identified specific observational signatures:
- Enhanced Growth: Stronger gravitational coupling ( $\mu > 1$ ) boosts the matter power spectrum and CMB lensing.
- ISW Suppression: The modified expansion history suppresses the late-time Integrated Sachs-Wolfe (ISW) effect, reducing power in the low- $\ell$ CMB tail.
- Peak Shifts: Altered expansion history shifts CMB acoustic peak positions to higher multipoles.
Neutrino Degeneracy: Quantified the degeneracy between the MG enhancement of growth and the suppression caused by massive neutrino free-streaming.

4. Results

A. Cosmological Tensions ( $H_0$ and $S_8$ )

Fixed Neutrino Mass ( $\Sigma m_\nu = 0.06$ eV):
- $H_0$ : The model prefers significantly higher expansion rates ( $H_0 \approx 74\text{--}78$ km/s/Mpc) compared to $\Lambda$ CDM ( $\approx 68$ km/s/Mpc), effectively alleviating the $H_0$ tension with local measurements.
- $S_8$ : The enhanced gravitational coupling increases the clustering amplitude ( $\sigma_8$ ). Consequently, the $S_8$ tension is worsened or remains unchanged, as the model predicts higher clustering than observed by weak lensing.
Varying Neutrino Mass ( $\Sigma m_\nu$ free):
- Allowing $\Sigma m_\nu$ to vary introduces a compensatory mechanism. The data prefers a large neutrino mass ( $\Sigma m_\nu \approx 0.7\text{--}0.8$ eV) to suppress the MG-enhanced growth.
- This reduces $\sigma_8$ and $S_8$ to values consistent with weak lensing ( $S_8 \approx 0.74$ ).
- However, this comes at the cost of requiring neutrino masses far exceeding current upper limits from $\Lambda$ CDM analyses ( $\Sigma m_\nu \lesssim 0.12$ eV).

B. Statistical Constraints

CMB Only (PLK18): The model shows a marginal improvement over $\Lambda$ CDM ( $\Delta \chi^2 \approx -4$ ), primarily driven by the preference for higher $H_0$ .
Combined Data (PBRS, PBRSD):
- The inclusion of late-time data (BAO, RSD, SNIa, DES) imposes tight constraints that largely remove any statistical improvement.
- $\Delta \chi^2$ becomes highly positive (worse fit) for the fixed mass case ( $\Delta \chi^2 \approx 84$ for PBRS).
- Even with free neutrino mass, the DIC indicates that the model is disfavored due to the complexity of requiring large neutrino masses without a significant gain in goodness-of-fit.

5. Significance and Conclusion

Viability: The inverse non-metricity $f(Q)$ model is a minimal yet strongly constrained alternative to dark energy. While it conceptually offers a geometric route to acceleration without extra parameters, it fails to provide a statistically superior fit to the full suite of cosmological data compared to $\Lambda$ CDM.
Trade-off: The model can alleviate the $H_0$ tension but exacerbates the $S_8$ tension unless compensated by unphysically large neutrino masses.
Future Prospects: The study highlights that future data with tighter control over late-time systematics and more precise neutrino mass constraints will be critical in testing this class of non-metricity models. The current results suggest that while the model is theoretically interesting, it is not currently supported by observational evidence as a viable replacement for $\Lambda$ CDM.

In summary, the paper concludes that while inverse non-metricity gravity offers an elegant geometric explanation for late-time acceleration, its specific phenomenological predictions (enhanced growth) are in conflict with large-scale structure observations unless fine-tuned with massive neutrinos, rendering it statistically disfavored against the standard model.