Comparing measures of the Hubble and BAO tensions in $Λ$CDM and possible solutions in $f(Q)$ gravity

This paper evaluates whether $f(Q)$ gravity and phenomenological models can resolve the Hubble tension while remaining consistent with DESI DR2 BAO data, finding that while an exponential $f(Q)$ model slightly alleviates parameter space tensions, no theoretically motivated solution successfully reconciles the Hubble tension with all observational datasets, particularly local $H_0$ measurements.

The Gist

Imagine the universe is a giant, expanding balloon. For decades, cosmologists have had a very specific, well-tested recipe for how this balloon should inflate, called the $\Lambda$CDM model. It's like a trusted map that tells us exactly how fast the balloon is growing right now (the Hubble constant, or $H_0$) and how the air inside is distributed.

But recently, two different groups of explorers have looked at the balloon and reported two very different speeds.

1. The "Early Universe" Team: They look at the baby picture of the universe (the Cosmic Microwave Background) and calculate the speed should be about 67 km/s/Mpc.
2. The "Local Universe" Team: They measure the speed of nearby galaxies and find it's about 73 km/s/Mpc.

This difference is the Hubble Tension. It's like if one group said the car was doing 60 mph and the other said 70 mph, and both were sure their speedometers were perfect.

To make matters worse, there's a second puzzle called the BAO Tension. The universe has "standard rulers" imprinted in it (like the rings on a tree trunk) that tell us how far away things are. The new, super-precise measurements from the DESI telescope suggest these rulers don't quite match the "Early Universe" map either.

The Proposed Solution: Rewriting the Rules of Gravity

The authors of this paper asked: "What if our map isn't wrong, but the rules of the road (gravity) are different than we thought?"

Instead of assuming the universe is filled with mysterious "Dark Energy" (a invisible fluid pushing the balloon), they tested a theory called $f(Q)$ gravity.

• The Analogy: Imagine General Relativity (our current gravity theory) is like a trampoline where heavy balls bend the fabric. $f(Q)$ gravity suggests the trampoline fabric itself has a slightly different texture or "stiffness" that changes over time, causing the universe to accelerate without needing a mysterious fluid.

They tested three different "flavors" of this new gravity theory:

1. Logarithmic: A slow, steady change.
2. Exponential: A rapid, sharp change.
3. Hyperbolic Tangent: A smooth S-curve change.

The Results: A Mixed Bag

Here is what they found, using simple terms:

1. The "Exponential" Gravity Model is the Hero (So Far)
The Exponential $f(Q)$ model was the only one that managed to fix the Hubble Tension without breaking the rules of the early universe. It successfully predicted a faster expansion rate today (around 72 km/s/Mpc) that matches the local measurements.

• The Catch: Even though it fixed the speed, it still had a slight "wobble" when compared to the new DESI ruler measurements. It didn't perfectly solve the BAO tension, but it was the closest we got to a "theoretically motivated" solution.

2. The "Logarithmic" and "Tangent" Models Failed
These two models tried to fix the speed, but in doing so, they made the universe look completely wrong compared to the new ruler measurements. They created a massive mismatch, like trying to fit a square peg in a round hole.

3. Adding a "Cosmological Constant" (The Cheat Code)
The authors tried adding a tiny bit of the old "Dark Energy" back into these new gravity models (like adding a little bit of the old map to the new one).

• Result: It made the fit to the ruler data slightly better, but it ruined the theoretical beauty. It's like saying, "We fixed the engine, but we also need to add a jetpack to make it work." It works, but it defeats the purpose of finding a new engine.

4. The "Phenomenological" Models (The "Just-Do-It" Approach)
These models didn't try to be a new theory of gravity. They just added a flexible "fudge factor" to the math to force the numbers to match.

• Result: They fit the data very well mathematically. However, they are physically weird. They predict that the universe's energy density becomes negative (which is impossible in standard physics) and they require the universe to suddenly speed up very recently (just in the last few billion years). It's like a car that suddenly revs its engine only when you are 10 feet from the finish line. It fits the data, but it feels unnatural.

The Big Takeaway

The paper concludes that it is incredibly difficult to solve both the Hubble Tension and the BAO Tension at the same time.

• If you fix the speed (Hubble Tension), you often break the ruler measurements (BAO Tension).
• If you try to fix the ruler measurements, you often break the speed.

The Exponential $f(Q)$ model is the most promising "theoretical" candidate because it solves the speed problem and doesn't break the ruler measurements too badly. However, the fact that even this model struggles suggests that the solution might not be a simple tweak to gravity.

The Final Metaphor:
Imagine you are trying to tune a radio to a specific station.

• The Hubble Tension is the static noise.
• The BAO Tension is the song playing slightly out of tune.
• The $\Lambda$CDM model is the old radio that can't get rid of the static.
• The $f(Q)$ models are new radios. The Exponential one clears up the static and gets the song mostly in tune, but there's still a tiny buzz. The Phenomenological models are like a DJ who just plays the song at the right speed but ignores the fact that the radio is broken.

The authors suggest that the solution might lie in local anomalies (like us living in a giant, empty bubble in the universe) or that the "rules of the road" change in ways we haven't even imagined yet. For now, the mystery remains, but the Exponential $f(Q)$ model is the strongest lead we have.

Technical Summary

1. Problem Statement

The paper addresses two major cosmological tensions within the standard $\Lambda$CDM model:

• The Hubble Tension: A $\sim5\sigma$ discrepancy between the Hubble constant ($H_0$) inferred from the Cosmic Microwave Background (CMB) under $\Lambda$CDM ($\approx 67.4$ km/s/Mpc) and local distance-ladder measurements ($\approx 73-74$ km/s/Mpc).
• The BAO Anomaly: A tension between the Baryon Acoustic Oscillation (BAO) ruler measurements (specifically from the DESI DR2 dataset) and the expansion history predicted by $\Lambda$CDM calibrated to CMB data. This manifests as a $\sim2.65\sigma$ tension in the $(\Omega_m, H_0 r_d)$ parameter space and a $\sim1.66\sigma$ tension in the data space.

The authors investigate whether $f(Q)$ gravity (symmetric teleparallel gravity where gravity is driven by non-metricity $Q$ rather than curvature) can resolve these tensions simultaneously without introducing new physics at early times (pre-recombination). They also compare these theoretically motivated models against flexible phenomenological models.

2. Methodology

Theoretical Framework:
The study utilizes the $f(Q)$ gravity formalism in the symmetric teleparallel framework (where curvature and torsion are zero). The authors focus on Connection III (the coincident gauge), which simplifies the modified Friedmann equations. They test three specific functional forms for $f(Q)$:

1. Logarithmic: $f(Q) = \frac{Q}{8\pi G} - \alpha \ln(Q/Q_0)$
2. Exponential: $f(Q) = \frac{Q}{8\pi G} \exp(\lambda Q_0/Q)$
3. Hyperbolic Tangent: $f(Q) = \frac{Q}{8\pi G} + \alpha \tanh(Q_0/Q)$

They also test variants of these models assisted by a cosmological constant ($\Lambda$) and three phenomenological models that add flexible decay terms to the standard $H(z)$ (Exponential, Hyperbolic Secant, and Polynomial Decay).

Data and Constraints:
The models are constrained using a comprehensive dataset:

• Local $H_0$: Four model-independent determinations (SH0ES, Megamaser, TRGB-SBF, Type II SN).
• DESI DR2 BAO: The latest BAO measurements providing constraints on $D_H/r_d$, $D_M/r_d$, and $D_V/r_d$.
• CMB: Compressed likelihood from Planck 2018 + SPT-3G + ACT DR6 (constraining $\theta_*$, $\omega_b$, $\omega_{bc}$).
• Cosmic Chronometers (CC): 32 $H(z)$ measurements.

Analysis Techniques:

• Bayesian Inference: Used the nested sampling algorithm dynesty to compute posterior probabilities and Bayesian Evidence ($Z$).
• Tension Quantification: The authors employ two distinct methods to measure BAO tension:
  1. Data Space: Predicting DESI measurements using CMB-only constraints.
  2. Parameter Space: Comparing the $(\Omega_m, H_0 r_d)$ contours from CMB-only vs. BAO-only fits.
  3. MDPS Tension: The maximum of the Data Space and Parameter Space tensions is used as the primary metric for anomaly significance.

3. Key Contributions

• Systematic Comparison: Provides a rigorous comparison of specific $f(Q)$ models against both the Hubble and BAO tensions, a combination often overlooked in favor of solving just one.
• Dual Tension Metrics: Explicitly distinguishes between data-space and parameter-space tensions, arguing that the latter is a more robust indicator of systematic deviations from $\Lambda$CDM.
• Role of the Cosmological Constant: Investigates how adding a $\Lambda$ term to $f(Q)$ models affects the fit, highlighting the trade-off between theoretical motivation (purely geometric acceleration) and observational fit.
• Phenomenological Benchmarking: Uses flexible phenomenological models to establish the "best-case scenario" for background solutions, revealing that even highly flexible models struggle to fit DESI data without systematic residuals.

4. Key Results

Hubble Tension Resolution:

• Most $f(Q)$ models (Log, Exp, Tanh) and phenomenological models successfully resolve the Hubble tension, predicting $H_0 \approx 72\text{--}74$ km/s/Mpc.
• Models assisted by $\Lambda$ (e.g., Log+$\Lambda$) predict lower $H_0$ ($\approx 71$ km/s/Mpc), partially alleviating but not fully solving the tension.

BAO Consistency (The Critical Finding):

• Log and Tanh Models: Fail significantly. When calibrated with CMB alone, they exhibit massive BAO tensions ($>4\sigma$ in parameter space, up to $7.55\sigma$ for Log).
• Exponential ($f(Q)$ Exp) Model: This is the only theoretically motivated model that performs comparably to $\Lambda$CDM regarding BAO.
  • It reduces the parameter space BAO tension slightly to 2.56$\sigma$ (compared to 2.65$\sigma$ for $\Lambda$CDM).
  • However, its data space tension remains similar to $\Lambda$CDM.
  • It achieves a high Bayesian Evidence ($\ln B \approx 16.6$), strongly preferred over $\Lambda$CDM.
• Models with $\Lambda$: Adding a cosmological constant to $f(Q)$ models generally improves the fit to DESI data (reducing $\chi^2$) but undermines the theoretical motivation of $f(Q)$ as an alternative to dark energy. Furthermore, when constrained by CMB+$H_0$+CC, these hybrid models show tensions $>3\sigma$.

Phenomenological Models:

• The "Phen, exp" and "Phen, sech" models fit all data well with low MDPS tensions ($\sim 2.1\text{--}2.2\sigma$).
• Crucial Caveat: Despite low statistical tension, these models systematically overpredict isotropically averaged BAO distances ($D_V/r_d$) at all redshifts compared to DESI DR2. This suggests the "solution" is an artifact of parameter uncertainty rather than a true physical resolution.
• These models predict a singularity in the effective dark energy Equation of State (EoS), violating the weak energy condition, rendering them unphysical within General Relativity.

Age of the Universe:

• All successful models predict a younger universe ($t_0 \approx 13.60\text{--}13.74$ Gyr) compared to Planck $\Lambda$CDM ($13.8$ Gyr) due to the higher $H_0$.
• This creates a mild tension ($2\text{--}3\sigma$) with the ages of the oldest Galactic stars and globular clusters, though it remains within the $1\sigma$ limits of some recent globular cluster studies.

5. Significance and Conclusions

• Difficulty of Joint Solutions: The paper concludes that finding a background solution that simultaneously resolves the Hubble tension and remains consistent with DESI DR2 BAO data is extremely difficult.
• The "Best" Candidate: The Exponential $f(Q)$ model emerges as the strongest candidate. It is theoretically motivated, solves the Hubble tension, and does not worsen the BAO tension compared to $\Lambda$CDM. However, it does not reduce the tension below the $\Lambda$CDM baseline.
• Nature of the Anomaly: The fact that $\Lambda$CDM calibrated with CMB+$H_0$+CC (ignoring BAO) predicts BAO data with negligible tension ($0.53\sigma$) suggests that the local $H_0$ measurements and CMB are mutually consistent approximations of the true expansion history. The BAO anomaly likely arises from a systematic deviation in the expansion history at low redshift ($z \lesssim 0.5$) that standard background modifications struggle to capture without overpredicting distances.
• Future Directions: The systematic residuals in the best-fitting models suggest that the solution to the Hubble and BAO tensions may not lie in simple background modifications (like $f(Q)$ or dynamical dark energy) but rather in perturbative effects (e.g., local voids) or new physics that alters the relationship between redshift and distance in a way not captured by standard $H(z)$ parametrizations.

In summary, while $f(Q)$ gravity offers a promising theoretical avenue, the specific models tested (except the Exponential one) fail to satisfy BAO constraints. Even the successful Exponential model does not fully eliminate the BAO anomaly, highlighting the robustness of the tension and the need for more complex physical mechanisms beyond simple background expansion history modifications.