Modified Teleparallel $f(T)$ Gravity, DESI BAO and the $H_0$ Tension

This paper investigates three late-time $f(T)$ gravity models using DESI BAO and other cosmological data to determine if torsional modifications can alleviate the $H_0$ tension, finding that while phantom-like regimes can partially shift the Hubble constant toward local measurements, none of the models are statistically preferred over the standard $\Lambda$CDM model.

The Gist

The Big Problem: The Universe is Splitting Its Personality

Imagine the universe is a giant, expanding balloon. For a long time, scientists have been trying to measure exactly how fast this balloon is inflating. This speed is called the Hubble Constant ($H_0$).

Here's the trouble: We have two different ways to measure it, and they give us two completely different answers.

1. The "Baby Picture" Method: We look at the Cosmic Microwave Background (the afterglow of the Big Bang). Based on this, the universe should be expanding at about 68 km/s/Mpc.
2. The "Adult Photo" Method: We look at nearby supernovae (exploding stars) and measure their brightness. Based on this, the universe is expanding much faster, at about 73 km/s/Mpc.

This disagreement is called the $H_0$ Tension. It's like if you measured your height as a child and got 5 feet, but measured it as an adult and got 6 feet, and you couldn't figure out if you grew too fast or if your ruler was broken.

The Proposed Solution: A New Set of Rules for Gravity

The authors of this paper ask: "What if our current rules for gravity (General Relativity) are slightly incomplete?"

They explore a theory called $f(T)$ Gravity.

• The Analogy: Think of General Relativity as a map drawn on a flat piece of paper. It works great for most things. But $f(T)$ gravity suggests that maybe the map needs a little "twist" or "torsion" added to it, like if the paper itself had a slight spiral in it.
• In this theory, gravity isn't just about the curvature of space (like a bowling ball on a trampoline); it's also about the twist (torsion) of space.

The authors created three specific versions of this "twisted" gravity to see if fixing the rules could make the two measurements of the universe's speed agree with each other.

The Three Experiments (The Models)

They tested three different mathematical "recipes" for this twisted gravity:

1. Model 1 (The Phantom): This version makes the universe expand faster than expected at late times. It's like adding a hidden turbocharger to the universe's engine.
2. Model 2 (The Quintessence): This version makes the universe expand slower than expected. It's like putting a brake on the engine.
3. Model 3 (The New Phantom): Similar to Model 1, this also acts like a turbocharger, speeding things up.

What Happened When They Ran the Numbers?

The team fed these models into a supercomputer along with the latest data from:

• Supernovae (The "Adult Photos")
• DESI (A massive survey of galaxy positions)
• Planck (The "Baby Picture" of the early universe)
• Galaxy Clumping (How matter is gathering together)

Here is what they found:

1. The "Turbo" Models (Model 1 & 3)

These models successfully pushed the "Baby Picture" measurement (the slow 68 number) up toward the "Adult Photo" measurement (the fast 73 number).

• The Good News: They helped solve the $H_0$ tension! The numbers got closer to agreeing.
• The Bad News: To make the speed match, these models broke something else. They messed up the "clumping" of matter. It's like fixing the speedometer on your car, but now the engine is overheating. The universe looks too "smooth" in these models compared to what we actually see in galaxy surveys.

2. The "Brake" Model (Model 2)

This model pushed the speed down even further.

• The Result: It made the disagreement between the "Baby" and "Adult" measurements worse. It's like trying to fix a speedometer by slowing the car down even more.

The Final Verdict: No Free Lunch

The authors ran a statistical check (like a judge weighing the evidence) to see which theory is the "best."

• The Winner: The standard model ($\Lambda$CDM), which uses Einstein's original rules, still wins the award for being the simplest and most accurate overall.
• The Losers: None of the three "twisted gravity" models were good enough to replace the standard model. While they could fix the speed problem, they created new problems with how galaxies clump together.

The Big Takeaway

Think of the universe like a delicate ecosystem. If you tweak one part (the expansion speed) to fix a problem, you often disturb another part (how galaxies form).

In simple terms:
The authors tried to fix the universe's speedometer by adding a "twist" to gravity.

• Two of the twists made the speedometer agree with local measurements, but they broke the engine (galaxy formation).
• One twist made the speedometer disagree even more.
• Conclusion: While these ideas are fascinating and show us how the universe could be different, the simplest explanation (Einstein's original gravity) still fits the data best. However, the study proves that if we do need new physics, it will likely have to be more complex than just a simple twist, because fixing one tension seems to create another.

Technical Summary

1. Problem Statement

The paper addresses the persistent Hubble tension ($H_0$ tension), the significant discrepancy between early-Universe determinations of the Hubble constant (derived from Cosmic Microwave Background (CMB) data, yielding $H_0 \approx 67-68$ km s$^{-1}$ Mpc$^{-1}$) and late-Universe local measurements (derived from the distance ladder, yielding $H_0 \approx 73$ km s$^{-1}$ Mpc$^{-1}$).

The authors investigate whether late-time modifications to gravity within the teleparallel framework can resolve or alleviate this tension. Specifically, they focus on $f(T)$ gravity, a theory where the gravitational action is an arbitrary function of the torsion scalar $T$, rather than the Ricci scalar $R$ used in General Relativity (GR). The goal is to determine if specific $f(T)$ parametrizations can shift the inferred $H_0$ from CMB/BAO data toward local values without violating other cosmological constraints, such as the growth of structure ($S_8$).

2. Methodology

Theoretical Framework

• $f(T)$ Gravity: The authors utilize the teleparallel equivalent of General Relativity (TEGR), where gravity is described by torsion rather than curvature. The action is generalized to $S = \frac{1}{16\pi G} \int d^4x \, e \, f(T) + S_m$.
• Cosmological Background: They assume a spatially flat FLRW universe. The torsion scalar is $T = -6H^2$. The modified Friedmann equations are recast to include an effective torsional fluid with energy density $\rho_T$ and pressure $p_T$, characterized by an equation of state $w_T$.
• Perturbations: They analyze linear scalar perturbations in the quasi-static subhorizon limit. In $f(T)$ gravity, the effective gravitational coupling is modified to $G_{\text{eff}} = G/f_T$ (where $f_T = df/dT$), while the gravitational slip parameter remains unity ($\eta = 1$) at linear order. The growth of matter density contrast $\delta_m$ is governed by a modified growth equation involving $G_{\text{eff}}$.

Models Considered

Three specific $f(T)$ parametrizations are analyzed, all designed to recover TEGR (standard GR) at early times and deviate only at late epochs:

1. Model 1 ($f_1$): $f_1(T) = T e^{\lambda_1 T_0/T}$. This model exhibits phantom-like behavior ($w_T < -1$) and enhances the effective gravitational coupling ($G_{\text{eff}} > G$).
2. Model 2 ($f_2$): $f_2(T) = T + T_0 e^{-\lambda_2 T_0/T}$. This model exhibits quintessence-like behavior ($w_T > -1$) and weakens gravity ($G_{\text{eff}} < G$).
3. Model 3 ($f_3$): $f_3(T) = T + \lambda_3 T_0 [1 - e^{-T_0/T}]$. A novel model similar to Model 1, also exhibiting phantom-like behavior ($w_T < -1$) and $G_{\text{eff}} > G$.

Data and Statistical Analysis

The models are constrained using a Bayesian Markov Chain Monte Carlo (MCMC) analysis (via the Cobaya framework) against a comprehensive dataset:

• Supernovae (SN): Pantheon+ (unanchored, calibrated with a local $H_0$ prior).
• Baryon Acoustic Oscillations (BAO): DESI Data Release 2 (DR2), covering a wide redshift range.
• Cosmic Microwave Background (CMB): Compressed distance priors from Planck.
• Redshift-Space Distortions (RSD): A compilation of $f\sigma_8(z)$ measurements to probe structure growth.

Model performance is evaluated using the Corrected Akaike Information Criterion ($\Delta AIC_C$) to compare the $f(T)$ models against the standard $\Lambda$CDM model.

3. Key Contributions

1. Systematic Comparison of $f(T)$ Dynamics: The paper explicitly links the phenomenological behavior of $f(T)$ models to their effective torsional dynamics. It demonstrates that phantom-like regimes ($w_T < -1$) tend to increase the inferred $H_0$, while quintessence-like regimes ($w_T > -1$) decrease it.
2. DESI DR2 Integration: It is one of the first studies to confront $f(T)$ gravity with the latest DESI DR2 BAO data, providing updated constraints on late-time expansion.
3. Trade-off Analysis: The study highlights a critical trade-off: models that successfully shift $H_0$ toward local measurements often exacerbate tensions in the matter sector (specifically $S_8$ and $\Omega_m$), and vice versa.
4. Growth of Structure Constraints: By including RSD data, the authors show how the modified effective gravitational coupling ($G_{\text{eff}}$) in $f(T)$ gravity alters the growth rate of structures, providing a distinct signature compared to standard dark energy models.

4. Results

• Impact on $H_0$:
  • Models 1 and 3 (Phantom-like): These models shift the $H_0$ value inferred from BAO and CMB data upwards to approximately 71.6–72.0 km s$^{-1}$ Mpc$^{-1}$, bringing them closer to local measurements (e.g., Riess et al.).
  • Model 2 (Quintessence-like): This model shifts $H_0$ downwards to $\approx 65.7$ km s$^{-1}$ Mpc$^{-1}$, worsening the tension with local measurements.
• Impact on Matter Density ($\Omega_m$) and Clustering ($S_8$):
  • The shift in $H_0$ is accompanied by a redistribution of tension. Models 1 and 3, while improving $H_0$, predict lower values for $\Omega_m$ and $S_8$ compared to $\Lambda$CDM when fitted to RSD data.
  • Model 2 predicts higher $S_8$, which could potentially alleviate the $S_8$ tension (discrepancy between CMB and weak lensing) but at the cost of worsening the $H_0$ tension.
• Statistical Viability:
  • Despite the ability of Models 1 and 3 to partially alleviate the $H_0$ tension, the global statistical comparison (using $\Delta AIC_C$) shows that none of the three $f(T)$ models are preferred over $\Lambda$CDM when using the full combined dataset (SN+BAO+CMB+RSD).
  • The $\Delta AIC_C$ values are significantly positive (ranging from 39.0 to 46.3), indicating "decisive evidence" against the $f(T)$ extensions in favor of the standard $\Lambda$CDM model.
• Reconstruction of $w_T(z)$: The MCMC reconstruction confirms that the data tightly constrain the effective equation of state. Models 1 and 3 are statistically confirmed to reside in the phantom regime ($w_T < -1$), while Model 2 remains in the quintessence regime ($w_T > -1$).

5. Significance and Conclusion

The paper concludes that while minimal one-parameter $f(T)$ extensions cannot simultaneously resolve both the $H_0$ and $S_8$ tensions, they serve as a powerful theoretical laboratory. The results demonstrate that late-time torsional modifications can non-trivially redistribute cosmological tensions between the background expansion sector and the growth of structure sector.

• Phantom-like torsion acts as a mechanism to boost the expansion rate (raising $H_0$) but enhances structure growth (lowering $S_8$), creating a new tension in the clustering sector.
• Quintessence-like torsion suppresses expansion (lowering $H_0$) but suppresses growth (raising $S_8$), potentially helping the $S_8$ tension but worsening $H_0$.

The authors suggest that resolving these tensions simultaneously may require more general teleparallel scenarios, such as extended torsional Lagrangians, non-minimal couplings, or additional degrees of freedom, rather than simple one-parameter $f(T)$ models. The study underscores the importance of combining background (expansion) and perturbation (growth) data to fully test modified gravity theories.