Realization of quintom dark energy after DESI DR2 in Nieh-Yan modified teleparallel gravity

The Gist

The Big Problem: The "Impossible Crossing"

Imagine the universe is a car speeding up (accelerating expansion). Scientists think this is caused by "Dark Energy," a mysterious fuel. For a long time, they thought this fuel was a constant, like a steady cruise control setting.

However, new data from the DESI telescope suggests the fuel is changing. It's not just a constant; it's "quintom" dark energy. This means its behavior is shifting across a specific "forbidden line" (where the equation of state $w = -1$).

• Quintom-A: The fuel starts on one side of the line and crosses to the other.
• Quintom-B: The fuel starts on the other side and crosses back. (Current data suggests the universe is doing this "Quintom-B" crossing).

The Catch: In standard physics (Einstein's gravity), trying to build a model where this fuel crosses that line is like trying to drive a car through a wall. The math breaks down. The "fuel" becomes unstable, causing the universe to rip apart or behave in impossible ways (mathematical "ghosts" and "gradient instabilities"). It's a "No-Go" zone.

The Solution: A New Road (Teleparallel Gravity)

The authors of this paper propose a new way to drive through that wall. Instead of using standard Einstein gravity, they use a different framework called Teleparallel Gravity.

Think of standard gravity as describing the universe by the curvature of a rubber sheet (like a bowling ball on a trampoline). Teleparallel gravity describes it differently: instead of curvature, it uses torsion (twisting). Imagine the universe isn't a smooth sheet, but a twisted rope.

In this new framework, the authors introduce a special "coupling" (a connection) between the Dark Energy and a specific mathematical feature called the Nieh-Yan density.

The Magic Trick: Removing the Problematic Part

Here is the core discovery, explained with an analogy:

Imagine you are trying to balance a spinning top (Dark Energy) on a table. In the old rules (Einstein gravity), if you try to make the top spin fast enough to cross the "forbidden line," it starts wobbling violently and falls over (instability).

The authors' new method adds a special clamp (the Nieh-Yan coupling).

1. The Background is Safe: The clamp doesn't stop the top from spinning or changing speed. The overall history of the universe (the "background evolution") looks exactly the same as before. The universe still accelerates, and the Dark Energy still crosses the line.
2. The Wobble is Gone: However, the clamp locks the wobbling (the perturbations) in place. It forces the wobble to be zero.

By forcing the "wobble" to vanish, the mathematical instability disappears. The Dark Energy is no longer a "dynamic" variable that can go crazy; it becomes a fixed constraint. The authors call this removing the Dark Energy perturbation from the "menu of dynamical degrees of freedom."

In short: They found a way to let the Dark Energy cross the forbidden line without the universe exploding, by using a new type of gravity that "locks" the dangerous fluctuations.

The "Smoking Gun": Parity Violation

The paper also points out a unique side effect of this new gravity model.

Imagine you have a pair of gloves: a left-handed one and a right-handed one. In most physics, these gloves behave exactly the same way. But in this new model, the "twisted" nature of gravity (torsion) treats them differently.

When gravitational waves (ripples in spacetime) travel through the universe in this model:

• The "left-handed" ripples travel at a slightly different speed than the "right-handed" ripples.
• This is called parity violation (breaking the symmetry between left and right).

The paper claims this is a unique prediction. If future experiments detect that gravitational waves have different speeds depending on their "handedness," it would be strong evidence that this specific "Nieh-Yan" gravity model is real.

Summary of the Paper's Claims

1. The Problem: Standard physics says Dark Energy cannot cross the $w = -1$ line without causing instability.
2. The Fix: By using Teleparallel Gravity and coupling Dark Energy to the Nieh-Yan density, the authors show that the background universe can cross the line safely.
3. The Mechanism: The coupling acts as a constraint that forces the dangerous "wobbles" (perturbations) of Dark Energy to be zero. The universe evolves smoothly, but the unstable fluctuations are mathematically removed.
4. The Prediction: This model predicts that gravitational waves will exhibit parity violation (left and right ripples travel at different speeds), which could be tested by future gravitational wave detectors.

The authors conclude that this provides a viable, stable way to model the "Quintom-B" behavior suggested by recent DESI data, without the mathematical disasters that plagued previous attempts.

Technical Summary

Technical Summary: Realization of Quintom Dark Energy in Nieh-Yan Modified Teleparallel Gravity

Problem Statement
Recent observational data, particularly from the DESI Collaboration (Data Release 2), indicate a preference for "quintom" dark energy, where the equation of state (EoS) parameter $w$ evolves across the cosmological constant boundary ($w = -1$). Specifically, current data favor the "quintom-B" scenario, where $w$ transitions from $w > -1$ to $w < -1$. However, theoretical models based on single perfect fluids or single kessence-like scalar fields minimally coupled to Einstein gravity face a "no-go" theorem. In these standard frameworks, crossing $w = -1$ inevitably leads to perturbative instabilities:

1. Gradient Instability: Arising from a negative squared sound speed ($c_s^2 < 0$) when the EoS crosses $-1$.
2. Ghost Instability: Arising from a wrong-sign kinetic term (negative $z^2$) in the quadratic action, particularly in quintom-B scenarios.

These instabilities render the perturbations unphysical, preventing the construction of viable single-component quintom models within standard General Relativity (GR).

Methodology
The authors propose a solution within the framework of Teleparallel Gravity, specifically extending the GR-equivalent teleparallel theory by coupling dark energy to the Nieh-Yan density. The modified action, termed Nieh-Yan Modified Teleparallel Gravity (NYTG), includes a coupling term between the dark energy scalar field $\phi$ and the Nieh-Yan density ($T_{\lambda\mu\nu}\tilde{T}^{\lambda\mu\nu}$).

The study analyzes two specific realizations of dark energy within this NYTG framework:

1. Single Perfect Fluid: Modeled using the standard fluid action with particle number density flux $j^\mu$.
2. Single Kessence-like Scalar Field: Modeled with a Lagrangian $L(X, \phi)$ dependent on the kinetic term $X$ and the field $\phi$.

The analysis proceeds by:

• Deriving background evolution equations in a spatially flat Friedmann-Robertson-Walker (FRW) universe.
• Performing a linear cosmological perturbation analysis using the tetrad formalism and the Weitzenböck gauge.
• Constructing the quadratic action for scalar perturbations to identify dynamical degrees of freedom and check for stability.
• Investigating the tensor sector to determine implications for gravitational waves.

Key Results and Contributions

1. Elimination of Perturbative Instabilities:
   The central finding is that the coupling to the Nieh-Yan density introduces an additional constraint equation ($H\delta\phi^I + \phi'\Psi = 0$) that forces the gauge-invariant dark energy perturbation ($\zeta_1$) to vanish.

   • Consequently, the dark energy perturbation is removed from the menu of dynamical degrees of freedom.
   • The quadratic action for scalar perturbations reduces to a form containing only the matter sector degrees of freedom ($\zeta_2$).
   • This mechanism successfully circumvents both gradient and ghost instabilities, allowing for a stable realization of quintom behavior (including the quintom-B scenario) using a single fluid or single scalar field.

2. Background Evolution:
   The authors demonstrate that the Nieh-Yan coupling term vanishes in the homogeneous and isotropic background. Therefore, the background evolution of the universe remains identical to that of standard teleparallel gravity (and effectively GR). The authors construct explicit "toy models" for both fluid and scalar field scenarios that reproduce the observed quintom-B behavior (crossing $w=-1$ at $z \approx 0.44$) without altering the background dynamics.

3. Parity Violation in Gravitational Waves:
   The paper derives the quadratic action for tensor perturbations (gravitational waves). The Nieh-Yan coupling introduces a term proportional to $\epsilon_{ijk}h^T_{il}\partial_j h^T_{kl}$, which breaks parity symmetry.

   • This leads to velocity birefringence: gravitational waves with left-handed ($L$) and right-handed ($R$) circular polarizations propagate with different phase velocities ($v_p^A \approx 1 + p_A a M_{PV} / 2k$).
   • This parity violation is presented as a unique theoretical prediction of the mechanism, distinguishing it from other non-minimal coupling models.

Significance and Claims
The paper claims to provide a consistent theoretical mechanism to realize quintom dark energy with a single component, a feat previously obstructed by no-go theorems in minimal Einstein gravity. By shifting the framework to teleparallel gravity with a specific Nieh-Yan coupling, the authors show that the inherent instabilities of quintom models can be avoided not by adding new degrees of freedom, but by constraining the existing dark energy perturbation to zero.

The authors emphasize that this approach:

• Allows model builders to focus solely on background evolution to construct quintom models with confidence.
• Offers a testable prediction via parity violation in gravitational waves, which can be constrained by future observations (e.g., binary black hole mergers).
• Is specific to the linear perturbation theory around a spatially flat FRW background; the authors note that the non-propagating nature of the scalar perturbation may not hold for closed or open universes or beyond linear theory.

The work does not claim to definitively prove the nature of dark energy but offers a viable theoretical pathway to accommodate recent DESI DR2 observations within a modified gravity framework that remains consistent with current stability requirements.