Holographic Dark Energy with Hubble Radius as an Infrared Cutoff in Einstein-Cartan Gravity

This paper investigates non-interacting holographic dark energy with the Hubble radius as an infrared cutoff within Einstein-Cartan gravity, demonstrating that a Weyssenhoff spin fluid naturally induces a torsion scalar scaling as $\Phi \sim a^{-3}$, which drives cosmic acceleration, allows the dark energy equation of state to cross the phantom divide, and modifies the cosmic distance duality relation while remaining consistent with recent DESI observations.

The Gist

The Big Picture: Fixing the Universe's Engine

Imagine the universe is a giant car. For a long time, physicists thought the engine (gravity) was described perfectly by Einstein's General Relativity. But recently, we've noticed the car is speeding up (accelerating expansion), and the "fuel" causing this speed-up is a mystery called Dark Energy.

The standard model of cosmology (the "Lambda-CDM" model) tries to explain this with a "cosmological constant," but it has some major glitches, like why the fuel amount is so weirdly tuned.

This paper proposes a new mechanic for the engine. It suggests that the universe isn't just a smooth fabric (spacetime) but has a hidden "twist" or "kink" in it, called torsion. This twist comes from the fact that tiny particles (like electrons) have an intrinsic spin, much like a spinning top. The authors combine this idea with a concept called Holographic Dark Energy to see if they can explain the universe's acceleration without breaking the rules of physics.

The Core Idea: The "Twist" in Spacetime

In standard Einstein gravity, spacetime is like a smooth sheet of rubber. If you roll a marble on it, it follows a straight line unless the sheet curves.

In Einstein-Cartan gravity (the theory used in this paper), the rubber sheet can also twist.

• The Analogy: Imagine a spiral staircase. In normal gravity, you just walk up the steps. In this twisted gravity, the steps themselves are spiraling.
• The Source: This twist isn't random; it's caused by the "spin" of matter. Just as a spinning top has angular momentum, the universe's matter has a collective spin that creates this torsion.

The Problem They Solved: The "Guessing Game"

In previous attempts to use this "twist" to explain Dark Energy, scientists had to make a big, unproven guess (an "ansatz") about how the twist behaves as the universe expands. They essentially said, "Let's assume the twist gets weaker exactly like this."

This paper's breakthrough: They didn't guess. They derived the behavior of the twist from first principles (the fundamental laws of physics).

• The Result: They found that the twist (called the torsion scalar) naturally fades away as the universe grows, shrinking at the exact same rate as regular matter (like dust or gas).
• The Metaphor: Imagine a crowd of people spinning in a room. As the room gets bigger, the people spread out. The authors proved mathematically that the "spinning energy" of the crowd naturally dilutes exactly as the room expands, without needing to force a rule on it.

How It Explains the Acceleration

The authors applied this twist to a specific model of Dark Energy called Holographic Dark Energy.

• The Setup: Think of the universe as a hologram. The amount of energy in a region depends on the size of its boundary (like the surface area of a box).
• The Twist's Effect: In a normal universe, this model fails to explain why the universe is speeding up unless you add a mysterious interaction between Dark Matter and Dark Energy.
• The Solution: The "twist" (torsion) acts like a hidden lever. Even if Dark Matter and Dark Energy don't talk to each other, the presence of the twist changes the "pressure" of Dark Energy.
  • It pushes the equation of state (a measure of how "pushy" the energy is) into negative values.
  • The Result: This negative pressure is exactly what is needed to make the universe accelerate. It works even when the twist is very weak (which is good, because we haven't seen strong evidence of it yet).

The "Phantom" Crossing

One of the most interesting findings is about the "Phantom Divide."

• The Concept: In physics, there's a speed limit for how "negative" the pressure of Dark Energy can be. Crossing this line (into "phantom" territory) usually causes wild, unstable predictions.
• The Paper's Claim: Their model allows the universe to cross this line naturally and safely, but only if the twist is within a specific, weak range. This matches recent observations (like those from the DESI instrument) which suggest the universe's acceleration might be slowly weakening over time.

The "Ruler" Problem: Measuring the Universe

If spacetime is twisted, does it mess up how we measure distances?

• Redshift (The Color Change): The authors checked if the twist changes the color of light coming from distant stars (redshift).
  • The Finding: Surprisingly, no. The relationship between how fast the universe is expanding and the color of the light remains the same as in standard Einstein gravity. The twist cancels itself out in this specific measurement.
• Distance (The Ruler): However, the twist does change how we relate two different ways of measuring distance:
  1. Luminosity Distance: How bright a star looks (like a lightbulb).
  2. Angular Diameter Distance: How big a star looks (like a coin held at arm's length).
• The Finding: In a twisted universe, the standard rule connecting these two distances is slightly off. The authors derived a new formula that includes a "deviation parameter" (let's call it $\eta$).
  • The Metaphor: Imagine looking at a lighthouse through a slightly warped glass. The light's brightness and its apparent size don't match the standard rulebook anymore. The paper provides the math to calculate exactly how much the glass is warped.

Summary of Claims

1. No Guessing: They derived the behavior of the cosmic "twist" (torsion) from fundamental laws, solving a previous problem where scientists had to guess its behavior.
2. Acceleration: This twist allows a specific type of Dark Energy (Holographic) to cause the universe to accelerate, even without Dark Matter and Dark Energy interacting.
3. Weak Twist: The effect is small enough to be consistent with current observations (we haven't seen a huge twist yet), but strong enough to explain the acceleration.
4. Changing Acceleration: The model predicts that the universe's acceleration is slowly slowing down, which matches very recent data.
5. New Measurement Rule: While the "color" of light (redshift) follows the old rules, the relationship between how bright and how big distant objects appear has a new, slight correction due to the twist.

What this means for the future: The authors aren't claiming this is the final answer, but they have built a solid theoretical framework. They have provided the specific mathematical tools (the new distance formula) that astronomers can use to test this idea against real telescope data in the future.

Technical Summary

Technical Summary: Holographic Dark Energy with Hubble Radius as an Infrared Cutoff in Einstein-Cartan Gravity

Problem Statement
The standard $\Lambda$CDM model, while successful, faces significant theoretical challenges, including the vacuum energy fine-tuning problem and the cosmic coincidence problem. Furthermore, recent observational tensions (Hubble and $S_8$ tensions) and data from the Dark Energy Spectroscopic Instrument (DESI) suggesting a dynamic, weakening cosmic acceleration require new physical interpretations. In the context of Holographic Dark Energy (HDE), models utilizing the future event horizon as the infrared (IR) cutoff suffer from causality and circular logic issues. Conversely, non-interacting HDE models using the Hubble radius as the IR cutoff fail to produce cosmic acceleration within General Relativity (GR). Additionally, previous attempts to incorporate torsion into HDE models often relied on ad hoc ansatzes for the torsion scalar, lacking a rigorous derivation from first principles.

Methodology
The authors investigate non-interacting HDE with the Hubble radius as the IR cutoff within the framework of Einstein-Cartan (EC) gravity, a theory that relaxes the torsion-free condition of GR to incorporate the intrinsic spin of matter.

1. Derivation of Field Equations: The Einstein-Cartan equations and Cartan equations are derived from the action principle. The authors recast the equations into an "Einstein-like" form ($G_{\mu\nu} = \kappa T_{\mu\nu}$), where the canonical energy-momentum tensor $T_{\mu\nu}$ (incorporating spin effects) is identified as the physical source, rather than the standard energy-momentum tensor $\tilde{T}_{\mu\nu}$.
2. Cosmological Setup: A Weyssenhoff spin fluid is adopted to model matter. By imposing the Frenkel condition and averaging over random spin orientations, the authors derive Friedmann-like equations that include a torsion scalar $\Phi$.
3. Scaling Behavior: Instead of assuming a form for the torsion scalar, the authors derive its scaling behavior self-consistently using the continuity equation and the relation between spin density and particle number density in an adiabatically expanding universe.
4. Light Propagation: The study analyzes light propagation in spacetimes with torsion using the geometric optics approximation. This involves examining the reciprocity theorem and the relationship between luminosity distance ($d_L$) and angular diameter distance ($d_A$) to determine how torsion modifies standard cosmological distance relations.

Key Contributions and Results

• Resolution of the Ansatz Problem: The paper derives the scaling behavior of the torsion scalar as $\Phi \sim a^{-3}$ without introducing any ad hoc ansatz. This behavior is shown to be matter-like, a direct consequence of the non-propagating nature of torsion in EC gravity.
• Cosmic Acceleration without Interaction: In the absence of interactions between dark matter and dark energy, the torsion scalar shifts the equation of state (EoS) for holographic dark energy ($\omega_X$) from the dust-like value ($\omega_X = \omega_m$) found in standard HDE with a Hubble cutoff toward negative values. This shift enables cosmic acceleration even in the weak torsion regime ($|\Phi/H| < 1$).
• Dynamical Equation of State: The model predicts a dynamical EoS where $\omega_X$ can approach $-1$ and cross the phantom divide ($\omega_X < -1$) depending on the free parameter $d$ and the ratio $|\Phi/H|$. Crucially, the time derivative of the EoS is positive ($\dot{\omega}_X > 0$), indicating that cosmic acceleration is gradually weakening. This result is presented as potentially consistent with recent DESI observations.
• Modified Distance Duality Relation: The study demonstrates that while the standard relation between redshift and the scale factor is preserved, the cosmic distance duality relation is modified in the presence of torsion. The relation becomes $d_L = d_A(1+z)^2(1+\eta)$, where the deviation parameter $\eta$ is determined by the integral of the torsion scalar over time: $\eta \sim \int_{t_S}^{t_O} dt \, a^{-3}$.
• Weak Equivalence Principle: The analysis clarifies that while totally antisymmetric torsion preserves the weak equivalence principle, general torsion (which includes the scalar component considered here) violates it, preventing the existence of a local inertial frame in the general case.

Significance
The paper claims to provide a theoretically complete framework for non-interacting HDE with the Hubble radius as the IR cutoff, resolving the causality and circular logic problems associated with future event horizon models. By deriving the torsion scalar's behavior from the action principle rather than assuming it, the model offers a more robust theoretical basis. The results suggest that the subtle effects of torsion, encoded in the scaling of the deviation parameter $\eta$, could be implicitly detectable in direct observations such as Type Ia supernovae (SNIa) and Baryon Acoustic Oscillations (BAO). Ultimately, the work supports the feasibility of this EC gravity-based model as a viable alternative to explain the observed dynamics of dark energy and prepares a theoretical foundation for future likelihood analyses.