Non-interacting holographic dark energy with Torsion via Hubble Radius

This paper demonstrates that incorporating a time-dependent torsion scalar into a non-interacting holographic dark energy model with the Hubble radius as an infrared cut-off successfully drives cosmic acceleration and yields an equation of state distinct from the cosmological constant, thereby resolving limitations found in earlier interacting models.

The Gist

The Big Picture: A Cosmic Mystery

Imagine the universe is a giant, expanding balloon. For a long time, scientists thought the balloon was expanding at a steady or slowing pace. But then, they discovered it's actually speeding up (accelerating). Something invisible is pushing it faster. We call this invisible pusher Dark Energy.

In the standard model of physics (General Relativity), this pusher is often treated as a "Cosmological Constant"—a fixed, unchanging force. However, this creates a problem. When scientists try to explain this acceleration using a specific rule called the Holographic Principle (which suggests the universe's information is like a 2D hologram projected onto a 3D surface), they hit a wall.

The Wall: To make the math work with the Holographic Principle, they usually have to assume Dark Energy and Dark Matter are "holding hands" and exchanging energy (interacting). But there is no evidence they are doing this. If they don't interact, the standard math says the universe shouldn't be accelerating.

The New Idea: Adding "Twist" to Space

This paper proposes a solution by changing the shape of the stage itself. The authors use a theory called Einstein-Cartan theory.

• Standard Physics (General Relativity): Imagine space is like a smooth, flat trampoline. Objects roll on it based on how heavy they are.
• This Paper's Physics (Einstein-Cartan): Imagine space is like a trampoline made of a slightly twisted fabric. This "twist" is called Torsion.

What causes the twist?
In Einstein-Cartan theory, torsion is generally associated with the spin of matter — a spinning top has angular momentum, and at the microscopic level every fundamental particle carries spin. The authors do NOT claim to have measured how all the spinning matter in the universe adds up at cosmological scales (no one has done that). Instead, they take a pragmatic shortcut: they postulate a simple time-dependent shape for the twist (an ansatz) motivated by what spin would broadly do, and then they work out what cosmology that ansatz produces. The twist in their model is a deliberately phenomenological stand-in for whatever real spin-driven torsion would look like, not a direct calculation from established macroscopic spin data.

The Experiment: Testing the Twist

The authors asked: If we add this tiny "twist" to the universe, can we explain the acceleration without forcing Dark Energy and Dark Matter to interact?

They tested three scenarios for how this "twist" (Torsion) behaves:

1. Steady Twist: The twist is constant everywhere.
   • Result: This acts just like normal matter. It doesn't cause acceleration. Fail.
2. Constant Twist: The twist strength is fixed but doesn't change over time.
   • Result: It can cause acceleration, but it's a bit of a "maybe." Maybe.
3. Time-Dependent Twist (the authors' chosen ansatz): The twist is allowed to vary as the universe evolves, with a chosen time-dependence motivated by — but not directly derived from — the idea that spinning matter contributes to torsion.
   • Result: Success! Even a very weak twist is enough to make the universe accelerate.

The "Holographic" Trick

The paper focuses on a specific rule for Dark Energy called the Hubble Radius cut-off. Think of the Hubble Radius as the "horizon" of the observable universe—the limit of how far we can see.

• The Old Problem: In standard physics, using this horizon as a limit for Dark Energy only works if Dark Energy and Dark Matter are interacting. If they aren't, the math breaks, and the universe doesn't accelerate.
• The New Solution: By adding the "twist" (Torsion), the math suddenly works! The twist acts like a hidden lever that allows the universe to accelerate even if Dark Energy and Dark Matter are completely separate (non-interacting).

The Results: A Slight Difference

The authors calculated the "equation of state" for this new Dark Energy. This is a number (let's call it $\omega$) that tells us how the energy behaves.

• A value of -1 represents the standard "Cosmological Constant" (a rigid, unchanging force).
• Their model with Torsion gives a value between -1 and -0.778.

What does this mean?
It means the "twisted" Dark Energy behaves slightly differently from the standard constant. It is dynamic—it changes slightly over time rather than being a frozen, unchanging number. However, because the "twist" in our current universe is very weak, this difference is subtle.

The Conclusion

The paper concludes that:

1. The authors show that, within this framework, one does not strictly need to assume an interaction between Dark Energy and Dark Matter to obtain cosmic acceleration when the Hubble radius is used as the holographic cut-off — opening up a possible direction toward addressing late-time cosmological tensions.
2. By introducing a phenomenological time-dependent twist in space (whose time-dependence is motivated by spin effects), the Hubble Radius becomes a viable way to calculate Dark Energy within this specific theoretical framework.
3. This may help address major logical problems (avoiding "circular logic" and causality issues) that plagued previous models, offering a possible direction toward resolving them.

In short: The universe is accelerating not just because of a mysterious constant force, but perhaps because the fabric of space itself has a microscopic "twist" whose phenomenological time-dependence is motivated by spin effects, allowing the universe to expand faster without needing a secret handshake between Dark Energy and Dark Matter.

Technical Summary

Technical Summary: Non-interacting Holographic Dark Energy with Torsion via Hubble Radius

Problem Statement
The origin of cosmic acceleration remains a primary challenge in modern cosmology. While the $\Lambda$CDM model interprets this via a cosmological constant, recent data indicating the "Hubble tension" suggest the need for new dark energy models beyond standard General Relativity (GR). A specific theoretical hurdle exists within Holographic Dark Energy (HDE) models: when the Hubble radius is used as the infrared (IR) cut-off, the model fails to produce cosmic acceleration in a non-interacting scenario within standard GR. Previous attempts to resolve this using the Hubble radius required assuming an interaction between dark energy and dark matter, a feature lacking observational confirmation and introducing a "non-interacting limit" gap. Furthermore, while the Einstein-Cartan theory extends GR by introducing torsion — a geometric degree of freedom theoretically associated with matter spin at the microscopic level — its cosmological implications for HDE remain under-explored, particularly regarding whether torsion can facilitate a viable non-interacting HDE model using the Hubble radius.

Methodology
The authors reconstruct a holographic dark energy model within Friedmann cosmology incorporating a torsion scalar, derived from the Einstein-Cartan theory. The study assumes no interaction between dark energy and dark matter.

1. Theoretical Framework: The authors utilize the Einstein-Cartan equations, where the torsion tensor is defined as the antisymmetric part of the affine connection. They adopt a specific form of the torsion tensor preserving the cosmological principle, characterized by a time-dependent torsion scalar $\phi(t)$.
2. IR Cut-off: The Hubble radius ($L = H^{-1}$) is set as the IR cut-off for the holographic constraint, $\rho_X = 3d^2 M_p^2 L^{-2}$.
3. Scenarios: Due to the lack of direct observational data on torsion, the authors analyze three distinct scenarios for the torsion scalar $\phi$:
   • (i) Steady-state torsion ($\phi/H$ is constant).
   • (ii) Constant torsion scalar ($\phi$ is constant).
   • (iii) Time-dependent torsion scalar linked to matter density with spin. For this case, they propose an ansatz where $\phi(t) \propto \rho_m(t)$, motivated by the physical source of torsion being matter spin.
4. Derivation: The authors derive the continuity equations for the total energy density and split them into dark energy and matter components. They then derive the equation of state (EoS), $\omega_X$, for the holographic dark energy in terms of the free parameter $d$, the Hubble parameter $H$, the deceleration parameter $q$, and the torsion scalar $\phi$.

Key Contributions and Results

• Resolution of the Non-Interacting Hubble Limit: The primary contribution is demonstrating that introducing torsion allows the Hubble radius to serve as a viable IR cut-off for holographic dark energy without requiring an interaction between dark energy and dark matter. In standard GR ($\phi=0$), setting the Hubble radius as the cut-off leads to $\omega_X = \omega_m$, preventing acceleration. The presence of the torsion scalar breaks this degeneracy, allowing $\omega_X \neq \omega_m$ and enabling cosmic acceleration.
• Analysis of Scenarios:
  • Steady-state torsion: This scenario results in a constant density parameter $\Omega_X$, causing the dark energy EoS to behave like matter, failing to explain cosmic acceleration.
  • Constant torsion scalar: This scenario permits both accelerating and non-accelerating solutions depending on the values of $\phi$ and $d$.
  • Time-dependent torsion scalar: This is the most promising regime. Assuming a positive ratio $\phi_0/H_0$, the model yields an equation of state within the range $-1 < \omega_X^0 < 0$.
• Equation of State Minima: For the time-dependent case, the authors identify minima for the current equation of state, $(\omega_X^0)_{\text{min}}$. As the free parameter $d$ varies from 1 to 0.654, $(\omega_X^0)_{\text{min}}$ lies in the range $-1 < (\omega_X^0)_{\text{min}} < -0.778$.
• Behavior near $d \approx 1$: Focusing on $d \approx 1$ (consistent with thermodynamic and quantum perspectives in literature), the model predicts a very weak torsion contribution. In this limit, the equation of state exhibits behavior slightly different from a pure cosmological constant ($\omega = -1$), remaining dynamic rather than static.

Significance and Claims
The paper claims that this approach provides a "non-interacting limit" that is absent in earlier interacting models utilizing the Hubble radius as an IR cut-off. By incorporating torsion, the model avoids the causality and circular logic issues often associated with HDE models that rely on the future event horizon as the IR cut-off. The authors assert that even very weak torsion is sufficient to induce cosmic acceleration in this framework.

The significance lies in the theoretical possibility of reconciling the Hubble radius as an IR cut-off with observed cosmic acceleration without invoking unconfirmed interactions between dark sectors. The authors are careful to note that this framework does not by itself resolve the Hubble tension or constitute an established alternative to $\Lambda$CDM; it offers a possible theoretical direction toward addressing late-time cosmological tensions, with the specific phenomenological torsion ansatz playing the role of a working hypothesis rather than a directly measured macroscopic spin distribution. They conclude that if a more explicit relationship between torsion and matter spin were established, it could provide stronger theoretical support for a feasible holographic dark energy model.