Dynamical Resolution of the Cosmic Coincidence Problem in Non-Interacting Holographic Dark Energy via Einstein-Cartan Torsion

This paper demonstrates that Einstein-Cartan torsion provides a geometric mechanism to dynamically resolve the cosmic coincidence problem in non-interacting holographic dark energy by enabling a viable density ratio and cosmic acceleration without requiring phenomenological dark sector interactions or fine-tuning.

The Gist

The Big Mystery: The "Cosmic Coincidence"

Imagine you are walking into a room and you see two people: one is a giant, and the other is a tiny mouse. You might think, "Wow, that's a weird coincidence that they are both here right now."

In our universe, we have a similar mystery. We have Matter (stars, planets, you, me) and Dark Energy (a mysterious force pushing the universe apart). For a long time, scientists thought Matter would eventually win the race and dominate everything, or Dark Energy would win immediately and blow everything apart.

But here is the weird part: Right now, they are almost equal. They are roughly the same size. The paper calls this the "Cosmic Coincidence Problem." Why are they balanced exactly at the moment we are here to look at it? It feels like a lucky break, or a "fine-tuned" setup.

The Old Way of Fixing It: The "Handshake"

To explain this balance, many scientists tried to say that Matter and Dark Energy are holding hands and talking to each other. They imagined a secret "interaction" where Matter slowly leaks into Dark Energy (or vice versa) to keep them balanced.

Think of it like two water tanks connected by a hose. If one gets too full, water flows to the other to keep them equal. The problem with this idea is that we have never seen this "hose" (interaction) in real life. It feels like a made-up rule just to make the math work.

The New Idea: The "Twist" in Space

This paper proposes a different solution. Instead of a secret handshake between Matter and Dark Energy, the authors suggest the universe itself has a hidden twist.

In the standard theory of gravity (General Relativity), space is like a smooth trampoline. But in this paper, they use Einstein-Cartan theory, which says space can actually twist or spin because of the "spin" of particles inside it.

Think of space not as a flat sheet, but as a spiral staircase.

• The Twist (Torsion): As the universe expands, this spiral staircase gets tighter and tighter.
• The Effect: This twisting motion changes how Matter and Dark Energy behave, even though they aren't talking to each other.

How the "Twist" Solves the Problem

The authors show that this "twist" acts like a natural regulator.

1. It Makes the Ratio Change: In the old "smooth space" models, if you didn't have a secret handshake, the ratio of Matter to Dark Energy would stay frozen forever. But with the "twist," the ratio starts to evolve. It's like a clock that ticks. The twist makes the balance between Matter and Dark Energy change over time naturally.
2. It Creates Acceleration: The twist also pushes the Dark Energy to become more "negative" (more repulsive). This is what causes the universe to speed up its expansion (accelerate), which is what we observe today.
3. No Tuning Required: Usually, scientists have to "tune" their models like a radio dial to get the numbers to match what we see. The authors found that with this "twist," the universe naturally lands on the right balance (where Matter and Dark Energy are comparable) without needing to fiddle with the settings.

The "Weak Twist" Zone

The paper calculates that for this to work, the "twist" needs to be weak but present.

• If there is no twist, the balance is stuck and doesn't explain why we are here.
• If the twist is too strong, the math breaks.
• But in the "Goldilocks zone" (a weak, nonzero twist), the universe naturally evolves to the point where Matter and Dark Energy are roughly equal today.

The Bottom Line

The paper claims that we don't need to invent a mysterious "interaction" between Dark Matter and Dark Energy to explain why they are balanced today. Instead, the geometry of space itself—specifically a property called torsion (a kind of cosmic twist)—does the work for us.

It's like saying the reason two runners are neck-and-neck at the finish line isn't because they are holding hands, but because the track itself is curving in a way that naturally brings them together. This offers a "geometric" solution to a problem that previously required "phenomenological" (made-up) fixes.

Technical Summary

Technical Summary: Dynamical Resolution of the Cosmic Coincidence Problem in Non-Interacting Holographic Dark Energy via Einstein–Cartan Torsion

Problem Statement
The paper addresses the "cosmic coincidence problem," which questions why the matter density ($\rho_m$) and dark energy density ($\rho_X$) are of the same order of magnitude at the current epoch. In the context of holographic dark energy (HDE) models using the Hubble radius as the infrared (IR) cutoff, a specific difficulty arises in General Relativity (GR): without an interaction between dark sectors, the equation of state for dark energy ($\omega_X$) coincides with that of matter ($\omega_m$), preventing cosmic acceleration. To resolve this in GR, phenomenological interaction terms ($Q \neq 0$) are typically introduced. However, such interacting models suffer from limitations, including the absence of a non-interacting limit, a constant density ratio ($r = \rho_m/\rho_X = \text{constant}$) that precludes a dynamical resolution of the coincidence problem, and potential conflicts with observational tensions (Hubble and $S_8$).

Methodology
The authors investigate the HDE model within the framework of Einstein–Cartan (EC) gravity, which extends GR by incorporating spacetime torsion. The torsion tensor is algebraically related to the intrinsic spin of matter rather than introduced as a phenomenological dark-sector coupling.

1. Framework: The study utilizes a flat FLRW metric with a torsion scalar $\Phi(t)$ compatible with the cosmological principle, defined via the torsion tensor $S_{\mu\nu\rho} = \Phi(t)h_{\rho[\mu}u_{\nu]}$.
2. Field Equations: The Friedmann-like equations are derived from the EC action, yielding a modified first Friedmann equation: $\Omega_m + \Omega_X - (\Phi/H)^2 = 1$.
3. Scaling Behavior: A self-consistent evolution equation for the torsion scalar is derived ($\dot{\Phi} + 3H\Phi = 0$), leading to the scaling behavior $\Phi \sim a^{-3}$ without ad hoc assumptions.
4. Non-Interacting Case: The analysis strictly assumes no phenomenological interaction between dark matter and dark energy ($Q=0$). The continuity equations for matter and dark energy are solved separately.
5. HDE Constraint: The HDE density is defined as $\rho_X = 3d^2 M_p^2 H^2$, where $d$ is a free parameter and $L=H^{-1}$ is the IR cutoff.

Key Contributions and Results
The paper demonstrates that the geometric contribution of the torsion scalar $\Phi$ fundamentally alters the dynamics of the HDE model in the absence of dark sector interactions:

• Dynamical Density Ratio: In standard GR with the Hubble cutoff and $Q=0$, the density ratio $r$ is constant. In the EC framework, the torsion contribution renders $r$ time-dependent even when $Q=0$. The evolution of the ratio is governed by:
  $$\frac{\dot{r}}{r} = -2H (2 - q) \frac{(\Phi/H)^2}{1 - d^2 + (\Phi/H)^2}$$
  where $q$ is the deceleration parameter.
• Cosmic Acceleration: The torsion term shifts the equation of state for dark energy toward negative values. The derived equation of state is:
  $$\omega_X = \omega_m - \frac{2}{3}(2-q) \frac{(\Phi/H)^2}{1 - d^2 + (\Phi/H)^2}$$
  This shift allows for cosmic acceleration ($\omega_X < -1/3$) without requiring an interaction term.
• Resolution of the Coincidence Problem:
  • Dynamical Evolution: The authors show that $\dot{r} < 0$ is satisfied both during the matter-dominated era ($q_{MD} = 0.5$) and at the current epoch ($q_0 \simeq -0.51$). This indicates that the density ratio can dynamically decrease from larger values in the past to an order-unity value today, providing a dynamical resolution to the coincidence problem.
  • Parameter Constraints: For the observed current density ratio $r_0 \simeq 0.429$ and the observationally favored equation of state range $-1 \leq \omega_0^X \lesssim -0.9$, the holographic free parameter is constrained to $0.913 \lesssim d \lesssim 0.924$.
  • Weak Torsion Regime: Crucially, in the weak torsion regime ($0.316 \lesssim |\Phi_0/H_0| < 1$), the current density ratio $r_0$ remains in the order-unity range across the entire allowed interval of the free parameter ($0.837 \lesssim d < 1$). This suggests that the observed order-unity value of $r_0$ is a consequence of torsion-induced dynamics rather than a result of fine-tuning the parameter $d$.

Significance
The paper claims that Einstein–Cartan torsion provides a geometric mechanism that replaces the need for phenomenological dark sector interactions. By utilizing the intrinsic spin-torsion coupling, the model achieves three goals simultaneously:

1. It drives cosmic acceleration in a non-interacting HDE model with the Hubble radius as the IR cutoff.
2. It renders the matter-to-dark-energy density ratio time-dependent, allowing for a dynamical resolution of the cosmic coincidence problem.
3. It realizes the observed order-unity density ratio without requiring the fine-tuning of the holographic free parameter $d$, provided the torsion is weak but non-zero.

The authors conclude that this approach offers a path to resolving the coincidence problem that is closer to a fundamental theory (via the classical limit of quantum gravity passing through EC theory) than phenomenological interaction models, while avoiding the conceptual difficulties associated with circular logic or causality problems found in other HDE formulations.