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

Dynamical Resolution of the Cosmic Coincidence Problem in Einstein-Cartan Holographic Dark Energy

Background and Motivation

The cosmic coincidence problem refers to the observation that the energy densities of dark matter and dark energy are comparable at the current cosmological epoch, despite their typically distinct scaling behaviors. Traditional $\Lambda$CDM frameworks typically address this with a cosmological constant, but the associated fine-tuning and coincidence issues motivate alternative models such as holographic dark energy (HDE).

HDE, based on the holographic principle, constrains dark energy density via an infrared (IR) cutoff. When the Hubble radius serves as this cutoff, non-interacting HDE models within standard GR yield a dust-like equation of state, unable to drive cosmic acceleration. Interacting HDE models introduce phenomenological couplings between dark matter and dark energy to overcome this, but such couplings lack fundamental motivation and observational evidence, and pose complications with concurrent resolutions of cosmological tensions.

Einstein-Cartan (EC) gravity, incorporating spacetime torsion, offers a geometric extension of GR where torsion is sourced by intrinsic spin. The model analyzed in this paper (2605.27973) exploits a torsion scalar $\Phi$ that self-consistently scales as $a^{-3}$. The goal is to demonstrate that EC torsion enables non-interacting HDE models with the Hubble radius cutoff to both drive acceleration and yield a time-dependent density ratio, providing a dynamical mechanism for resolving the coincidence problem.

Theoretical Framework

The EC theory modifies cosmological dynamics through the torsion tensor $S_{\mu\nu}^{\ \ \rho}$, which, under the cosmological principle, is characterized by a time-dependent scalar $\Phi(t)$. The Weyssenhoff spin fluid formalism links torsion to the spin of matter, yielding generalized Friedmann equations:

$H^2 = \frac{1}{3 M_p^2} (\rho - 3 M_p^2 \Phi^2)$

$$\dot{H} + H^2 = -\frac{1}{6 M_p^2} (\rho + 3p - 12 M_p^2 \Phi^2)$$

The model adopts the Hubble radius $L = H^{-1}$ for the IR cutoff, leading to a holographic dark energy density $\rho_X = 3d^2 M_p^2 H^2$. The torsion scalar evolves via $\dot{\Phi} + 3H\Phi = 0$, implying $\Phi$0 without ad hoc assumptions.

Dynamical Density Ratio and Cosmic Acceleration

In standard non-interacting HDE (GR), the ratio $\Phi$1 is constant, precluding any natural explanation for its current order-unity value. EC torsion modifies this fundamentally:

$\Phi$2

Thus, $\Phi$3 becomes explicitly time-dependent even without a dark sector interaction term ($\Phi$4), allowing for dynamical evolution from large values in earlier epochs to the observed current value.

The torsion scalar also alters the equation of state for HDE:

$\Phi$5

This enables $\Phi$6 to be shifted toward negative values, facilitating cosmic acceleration in the absence of interaction. The model quantitatively demonstrates compatibility with observational constraints for the favored range $\Phi$7 and current density ratio $\Phi$8, yielding a restricted interval $\Phi$9 for the holographic free parameter (Figure 1).

Figure 1: The allowed range $a^{-3}$0 satisfies both the observed density ratio $a^{-3}$1 and the favored equation of state for dark energy, $a^{-3}$2.

Dynamical Evolution and Order-Unity Density Ratio

The model tracks the evolution of $a^{-3}$3 via

$a^{-3}$4

For $a^{-3}$5, both in the matter-dominated era ($a^{-3}$6) and at the current epoch ($a^{-3}$7), $a^{-3}$8 decreases over time, supporting dynamical resolution of the coincidence problem. The observed order-unity value of $a^{-3}$9 is not a consequence of fine-tuning $S_{\mu\nu}^{\ \ \rho}$0, but arises naturally from the weak torsion regime, $S_{\mu\nu}^{\ \ \rho}$1 (Figure 2).

Figure 2: The yellow region indicates $S_{\mu\nu}^{\ \ \rho}$2 for weak torsion $S_{\mu\nu}^{\ \ \rho}$3, while intersections in the blue inset correspond to simultaneous satisfaction of $S_{\mu\nu}^{\ \ \rho}$4 and $S_{\mu\nu}^{\ \ \rho}$5 for $S_{\mu\nu}^{\ \ \rho}$6.

Implications and Future Directions

This analysis demonstrates that EC torsion provides a geometric origin for the dynamical density ratio, obviating the need for phenomenological interaction terms. The results have the following implications:

• Resolution of the Coincidence Problem: The observed density ratio is dynamically achieved, not artificially fixed by parameter tuning.
• Cosmic Acceleration: The negative shift of $S_{\mu\nu}^{\ \ \rho}$7 from torsion allows for acceleration in non-interacting HDE.
• Parameter Space: Observational constraints are satisfied within a well-defined free parameter range, subject to weak torsion.
• Dark Sector Models: EC gravity shapes cosmological dark energy phenomenology without explicit coupling, potentially avoiding issues with $S_{\mu\nu}^{\ \ \rho}$8 and $S_{\mu\nu}^{\ \ \rho}$9 tensions in interacting models.

Future work may involve:

• Constraining torsion effects through cosmological observations (e.g., CMB, large-scale structure).
• Exploring the interplay between torsion, early universe dynamics, and inflation.
• Investigating possible quantum gravity connections and phenomenological signatures of EC torsion in the dark sector.

Conclusion

Einstein-Cartan gravity, with a cosmologically consistent torsion scalar, enables non-interacting holographic dark energy models to dynamically resolve the cosmic coincidence problem and drive cosmic acceleration without phenomenological interaction terms. The current density ratio and equation of state requirements are met within a narrow yet natural parameter regime and weak torsion. Theoretical and observational implications indicate that geometric torsion offers a viable mechanism for addressing fundamental cosmological questions and motivates further exploration of spin-torsion effects in cosmic evolution.