Torsion-Induced Modification to Friedmann Equations in $AdSL_{4}$ Gauged Gravity

1. Problem Statement

Modern cosmology faces challenges in explaining early-universe inflation and late-time cosmic acceleration without resorting to exotic matter sources or fine-tuned cosmological constants. While gauge-theoretic extensions of General Relativity (GR), such as Maxwell gravity and AdSL4 (Anti-de Sitter-Lorentz) gravity, offer promising frameworks, previous studies often neglected or only partially addressed the role of torsion.

Torsion, arising from the antisymmetric part of the affine connection, is a natural feature of gauge theories of gravity but is typically absent in standard Riemannian GR. The authors aim to investigate how the antisymmetric tensorial gauge field ($B_{ab}$) inherent to the AdSL4 algebra induces torsion and subsequently modifies the Friedmann equations in a spatially homogeneous and isotropic (FLRW) universe. Specifically, they seek to determine if this geometric structure can drive cosmic acceleration or mimic dark energy/cosmic string dynamics without introducing external matter fields.

2. Methodology

A. Theoretical Framework: AdSL4 Gauge Theory

The authors formulate gravity as a gauge theory based on the AdSL4 algebra, a semi-simple extension of the Poincaré group. The algebra includes:

• $P_a$: Translation generators.
• $M_{ab}$: Lorentz rotation generators.
• $Z_{ab}$: Additional tensorial generators (AdSL4 symmetry).

The gauge connection 1-form is defined as:
$$A = e^a P_a - \frac{1}{2}\omega^{ab} M_{ab} + \frac{1}{2}B^{ab} Z_{ab}$$
where $e^a$ is the vielbein, $\omega^{ab}$ is the spin connection, and $B^{ab}$ is the new antisymmetric gauge field.

B. Geometric Construction and Torsion

The curvature 2-forms are derived, revealing that the presence of $B^{ab}$ modifies the torsion. The authors introduce a shifted connection $\tilde{\omega}^{ab} = \omega^{ab} - \mu B^{ab}$ (where $\mu$ is a coupling constant).

• Key Relation: The contorsion 1-form $K^a_{\ b}$ is directly identified with the gauge field: $K^a_{\ b} = \mu B^a_{\ b}$.
• Torsion Equation: The field equations imply that the total torsion $T^a$ is non-zero and sourced by $B^{ab}$: $T^a = \mu B^a_{\ c} \wedge e^c$.
• Shifted Curvature: A shifted curvature $\mathcal{J}^{ab}$ is constructed to ensure the action remains invariant under the subalgebra $\{M, Z\}$.

C. Action and Field Equations

The gravitational action is constructed using the shifted curvature $\mathcal{J}^{ab}$, analogous to the Einstein-Hilbert action:
$$S_{EHJ} = \frac{1}{2\kappa} \int \epsilon_{abcd} \mathcal{J}^{ab} \wedge e^c \wedge e^d$$
Using the Palatini formalism (varying the action with respect to the independent vielbein $e^a$ and the shifted connection $\tilde{\omega}^{ab}$), the authors derive:

1. Torsion Equation: Confirms $B^{ab}$ is the source of torsion.
2. Generalized Einstein Equation:
   $$R^\mu_{\ \alpha} - \frac{1}{2}R\delta^\mu_{\ \alpha} - \Lambda \delta^\mu_{\ \alpha} = T^B_{\ \mu\alpha}$$
   Here, the cosmological constant $\Lambda = 6\mu\lambda$ arises intrinsically from the algebra structure, and $T^B_{\ \mu\alpha}$ is the effective stress-energy tensor generated by the $B^{ab}$ gauge field.

D. Cosmological Application (FLRW)

The authors apply this framework to a flat FLRW metric ($ds^2 = dt^2 - a(t)^2 d\vec{x}^2$). To maintain spatial homogeneity and isotropy, they propose specific ansätze for the gauge field components $B_{\mu}^{ab}$ involving two time-dependent scalar fields, $\psi(t)$ and $\zeta(t)$:

• $B^{0s}_\mu \propto \delta^s_i \psi(t)$
• $B^{rs}_\mu \propto \epsilon^{rs}_i \zeta(t)$

3. Key Results

A. Modified Friedmann Equations

Substituting the ansatz into the generalized Einstein equations yields the modified Friedmann equations:

1. Hubble Equation:
   $$H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{2\mu^2}{a^2}(\psi^2 - \zeta^2) + \frac{\Lambda}{3}$$
2. Acceleration Equation:
   $$\frac{\ddot{a}}{a} = \frac{\Lambda}{3}$$

Crucial Finding: The torsional contributions (terms involving $\psi$ and $\zeta$) cancel out in the acceleration equation. The cosmic acceleration is driven purely by the effective cosmological constant $\Lambda$ derived from the AdSL4 algebraic structure. The torsion terms only modify the expansion rate (the Hubble parameter $H$) but do not directly cause acceleration or deceleration in the isotropic limit.

B. Effective Equation of State

The gauge field $B^{ab}$ behaves as a perfect fluid with effective energy density $\rho_B$ and pressure $P_B$:
$$\rho_B = \frac{3\mu^2}{4\pi G a^2}(\psi^2 - \zeta^2), \quad P_B = -\frac{\mu^2}{4\pi G a^2}(\psi^2 - \zeta^2)$$
This leads to an effective equation-of-state (EoS) parameter:
$$\omega_B = \frac{P_B}{\rho_B} = -\frac{1}{3}$$

C. Source of the Cosmological Constant

The scalar fields $\psi(t)$ and $\zeta(t)$ are shown to act as the physical source of the cosmological constant. Solving the scale factor equation $a(t) = C_1 e^{kt} + C_2 e^{-kt}$ (where $k=\sqrt{\Lambda/3}$) allows the authors to express $\Lambda$ explicitly in terms of the gauge field amplitudes and integration constants:
$$\Lambda = -\frac{3\mu^2(\psi^2 - \zeta^2)}{2C_1 C_2}$$

4. Significance and Implications

1. Geometric Origin of Dark Energy: The paper demonstrates that a cosmological constant and cosmic acceleration can emerge purely from the gauge-theoretic structure of enlarged spacetime symmetries (AdSL4), without introducing exotic matter fields by hand.
2. Cosmic String Analogy: The derived EoS parameter $\omega_B = -1/3$ is significant. In standard cosmology, this value corresponds to a network of cosmic strings. This suggests that the dynamical components of the AdSL4 gauge field mimic the gravitational effects of topological defects formed during symmetry-breaking phase transitions.
3. Torsion as a Dynamical Modifier: While torsion does not drive acceleration in the isotropic FLRW background (unlike some other torsion cosmologies), it modifies the expansion history and provides a mechanism for generating an effective cosmological constant.
4. Unification of Symmetry and Dynamics: The work bridges the gap between algebraic extensions of the Poincaré group and observable cosmological dynamics, showing how the "extra" generators ($Z_{ab}$) manifest as physical torsion and effective energy-momentum.

Conclusion

Cebecioğlu and Kibaroğlu establish that AdSL4-gauged gravity provides a consistent extension of GR where torsion is naturally induced by an antisymmetric gauge field. In a cosmological setting, this framework yields a de Sitter-like acceleration driven by an intrinsic cosmological constant, while the gauge field itself behaves as a fluid with an equation of state $\omega = -1/3$, linking the theory to cosmic string dynamics. This offers a novel geometric pathway to understanding the effective energy budget of the universe.