Geometric origin of the cosmological constant from Einstein-Chern-Simons gravity compactified to four dimensions

Abstract: We present a model in which the cosmological constant emerges as a purely geometric effect from the four-dimensional compactification of five-dimensional Einstein-Chern-Simons gravity. The compactification of the extra dimension generates an effective cosmological constant $Λ$ depending on the compactification radius $r_c$, the coupling parameter $l$, and the trace $\tilde{h}$ of the compactified field $h^a$, rather than being introduced as a free parameter. The resulting field equations are structurally equivalent to those of General Relativity with a cosmological constant, so all known vacuum solutions -- Schwarzschild--de Sitter, Kerr--de Sitter, and FLRW spacetimes -- remain valid. As a concrete application, we derive the Kottler (Schwarzschild--de Sitter) black hole solution. We identify two dynamical regimes. In the weak-field regime, $Λ\propto l^{2}\tilde{h}/r_{c}^{3}$, whose sign is controlled by $l^2\tilde{h}$, requiring fine-tuning to reproduce $Λ{\text{obs}} \approx 10^{-52}\,\text{m}^{-2}$. In the strong-field regime, dependence on $l$ and $\tilde{h}$ cancels algebraically, yielding $Λ\approx 3/(4r{c}^{2})$ independently of the Chern-Simons coupling. This regime naturally reproduces $Λ{\rm obs}$ for $r{c} \approx 0.78\,H_{0}^{-1} \approx 8.2 \times 10^{25}\,\text{m}$, without fine-tuning. The Bekenstein-Hawking entropy of the cosmological horizon gives $S_{\rm cosm} = 4πk_B r_c^2/l_{\rm Pl}^2 \sim 10^{122}\,k_B$, consistent with the Gibbons-Hawking result and admitting a direct geometric interpretation in terms of $r_c$. This framework geometrically reframes the cosmological constant problem: rather than asking why $Λ$ is small, one asks why $r_c$ is large -- a reformulation compatible with a large extra dimension without violating established gravitational tests.

• The paper establishes that compactification of 5D Einstein–Chern–Simons gravity yields a 4D cosmological constant expressed in terms of the compactification radius and Chern–Simons coupling.
• The paper employs a Randall–Sundrum ansatz to reduce the theory, ensuring standard general relativity vacuum solutions are preserved.
• The paper identifies distinct weak and strong-field regimes, demonstrating that the observed dark energy magnitude arises without fine-tuning.

Geometric Emergence of the Cosmological Constant from 5D Einstein–Chern–Simons Gravity

Overview

This paper establishes a geometrical mechanism for the origin of the four-dimensional cosmological constant ($\Lambda$) via compactification from a five-dimensional Einstein–Chern–Simons (EChS) gravity theory. By compactifying one spatial dimension (with radius $r_c$), a cosmological constant term emerges in the resulting four-dimensional action, not as an arbitrary parameter but as an explicit function of $r_c$, a Chern–Simons coupling ($l$), and the trace $\tilde{h}$ of a compactified auxiliary field. This result reframes $\Lambda$ as a geometric, rather than ad hoc or quantum-vacuum, parameter and reveals a regime where its value is determined solely by $r_c$—accounting naturally for the observed magnitude of dark energy.

Einstein–Chern–Simons Gravity and Compactification Construction

The parent theory is a 5D Chern–Simons gauge theory constructed for the (A)dS algebra. The bulk action is manifestly background-independent and built from the vielbein $e^a$, spin connection $\omega^{ab}$, and extra gauge fields $h^a$ and $r_c$0. Compactification is performed via a Randall–Sundrum geometric ansatz, yielding an effective 4D action that generalizes Einstein gravity with additional terms coupling the 4D Ricci tensor and curvature to the compactified field $r_c$1.

A critical step is ensuring the 4D theory admits all standard vacuum solutions of GR with $r_c$2 despite its geometric origin. The authors impose an ansatz for $r_c$3, proportional to the 4D metric, guaranteeing that the field equations structurally reduce to Einstein equations with a cosmological constant.

Determination and Dynamics of the Cosmological Constant

The compactification yields an explicit formula:

$r_c$4

where $r_c$5 is a function of $r_c$6. This mapping exposes two dynamical regimes:

1. Weak-field regime: When $r_c$7, $r_c$8 recovers the familiar proportionality to $r_c$9, demanding fine-tuning among the parameters to match the observed small value of the cosmological constant.
2. Strong-field regime: When $r_c$0, an algebraic cancellation occurs, and

$r_c$1

This regime renders $r_c$2 independent of the Chern–Simons sector ($r_c$3, $r_c$4), with the compactification radius $r_c$5 as the sole controlling parameter. For $r_c$6 on the order of $r_c$7 m (very close to the current Hubble radius), this reproduces the observed value $r_c$8 m$r_c$9 without fine-tuning.

The sign of $l$0 is controlled by the sign of $l$1, allowing both de Sitter and Anti–de Sitter branches.

Solution Structure and Consistency with Experimental Gravity

All vacuum solutions of Einstein’s equations with $l$2—such as Schwarzschild–de Sitter, Kerr–de Sitter, and FLRW cosmologies—exist as solutions to the compactified theory. The explicit static, spherically symmetric solution reduces to the Kottler (Schwarzschild–de Sitter) black hole with $l$3 as above.

Importantly, this approach is consistent with gravitational tests, because the extra dimension’s signature appears exclusively as a cosmological constant and does not induce modifications to Newtonian or post-Newtonian gravity at observable scales. The mechanism is robust even if the compactification radius is as large as the Hubble scale, consistent with the logic of the Randall–Sundrum scenario.

Thermodynamic Implications and Geometric Reformulation

A direct consequence lies in the interpretation of the de Sitter horizon entropy:

$l$4

for $l$5, which matches the Gibbons–Hawking formula for de Sitter entropy. This result provides a geometric-statistical meaning to the cosmological horizon entropy in terms of the compactification radius.

The cosmological constant problem, typically framed as an enormous fine-tuning of quantum vacuum energy, is reframed in this model: the small observed value of $l$6 points instead to a very large compactification radius. The fine-tuning issue is recast as the question of why $l$7 m—a geometric rather than energetic puzzle.

Implications for Cosmology and Theoretical Perspectives

The framework embeds $l$8CDM directly: all the equations and phenomenology of cosmic acceleration (expansion history, structure formation) are preserved. However, the theoretical underpinning is altered: the effective cosmological constant is a prediction of the compactification scenario, not a free parameter.

The model allows for, but does not require, a dynamical $l$9 (via an evolving $\tilde{h}$0), suggesting possible directions for connecting early universe inflation (high $\tilde{h}$1) to the present vacuum. However, in the present work, $\tilde{h}$2 is constant.

In future work, a time-dependent compactification radius or scalar field could catalyze scenarios for dynamical dark energy, with cosmological evolution across different branches of the model's parameter space.

Conclusion

The compactification of 5D Einstein–Chern–Simons gravity yields a geometric origin for the 4D cosmological constant, with its value set naturally by the size of the extra dimension. In the relevant strong-field limit, the observed value of dark energy is explained without parameter fine-tuning, and the standard solutions of GR remain intact. This shifts the cosmological constant problem into a geometric context and provides a concrete, technically consistent route to connect extra dimensions with cosmic acceleration. Further developments, such as time-dependent compactification and constraints on large extra dimensions, should be explored for their potential to connect with the physics of early universe and quantum gravity.