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 $\Lambda$ 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, $\Lambda \propto l^{2}\tilde{h}/r_{c}^{3}$, whose sign is controlled by $l^2\tilde{h}$, requiring fine-tuning to reproduce $\Lambda_{\text{obs}} \approx 10^{-52}\,\text{m}^{-2}$. In the strong-field regime, dependence on $l$ and $\tilde{h}$ cancels algebraically, yielding $\Lambda \approx 3/(4r_{c}^{2})$ independently of the Chern-Simons coupling. This regime naturally reproduces $\Lambda_{\rm obs}$ 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 $S_{\rm cosm} = 4\pi k_B r_c^2/l_{\rm Pl}^2 \sim 10^{122}\,k_B$, 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 $\Lambda$ is small, one asks why $r_c$ is large -- a reformulation compatible with a large extra dimension without violating established gravitational tests.