A New Formula for the Cosmological Constant: Λ = (α⁴/4π)(m_e/m_Pl)⁵ / ℓ_Pl²

Abstract

We present a new formula for the cosmological constant:

where $\alpha$ is the fine structure constant, $m_e$ the electron mass, $m_{\rm Pl} = \sqrt{\hbar c/G}$ the Planck mass, and $\ell_{\rm Pl} = \sqrt{\hbar G/c^3}$ the Planck length. The formula predicts $\Lambda = 1.110 \times 10^{-52}$ m⁻², within 1.9% of the Planck 2018 observed value — at the edge of the current 1σ observational uncertainty. The exponent structure is different from that of all previously known formulas, including the Beck–Nottale–Boehmer–Harko result, which has 25% error. The vacuum energy density implied by the formula admits a suggestive algebraic decomposition: $\rho_{\rm vac} = \tfrac{1}{2} \cdot m_e^4/(16\pi^2) \cdot \alpha^4 \cdot m_e/m_{\rm Pl}$. We identify each factor with a physical scale: $m_e^4/(16\pi^2)$ has the form of a one-loop electron vacuum energy; $\alpha^4$ matches the order of the first genuinely nonlinear QED vacuum effect (light-by-light scattering); and $m_e/m_{\rm Pl}$ is a gravitational suppression factor corresponding to a single graviton vertex. The implied dark energy scale is 2.25 meV, matching the observed ~2.3 meV. Unlike Dirac-type large-number coincidences, the formula involves only true constants ($G$, $m_e$, $\alpha$, $c$, $\hbar$). Four additional predictions follow: a strictly constant equation of state $w = -1$ (a sharp test of the present DESI 2024 BAO hint of dark energy evolution, to be resolved by DESI Y3/Y5), no time-variation of $\alpha$ (consistent with ESPRESSO), a selection rule that suppresses heavier-lepton contributions, and sub-percent testability of the $\Lambda$ value itself by DESI Y3 and Euclid within 2–3 years.

DOI

https://doi.org/10.5281/zenodo.19980707