Eur. Phys. J. C (2026) 86: 373
https://doi.org/10.1140/epjc/s10052-026-15613-3

Regular Article - Theoretical Physics

The pseudo-complex FLRW model and the time evolution of the Hubble parameter

¹ Physics Department, University of Mentouri, Constantine 1, B. P. 325, Ain El Bey Way, 025017, Constantine, Algeria
² Mathematical and Subatomic Physics Laboratory (LPMPS), University of Mentouri, Constantine 1, Constantine, Algeria
³ Instituto de Ciencias Nucleares, UNAM, A.P. 70-543, Circuito Exterior, C.U., 04510, Mexico City, Mexico
⁴ Frankfurt Institute for Advanced Studies, J. W. von Goethe University, Hessen, Germany
⁵ Department of Physics, San Diego State University, 92182, SanDiego, CA, USA
⁶ Department of Physics, University of California at SanDiego (UCSD), 92093, La Jolla, CA, USA
⁷ International Center for Relativistic Astrophysics Network (ICRANet), 65122, Pescara, Italy
⁸ Instituto de Física, Universidade Federal do Rio Grandedo Sul (UFRGS), 91501-970, Porto Alegre, Brazil

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Received: 6 February 2026
Accepted: 22 March 2026
Published online: 11 April 2026

Abstract

The pseudo-complex version of the Friedmann–Lemaître–Robertson–Walker model (pcFLRW) is presented within the framework of pseudo-complex General Relativity (pcGR). In this approach, dark energy emerges as a geometric consequence of the pseudo-complex structure, leading to a specific functional form for the Hubble parameter H(z) characterized by a single geometric parameter $\beta$. This parameter governs the effective dark-energy equation of state via $p_{\mathrm{\Lambda }}=-\beta {\epsilon }_{\mathrm{\Lambda }}$ and is directly linked to the present-day time derivative of the Hubble parameter through ${\dot{H}}_0=\frac{3}{2}(\beta -1)H_0^2$. Using recent DESI BAO data, we constrain $\beta =1.0426\pm 0.0144$, which yields a positive ${\dot{H}}_0\simeq (0.94\pm 0.32)\times {10}^{-17}(\mathrm{km}/{\mathrm{s}}^2)/\mathrm{Mpc}$. This contrasts with the $\mathrm{\Lambda }$CDM prediction, where ${\dot{H}}_0$ is negative (${\dot{H}}_0\approx -0.45H_0^2$ for standard parameters), indicating that in pcGR the expansion rate is increasing with time while in $\mathrm{\Lambda }$CDM it decreases. The best-fit value also implies a deceleration parameter $q=-0.9361\pm 0.0216$. Using the exact Sandage–Loeb relation, the predicted redshift drift over 20 years for a source at $z=4$ is $\mathrm{\Delta }v\simeq -11.1\mathrm{cm}{\mathrm{s}}^{-1}$, in close agreement with the $\mathrm{\Lambda }$CDM prediction but arising from a distinct geometric origin. Thus, the non-vanishing and positive ${\dot{H}}_0$ in pcGR provides a clear and testable target for future high-precision spectroscopic observations.