⚛️ general relativity

The pseudo-complex Friedmann Lemaitre Robertson Walker model and the time dependence of the Hubble constant

This paper presents a pseudocomplex General Relativity (pcGR) version of the FLRW model where dark energy emerges geometrically, yielding a time-dependent Hubble parameter and a non-zero Hubble acceleration that fits recent DESI BAO data and predicts a redshift drift consistent with $\Lambda$ CDM while offering a distinct, testable geometric signature.

Original authors: L. Maghlaoui, P. O. Hess, F. Weber, C. A. Zen vasconcellos

Published 2026-01-15

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CC BY 4.0

Original authors: L. Maghlaoui, P. O. Hess, F. Weber, C. A. Zen vasconcellos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, expanding balloon. For decades, scientists have been trying to measure exactly how fast this balloon is inflating. This speed is called the Hubble Constant.

Here's the problem: When scientists look at the "baby picture" of the universe (the Cosmic Microwave Background), they get one speed. When they look at the "adult picture" (supernovas and nearby galaxies), they get a slightly faster speed. This disagreement is known as the Hubble Tension, and it's been a major headache for cosmologists.

This paper proposes a new way to solve that headache using a mathematical trick called Pseudo-Complex General Relativity (pcGR). Here is the breakdown in simple terms:

1. The New Mathematical "Glasses"

Standard physics (General Relativity) treats space and time as a smooth, 4-dimensional fabric. The authors suggest we should look at the universe through a different pair of mathematical glasses. They add a "pseudo-complex" layer to the coordinates.

Think of it like this:

Standard View: The universe is a flat sheet of paper.
pcGR View: The universe is like a sheet of paper that has a hidden, microscopic "texture" or "grain" to it, similar to how a piece of wood has a grain you can't see with the naked eye but affects how it bends.

This "grain" is based on a minimal length (the smallest possible distance in the universe, like the Planck length). This tiny, geometric feature changes how gravity works on the largest scales.

2. Dark Energy is Geometry, Not a Mystery Substance

In the standard model, "Dark Energy" is treated like a mysterious fluid or a constant force (the Cosmological Constant, $\Lambda$ ) that pushes the universe apart.

In this new model, Dark Energy isn't a substance at all. It is a side effect of that microscopic "grain" in the geometry of space-time.

Analogy: Imagine you are walking on a trampoline. If the trampoline is perfectly smooth, you roll easily. But if the trampoline has a specific, tiny pattern woven into the fabric, your path changes slightly, and you might feel a "push" even though no one is pushing you. That "push" is what we call Dark Energy in this model. It's just the shape of the trampoline (space-time) reacting to its own texture.

3. The Hubble Constant is Actually a "Hubble Variable"

Because this "geometric push" comes from the structure of space-time itself, the authors argue that the expansion rate of the universe isn't actually constant. It changes very slowly over time.

The Old Idea: The Hubble Constant is like a speedometer stuck on 70 mph. It never changes.
The New Idea: The Hubble "Constant" is more like a car accelerating. It's currently at 70 mph, but it is very, very slowly speeding up.

The paper calculates exactly how much it is speeding up. They found a specific number (called $\beta$ ) that fits the data perfectly. This number tells us that the universe's expansion is accelerating slightly more than the standard model predicts.

4. Testing the Theory with a Cosmic Ruler

To check if their theory is right, the authors used data from DESI (Dark Energy Spectroscopic Instrument).

The Analogy: Imagine trying to measure the size of a room by looking at how sound waves bounce off the walls. In the early universe, sound waves bounced around, leaving a specific "fingerprint" or "ruler" (called Baryon Acoustic Oscillations) imprinted on the distribution of galaxies.
The Result: The authors took this cosmic ruler and measured the distances between galaxies at different times in the past. They compared these measurements to their new "geometric" model.
The Outcome: The model fit the data perfectly. In fact, the data suggests the universe is expanding slightly faster than the standard "constant" model predicts, which aligns with the idea that the expansion rate is changing over time.

5. What This Means for the Future

The paper makes a very specific prediction for the future:

The Prediction: If we watch a distant galaxy for 20 years, its light will shift in color (redshift) by a tiny, tiny amount.
The Number: The authors predict a shift of about -11.1 centimeters per second over 20 years for a galaxy very far away.
The Catch: This is incredibly small (about the speed of a slow snail). However, the paper argues that because this change is a direct result of the geometry of space, it is a real, testable fact. Future telescopes (like the ELT) might be able to measure this tiny shift, proving that the Hubble "Constant" is indeed changing.

Summary

The paper suggests that the "Hubble Tension" (the disagreement between different measurements) might be solved if we stop thinking of the universe's expansion as a fixed speed and start thinking of it as a slowly accelerating process caused by the fundamental texture of space-time itself. Dark energy isn't a mysterious fluid; it's just the universe's geometry doing what it naturally does. The data from DESI supports this idea, and future telescopes could confirm it by watching the universe's expansion speed change in real-time.

Technical Summary: The pseudo-complex Friedmann–Lemaître–Robertson–Walker model and the time dependence of the Hubble constant

Problem Statement
The paper addresses the "Hubble tension," a persistent discrepancy exceeding $4\sigma$ between the Hubble constant ( $H_0$ ) derived from early-universe Planck data ( $67.4 \pm 0.5$ km/s/Mpc) and local distance-ladder measurements from Type Ia supernovae ( $73.30 \pm 1.04$ km/s/Mpc). The authors argue that in the standard $\Lambda$ CDM model, this tension represents a crisis of constants. They propose reframing the issue as a question of cosmic dynamics by introducing a time-dependent Hubble parameter ( $\dot{H}_0 \neq 0$ ) derived from a modified geometric framework, rather than simply adjusting the value of $H_0$ .

Methodology
The authors utilize pseudo-complex General Relativity (pcGR), an algebraic extension of GR where coordinates are extended from 4D to 8D ( $X^\mu = x^\mu + I y^\mu$ , with $I^2=1$ ). This extension introduces a minimal length scale ( $l$ , interpreted as the Planck length) without violating Lorentz/Poincaré symmetry.

Theoretical Framework: Within pcGR, the Friedmann–Lemaître–Robertson–Walker (pcFLRW) model is formulated. The geometric constraints of the pseudo-complex structure lead to an effective dark energy fluid. Unlike the standard cosmological constant $\Lambda$ , this dark energy is characterized by an equation of state $p_\Lambda = -\beta \varepsilon_\Lambda$ , where $\beta$ is a geometric parameter.
Key Relation: The model derives a direct link between the parameter $\beta$ and the time derivative of the Hubble parameter:
$\beta = 1 + \frac{2}{3} \frac{\dot{H}_0}{H_0^2}$
This implies that $\beta \neq 1$ necessitates a non-zero $\dot{H}_0$ .
Data Analysis: The authors constrain the parameter $\beta$ using the latest DESI (Dark Energy Spectroscopic Instrument) Baryon Acoustic Oscillation (BAO) data. They utilize a dataset of 13 correlated measurements (involving $D_V/r_d$ , $D_M/r_d$ , and $D_H/r_d$ ) across various redshifts. A multivariate Gaussian likelihood analysis is performed using the full DESI mean vector and the $13 \times 13$ covariance matrix to minimize the $\chi^2$ statistic.

Key Contributions and Results

Best-Fit Parameter: The statistical analysis yields a best-fit value of $\beta = 1.0426 \pm 0.0144$ . The reduced chi-squared value is $\chi^2_\nu \approx 1.053$ , indicating an excellent fit to the DESI data.
Hubble Acceleration: Using the best-fit $\beta$ and a standard $H_0 = 67.4$ km/s/Mpc, the authors calculate a non-vanishing time derivative of the Hubble constant:
$\dot{H}_0 \simeq (0.94 \pm 0.32) \times 10^{-17} \text{ (km/s}^2\text{)/Mpc}$
This corresponds to $\dot{H}_0 \approx (0.30 \pm 0.10) \times 10^{-36} \text{ s}^{-2}$ .
Deceleration Parameter: The model predicts a deceleration parameter of $q = -0.9361 \pm 0.0216$ . This is significantly more negative than the standard $\Lambda$ CDM prediction ( $q \approx -0.55$ ), suggesting a more rapid acceleration of the universe in the pcGR framework.
Equation of State: The effective dark energy equation of state parameter is $w_\Lambda = -\beta \approx -1.0426$ . While this places the model in the "phantom" domain ( $w < -1$ ), the authors emphasize that in pcGR, this arises from geometric constraints rather than an exotic matter field with negative kinetic energy, thereby avoiding the ghost and gradient instabilities typically associated with phantom scalar fields.
Redshift Drift: Using the exact Sandage–Loeb relation, the model predicts a redshift drift of $\Delta v \simeq -11.1$ cm s $^{-1}$ over 20 years for a source at $z=4$ . This is in close agreement with the $\Lambda$ CDM prediction of $-11.0$ cm s $^{-1}$ , with a difference of only $\sim 0.1$ cm s $^{-1}$ .

Significance and Claims
The paper claims that the pseudo-complex extension of General Relativity provides a consistent cosmological framework where dark energy emerges as a geometric consequence, naturally introducing dynamics into the expansion history.

Reframing the Tension: The authors assert that pcGR does not resolve the Hubble tension by shifting the value of $H_0$ to match local measurements, but rather by transforming the problem into one of dynamics ( $\dot{H}_0 \neq 0$ ).
Testability: The non-vanishing $\dot{H}_0$ is presented as a direct geometric prediction, offering a concrete, testable target for future high-precision spectroscopic observations. The authors specifically cite the ANDES spectrograph on the ELT and the SKA as facilities capable of measuring the redshift drift with sufficient precision to distinguish between the pcGR prediction and the $\Lambda$ CDM baseline over the coming decade.
Geometric Origin: The study highlights that the deviation from $\Lambda$ CDM (specifically the phantom-like equation of state) is rooted in the pseudo-complex geometry and the minimal length scale, distinguishing it from phenomenological models that introduce new dynamical degrees of freedom.