The discovery that the universe’s expansion is accelerating represents one of the most significant achievements of modern observational cosmology. The evidence emerged in the late 1990s from type Ia supernova (SNe Ia) measurements, which showed that distant events appeared dimmer than expected in a uniformly expanding universe [6] [7] .

This result implied that the Hubble parameter, expressed as a function of redshift $H\left(z\right)$, decreases less rapidly at low redshifts ( $z\lesssim 0.5$) than predicted by a matter-dominated model, indicating a phase of accelerated expansion.

In the standard ΛCDM cosmology, the expansion rate of the universe at cosmic time t is given by

$H\left(t\right)=\frac{1}{a\left(t\right)}\cdot \frac{\text{d}a}{\text{d}t}$

where a(t) is the scale factor of the universe and t denotes cosmic time.

To account for the supernovae evidence, as well as other cosmological probes such as the cosmic microwave background (CMB) anisotropies [8] [9] and baryon acoustic oscillations [10] [11] , the ΛCDM model introduces a dominant dark energy component, commonly identified with Einstein’s cosmological constant Λ. In this framework, the present-day energy density of the universe is dominated by dark energy ( ${\Omega }_{\Lambda }\simeq 0.69$) and cold dark matter ( ${\Omega }_m\simeq 0.31$), according to the most recent Planck results [9] . The accelerated expansion is thus explained by a vacuum energy term with negative pressure, corresponding to an equation-of-state parameter $w=p/\rho \simeq -1$.

While the ΛCDM model explains the observed acceleration by introducing the cosmological constant Λ, its physical origin remains unknown. This has motivated a broad range of theoretical attempts to account for the phenomenon without postulating an unexplained vacuum energy. Two main strategies can be distinguished: 1) introducing new dynamical components, and 2) modifying the theory of gravity itself.

A first class of alternatives considers dynamical dark energy, in which the acceleration is driven by scalar fields evolving over cosmic time. Quintessence models describe a slowly rolling scalar field with a potential that produces negative pressure [12] [13] . Variants such as k-essence and phantom energy explore different kinetic terms or equations of state with $w<-1$ [14] [15] . While these models can reproduce the late-time acceleration, they require fine-tuning of initial conditions and potentials, raising concerns about predictivity.

A second class of proposals focuses on modifications of gravity at cosmological scales. Extended gravity theories generalize Einstein’s General Relativity by altering the gravitational Lagrangian. One of the most studied examples is f(R) gravity, where the Einstein-Hilbert action is replaced by a more general function of the Ricci scalar R [16] [18] [19] . These models can produce self-acceleration without invoking a cosmological constant and have been extensively investigated for their cosmological and astrophysical implications [1] [17] . Other frameworks include scalar-tensor theories, such as Brans-Dicke gravity [20] , Horndeski theories [21] [33] , and Gauss-Bonnet extensions.

More recently, massive gravity and bimetric theories have been explored as possible explanations for cosmic acceleration [22] . These approaches modify the graviton’s properties or introduce additional metrics, leading to an effective late-time repulsive effect. Similarly, higher-dimensional frameworks such as braneworld cosmologies [23] generate acceleration from modifications of gravity in extra dimensions.

Despite their diversity, all these alternative frameworks share the goal of addressing the dark energy problem without invoking a pure cosmological constant. However, they typically come at the cost of introducing new dynamical fields, parameters, or degrees of freedom. Furthermore, they are strongly constrained by both cosmological observations and local gravity tests, which often limit the allowed parameter space [24] [25] . Ultimately, their viability depends on the ability to distinguish them from General Relativity through precise experimental tests, such as the interferometric detection of gravitational waves [26] .

The aim of the following section is to briefly review the foundations of the Four-Dimensional Electromagnetic Universe (4DEU) framework, so as to provide the necessary background for the subsequent analysis. Table 1 summarizes the main alternative theoretical approaches to the dark energy problem.

The aim of this section is to briefly review the foundations of the Four-Dimensional Electromagnetic Universe (4DEU) framework, so as to provide the necessary background for the subsequent analysis.

The Four-Dimensional Electromagnetic Universe (4DEU) framework postulates that the cosmos is not a (3 + 1)-dimensional spacetime with time as an abstract coordinate, but a real four-dimensional Euclidean hypersphere. Within this framework, space is postulated to be quantized, with the Planck length representing the fundamental minimum unit [3] . In this picture, the three ordinary spatialdimensions correspond to the curved 3D hypersurface where we live, while the fourth real spatial dimension is perceived by observers as time. The universe expands uniformly at rate c along this fourth dimension, denoted T, which acts as a privileged temporal coordinate. In the 4DEU framework, c does not represent a velocity in the conventional sense; rather, it should be regarded as the expansion rate of time real dimension and, more fundamentally, as a universal conversion factor between the units historically used to measure spatial and temporal intervals.

Accordingly, in the 4DEU framework the so-called Big Bang corresponds to the center of the 4D universe, while the temporal dimension represents its radius. From this postulate also follows the existence of a privileged reference frame centered on the Big Bang event. This is analogous to the use of cosmic time in the standard cosmological model, where the cosmic microwave background (CMB) provides the universal reference frame in which the universe appears isotropic.

Consequently, every entity moving with respect to the Big Bang must also possess a temporal velocity component of fixed magnitude c, corresponding to the constant expansion rate that governs everything in the 3D portion of the 4D universe. For example, electromagnetic waves, including light, have a four-velocity composed of a spatial component in 3D and a temporal component of constant value c [2] [3] .

Two further key postulates underlie the 4DEU framework.

The second postulate is the Restricted Holographic Principle (RHP), which establishes that any phenomenon occurring along the fourth spatial dimension is not directly accessible, but manifests in the 3D portion of the 4D universe, where we live, in a qualitatively different yet quantitatively proportional way.

The third postulate defines the Temporal Waves (TWs), stationary electromagnetic waves confined exclusively to the fourth spatial dimension, with a wavelength equal to four times the radius of the 4D universe. As a consequence of the Restricted Holographic Principle, properties such as mass, energy, and charge are interpreted as emergent effects of these TWs. TWs are at the very origin of the universe: at the onset of cosmic expansion, from the ever-existing quantum vacuum, four orthogonal TWs of opposite phases were generated, ensuring isotropy and overall charge neutrality. These waves are stable at all times, including the very beginning, and their properties evolve coherently with the growth of the 4D hypersphere [4] [5] .

TWs exert a negative radiation pressure directed outward, perpendicular to the 3D hypersurface where observers reside. This radiation pressure drives the uniform expansion of the 4D universe at rate c, playing in 4DEU the same role attributed to dark energy in the ΛCDM model.

The Restricted Holographic Principle explains how physical quantities such as time, mass, electric and magnetic charge, and gravity emerge as projections of TW dynamics.

The main consequences of the RHP are summarized in Table 2 .

From these postulates, the 4DEU framework leads to several major conclusions:

$T\left(z\right)=T_0\left(1+z\right)$

where $T_0$ denotes the present-day temperature of the CMB. In this framework, such a dependence follows directly from the 4D geometry without requiring any additional assumptions, thus finding direct confirmation in the observational data [4] [5] .

¹“4D Source” indicates the fundamental physical cause acting along the fourth spatial dimension of the real four-dimensional universe (the 4D hypersphere). Its effect appears in 3D as the corresponding observable manifestation. (a) Eq. 115 in [5] . The equivalent cosine representation is given in Eq. 80 of [3] . Here x indicates a point along the time dimension, ranging from the privileged coordinates $-R_t$ to $+R_t$, $E_{tw\left(x,R_t\right)}$ indicate the intensity of the electric field of a TW at point x (along the radius at time T $\left(R_t\right)$ of the 4D universe), and $E_{0\left[tw\left(R_t\right)\right]}$ the maximum absolute amplitude of the electric field. (b) Eq. 2.23 in [4] .

hyperspherical shell of the 4D universe which, because it does not exert gravitational attraction on itself, cannot decelerate the expansion of the 4D universe [2] .

Building on these foundations, we next apply the 4DEU framework to the evolution of the Hubble parameter and directly compare its predictions with model-independent observational data.

In most alternative frameworks, the dark energy problem is addressed either by introducing new dynamical components, such as scalar fields [12] [13] , or by modifying gravity on cosmological scales [1] [17] . These approaches can reproduce the observed late-time acceleration and are generally constructed to remain consistent with the wealth of cosmological observations [9] , but they do so at the cost of introducing additional fields, free parameters, or degrees of freedom. Their viability ultimately depends on the possibility of distinguishing them from General Relativity through high-precision experimental tests—for example, the interferometric detection of gravitational waves, which provides a direct empirical discriminator between GR and alternative gravity theories [26] .

By contrast, the 4DEU framework explains the apparent acceleration without invoking new dynamical entities or altering the Einstein-Hilbert action. Its only conceptual difference with respect to General Relativity is an additional simplification: in 4DEU, time is treated as a privileged linear coordinate rather than a curved one. In this context, we denote by $t_{rel}$ the relative time, i.e., the time that an observer assigns to a clock located in another reference frame, either in a different gravitational potential or in relative motion. This quantity corresponds to the coordinate time in Special and General Relativity and, in most cases, coincides with the proper time $\tau$. By contrast, the privileged time T coincides with the locally measured proper time of each observer, advancing identically for all regardless of motion or gravity [4] [5] . The relation between T and $t_{rel}$ is given by the standard expressions:

in a gravitational field,

$t_{rel}=T\sqrt{\frac{1-\frac{2GM}{r^{\prime }{c}^2}}{1-\frac{2GM}{r{c}^2}}}$

and in inertial relative motion,

$t_{rel}=\frac{T}{\sqrt{1-\frac{v^2}{c^2}}}$

As a consequence, the 4DEU reproduces the same weak-field results as GR, including gravitational redshift, light deflection, Shapiro delay, and the perihelion precession of Mercury [4] [5] .

The results presented here show that the linear 4DEU prediction (Eq.1) for the Hubble parameter $H\left(z\right)$ matches model-independent datasets of type Ia supernovae and cosmic chronometers with high level of statistical significance. In particular, the statistical consistency is markedly higher when adopting the Planck-CMB value of the Hubble constant ( $H_0=67.4\pm 0.5$ km/s/Mpc) than when using the local distance-ladder estimate ( $H_0\approx 73$ km/s/Mpc). This behavior indicates that the value near 67 km/s/Mpc should be regarded as the physically meaningful one, while the higher local values may be influenced by systematics that are not yet fully understood.

Crucially, within the 4DEU framework, the observed late-time acceleration no longer requires the introduction of a dark energy component: it emerges naturally as a geometric projection effect of uniform 4D expansion. By Occam’s razor, the combination of stronger statistical agreement with independent datasets and the conceptual simplicity resulting from eliminating an otherwise unknown form of energy strongly favors the solution with $H_0\approx 67$ solution.

Moreover, other observational puzzles—such as the CMB temperature-redshift relation, the unexpectedly advanced ages of galaxies detected by JWST at $z\gtrsim 10$, and the thermodynamic balance of the Universe—are also consistent with the 4DEU framework, as discussed in previous works by Maglione [2] - [5] . Taken together with the present analysis of the Hubble parameter, these results indicate that a broad set of independent astronomical datasets is already in good agreement with the predictions of the 4DEU framework. Further systematic analyses across a broader range of cosmological probes will be necessary, but the current evidence suggests that 4DEU represents a viable alternative to more complex dark-energy or extended-gravity models, without introducing extra degrees of freedom, additional parameters, new fields, or exotic energy components.

The author sincerely thanks the anonymous referee for their initial comments and suggestions, which contributed to improving the clarity and completeness of the manuscript. In particular, the invitation to discuss the dark energy problem more extensively and to compare it with extended gravity approaches proved very helpful.

The Author is also grateful for the final evaluation: although the work was considered speculative and unorthodox, the referee’s open-mindedness in recommending its publication has given this research the opportunity to be considered within the scientific debate.

For completeness, we derive Equation (1) within the 4DEU framework in a self-contained way.

Starting from the definition of the expansion rate:

$H\left(t\right)=\frac{1}{a\left(t\right)}\frac{\text{d}a}{\text{d}t}=\frac{1}{R_t}\frac{\text{d}{R}_t}{\text{d}t}$ (A1)

where $a\left(t\right)$ denotes the scale factor; $H\left(t\right)$, the Hubble parameter at cosmic time $t$; $R_t$, the 4D radius at time $t$.

The uniform expansion along the time dimension in 4DEU Theory is described by Eq.1 in [3] :

$\frac{\text{d}{R}_t}{\text{d}t}=c$ (A2)

Combining (A1) with (A2) gives:

$H\left(t\right)=\frac{c}{R_t}$ (A3)

Equivalently, derived here,

$R_t=\frac{c}{H\left(t\right)}$ (A4)

Denoting $R_z$ as the 4D radius of the universe at redshift $z$ and $H\left(z\right)$ as the Hubble parameter at redshift $z$, Eq. (A4) becomes:

$R_z=\frac{c}{H\left(z\right)}$ (A4a)

At the present epoch the scale factor equals unity, $a_0\equiv a\left(t_0\right)=1$, and the redshift-radius relation in 4DEU framework is Eq.B.5 in [3] :

$1+z=\frac{R_0}{R_z}$ (A5)

where $z$ indicates redshift and $R_0$ represents the present day radius of 4D universe.

Isolating $R_z$ from (A5) we have:

$R_z=\frac{R_0}{1+z}$ (A6)

Here $R_z$ denotes the 4D radius at redshift $z$. For example, at the present epoch ( $z_0=0$), $R_{z_0}=R_0$.

Evaluating (A4a) at the present epoch ( $z_0=0$) yields the present-day radius in terms of the Hubble parameter,

$R_0=\frac{c}{H_0}$ (A6a)

where $H_0$ is the Hubble parameter today, that is at $z=0$.

Finally, combining (A4a), (A6) and (A6a), we have:

$\frac{c}{H\left(z\right)}=\frac{R_0}{1+z}=\frac{c}{H_0}\frac{1}{1+z}$ (A7)

which yields Equation (1) of the main text:

$H\left(z\right)=H_0\left(1+z\right)$