The size of the universe and the speed at which it expands may be linked by a quantum uncertainty that changes the way cosmic expansion is calculated. A new theoretical model proposes that this uncertainty could add a geometric correction to the Friedmann equation, allowing the same framework to produce a late-time form of dark energy without introducing a new particle or field.
The proposal starts with a simple change to the mathematical description of an expanding universe.
In the standard treatment of cosmology, the universe is described by a scale factor, usually written as a, that tracks how its size changes with time. The corresponding rate of change is related to the Hubble rate, H. In canonical quantum cosmology, the scale factor and its canonical momentum obey the usual quantum commutation relation.
Instead, the model introduces a nonzero commutation relation directly between the scale factor and its rate of expansion. The proposed relation is
[â, â̇] = −iβ â²[1 + (â/a₀)ⁿ],
where β sets the strength of the deformation, a₀ is a crossover scale, and n is a free exponent. The relation is intended as a cosmological counterpart to the quantum uncertainty principle. It means that the size of the universe and its expansion rate cannot be simultaneously specified with arbitrary precision.
The construction is different from generalized uncertainty principle models and other forms of noncommutative cosmology discussed in the research. Those approaches modify other relationships, such as the one between the scale factor and its canonical momentum. Here, the velocity-configuration relation itself is deformed.
The distinction matters because it changes where the resulting correction appears in the equations governing cosmic expansion. Instead of behaving like an additional energy density on the right side of the Friedmann equation, the correction appears on the left side as a geometric contribution to the expansion rate.
The uncertainty changes the Friedmann equation
The central mathematical result is a modified Friedmann equation:
H² + β²[1 + (a/a₀)ⁿ]² = 8πGρ/3 + Λc²/3.
The extra term represents what the model describes as an irreducible quantum variance in the Hubble rate. It arises from the uncertainty associated with the deformed commutation relation.
The model’s derivation begins with the modified commutation relation, constructs a self-adjoint operator for the expansion rate, derives the corresponding momentum operator, and then applies the uncertainty relation to the Hamiltonian constraint. The resulting correction is therefore not inserted separately into the Friedmann equation. It follows from the proposed kinematic structure.
The correction can be separated into three pieces. One is constant and shifts the effective cosmological constant. The other two vary as powers of the scale factor. They can be written as terms with effective equations of state
w₁ = −1 − n/3
and
w₂ = −1 − 2n/3.
Their behavior depends on the sign and size of n.
The model therefore produces two especially different cosmological regimes. For sufficiently negative n, the correction can become important when the universe is very small. For positive n, it becomes increasingly important as the scale factor grows.
That second case is the one that provides the model’s late-universe prediction.
Positive n produces an effective dark energy component
In the cosmological regime considered in the study, the crossover scale is of order the present Hubble length and β is much smaller than the present Hubble rate. At late times, when matter and radiation have diluted, the expansion equation becomes
H² = Λeffc²/3 − 2β²(a/a₀)ⁿ − β²(a/a₀)²ⁿ.
For n greater than zero, both power-law terms grow with the scale factor. They progressively reduce H² and behave as an effective dark-energy contribution with an equation of state greater than −1.
The combined effective dark-energy equation of state is
w_eff = −1 + [2nε²aⁿ(1 + aⁿ)]/[3ΩDE(a)],
when the present-day scale factor is set to 1 and β is written as εH₀.
For positive n, the model gives w_eff > −1 at every redshift. It also predicts that the effective equation of state approaches −1 at high redshift rather than approaching a fixed value different from −1.
This produces a specific expansion history. The deviation from ΛCDM follows a power law in redshift rather than the form used in the commonly employed Chevallier-Polarski-Linder parameterization. The study identifies the shape of H(z) as the key way to distinguish the proposed model from that parameterization.
Current observations do not yet separate the model from ΛCDM
The study compares example models with measurements of the Hubble rate from cosmic chronometers and baryon acoustic oscillations.
For the illustrated case with ε = 0.1 and a₀ equal to the present-day scale factor, the models with n = 1, 2, 3 and 4 remain extremely close to the ΛCDM prediction. The calculated difference is about −1.0 km s⁻¹ Mpc⁻¹ at redshift zero in these examples and decreases toward higher redshift.
The study notes that this difference is more than an order of magnitude smaller than the typical uncertainty of the cosmic-chronometer measurements used in the comparison. It also remains below the precision of the current BAO measurements considered there.
As a result, the model is observationally indistinguishable from ΛCDM with the present data used in the analysis. The study identifies future measurements with sub-percent precision, including those expected from DESI Year 5 and Euclid, as potentially capable of distinguishing the predicted H(z) shape.
The model also has a particular relationship to the dynamical-dark-energy preference reported from DESI Year 1 BAO data. For positive n, it predicts w₀ greater than −1 and wₐ less than zero, the same signs as the ranges favored in that DESI analysis. But the study emphasizes that the reported DESI preference has been questioned and that the decisive test for this model would come from the detailed shape of H(z), rather than from those two parameters alone.
The model is also constrained by distance measurements. For the cosmological regime and values of |n| above roughly 0.5, the study finds that consistency at the percent level with current BAO and supernova data requires ε² to be no larger than order 10⁻² when a₀ is of order H₀⁻¹.
The same relation can produce a bounce in the early universe
The proposed deformation does more than modify late-time expansion. Its behavior changes substantially when n is negative.
For n less than −2, the noncommutative correction grows faster than the radiation contribution as the scale factor approaches zero. The result can be a classically forbidden region at very small scale factor and a classical bounce instead of a singular beginning.
The bounce scale is
a_bounce = a₀[−2/(n + 2)]¹⁄ⁿ.
The special case n = −4 places the bounce exactly at the crossover scale a₀. In the Planck-scale version of the model, the relevant scales can then also be of Planck order.
The behavior is different for positive n. In that case, the noncommutative correction grows with the scale factor during radiation domination. The expansion eventually reaches a maximum scale factor, where H² becomes zero. The classical solution then recollapses rather than passing through a bounce.
For positive n, the model does not classically remove the Big Bang singularity. The allowed region ends at the maximum scale factor, and in the minimal model there is no second classical branch beyond the forbidden region.
The intermediate range, −2 ≤ n ≤ 0, does not resolve the Big Bang singularity in the model.
The model does not introduce a new dark-energy particle
One feature of the proposal is that the additional contribution to cosmic expansion does not come from a new matter field.
The correction is geometric. The matter action, kinetic terms and equations of motion remain those of standard cosmology. The study also finds that, at leading order in β/H, the deformation does not enter the matter perturbation equations, leaving the scalar-perturbation sound speed unchanged. The authors describe these as qualitative arguments and note that a rigorous analysis of the perturbed action remains to be done.
The individual power-law pieces can formally be assigned equations of state below −1 when n is positive, but the combined effective dark energy in the model has w_eff > −1. The study therefore does not identify a ghost associated with the effective dark-energy behavior because no new scalar field with a wrong-sign kinetic term has been introduced.
The model also leaves the primordial power spectrum effectively standard in the cosmological regime. At horizon crossing, the noncommutative suppression is negligible when β²/H² is much smaller than 1, so the study concludes that the deformation does not significantly alter scales accessible to cosmic microwave background experiments.
The proposed quantum effect may be tied to the cosmic horizon
The interpretation of the deformation is not restricted to the Planck scale.
The study argues that the commutation relation can apply at any value of the scale factor, with the scale at which its effects become important determined by β and a₀. This allows the same mathematical structure to describe a Planck-scale early universe and a cosmological regime in which the relevant scale is comparable to today’s horizon.
A proposed interpretation connects β to the cosmological horizon through a relation of the form β ∼ c/a_horizon. Under this interpretation, a Planck-sized horizon would correspond to a Planck-scale value of β, while a horizon of order H₀⁻¹ today would give β of order H₀.
The authors also discuss a possible holographic motivation for this horizon-scale interpretation. They propose that imposing β = c/a₀ would reduce the model from three parameters to two, with the deformation strength determined by the crossover scale. But this remains an interpretation of the proposed framework rather than a derivation of the underlying microscopic theory.
The microscopic origin of the commutation relation remains unknown. The study points to several quantum-gravity programs that contain structures with qualitative similarities, including group field theory, polymer and loop quantization, string T-duality and modified horizon thermodynamics. None of these approaches, however, derives the proposed relation exactly.
Several important problems remain unresolved
The model has a significant fine-tuning problem in its Planck-scale form.
The constant β² contribution shifts the effective cosmological constant according to
Λ_eff = Λ − 3β²/c².
For β of Planck scale, this requires a 122-decimal-place cancellation, so the model does not solve the cosmological constant problem. The study identifies the cosmological regime, with β much smaller than H₀, as an alternative in which this fine-tuning is not required.
There is also a structural limitation for positive n. The study finds that no single choice of a₀ and β can simultaneously provide a Planck-scale early-universe turning point, an observable late-universe dark-energy signal and expansion consistent with Big Bang nucleosynthesis through the present epoch. The positive-n family is therefore treated as either a late-universe dark-energy model or an early-universe quantum-cosmology model, rather than both at once.
The model does not resolve the H₀ tension either. Because the noncommutative contribution is positive on the left side of the Friedmann equation, it reduces the expansion rate available for a given energy budget. In the example discussed in the study, ε = 0.1 lowers the inferred H₀ relative to ΛCDM by roughly 1.5 percent for fixed Planck-anchored matter density and sound horizon. The study notes that this moves in the opposite direction from the late-universe SH0ES result.
Other questions remain open. The stability of the proposed classical bounce under metric and matter back-reaction has not been established. The equations become non-perturbative near the bounce, and a systematic Bayesian analysis using future DESI, Euclid and LSST data has not yet been carried out. The proposed commutation relation is also not covariant under the full group of spacetime diffeomorphisms, leaving a fully gauge-invariant formulation as an open problem.
The central unresolved issue is the physical origin of the new algebra itself. The study proposes several possible directions, but none currently provides a complete microscopic derivation of the relation.
The model therefore presents a specific way in which quantum kinematics could alter cosmic expansion without adding a new particle or field. In its positive-n regime, the predicted effect is a characteristic modification of H(z) that remains too small to distinguish from ΛCDM with current measurements. Future measurements of the expansion history are identified as the principal observational test of the proposal.
The study was published in Physical Review D.
