Zero-Point Length Cosmology

• Zero-Point Length Cosmology is a framework that integrates a minimal, fundamental length scale from quantum gravity to regulate singularities.
• It modifies key equations like the Friedmann and Raychaudhuri equations by incorporating quantum and thermodynamic corrections.
• The model offers observable predictions affecting primordial perturbations, gravitational waves, and emergent universe scenarios.

A zero-point length cosmology is a framework in which the existence of a minimal, fundamental length scale—arising from quantum gravity, string T-duality, or generalized uncertainty principles—is incorporated directly into the gravitational sector and cosmological dynamics. The zero-point length, typically denoted $\ell_0$ or $l_0$, acts as a regulator of both geometric and field-theoretic divergences, modifying the classic equations of general relativity, the thermodynamics of horizons, early-universe singularity structure, quantum cosmology, and the macro- and microphysical behavior of the cosmos at Planckian or sub-Planckian scales.

1. Fundamental Origin: Minimal Length from Quantum Gravity

The concept of zero-point length arises in several quantum gravity formalisms, including string theory (notably via T-duality), deformations of the Heisenberg algebra via generalizations such as the @@@@3@@@@, and in path-integral duality approaches. These frameworks consistently predict a universal lower bound for operationally defined spatial intervals, encoded as a minimal length

$$\ell_0 \sim \sqrt{\hbar G / c^3} \sim \ell_{\rm Pl}\,,$$

though phenomenological bounds may allow for $\ell_0 \gg \ell_{\rm Pl}$ (Sooraki et al., 2 Dec 2025). In string theory, the propagator regularization $G(k) = -l_0 / [\sqrt{k^2} K_1(l_0 \sqrt{k^2})]$ and the associated replacement of the Newtonian potential by

$\phi(r) = -\frac{M}{\sqrt{r^2 + \ell_0^2}}$

renders the potential finite at the origin and signals the breakdown of the notion of a spacetime “point” at sub-$\ell_0$ distances (Sooraki et al., 2 Dec 2025, Luciano et al., 2024).

In quantum cosmology and minisuperspace models, GUP-induced deformations of the canonical commutator, such as

$$[\hat X, \hat P] = i\hbar (1 - \beta \hat P^2)^{-1}$$

with $\beta > 0$, force a minimum position uncertainty

$$\Delta x_{\min} = \hbar \sqrt{3\beta} \equiv \ell_0\,,$$

which is nonzero for all physical states (Gusson et al., 2020).

2. Modified Horizon Entropy and Thermodynamics

The horizon entropy-area relation, fundamental to the thermodynamic derivation of gravitational field equations, acquires explicit zero-point length corrections. The entropy of a spherical horizon of radius $R$ is generally modified to (Sooraki et al., 2 Dec 2025, Sheykhi et al., 2024, Jusufi et al., 2022)

$$S_h(R, \ell_0) = \pi R^2 (1 + \ell_0^2/R^2)^{-1/2} + 3\pi \ell_0^2 (1 + \ell_0^2/R^2)^{-1/2} - 3\pi \ell_0^2 \ln[R + \sqrt{R^2 + \ell_0^2}]$$

with the infinitesimal change

$$dS_h = 2\pi R \left(1 + \ell_0^2 / R^2 \right)^{-3/2} dR\,.$$

Expanding the entropy in the regime $\ell_0^2 / R^2 \ll 1$, one finds leading-order logarithmic corrections,

$$S(A) = \frac{A}{4} - \frac{3 \ell_0^2}{4} \ln \left( \frac{A}{4\pi} \right) + \cdots\,,$$

reminiscent of quantum-gravity expectations (Jusufi et al., 2022).

The apparent horizon’s temperature and work term also receive corrections, which modify the first law of thermodynamics at the horizon, forming the basis for the derivation of modified Friedmann equations (Sheykhi et al., 2024, Luciano et al., 2024).

3. Modified Friedmann and Raychaudhuri Equations

The principal cosmological effect of zero-point length is a deformation of the Friedmann dynamics. Applying the first law or entropic-force prescription on the apparent horizon yields a modified Friedmann equation, for $k = 0$ and vanishing cosmological constant (Sooraki et al., 2 Dec 2025, Luciano et al., 2024, Jusufi et al., 2022): $$H^2 - \frac{3}{4} \ell_0^2 H^4 = \frac{8\pi}{3} \rho\,,$$ or, to leading order in $\ell_0^2$,

$$H^2 \simeq \frac{8\pi}{3} \rho \Bigl[ 1 + 2\pi \ell_0^2 \rho \Bigr]\,.$$

The quadratic term induces a high-energy correction analogous to that in braneworld (Jusufi et al., 2022) or loop quantum cosmology scenarios, with a sign depending on the specifics of the underlying GUP deformation.

The Raychaudhuri equation for geodesic congruences is similarly modified when expressed in terms of the q–metric, resulting in the replacement (Chakraborty et al., 2019)

$$\frac{d\theta_q}{d\tau} = -\frac{1}{3} \theta_q^2 - R_{ab} u^a u^b - 3 \frac{d}{dS} \ln \Delta,$$

where $S^2 = \sigma^2 + L_0^2$ is the smeared geodesic interval and $\Delta$ is the Van Vleck determinant. In homogeneous FRW universes, this leads to an effective repulsive correction growing in significance at high densities, yielding upper bounds on attainable $\rho$ and a cosmic bounce (Chakraborty et al., 2019, Ali et al., 2014).

4. Quantum Cosmology and Wheeler–DeWitt Modification

Zero-point length modifies the quantum cosmological phase space, reflected in the structure of the Wheeler–DeWitt equation, wavefunction properties, and the associated spectra. For example, under the Pedram GUP, the minisuperspace momentum receives higher-derivative corrections,

$$\hat{P}_a \rightarrow -i\hbar [ \partial_a + \beta \partial_a^3 + \beta^2 \partial_a^5 + \cdots ]$$

so that the WDW equation becomes, for the small $a$ limit (Gusson et al., 2020)

$$\phi^{\prime\prime} + 2\beta \phi^{(4)} + 3\beta^2 \phi^{(6)} + 24\omega a^4 \phi = 0.$$

The physical wavefunction is projected onto “maximally localized” quasi–position states. The minimal spread of these states is set by the zero-point length, and their spectral properties reflect the discretization induced by the Hamiltonian constraint algebra with su(1,1) Casimirs. The entropy calculated in this framework becomes a Dirac observable and matches holographic predictions for large quantum numbers (Gusson et al., 2020, Jalalzadeh et al., 2014).

5. Singularity Resolution and Bounce Scenarios

A generic outcome of introducing a zero-point length in cosmological models is the nonsingular evolution of the early universe. The repulsive $\ell_0$-dependent terms in the Friedmann or Raychaudhuri equations prevent the divergence of the Hubble parameter or matter density:

• The expansion scalar and higher curvature invariants remain finite as $t \to 0$ (Sheykhi et al., 2024, Jusufi et al., 2022, Chakraborty et al., 2019).
• The critical density for bounce is set by $\rho_{\rm c} \sim 1/ (2\pi \ell_0^2)$.
• The minimal scale factor at the bounce is $a_{\min} \sim \ell_0$ (Jusufi et al., 2022, Ali et al., 2014).
• In quantum cosmology, physical states cannot be localized at $a = 0$; the wavefunction vanishes or is regularized near the classical singularity (Gusson et al., 2020).

In anisotropic cosmologies (e.g., Bianchi I/II), the GUP-induced zero-point length suppresses chaotic mixmaster behavior and enables isotropization (Ali et al., 2014).

6. Cosmological Phenomenology and Observational Constraints

Zero-point length corrections modify cosmological observables in several regimes:

• Early-universe baryogenesis: The correction to the Friedmann equation induces a nonzero time derivative of the Ricci scalar during radiation era, enabling gravitational baryogenesis with

$\eta \propto \ell_0^2 T_D^9 / M_{\rm Pl}^7$

and bounding $\ell_0 \lesssim 7.1 \times 10^{-33}$ m ($\sim 440 \ell_{\rm Pl}$) (Sooraki et al., 2 Dec 2025).

• Primordial perturbations and inflation: The presence of $\ell_0$ leads to departures from perfect scale invariance at high $k$, naturally produces a broken-power-law for the scalar power spectrum, and modifies the tensor-to-scalar ratio and tilt (Luciano et al., 2024). The magnitude of these corrections is constrained to be small, $\ell_0 \sim \mathcal{O}(1)\ell_{\rm Pl}$, by Planck and BICEP/Keck data.
• Gravitational waves: Running of $\ell_0$ with energy can amplify the primordial GW background at high frequency, potentially making departures from GR accessible to space-based detectors if $\ell_0$ grows at high scale (Luciano et al., 2024).

At late times,

• Corrections mildly increase the cosmic age,
• Delay the onset of accelerated expansion (Sheykhi et al., 2024),
• Provide a possible quantum-gravity-based shift in the Hubble parameter, addressing the Hubble tension (Jusufi et al., 2022).

7. Non-Singular Emergent and Static Cosmologies

Zero-point length cosmology enables the construction of non-singular emergent-universe models wherein the universe is past-eternal and remains in a stable, static (Einstein Static Universe, ESU) phase before a controlled departure to inflation. The stability of the ESU, its spectral properties under scalar, vector, and tensor perturbations, and the impact of additional energy components (topological defect networks, etc.) can be systematically analyzed within the zero-point length-modified Friedmann system: $$X - \alpha X^2 = \frac{1}{3} \rho_{\rm tot}\,, \qquad X = H^2 + k / a^2, \quad \alpha \propto \ell_0^2$$ The existence and stability windows of the ESU depend crucially on the interplay between $\ell_0$, the equation of state $\omega$, topological defect parameters, and spatial curvature (Bhuyan et al., 2024). Crossing specific parameter boundaries triggers a graceful exit from the static phase into inflation. These constructions realize a singularity-free “emergent universe” consistent with thermodynamic stability (Bhuyan et al., 2024).

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In sum, zero-point length cosmology operationalizes the quantum-gravitational hypothesis of a fundamental spacetime discreteness at length scale $\ell_0$ into explicit modifications of gravitational, thermodynamic, and quantum-cosmological structures, leading to regularized early-universe evolution, calculable departures from standard cosmological observables, a direct bridge between Planck-scale/UV physics and macroscopic cosmology, and new possibilities for observational probes of quantum gravity in the cosmic microwave background, primordial element abundances, gravitational wave spectra, and large-scale cosmic parameters (Sooraki et al., 2 Dec 2025, Sheykhi et al., 2024, Gusson et al., 2020, Jusufi et al., 2022, Chakraborty et al., 2019, Ali et al., 2014, Luciano et al., 2024, Bhuyan et al., 2024, Jalalzadeh et al., 2014, Pesci, 2020).