In our previous work [1], the relation between the Planck length ℓ p and the Big Bounce mechanism is multifaceted and deeply rooted in the statistical nature of the pre-geometric phase. The Planck length does not appear as a fundamental constant that directly quantizes space, but rather as an emergent geometric scale arising from the saturation of velocity states of primary particles.

Planck Length as a Coarse-Graining Scale

In deriving the amplitude of density perturbations, we use the estimate

where V patch denotes the volume of the causal region at the bounce, and ℓ p represents the fundamental coarse-graining scale in phase space, i.e., the minimal volume within which primary particles can be treated as statistically independent.

This implies that the Planck scale emerges naturally from the discrete structure of velocity space (via ε ) and the total mass of the universe M U , as introduced in [2] and consistent with [3], rather than being imposed as an independent postulate.

Critical Bounce Density in Terms of ℓ p

In Ref. [1], Equation (49), we derived the critical density in loop quantum cosmology as

This relation is crucial: the Planck length directly controls the value of ρ c , and that is an important reinterpretation. When the energy density reaches ρ c , the available velocity states become saturated and the universe undergoes a bounce.

Thus, the Planck length is not an independent constant, but rather a consequence of velocity quantization ( ε ) and the total cosmological mass M U .

RelationbetweenEquilibrium Scale and Planck Length

In Appendix C, the spectral index n s is related to

where ℓ e q is the equilibrium scale in the pre-geometric phase. A numerical estimate ℓ e q ~ 30 ℓ p yields n s ≈ 0.97 , in agreement with Planck data.

This indicates that the Planck length serves as a reference scale for all other length scales in the early universe.

Physical Interpretation: Planck Length as Maximal Information Density

Within this framework, ℓ p is not a minimal length in the geometric sense, but rather the scale at which the collective statistics of primary particles transitions from kinetic to hydrodynamic behavior.

The cell defined by the Planck length represents the maximum information density that emergent space-time can handle.

When the energy density approaches ρ c , the mean squared velocity 〈 v ′ 2 〉 reaches its maximum, and the effective cell size becomes ℓ p . The Big Bounce thus acquires a statistical-mechanical interpretation: it corresponds to a state in which no additional velocity levels can be populated.

Summary

In [1], the Planck length is an emergent scale arising from velocity quantization and global cosmological parameters. It determines the size of coarse-grained cells, the critical bounce density, and the equilibrium scale, providing a description of the bounce without singularities and without free parameters.

In the previous analysis, the fundamental scale associated with the bounce was taken to be of order the Planck length, R bounce ~ O ( ℓ p ) . However, cosmological observables today originate from a much larger comoving region, corresponding to the present-day observable universe.

The radius of the observable universe is

so that

The comoving volume at the bounce is

Using the estimate

we obtain

Thus, the entire observable universe originates from a single Planck-scale comoving cell.

The robustness of this result under variations of the scaling assumptions is analyzed in AppendixA.

This provides a natural realization of an ultralocal initial condition and supports a statistical-mechanical interpretation in which the initial state corresponds to a minimal set of microstates.

The entropic implications of this result are discussed in AppendixB, where it is shown that this naturally leads to a low-entropy initial state and a well-defined arrow of time.

Furthermore, this framework allows a reinterpretation of previously proposed superluminal transitions as emergent phenomena associated with the reorganization of microscopic degrees of freedom near the bounce.

A key question is whether the result

can provide a viable origin for primordial fluctuations observed in the cosmic microwave background (CMB).

All observable modes originate from a single Planck-scale region, implying a highly constrained initial quantum state. As the universe expands, these fluctuations are stretched to cosmological scales, similarly to standard scenarios [4][5].

The observed amplitude is

(8)

as measured by Planck [6].

The synthesis of the PPH framework with the emergent nature of the Planck scale yields several robust results:

1) Emergence of the Planck Scale: The Planck length is derived as a consequence of the fundamental velocity quantum ε and the total mass-energy M U . It serves as the scale of velocity-state saturation, defining the critical density ρ c ≈ 0.06 ρ Planck at which the Big Bounce occurs.

2) The Single-Cell Origin: Scaling the present-day observable radius R obs ~ 10 26   m back to the bounce epoch leads to the robust result:

This indicates that the causal connectivity of the universe is established within a single pre-geometric cell, governed by superluminal transport v p ≫ c in ℳ ′ .

3) Statistical Origin of n s : The scalar spectral index is determined by the dynamical growth of the correlation length L ( a ) ~ a 1 + δ and the statistical suppression factor γ . Using the PPH-derived range γ ~ 0.01   -   0.05 and δ ~ 0.1 , we obtain:

aligning precisely with Planck 2018 data without fine-tuning scalar potentials.

4) Thermodynamic Evolution: The initial entropy S initial ~ k B (corresponding to Ω ~ O ( 1 ) ) evolves to the current value S today ~ 10 100 k B . This massive expansion of the accessible phase space provides a first-principles explanation for the arrow of time.

In this work, we have closed the conceptual circle between the microscopic Hypothesis of Primary Particles and the macroscopic structure of the universe. The Planck length ℓ p has been stripped of its status as an irreducible postulate and reformulated as an emergent boundary of information density. It represents the “saturation point” of the pre-geometric velocity ensemble, where the discrete nature of ε manifests as a minimal geometric volume.

The discovery that the observable universe originates from a single Planck-scale comoving cell carries profound implications. It eliminates the need for an exponential expansion phase (inflation) to explain homogeneity, as the necessary connectivity is provided by the superluminal nature of primary particles in the pre-emergent phase. Furthermore, the “single-cell origin” provides a definitive answer to the problem of initial conditions: the universe began in a state of maximal simplicity and minimal entropy ( S ~ k B ), rendering the arrow of time a natural consequence of the subsequent growth of effective degrees of freedom.

By reinterpreting primordial fluctuations as statistical noise of a finite ensemble, rather than vacuum fluctuations of an abstract field, the PPH framework offers a more grounded, kinetic origin for cosmological structure. The agreement between our derived spectral index n s and CMB observations suggests that spacetime geometry is a statistical fixed point of a deeper, underlying transport dynamics.

Ultimately, this framework suggests that the Big Bounce is not merely a geometric event, but a phase transition in information processing, where the saturated states of primary particles reorganize into the Lorentzian spacetime and the causal structure we observe today.

If

then

which remains O ( 1 ) for α ~ 1 .

Thus, the result is robust against order-one uncertainties.

The initial number of microstates is

giving

Today,

This large increase naturally explains the arrow of time.

The expansion corresponds to a rapid growth of accessible degrees of freedom, consistent with a statistical-mechanical interpretation.

In this Appendix we introduce a minimal dynamical model for the evolution of the effective correlation length L ( a ) , which allows for an explicit estimate of the spectral index.

In this Appendix we provide a microscopic interpretation of the parameter γ introduced in the main text, relating it to statistical fluctuations within the PPH framework.