In this work, we develop a statistical-mechanical realization of the Primary Particle Hypothesis (PPH), in which spacetime, Lorentz invariance, and primordial cosmological perturbations emerge from the collective dynamics of a pre-geometric ensemble of primary particles. Understanding whether spacetime is fundamental or emergent remains one of the central problems of modern theoretical physics. General Relativity provides a highly successful macroscopic description of gravitational dynamics, yet its singular behavior at early times and its incompatibility with quantum theory strongly suggest that spacetime geometry may arise from a deeper microscopic structure.

Several approaches to quantum gravity, including Loop Quantum Cosmology (LQC) [1][2], replace the initial singularity with a non-singular bounce. However, the microscopic origin of spacetime, the emergence of Lorentz invariance, and the physical meaning of the critical density remain open questions.

In this work, we develop an extended mathematical formulation of the Primary Particle Hypothesis, in which spacetime and its causal structure emerge from a pre-geometric manifold ℳ ′ populated by primary particles with a fundamental invariant velocity scale v p > c .

The key idea of the framework is that Lorentz invariance is not fundamental, but arises as a universal statistical property of coarse-grained microscopic dynamics. Using transport theory and the central limit theorem, we show that the Minkowski metric and the invariant speed c emerge as collective properties determined solely by the second moment of the underlying velocity distribution.

This perspective leads to a hierarchy of causal structures: a fundamental causal cone governed by v p and an emergent Lorentz cone governed by c . Superluminal microscopic dynamics is therefore compatible with macroscopic relativistic causality.

A central feature of the model is the introduction of a discrete velocity spectrum, which provides a direct microphysical origin for cosmological observables. In particular:

The critical density of Loop Quantum Cosmology is derived from velocity-state saturation,Scalar perturbations arise from finite-number statistical fluctuations, yielding δ ρ / ρ ~ 10 − 5 ,A nearly scale-invariant power spectrum with a small red tilt emerges without inflation,Tensor modes originate from quadrupole anisotropies of the velocity ensemble.

The framework therefore provides a statistically grounded alternative to inflation in which cosmological structure is a direct consequence of microscopic transport dynamics.

Importantly, the model leads to clear observational signatures, including a violation of the inflationary consistency relation and a nearly scale-invariant tensor spectrum. These predictions make the framework directly testable with upcoming CMB observations.

Finally, time itself emerges as a relational quantity associated with collective dynamics, reinforcing the pre-geometric nature of the theory.

The goal of this work is to establish a mathematically consistent bridge between microscopic velocity dynamics and macroscopic spacetime structure, thereby providing a unified statistical foundation for relativity and cosmology.

We postulate a pre-spacetime manifold ℳ ′ endowed with coordinates ( t ′ , r ′ ) and characteristic velocity scale v p > c .

The invariant interval in pre-spacetime is defined as

It is important to emphasize that Equation (1) should not be interpreted as a physical spacetime metric, but rather as a kinematic quadratic form defining the causal ordering in the pre-geometric manifold ℳ ′ . The Lorentzian metric of emergent spacetime arises only after statistical coarse-graining, as shown in Section 2.2.

Primary particles follow trajectories

The distribution function f ( r ′ , v ′ , t ′ ) satisfies the collisionless transport equation:

These particles are not bound by the Lorentz invariant, because we introduced a new transformation factor (Section 3).

Fundamental dynamical equation

The fundamental dynamical object of the PPH framework is the distribution function f ( r ′ , v ′ , t ′ ) , whose evolution governs both microscopic transport and macroscopic emergence.

While Equation (3) describes collisionless dynamics, the discrete structure of the velocity spectrum naturally induces stochastic transitions between neighboring velocity states (see AppendixA). This leads to an effective coarse-grained evolution equation of Fokker-Planck type:

Equation (4) constitutes the fundamental equation of the PPH framework. All emergent structures, including spacetime geometry, causal structure, and cosmological perturbations, arise from its statistical solutions.

In this sense, spacetime is not a fundamental entity but an effective description of the collective dynamics encoded in f .

Main result: The motion of primary particles generates an emergent space-time with effective velocities with an upper limit velocity c .

Implication: Standard particles and physical laws arise from the collective behavior of primary particles.

Between inertial frames S and S ′ in ℳ ′ moving with velocity v , we define the transformation:

where

and we obtained this transformation similarly to Equation (14), Ref. [3].

For v p → c , standard Lorentz transformations are recovered.

Main result: These transformations remain consistent for velocities higher than c , thus extending standard relativity.

We define emergent spacetime coordinates as coarse-grained averages:

where averaging is performed over the distribution f .

Effective limiting speed in emergent spacetime:

The emergent metric tensor is obtained via statistical averaging:

Fast equilibration in pre-spacetime occurs over scale

with v p ≫ c , ensuring causal contact before emergence.

The resulting density contrast satisfies

consistent with observed CMB anisotropies (see Equation (27), AppendixA and Equation (29)).

Main result: This rapid dynamics creates a quasi-isotropic distribution that, upon emergence into our spacetime, appears as a homogeneous CMB.

In the PPH framework, both scalar and tensor perturbations arise from statistical fluctuations of the discrete velocity distribution of primary particles. Scalar modes originate from density fluctuations, while tensor modes arise from fluctuations of the traceless anisotropic stress tensor.

Scalar fluctuations

The energy density is given by Equation (25)

For a coarse-grained region containing N eff statistically independent microscopic cells, the central limit theorem implies Equation (27),

Near the bounce, taking

is obtained as in Equation (30)

in agreement with the observed amplitude of CMB anisotropies.

Tensor fluctuations from multipole statistics

Tensor modes are sourced by the traceless part of the stress tensor

The tensor component corresponds to the quadrupole ( ℓ = 2 ) moment of the angular distribution of velocities.

Let the angular distribution be expanded in spherical harmonics:

For an almost isotropic distribution, the multipole coefficients satisfy

where N ang is the effective number of independent angular cells contributing to isotropization.

Since tensor modes are sourced by the quadrupole,

(40)

PPH prediction for the tensor-to-scalar ratio

An important feature of the PPH framework is the existence of a consistency relation between scalar and tensor perturbations, analogous in spirit to the inflationary consistency relation r = − 8 n t .

In the PPH model, scalar and tensor perturbations originate from different statistical aspects of the same microscopic system. Scalar fluctuations arise from volumetric density fluctuations of primary particles, while tensor modes originate from the quadrupole anisotropy of the velocity distribution.

The scalar power scales as

(41)

where N eff is the effective number of independent microscopic cells contributing to a coarse-grained region.

Tensor modes arise from angular multipole fluctuations, leading to

(42)

where N ang denotes the number of independent angular cells participating in the isotropization of the velocity distribution.

The tensor-to-scalar ratio therefore satisfies

(43)

This relation constitutes a characteristic prediction of the PPH framework.

Efficient angular mixing implies N ang ≫ N eff , which naturally suppresses tensor modes.

Using the representative estimates

which should be understood as an order of magnitude characterization rather than a precise prediction, we obtain

with stronger isotropization yielding

This result provides a direct microphysical interpretation of tensor suppression in terms of the statistical structure of the primary particle ensemble.

Key predictions

The statistical structure of the PPH model therefore predicts:

Scalar amplitude δ ρ / ρ ~ 10 − 5 ,Nearly scale-invariant scalar and tensor spectra,Tensor-to-scalar ratio r suppressed by angular isotropization,Absence of the slow-roll consistency relation.

The suppression of r is controlled by the ratio between volumetric and angular statistical degrees of freedom, providing a clear physical interpretation and a falsifiable prediction.

Consistency with LQC bounce dynamics

In effective Loop Quantum Cosmology, the modified Friedmann equation reads

Within the PPH framework, the critical density is derived from the quantized velocity scale:

When we replace ( M U m P ) 2 with c 2 ε from Equation (24) we obtain:

The agreement with the standard LQC value should be interpreted at the level of scaling and order of magnitude, rather than as an exact numerical derivation.

The bounce occurs when ρ = ρ c , corresponding physically to saturation of accessible velocity states:

Hence, the LQC bounce condition acquires a microscopic interpretation: it corresponds to the maximal occupation of discrete velocity levels. The non-singular evolution is therefore not merely a geometric artifact of quantum gravity corrections, but a statistical consequence of the finite velocity spectrum of primary particles.

This establishes a closed bridge between PPH microphysics and effective LQC dynamics, linking:

quantized velocity levels ( ε ),statistical density fluctuations ( δ ρ / ρ ),and the critical bounce density ( ρ c ).

Comment: The critical density ρ c at which the Big Bang occurs is usually treated as a free parameter in the effective LQC. However, in the PPH framework, we can derive ρ c by considering the interaction between the fundamental velocity quantum ε and the global mass-energy content of the universe M U . Substituting Equation (24) into the modified Friedmann dynamics, we find that the big bang is triggered exactly when the energy density reaches a threshold defined by the saturation of the velocity state of the primary particles ρ = ρ c . This leads to a non-singular evolution where the Planck scale density is restored as a natural consequence of the velocity quantization.

Main result: The model predicts an escape from the singularity (Big Bang → Big Bounce).

Implication: We can compare current observational constraints from the Planck 2018 full-mission analysis and combined Planck + BICEP/Keck ([6] and [7]) limits on primordial tensors with representative predictions of the PPH framework developed in this work. The PPH predictions are not single-point fits but representative ranges derived from the microscopic parameters ( ε , N eff , N ang and the equilibration scale ℓ e q ). Importantly, the scalar amplitude and tilt produced by finite-number statistical fluctuations and transport-induced equilibration lie naturally in the Planck-preferred region, while the tensor-to-scalar ratio is generically suppressed (typical r ~ 10 − 3   -   10 − 2 ) and therefore consistent with present upper limits. Therefore, these results motivate a full numerical implementation and MCMC confrontation described in AppendixB.

A crucial question for any alternative to inflation is whether it makes observational predictions that distinguish it from slow-roll scenarios.

Absence of the inflationary consistency relation

In standard single-field slow-roll inflation the tensor-to-scalar ratio and the tensor spectral index satisfy the well-known consistency relation

which follows directly from the slow-roll relations r = 16 ϵ and n t = − 2 ϵ [8]-[10].

In the PPH framework, scalar and tensor modes originate from distinct statistical aspects of the same pre-geometric system:

Scalar modes arise from density fluctuations, Tensor modes arise from anisotropic stress fluctuations.

As a result, we obtain

with no fundamental relation between r and n t .

Therefore, the inflationary consistency relation is generically violated in the PPH scenario. Also, the PPH expectation is that the frequency (temperature) of a hyperenergetic spherical wave of pre-geometric origin, from the very initial moment of exiting the singularity (Big Bang → Big Bounce), decreases sharply until the epoch of nucleogenesis as in Ref. [11].

Spectral structure

The scalar spectral index is given by

where α originates from finite equilibration effects in quantized velocity space, and an additional, cosmological estimate of the spectral tilt is shown in AppendixC.

The tensor spectrum remains approximately scale-invariant at leading order:

This separation of scalar and tensor tilt provides a clear observational discriminator.

Non-Gaussianity scaling

Since fluctuations arise from finite-number statistical effects, residual non-Gaussianity scales as

differing structurally from the slow-roll expectation.

Summary

The PPH framework predicts:

Violation of the inflationary consistency relation, Suppressed tensor-to-scalar ratio, Nearly scale-invariant tensor spectrum, Scalar tilt determined by transport dynamics, Distinct statistical origin of non-Gaussianity.

Future measurements of n t and r , such as [12] and [13], therefore provide a direct test of the model.

Primary falsifiable prediction

A central, model-independent prediction of the PPH framework is the following:

Prediction: A future detection of a tensor-to-scalar ratio r ≳ 10 − 3 together with a nearly scale-invariant tensor spectrum ( n t ≈ 0 ) would violate the inflationary consistency relation r = − 8 n t and provide strong evidence in favor of the PPH framework.

Conversely, a confirmed detection of a red tensor tilt satisfying r = − 8 n t would falsify the present statistical origin of tensor modes.

This establishes a direct and decisive observational test of the model.

Time is defined relationally:

Thus time arises from collective dynamics rather than being fundamental.

Main result: Time in our space results from the average dynamics of primary particles in prespace.

We have developed an extended mathematical formalism of the Primary Particle Hypothesis in which Lorentzian spacetime, relativistic causality, and cosmological perturbations emerge from a superluminal pre-geometric manifold governed by transport dynamics. Special relativity appears as a universal statistical limit, rather than a fundamental postulate.

The discrete velocity spectrum provides a microscopic derivation of the LQC critical density and predicts a non-singular Big Bounce. Scalar and tensor perturbations arise from finite-number statistical fluctuations, yielding Planck-compatible amplitude and tilt while generically violating inflationary consistency relations.

Time itself emerges relationally from collective dynamics in ℳ ′ . The resulting framework constitutes a microphysically grounded cosmological model with distinctive observational signatures, offering a testable alternative to inflationary paradigms. We expect that recent analyses of the cosmological signatures of the LiteBIRD and CMB-S4 missions will provide a clear discrimination between the PPH framework and inflationary models.

While several elements of the framework remain at the level of leading-order effective description, the internal consistency of the model and its clear observational signatures provide a well-defined basis for further quantitative development.

Because the velocity spectrum is discrete, statistical redistribution among neighboring velocity levels produces an effective diffusion in velocity space, leading to a Fokker-Planck type evolution of the distribution function [14]-[16].

The statistical fluctuations responsible for primordial density perturbations can be derived directly from the transport equation governing the distribution of primary particles.

Let the distribution function be decomposed as

where f 0 is the isotropic equilibrium distribution and δ f represents small fluctuations.

Substituting into the transport Equation (3)

and linearizing yields

From discrete velocity transitions to Fokker-Planck dynamics

The effective diffusion term appearing in the transport equation can be justified from the discrete structure of the velocity spectrum.

In the PPH framework, the velocity states are quantized as in Equation (23)

so that the distribution function evolves through transitions between neighboring velocity levels. The statistical evolution of the distribution can therefore be written in the form of a master equation

where W Δ n denotes the transition probability between adjacent velocity states.

Expanding the right-hand side in powers of Δ v leads to the Kramers-Moyal expansion

with coefficients determined by the transition moments of the underlying stochastic process.

Because the velocity quantum ε is extremely small, higher-order terms are strongly suppressed and the expansion can be truncated at second order. The evolution equation therefore reduces to the Fokker-Planck form

This result provides a microscopic justification for the diffusive term used in the main text and shows that the statistical fluctuations of the primary particle ensemble naturally obey a Fokker-Planck type dynamics (i.e. Equation (4)).

Also: Because the velocity spectrum is discrete, statistical redistribution among nearby velocity levels leads to an effective diffusion in velocity space. The transport equation therefore acquires a Fokker-Planck form, i.e. Equation (4)

For the perturbations this becomes

Fourier transforming in space leads to

The solution of this equation shows that the variance of coarse-grained density fluctuations scales as

which directly leads to

and that is the same as (27)

Thus the amplitude of primordial density perturbations emerges as a statistical consequence of the transport dynamics of the primary particle ensemble.

In the main text, approximate scale invariance was obtained under the assumption of nearly uniform occupation numbers f n for the discrete velocity levels. We now relax this assumption and allow for a mild scale dependence in the form

Such a deviation may arise from incomplete equilibration in the pre-emergent phase or from residual dynamical bias in the velocity distribution.

Modified variance

The mean squared velocity becomes

Approximating the sum by an integral up to n max , we obtain

The effective number of states contributing to a fluctuation mode of comoving wavenumber k scales as

Hence,

Power spectrum

The density contrast variance scales inversely with the effective number of independent cells,

The dimensionless power spectrum is

(73)

Substituting the above scaling yields

(74)

Therefore, the spectral index is

Magnitude of the tilt

For small α , the deviation from exact scale invariance is mild. Matching observational data,

implies

Thus, a slight suppression of higher velocity levels in the occupation distribution naturally produces a red spectral tilt consistent with current cosmological observations.

This result demonstrates that the PPH framework not only reproduces approximate scale invariance, but also accommodates a small spectral tilt through minimal deviations from uniformity in the discrete velocity distribution.

Transport-induced deviation from uniformity

The distribution function of primary particles satisfies the collisionless transport equation (3)

In a finite causal domain of size L , equilibration occurs over

Modes with higher discrete velocity v ′ n = n ε have shorter crossing time

Incomplete equilibration leads to a mild suppression of large- n occupation numbers, yielding

Relation to density power spectrum

As given in Equation (53),

We now estimate α from cosmological scales.

Equilibration length and horizon scale

Let the equilibration length in the pre-emergent phase be

Near the bounce, the characteristic geometric scale is Planckian:

A finite ratio between the horizon size and equilibration length generates residual scale dependence:

Numerical estimate

If equilibration occurs over a scale moderately larger than the Planck length, for example

then

This corresponds to

in excellent agreement with current CMB observations.

Interpretation

The small red tilt therefore arises from incomplete but nearly efficient equilibration in quantized velocity space. It is controlled by the ratio of the microscopic equilibration scale to the geometric horizon scale at the bounce.

No fine-tuning of parameters or scalar potentials is required. The observed deviation from exact scale invariance is a direct statistical imprint of the finite transport dynamics in the pre-geometric phase.

Statistical Origin of the Running of the Spectral Index

A further refinement of the PPH framework involves the scale dependence of the spectral tilt, known as the running of the spectral index, α s = d n s / d ln k . In our model, this parameter is not an independent constant but a direct probe of the “texture” of the primary particle distribution f n . If we relax the assumption of a strict power-law and allow the occupation exponent α to vary logarithmically with the velocity level n , we can write:

where β represents a second-order correction to the equilibration process in ℳ ′ . Given the inverse relationship between the microscopic state index and the

cosmological wavenumber, n ~ ( k ℓ p ) − 1 , the running is derived as:

Current observations from the Planck mission constrain α s ≈ − 0.006 ± 0.007. In the PPH context, the smallness of this value is naturally explained by the vast range of available velocity states ( n max ~ 10 122 ), which suppresses higher-order logarithmic deviations. A non-zero α s would therefore provide a direct measurement of the non-ideal statistical equilibration of the primary particle gas during the pre-emergent phase, offering a concrete observational test that distinguishes PPH from standard slow-roll inflation.

Discussion

The extended PPH framework presents a coherent extrapolation of special relativity rather than its breakdown. The invariant structure of the pre-spacetime manifold is governed by v p , while the Lorentz cone with limiting speed c emerges statistically as a coarse-grained fixed point. This hierarchical causal structure ensures global hyperbolicity at the fundamental level and preserves relativistic causality in the emergent phase. Superluminality is therefore not paradoxical but reflects the deeper invariant structure of ℳ ′ .

A central achievement of the model is the derivation of cosmological signatures directly from microphysics. The amplitude of scalar perturbations follows from central-limit scaling of discrete velocity occupation numbers, producing δ ρ / ρ ~ 10 − 5 without invoking vacuum fluctuations of a scalar field. The small red spectral tilt arises from finite equilibration effects governed by transport dynamics, yielding values consistent with Planck measurements. Tensor modes originate from quadrupole anisotropies of the velocity distribution and predict a suppressed tensor-to-scalar ratio, generically violating the slow-roll consistency relation.

The quantized velocity scale ε establishes a direct bridge between microscopic structure and the effective LQC Friedmann equation, providing a physical interpretation of the critical density as velocity-state saturation. In this sense, the Big Bounce acquires a statistical-mechanical origin.

Finally, the relational emergence of time highlights the pre-geometric nature of the framework: temporal ordering arises from collective dynamics rather than being fundamental. This statistical-mechanical perspective on spacetime suggests a new route toward quantum gravity in which Lorentz symmetry and cosmological structure are emergent phenomena. In this picture the Lorentz group emerges as the universal symmetry of the Gaussian fixed point of the underlying velocity ensemble.

We present a concise, comparative overview of the most important features that distinguish the standard inflationary scenario from our PPH framework in the Table A1.

Table A1. Comparison of cosmological paradigms: inflationary scenario vs. Primary Particle Hypothesis (PPH).

The exact expectations for future analysis of cosmological signatures are given in Appendix D.

The Primary Particle Hypothesis (PPH) provides a distinct set of signatures in the Cosmic Microwave Background (CMB) B-mode polarization that allow for a direct, model-independent test against the standard inflationary paradigm. With the upcoming LiteBIRD mission (expected launch ~2030), the precision in measuring the tensor-to-scalar ratio r will reach δ r < 0.001 , providing the ideal laboratory for testing the PPH prediction for the tensor-to-scalar ratio as in Section 6.