The physical description of the earliest Universe requires the simultaneous consideration of quantum mechanics, relativistic gravity, extremely high energies, and rapidly evolving characteristic length and time scales. Near the Planck epoch, the characteristic length and time scales are the Planck length and Planck time [1], respectively, and the conventional separation between classical gravitational dynamics and quantum dynamics is expected to become inadequate.

In this work, we investigate a phenomenological approach in which the evolution of momentum and energy is represented through a cascade-like scaling process. The idea is motivated by the appearance of cascades in nonlinear physical systems, where energy or another conserved quantity is transferred between different scales [2].

We introduce a momentum transport equation

This equation should be understood explicitly as a Euclidean phenomenological advection model, introduced to capture the kinematic effect of an expanding background within the cascade framework. It is not intended to represent a fully covariant general-relativistic fluid equation, nor is it derived directly from the Einstein field equations. In particular, the coordinates x i are ordinary spatial coordinates and the term H c x j acts as a prescribed expansion-induced advection velocity. The parameter H c is identified with the physical Hubble parameter H = R ˙ / R [3]. The first term represents local temporal evolution, while the second term describes transport associated with an expanding background.

The equation can be written in terms of a characteristic derivative,

so that

The objective of this paper is to examine the consequences of combining Equation (7) with a cascade representation of energy and momentum and with a proposed modified uncertainty relation.

We assume that energy and momentum can be represented by scale dependent power laws,

and

where C and D are dimensional constants and a , b , r , and w determine the cascade exponents [4]. Here C and D set the dimensional amplitudes (normalizations) of the energy and momentum cascade laws, respectively. Since E denotes an energy density and f has dimensions of inverse time,

Similarly, since P denotes a momentum density and k phys has dimensions of inverse length,

Thus, C and D do not determine the cascade exponents; rather, they provide the dimensional normalization required for the power-law representations of E and P .

For convenience, define

Equations (11) and (12) then become

and

For a wave-like excitation, the wavelength and period are related to wavenumber and frequency by

The frequency may also be expressed in terms of angular frequency,

For a relativistic excitation,

and therefore

Let R ( t ) denote the cosmological scale factor [5]. The physical and comoving wavenumbers [6] are related by

For a fixed comoving wavenumber,

Differentiation with respect to time gives

Introducing the Hubble parameter,

we obtain

Substitution into the momentum cascade relation

gives

Consequently,

Integration gives

or, using β = r / w ,

This relation represents the evolution of the cascade momentum associated with a fixed comoving mode [7].

Suppose the characteristic frequency scales according to

Using the cascade energy relation,

we obtain

Therefore,

or

Equations (32) and (37) provide the scale-factor evolution predicted by the assumed cascade laws.

Consider a power-law scale factor,

The Hubble parameter becomes

The momentum cascade then becomes

Similarly, the energy cascade becomes

For a radiation-dominated background [8],

so

Consequently,

and

These relations should be interpreted as consequences of the assumed cascade scaling and not as a replacement for the Friedmann equations [9].

We now introduce the proposed cascade-based uncertainty relations,

and

Here ε is a dimensionless parameter and λ and T represent characteristic wavelength and period. Equations (49) and (50) serve only as scale-dependent heuristic ansatzes rather than formal expressions derived from canonical field commutators.

Using

the relations become

and

If the characteristic spatial and temporal uncertainties are of the order of one wavelength and one period,

then

Thus,

and

The proposed characteristic balance

therefore requires

For a relativistic mode satisfying

Equation (59) gives

Thus, the characteristic form of the proposed uncertainty relations is compatible with the relativistic energy-momentum relation [10].

We can now compare the cascade expressions for energy and momentum.

From

and

the relativistic relation E ≃ c P requires

Since

for relativistic modes, we obtain

However, if the characteristic balance is written directly as

then

Using 2 π   f = c k gives again

If, rather than relativistic behavior, the cascade spectra are interpreted via a non-relativistic dispersion process, Equation (65) can take the general cascade coupling mechanism and transits into a linkage of f ∝ k a / ( a + b ) . The result is

Equation (70) is therefore a candidate constraint connecting the energy and momentum cascade exponents. Whether this relation survives a fundamental derivation requires specifying the underlying cascade dynamics.

The proposed transport equation is

The constant H c introduced in the present transport model should be distinguished from the physical, time-dependent Hubble parameter ( H ( t ) ) . By contrast, H c is a constant characteristic expansion rate introduced phenomenologically in the advection operator ( H c x j ∂ / ∂ x j ). It therefore represents a local or characteristic expansion rate associated with the proposed transport process, rather than the instantaneous cosmological expansion rate. The use of a constant H c allows the characteristic-scale analysis to isolate the momentum-transfer mechanism without requiring a specific cosmological evolution for R ( t ) . Consequently, H c should not be identified with H ( t ) unless an additional physical model is introduced that establishes such a correspondence. For an approximately isotropic configuration, we may consider a radial characteristic scale x and write

The equation becomes

At the characteristic scale

we estimate

Hence,

For relativistic modes, E ≃ c P , giving

Since

we have

Using

the characteristic equation becomes

This suggests a characteristic cascade timescale

Thus, within this phenomenological picture, the characteristic cascade transfer time is of the same order as the wave period.

The Planck quantities are

and

These quantities satisfy

and

The proposed uncertainty relations therefore become, at the Planck scale,

and

Consequently,

This result shows that the proposed relations naturally recover a Planck-scale quantity with dimensions of action. It does not, however, by itself establish that the standard Planck constant is fundamentally replaced.

For a radiation-dominated universe,

and hence

At the Planck epoch,

so

Consider a characteristic Planck-scale spatial variation,

The expansion term is then

Using

we find

Therefore,

Thus the cosmological expansion contribution and the energy-gradient contribution are of comparable order at the Planck scale.

This provides a dimensional motivation for investigating an equation in which expansion transport and energy gradients appear at the same characteristic order.

For clarity, the P P and E P used in this section are characteristic normalization scales for the local momentum and energy fields. They are not the single-particle Planck momentum and energy introduced in the preceding section. A strict density normalization would require corresponding Planck momentum-density and energy-density scales; here we retain the original normalization convention E P = P P l P / t P solely to obtain the dimensionless characteristic equation.

Introduce Planck-normalized variables

Since

the governing equation becomes

where

At the Planck epoch,

so the dimensionless equation is approximately

Equation (107) provides a convenient starting point for investigating the proposed dynamics near the Planck scale.

The proposed model can be interpreted as a coupling between three processes.

First, the term

describes local temporal evolution of the momentum field.

Second, the expansion term

describes the change associated with the stretching of physical length scales by cosmological expansion.

Third, the term

represents a momentum change produced by a spatial energy gradient. As such, the resulting equation has the schematic structure [11]. The cascade hypothesis adds a scale-dependent structure to E and P . Consequently, the model connects cosmological expansion with the transfer of momentum and energy between characteristic scales.

The results obtained above suggest several potentially important features of the cascade model.

The first is that an expanding universe naturally produces a time-dependent physical wavenumber,

When this relation is inserted into the proposed momentum cascade, the momentum evolves according to

The second feature is that a frequency cascade of the form

leads to

The third feature is the connection between the two modified uncertainty relations. When their characteristic spatial and temporal scales are identified with λ and T , respectively, one obtains

For relativistic modes this gives

The Planck-scale normalization is also suggestive. At the Planck scale,

and

Therefore the proposed uncertainty relations reproduce an action scale of order ℏ when evaluated at Planck scales.

However, this observation should not be interpreted as a proof that ℏ is emergent or that the conventional uncertainty principle is invalid. Such a conclusion would require a fundamental derivation from a microscopic theory.

Similarly, the replacement of the conventional Friedmann dynamics by Equation (5) is not justified by dimensional analysis alone. A complete theory would need to specify the underlying action, degrees of freedom, conservation laws, and coupling to the spacetime metric.

The present formulation is phenomenological. Several additional steps are required before it can be considered a fundamental model of the early Universe.

First, the equation

should be generalized to a covariant formulation in curved spacetime.

Second, the identification H c = H is a modeling convention for the homogeneous FLRW background and should be generalized or derived in a covariant formulation.

Third, the cascade exponents a / b and r / w need to be obtained from an underlying nonlinear interaction mechanism.

Fourth, the energy and momentum cascade should be formulated in terms of a well-defined spectral transfer function. This would allow the model to distinguish between a genuine transfer of energy between scales and simple redshifting caused by cosmological expansion.

Fifth, if the model is intended to describe the Planck epoch, it must ultimately be compatible with quantum gravity. In particular, the stress-energy tensor and the geometry of spacetime cannot generally be treated as independent classical quantities at the Planck scale.

Finally, the proposed uncertainty relations must be tested against the standard Heisenberg relations in regimes where conventional quantum mechanics is experimentally established.

A phenomenological cascade formulation for momentum and energy in an expanding early Universe has been developed. The central equation is

The cascade assumptions

combined with

lead to

If the characteristic frequency scales as

then

The proposed uncertainty relations,

lead, at the characteristic scale, to

For relativistic modes this is compatible with

At the Planck scale,

so that the proposed uncertainty products naturally acquire the Planck action scale.

The resulting framework suggests a possible connection between cosmological expansion, scale-dependent cascade dynamics, and Planck-scale uncertainty. Nevertheless, the present model remains phenomenological. Establishing it as a fundamental theory requires a covariant derivation, an underlying microscopic cascade mechanism, and consistency with general relativity and quantum mechanics.

Conceptualization; methodology; investigation; writing original draft preparation: Asghar Noormohammadi.