TITLE:
A Cascade-Based Momentum-Energy Transport Equation for the Planck and Primordial Universe
AUTHORS:
Asghar Noormohammadi
KEYWORDS:
Cosmological Expansion, Energy Cascade, Momentum Cascade, Planck Scale, Early Universe, Uncertainty Relations
JOURNAL NAME:
Journal of Modern Physics,
Vol.17 No.9,
September
22,
2026
ABSTRACT: We develop a phenomenological cascade model for the evolution of momentum and energy during the earliest stages of the Universe. The starting point is a momentum transport equation of the form
∂
P
i
∂t
+
H
c
x
j
∂
P
i
∂
x
j
=−
∂E
∂
x
i
, (1)where the second term represents the effect of cosmological expansion. The momentum and energy are assumed to obey cascade scaling relations
P=D
k
−r/w
,
E=C
f
−a/b
. (2)A modified uncertainty construction is introduced through
ΔP Δx=εpλ
,
ΔE Δt=εET
. (3)Under characteristic-scale estimates, these relations lead to
ΔP
Δt
~
ΔE
Δx
, (4)and, for relativistic modes, to the familiar energy-momentum relation
E≃cP
. When the physical wavenumber evolves as
k
phys
∝
R
−1
in an expanding universe, the momentum cascade gives
P∝
R
r/w
. If the characteristic frequency also scales as
f∝
R
−1
, the energy cascade gives
E∝
R
a/b
. Consistency with relativistic propagation then requires the exponent relation
a/b
=r/w
. At the Planck scale, the proposed transport equation becomes naturally dimensionless when normalized by the Planck length, time, energy, and momentum. The resulting framework provides a possible phenomenological connection between cascade dynamics, cosmological expansion, and Planck-scale uncertainty.