Exact Solution of the Cosmic Equation in Non-Equilibrium Dynamics: Cosmic Age, Time Variation of Physical Constants, and the Observational Difficulty
Abstract
We formulate the cosmic equation of motion in the Non-Equilibrium Dynamics (NED)
framework, including the time variation of mass, and derive its exact analytic solution.
Using the exact mass scaling M(R) = (Mnow/R2
now)R(R − ℓp) with R ≥ 2ℓp, the equation
of motion M′v + M a = F0 reduces to the total derivative form
d
dt
[
R(R − ℓp)R˙
]
= K, K =
F0R2
now
Mnow
.
Integrating twice yields the implicit solution
2R
3 − 3ℓpR
2 = 3Kt2 + 12ℓ
2
p
v0 t + 4ℓ
3
p
.
The explicit solution is obtained via Cardano’s formula:
R(t) = ℓp
2
+
3
√
A +
√
∆ + 3
√
A −
√
∆,
A =
3K
4
t
2 + 3ℓ
2
p
v0 t +
9ℓ
3
p
8
, ∆ = A
2 −
ℓ
6
p
64
.
This exact solution describes, without approximation, the entire cosmic history from the
Planck scale (R = 2ℓp) through a radiation-like era (R ∝ t
1/2
), a matter-dominated era
(R ∝ t
2/3
), and the eternal future (v → 0, a → 0). The cosmic age is given by
tnow =
−2ℓ
2
p
v0 +
√
4ℓ
4
p
v
2
0 −
Kc
3
K
,
which is essentially independent of the initial velocity v0. For Rnow = 4.4 × 1026 m, we
obtain tnow ≈ 1.39 × 1010 years, in agreement with observation. Introducing the time
variable t
′ = tnow + t, all physical quantities are expressed in the unified form
X(t) = Xnow (
tnow + t
tnow )n
,
where n is a constant specific to each quantity. This representation naturally explains why
the time variation of physical constants is not observed: the rate of change near the present is
extremely small (1/tnow ∼ 10−18 s
−1
), and moreover the measuring instruments themselves
scale with the same time variable, so that the variation cancels out in any measurement.
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BibTeX
@misc{ueda2026exact,
title = {{Exact Solution of the Cosmic Equation in Non-Equilibrium Dynamics: Cosmic Age, Time Variation of Physical Constants, and the Observational Difficulty}},
author = {takashi ueda},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23422841},
url = {https://zenodo.org/records/23422841},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Exact Solution of the Cosmic Equation in Non-Equilibrium Dynamics: Cosmic Age, Time Variation of Physical Constants, and the Observational Difficulty
AU - takashi ueda
PY - 2026
DA - 2026/10/16
PB - Zenodo
DO - 10.5281/zenodo.23422841
UR - https://zenodo.org/records/23422841
AB - We formulate the cosmic equation of motion in the Non-Equilibrium Dynamics (NED) framework, including the time variation of mass, and derive its exact analytic solution. Using the exact mass scaling M(R) = (Mnow/R2 now)R(R − ℓp) with R ≥ 2ℓp, the equation of motion M′v + M a = F0 reduces to the total derivative form d dt [ R(R − ℓp)R˙ ] = K, K = F0R2 now Mnow . Integrating twice yields the implicit solution 2R 3 − 3ℓpR 2 = 3Kt2 + 12ℓ 2 p v0 t + 4ℓ 3 p . The explicit solution is obtained via Cardano’s formula: R(t) = ℓp 2 + 3 √ A + √ ∆ + 3 √ A − √ ∆, A = 3K 4 t 2 + 3ℓ 2 p v0 t + 9ℓ 3 p 8 , ∆ = A 2 − ℓ 6 p 64 . This exact solution describes, without approximation, the entire cosmic history from the Planck scale (R = 2ℓp) through a radiation-like era (R ∝ t 1/2 ), a matter-dominated era (R ∝ t 2/3 ), and the eternal future (v → 0, a → 0). The cosmic age is given by tnow = −2ℓ 2 p v0 + √ 4ℓ 4 p v 2 0 − Kc 3 K , which is essentially independent of the initial velocity v0. For Rnow = 4.4 × 1026 m, we obtain tnow ≈ 1.39 × 1010 years, in agreement with observation. Introducing the time variable t ′ = tnow + t, all physical quantities are expressed in the unified form X(t) = Xnow ( tnow + t tnow )n , where n is a constant specific to each quantity. This representation naturally explains why the time variation of physical constants is not observed: the rate of change near the present is extremely small (1/tnow ∼ 10−18 s −1 ), and moreover the measuring instruments themselves scale with the same time variable, so that the variation cancels out in any measurement.
ER -