Exact Solution of the Cosmic Equation in Non-Equilibrium Dynamics: Cosmic Age, Time Variation of Physical Constants, Observational Difficulty, and a Falsifiable Prediction of Decelerating Expansion
Authors/Creators
Description
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.
1 Finally, we derive a falsifiable prediction: the present cosmic acceleration is
anow ≈ −7.6 × 10−10 m/s
2
,
i.e., the universe is decelerating. This is in direct conflict with the standard interpretation of
Type Ia supernovae, CMB, and BAO observations, which indicate accelerating expansion.
This prediction provides a decisive test of NED.