Unified Description of Inflation, Radiation Era, and Cosmic Age in Non-Equilibrium Dynamics: F0 = 7Mpc 2/(2ℓp), R ∝ t 2/3 , and the Mnow–Rnow Relation
Abstract
We present a unified description of the inflation era, radiation era, and cosmic age in
Non-Equilibrium Dynamics (NED) using the exact repulsive force
F0 =
7
2
Mpc
2
ℓp
,
which is derived from the balance functional H with v1 = v2 = 0, y = z = ℓp/2, r = R,
M1 = M2 = Mp, and D = 1. The equation of motion is
(R(R − ℓp)R˙)
′ = F0.
For R ≫ ℓp, this reduces to
R(t) ≈
(
21Mpc
2
4ℓp
)1/3
t
2/3 = 8.60 × 1014 t
2/3 m.
Using this single power law, we obtain:
• At the end of inflation (t ∼ 10−32 s): R ≈ 4.0 × 10−7 m, giving R/(2ℓp) ≈ 1.2 × 1028
,
which is about 123 times the standard inflation factor e
60 ≈ 1026
.
• At the end of the radiation era (t ∼ 5 × 104 years): R ≈ 1.2 × 1023 m, comparable to
the standard value.
• The cosmic age is
tnow =
√
4ℓpMnowRnow
21Mpc
2
,
which gives the relation
MnowRnow ≈ 1.21 × 1080 kg · m
for tnow = 1.38 × 1010 years.
We present a table of Mnow and Rnow values that reproduce the observed cosmic age. The
results show that NED can reproduce the inflation era, radiation era, and cosmic age with
a single power law R ∝ t
2/3
, without invoking R ∝ t
1/2 or exponential expansion.
Related
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BibTeX
@misc{ueda2026unified,
title = {{Unified Description of Inflation, Radiation Era, and Cosmic Age in Non-Equilibrium Dynamics: F0 = 7Mpc 2/(2ℓp), R ∝ t 2/3 , and the Mnow–Rnow Relation}},
author = {takashi ueda},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23565911},
url = {https://zenodo.org/records/23565911},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Unified Description of Inflation, Radiation Era, and Cosmic Age in Non-Equilibrium Dynamics: F0 = 7Mpc 2/(2ℓp), R ∝ t 2/3 , and the Mnow–Rnow Relation
AU - takashi ueda
PY - 2026
DA - 2026/10/22
PB - Zenodo
DO - 10.5281/zenodo.23565911
UR - https://zenodo.org/records/23565911
AB - We present a unified description of the inflation era, radiation era, and cosmic age in Non-Equilibrium Dynamics (NED) using the exact repulsive force F0 = 7 2 Mpc 2 ℓp , which is derived from the balance functional H with v1 = v2 = 0, y = z = ℓp/2, r = R, M1 = M2 = Mp, and D = 1. The equation of motion is (R(R − ℓp)R˙) ′ = F0. For R ≫ ℓp, this reduces to R(t) ≈ ( 21Mpc 2 4ℓp )1/3 t 2/3 = 8.60 × 1014 t 2/3 m. Using this single power law, we obtain: • At the end of inflation (t ∼ 10−32 s): R ≈ 4.0 × 10−7 m, giving R/(2ℓp) ≈ 1.2 × 1028 , which is about 123 times the standard inflation factor e 60 ≈ 1026 . • At the end of the radiation era (t ∼ 5 × 104 years): R ≈ 1.2 × 1023 m, comparable to the standard value. • The cosmic age is tnow = √ 4ℓpMnowRnow 21Mpc 2 , which gives the relation MnowRnow ≈ 1.21 × 1080 kg · m for tnow = 1.38 × 1010 years. We present a table of Mnow and Rnow values that reproduce the observed cosmic age. The results show that NED can reproduce the inflation era, radiation era, and cosmic age with a single power law R ∝ t 2/3 , without invoking R ∝ t 1/2 or exponential expansion.
ER -