Cyclic Cosmic Expansion in Non-Equilibrium Dynamics: Identification of the Initial Velocity v0 as the Contraction Velocity near R = 2ℓp, Past Contraction, Future Turning Points, Cosmic Age, and Repeated Cycles

We identify the initial velocity v0 in the exact solution of Non-Equilibrium Dynamics

(NED) as the contraction velocity near R = 2ℓp, and we propose the concrete value

v0 = −2.5 × 10130 m/s .

This identification implies that the universe was contracting in the past. Using the exact

solution

2R

3 − 3ℓpR

2 = 3Kt2 + 12ℓ

2

p

v0 t + 4ℓ

3

p

,

we derive the time at which the expansion velocity vanishes,

tR˙ =0 = −

2ℓ

2

p

v0

K

≈ 9.7 × 109

years,

the corresponding radius,

RR˙ =0 ≈ 1.69 × 1026 m,

and the time required for the universe to return to R = 2ℓp,

t

future

bounce = −

4ℓ

2

p

v0

K

≈ 1.94 × 1010 years.

We also compute the cosmic age with v0 included:

tnow =

−2ℓ

2

p

v0 +

√

4ℓ

4

p

v

2

0 −

Kc

3

K

,

where c = 4ℓ

3

p − 2R3

now + 3ℓpR2

now. For v0 = −2.5 × 10130 m/s, we obtain

tnow ≈ 2.14 × 1010 years ,

which is closer to the observed value 1.38 × 1010 years than the v0 = 0 result 6.4 × 109

years. With constant v0, the equation admits only two times at which R = 2ℓp: t = 0

and t = −4ℓ

2

p

v0/K. This means that the universe undergoes at most two bounces and does

not repeat periodically. However, if v0 (or F0) varies in time, the universe can undergo

repeated expansion–contraction cycles. Since the speed of light does not limit the expansion

or contraction of space in NED, the large value of v0 is not physically excluded.