01 — The Proslambenomenos Derivation#
From \(\Lambda\) to \(a_0\) in three multiplications.
This notebook verifies the numerical chain:
\[\Lambda \;\xrightarrow{c\sqrt{\cdot/3}}\; \nu_\Lambda \approx H_0 \;\xrightarrow{c/2\pi}\; a_0\]
using only measured values and no free parameters. Stdlib Python only.
import math
# Measured constants
c = 2.998e8 # m/s, speed of light
H0_km = 70.0 # km/s/Mpc, Hubble parameter (Planck 2018 + SH0ES midpoint)
Mpc = 3.086e22 # m per Mpc
H0 = H0_km * 1e3 / Mpc # convert to s^-1
# Cosmological constant from Planck 2018
# Omega_Lambda ≈ 0.685, H0 ≈ 67.4 km/s/Mpc => Lambda = 3 * Omega_Lambda * H0^2 / c^2
# but we use the direct measurement:
Lambda = 1.1056e-52 # m^-2
# MOND acceleration (McGaugh 2016)
a0_obs = 1.2e-10 # m/s^2
print(f"H0 = {H0:.3e} s^-1")
print(f"Lambda = {Lambda:.3e} m^-2")
print(f"a0 (observed) = {a0_obs:.3e} m/s^2")
H0 = 2.268e-18 s^-1
Lambda = 1.106e-52 m^-2
a0 (observed) = 1.200e-10 m/s^2
Step 1: \(\Lambda \to \nu_\Lambda\)#
The proslambenomenos frequency:
\[\nu_\Lambda = c\sqrt{\Lambda/3}\]
nu_Lambda = c * math.sqrt(Lambda / 3)
print(f"nu_Lambda = {nu_Lambda:.3e} s^-1")
print(f"H0 = {H0:.3e} s^-1")
print(f"Ratio nu_Lambda / H0 = {nu_Lambda / H0:.3f}")
print()
print(f"These agree to within a factor of {nu_Lambda/H0:.1f} — order unity.")
print(f"The proslambenomenos IS the Hubble rate, measured in different units.")
nu_Lambda = 1.820e-18 s^-1
H0 = 2.268e-18 s^-1
Ratio nu_Lambda / H0 = 0.802
These agree to within a factor of 0.8 — order unity.
The proslambenomenos IS the Hubble rate, measured in different units.
Step 2: \(\nu_\Lambda \to a_0\)#
The synchronization threshold acceleration:
\[a_0 = \frac{c \, \nu_\Lambda}{2\pi}\]
The \(2\pi\) comes from the Kuramoto critical coupling formula — it separates angular frequency from cycle frequency.
a0_pred = c * nu_Lambda / (2 * math.pi)
print(f"a0 (predicted) = {a0_pred:.3e} m/s^2")
print(f"a0 (observed) = {a0_obs:.3e} m/s^2")
print(f"Ratio pred/obs = {a0_pred / a0_obs:.3f}")
print()
print(f"Discrepancy: factor of {a0_pred/a0_obs:.2f}")
a0 (predicted) = 8.684e-11 m/s^2
a0 (observed) = 1.200e-10 m/s^2
Ratio pred/obs = 0.724
Discrepancy: factor of 0.72
Step 3: The alternative route — \(cH_0 / 2\pi\) directly#
a0_alt = c * H0 / (2 * math.pi)
print(f"cH0 / 2pi = {a0_alt:.3e} m/s^2")
print(f"a0 (observed) = {a0_obs:.3e} m/s^2")
print(f"Ratio = {a0_alt / a0_obs:.3f}")
print()
print("The two routes (Lambda -> nu -> a0) and (H0 -> a0) agree because nu_Lambda ≈ H0.")
print("This is the claim: they are the same frequency measured differently.")
cH0 / 2pi = 1.082e-10 m/s^2
a0 (observed) = 1.200e-10 m/s^2
Ratio = 0.902
The two routes (Lambda -> nu -> a0) and (H0 -> a0) agree because nu_Lambda ≈ H0.
This is the claim: they are the same frequency measured differently.
Step 4: The full chain — one frequency, three units#
print("The Proslambenomenos Chain")
print("=" * 50)
print()
print(f" Lambda = {Lambda:.4e} m^-2 (curvature)")
print(f" |")
print(f" | c * sqrt(./3)")
print(f" v")
print(f" nu_Lam = {nu_Lambda:.4e} s^-1 (frequency)")
print(f" H0 = {H0:.4e} s^-1 (expansion rate)")
print(f" |")
print(f" | c / 2pi")
print(f" v")
print(f" a0_pred = {a0_pred:.4e} m/s^2 (acceleration)")
print(f" a0_obs = {a0_obs:.4e} m/s^2 (observed)")
print()
print("Three constants. One frequency. Zero free parameters.")
The Proslambenomenos Chain
==================================================
Lambda = 1.1056e-52 m^-2 (curvature)
|
| c * sqrt(./3)
v
nu_Lam = 1.8200e-18 s^-1 (frequency)
H0 = 2.2683e-18 s^-1 (expansion rate)
|
| c / 2pi
v
a0_pred = 8.6840e-11 m/s^2 (acceleration)
a0_obs = 1.2000e-10 m/s^2 (observed)
Three constants. One frequency. Zero free parameters.
Step 5: Sensitivity — how much does \(H_0\) tension matter?#
print(f"{'H0 (km/s/Mpc)':>15} {'a0_pred (m/s^2)':>18} {'ratio to observed':>18}")
print("-" * 55)
for h0_val in [67.4, 69.0, 70.0, 72.0, 73.0, 74.0]:
h0_si = h0_val * 1e3 / Mpc
a0_h = c * h0_si / (2 * math.pi)
print(f"{h0_val:>15.1f} {a0_h:>18.3e} {a0_h/a0_obs:>18.3f}")
print()
print("The prediction is robust to the Hubble tension.")
print("All values within a factor of ~0.9 of observed a0.")
H0 (km/s/Mpc) a0_pred (m/s^2) ratio to observed
-------------------------------------------------------
67.4 1.042e-10 0.868
69.0 1.067e-10 0.889
70.0 1.082e-10 0.902
72.0 1.113e-10 0.928
73.0 1.129e-10 0.941
74.0 1.144e-10 0.953
The prediction is robust to the Hubble tension.
All values within a factor of ~0.9 of observed a0.
Summary#
The proslambenomenos \(\nu_\Lambda = c\sqrt{\Lambda/3}\) is the vacuum’s fundamental oscillation frequency. The Hubble rate \(H_0\) is this frequency. The MOND scale \(a_0 \approx cH_0/2\pi\) is the acceleration at which Kuramoto synchronization becomes subcritical.
Three measurements. One frequency. The chain \(\Lambda \to H_0 \to a_0\) requires only \(c\) and \(2\pi\).
See: proslambenomenos.md for the full derivation.