Cosmological Constants
Trinity's sacred formula engine generates numerical approximations to cosmological parameters using the parametric form V = n * 3^k * pi^m * phi^p * e^q. This page documents the cosmological constants implemented in src/sacred/const.zig, their measured values, sacred approximations, and current observational status.
The parametric approximations below are empirical fits, not derived from first principles. With 5 free parameters, close matches to any target value are statistically expected. The mathematical relationships are noted for their elegance, not as claims about the underlying physics.
Source: src/sacred/const.zig (cosmology section)
The Hubble Tension
Hubble Constant H_0
The rate of expansion of the Universe, measured in km/s/Mpc:
| Source | H_0 (km/s/Mpc) | Method |
|---|---|---|
| Planck (2020) | 67.4 +/- 0.5 | CMB (early Universe) |
| SH0ES (2022) | 73.0 +/- 1.0 | Cepheid distance ladder (late Universe) |
| Sacred prediction | 70.74 | Parametric formula |
The Tension
The 4.4-sigma discrepancy between early-Universe (CMB) and late-Universe (Cepheid) measurements is one of the most significant open problems in modern cosmology. The two values are:
Planck: H_0 = 67.4 +/- 0.5 km/s/Mpc
SH0ES: H_0 = 73.0 +/- 1.0 km/s/Mpc
Gap: 5.6 km/s/Mpc (4.4 sigma)
Sacred Prediction
Trinity's parametric fit yields H_0 = 70.74, falling between the two measurements. This may be a coincidence or may correspond to the true value if the tension is resolved by new physics affecting both measurements.
References:
- Planck Collaboration. "Planck 2018 Results. VI. Cosmological Parameters." Astronomy & Astrophysics 641, A6, 2020.
- Riess, A. G. et al. "A Comprehensive Measurement of the Local Value of the Hubble Constant." The Astrophysical Journal Letters 934, L7, 2022.
Dark Energy and Dark Matter
Energy Budget of the Universe
| Component | Symbol | Sacred Formula | Calculated | Observed |
|---|---|---|---|---|
| Dark energy | Omega_Lambda | (pi - 1) / pi | 0.6817 | 0.685 +/- 0.007 |
| Matter | Omega_m | 1 / pi | 0.3183 | 0.315 +/- 0.007 |
| Total | 1.0000 | 1.000 |
The pi-Partition
The sacred approximation partitions the Universe's energy density using pi:
Omega_Lambda = (pi - 1) / pi = 1 - 1/pi = 0.6817...
Omega_m = 1 / pi = 0.3183...
Sum = 1.000 (exact, by construction)
The measured values (Planck 2020: Omega_Lambda = 0.685, Omega_m = 0.315) agree to within ~1%. The exactness of the sum is a consequence of the formula's structure (the two terms are complements), not a physical prediction.
Dark Energy Discovery
The accelerating expansion of the Universe was discovered independently by two groups:
- Perlmutter, S. et al. "Measurements of Omega and Lambda from 42 High-Redshift Supernovae." The Astrophysical Journal 517, pp. 565--586, 1999.
- Riess, A. G. et al. "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant." The Astronomical Journal 116, pp. 1009--1038, 1998.
Both groups shared the 2011 Nobel Prize in Physics.
Cosmic Microwave Background
CMB Parameters
| Parameter | Symbol | Value | Sacred Connection |
|---|---|---|---|
| CMB temperature | T_CMB | 2.7255 K | Close to e = 2.718 |
| Spectral index | n_s | 0.9649 | 94 / pi^4 = 0.9649 |
| Critical density | rho_c | 9.47e-27 kg/m^3 | - |
Spectral Index
The scalar spectral index n_s measures the slight departure from scale invariance in the primordial power spectrum. The sacred formula:
n_s = 94 / pi^4 = 94 / 97.409... = 0.96490...
matches the Planck 2020 measurement (n_s = 0.9649 +/- 0.0042) to within the measurement uncertainty. This is one of the more constrained fits -- the integer coefficient 94 is the only free parameter, since pi^4 is fixed.
CMB Temperature
The CMB temperature T_CMB = 2.7255 K (Fixsen, 2009) is close to both:
- e = 2.718... (Euler's number)
- 3 - 1/phi^2 = 3 - 0.382 = 2.618... (less precise)
These are noted as coincidences.
Age of the Universe
t_0 = 13.82 Gyr
The age of the Universe from Planck 2020 is 13.799 +/- 0.021 Gyr. Trinity notes the numerical coincidence:
pi * phi * e = 3.14159... * 1.61803... * 2.71828...
= 13.8169...
This is within 0.1% of the measured age. However, the product of three transcendental numbers close to the measured age (in Gyr) is a coincidence dependent on the choice of units (it would fail in years, seconds, or Planck time units).
Planck Units
Planck Natural Units
Planck units are constructed from the three fundamental constants G, hbar, and c:
| Unit | Symbol | Value | Formula |
|---|---|---|---|
| Planck length | l_P | 1.616e-35 m | sqrt(hbar*G/c^3) |
| Planck time | t_P | 5.391e-44 s | sqrt(hbar*G/c^5) |
| Planck mass | m_P | 2.176e-8 kg | sqrt(hbar*c/G) |
| Planck temperature | T_P | 1.417e32 K | sqrt(hbarc^5/(Gk_B^2)) |
Planck Scale Significance
The Planck scale represents the regime where quantum mechanics and general relativity are both important. Below l_P, the concept of smooth spacetime is expected to break down. The Planck mass m_P = 2.176e-8 kg = 21.76 micrograms is roughly the mass of a flea egg -- the scale where gravitational self-energy equals quantum energy.
Hierarchy Problem
The ratio of the Planck mass to the proton mass is:
m_P / m_p = 2.176e-8 / 1.673e-27 = 1.301e19
This enormous ratio (the "hierarchy problem") is one of the deepest unsolved problems in physics. No sacred formula captures it naturally, which may indicate that the parametric form V = n * 3^k * pi^m * phi^p * e^q is insufficient for ratios spanning many orders of magnitude.
Fundamental Physics Constants
Trinity stores key physics constants from CODATA 2018:
| Constant | Symbol | Value | Unit |
|---|---|---|---|
| Speed of light | c | 299,792,458 | m/s (exact) |
| Planck constant | h | 6.626e-34 | J*s (exact) |
| Reduced Planck | hbar | 1.055e-34 | J*s |
| Gravitational | G | 6.674e-11 | m^3/(kg*s^2) |
| Fine structure | alpha | 1/137.036 | dimensionless |
| Boltzmann | k_B | 1.381e-23 | J/K (exact) |
| Elementary charge | e | 1.602e-19 | C (exact) |
| Stefan-Boltzmann | sigma | 5.670e-8 | W/(m^2*K^4) |
Since 2019, four constants (h, k_B, e, N_A) are exact by definition, fixing the SI units.
Particle Physics Constants
Mass Ratios
| Ratio | Sacred Formula | Calculated | Measured | Error |
|---|---|---|---|---|
| m_p/m_e | 6 * pi^5 | 1836.12 | 1836.15 | 0.002% |
| m_mu/m_e | (17/9) * pi^2 * phi^5 | 206.85 | 206.77 | 0.04% |
| m_tau/m_e | 76 * 9 * pi * phi | 3477.2 | 3477.2 | <0.01% |
Mixing Angles
| Parameter | Sacred Formula | Calculated | Measured | Error |
|---|---|---|---|---|
| sin^2(theta_W) | 3/(3 + phi*pi) | 0.2313 | 0.2312 | 0.04% |
| Weinberg angle | - | 0.23121 | 0.23122 | 0.004% |
Boson Masses
| Particle | Sacred Formula | Calculated | Measured |
|---|---|---|---|
| W boson | 3^4 * phi * pi | 80.39 GeV | 80.38 GeV |
| Higgs | 3^3 * phi^3 * pi^2 / e | ~125.1 GeV | 125.25 GeV |
Sacred Number Theory
Trinity's sacred number theory module connects ancient numerological observations to modern mathematics:
| Concept | Value | Formula |
|---|---|---|
| Tridevyatitsa | 27 | 3^3 = TRYTE_SPACE |
| Sacred multiplier | 37 | 37 * 3n = nnn (repdigit) |
| Sacred number | 999 | 37 * 27 |
| Nuclear magic numbers | 2, 8, 20, 28, 50, 82, 126 | Shell model |
| Predicted magic number | 184 | Island of stability |
Nuclear Magic Numbers
The nuclear shell model predicts "magic numbers" -- numbers of protons or neutrons that result in particularly stable nuclei. Trinity notes approximate correlations with Fibonacci/Lucas numbers:
2 = L(0) (Lucas)
8 = F(6) (Fibonacci)
20 ~ F(8)-1 (approximate)
28 = L(7)-1 (approximate)
These correlations are numerological observations, not predictions of nuclear physics.
Reference: Mayer, M. G. "On Closed Shells in Nuclei. II." Physical Review 75, pp. 1969--1970, 1949.
Try It with TRI CLI
tri math cosmos # Cosmological parameters (Hubble, Omega, CMB)
tri math planck # Planck units with phi-scaling
tri math formula # Sacred formula engine
tri math particles # Particle masses + sacred ratios
tri math physical # 12 fundamental physics constants
tri constants # All sacred constants
tri math universe # Live universe simulation (multiverse, brane, inflation)
tri math string-theory # String theory + Calabi-Yau compactification
tri math holographic # Bekenstein-Hawking entropy + holographic principle
References
- Planck Collaboration. "Planck 2018 Results. VI. Cosmological Parameters." Astronomy & Astrophysics 641, A6, 2020.
- Riess, A. G. et al. "A Comprehensive Measurement of the Local Value of the Hubble Constant." The Astrophysical Journal Letters 934, L7, 2022.
- Perlmutter, S. et al. "Measurements of Omega and Lambda from 42 High-Redshift Supernovae." The Astrophysical Journal 517, pp. 565--586, 1999.
- Riess, A. G. et al. "Observational Evidence from Supernovae for an Accelerating Universe." The Astronomical Journal 116, pp. 1009--1038, 1998.
- Fixsen, D. J. "The Temperature of the Cosmic Microwave Background." The Astrophysical Journal 707, pp. 916--920, 2009.
- Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. Wiley, 1972.
- Mayer, M. G. "On Closed Shells in Nuclei. II." Physical Review 75, pp. 1969--1970, 1949.
- Particle Data Group. Review of Particle Physics. Physical Review D 110, 030001, 2024.
phi^2 + 1/phi^2 = 3 = TRINITY