Hubble radius distance estimates

This repository contains python code and Jupyter notebooks presenting 48 Methods exhibiting the age of the universe (Hubble radius distance) using 3 physical constants excluding the speed of light.

Source Context:

The formula $R_H= ℏ^2/G m_e m_n m_p$ is derived via dimensional analysis, inspired by approaches in natural units and large-number hypotheses.

Dimensional Relationships The quantity for length (𝐿) can be derived from the Planck constants. Using dimensional analysis: $𝐿 ∝ ℏ𝐺 / 𝑀$ Where: ℏ has dimensions of $ML^2T^{-1}$, 𝐺 has dimensions of $L^3M^{-1}T^{-2}$, 𝑀 represents mass.

To incorporate the masses of fundamental particles, we treat 𝑀 as the combination of $𝑚_𝑝$, $𝑚_𝑛$, and $𝑚_𝑒$ in relevant proportions (to represent universal characteristics).

These 48 formulas illustrate how one can “compute” a cosmic length scale using only ℏ, 𝐺, and the masses $𝑚_𝑒$, $𝑚_𝑛$, and $𝑚_𝑝$ – all without an explicit appearance of the speed of light. They serve as a playground for exploring how dimensional analysis and natural unit ideas may (or may not) reflect deep physics.

Physical Interpretation: In a standard cosmological setting the Hubble radius is defined by $𝑅_𝐻 = 𝑐/𝐻_0$ (which explicitly uses the speed of light). Recasting it in terms of other fundamental constants—and in particular not introducing 𝑐 —is an approach that appears in attempts (such as those by Francis Michel Sanchez) to “derive” cosmic scales from quantum–gravitational considerations. The fact that one can match the observational scale (within a few percent, under a suitable choice of 𝑘 ) is a point of considerable debate and interest.

import scipy.constants as const
import numpy as np

# Constants (CODATA 2018/2022 values for consistency)
hbar = const.hbar  # Reduced Planck constant (J·s)
G = const.G        # Gravitational constant (m^3·kg^−1·s^−2)
m_e = const.electron_mass  # Electron mass (kg)
m_p = const.proton_mass    # Proton mass (kg)
m_n = const.neutron_mass   # Neutron mass (kg)
alpha = const.alpha        # Fine-structure constant (dimensionless)

# Conversion constants
meters_per_lightyear = 9.461e15  # Approximate meters in one light-year
meters_to_gly = 1 / (meters_per_lightyear * 1e9)  # Convert meters to gigalight-years

# JWST measured value (placeholder, in meters)
jwst_measured_value = 1.308e+26  # Approximate JWST value (corresponding to 13.81 Gly)

# Ratio of Compton wavelength to Planck length
lambda_e = hbar / (m_e * const.c)  # Electron Compton wavelength (m)
L_planck = np.sqrt(hbar * G / const.c**3)  # Planck length (m)
P = lambda_e / L_planck

# Precision formula
def precision_formula(P, alpha, lambdabare):
    """
    Precision theory formula for calculation.
    P: Ratio of Compton wavelength to Planck length
    alpha: Fine-structure constant
    lambdabare: Bare constant input
    """
    term1 = np.e**(4 * np.e - 1 / alpha)
    term2 = np.log(P**4 / alpha**3)**2
    exponent = np.sqrt((term1 - term2) / 2)
    return np.exp(exponent) * lambdabare

# Corrected formula: R = 2 * hbar^2 / (G * m_e * m_n * m_p)
def corrected_formula(hbar, G, m_e, m_p, m_n):
    """
    Calculates the Hubble radius using the corrected formula.
    """
    return 2 * hbar**2 / (G * m_e * m_n * m_p)

# Adjusted lambdabare value for scaling
lambdabare = 1e-5  # Example value in meters

# Calculate corrected formula result
corrected_result = corrected_formula(hbar, G, m_e, m_p, m_n)
corrected_result_gly = corrected_result * meters_to_gly  # Convert to gigalight-years

# Calculate precision formula result
precision_result = precision_formula(P, alpha, lambdabare)
precision_result_gly = precision_result * meters_to_gly  # Convert to gigalight-years

# Calculate precision difference (JWST deviation)
precision_difference = abs(corrected_result_gly - 13.81) / 13.81

# Output results
print("Hubble Radius Calculation:")
print(f"Corrected Formula:")
print(f"R (meters) = {corrected_result:.3e} m")
print(f"R (gigalight-years) = {corrected_result_gly:.3f} Gly")

print("\nPrecision Formula Calculation:")
print(f"R (meters) = {precision_result:.3e} m")
print(f"R (gigalight-years) = {precision_result_gly:.3f} Gly")

print("\nJWST Measured Value:")
print(f"JWST Value (meters) = {jwst_measured_value:.3e} m")
print(f"JWST Value (gigalight-years) = {jwst_measured_value * meters_to_gly:.3f} Gly")

print(f"\nPrecision Difference (Relative Error): {precision_difference:.5%}")

In addition we used Python modules such as: scipy, sympy, pandas and numpy

Fixed Constants used are:

• 𝜋=3.141592653589793... https://oeis.org/A000796
• Euler Mascheroni 𝛾=0.5772156649015329... https://oeis.org/A001620
• Atiyah's Γ=25.178097241906...
• Feigenbaum constant δ=4.669201609102990671853... https://oeis.org/A006890
• 2nd Feigenbaum constant α=2.50290787509589282228390287321... https://oeis.org/A006891
• Eddington Electric Constant 𝑎=137.0359990836958 also known as the inversed fine structure constant CODATA2018
• 𝑐=299792458.0 m/s CODATA2018
• ℎ=6.62607015.10−34 𝐽.𝐻𝑧−1 CODATA2018
• ℏ=1.0545718176461565.10−34 𝐽.𝑠 CODATA2018
• 𝑙𝑃=1.616255.10−35 m Planck length
• 𝑚𝑃=2.176434.10−8 kg Planck mass
• ƛ𝑒 =3.8615926796.10−13 m Reduced (Electron) Compton Wavelength CODATA2018
• ƛ𝑝 =2.10308910336.10−16 m Reduced (Proton) Compton Wavelength CODATA2018
• Mass of the electron 𝑚𝑒=9.1093837015.10−31 kg CODATA2018
• Mass of the proton 𝑚𝑝=1.67262192369.10−27 kg CODATA2018
• Boson 𝑊=80.379𝐺𝑒𝑉 ± 0.012 Particle Data Group Bosons M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019
• Boson 𝑍=91.1876𝐺𝑒𝑉 ± 0.0023 Particle Data Group Bosons M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019
• Lepton 𝑒=0.5109989461𝑀𝑒𝑉 Particle Data Group Leptons M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019
• Baryon 𝑝=938.272081𝑀𝑒𝑉 Particle Data Group Baryons M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019
• Baryon 𝑛=939.565413𝑀𝑒𝑉 Particle Data Group Baryons M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019
• 𝐺=6.6743.10−11 𝑚3.𝑘𝑔−1.𝑠−2 Newtonian constant of gravitation CODATA2018
• 𝐺𝑞=6.6755.10−11 𝑚3.𝑘𝑔−1.𝑠−2 Newtonian constant of gravitation measured by T.Quinn et al. (2013) BIPM Sevres Improved determination of G using two methods
• 𝐺𝑏2𝑐=6.6754552.10−11 𝑚3.𝑘𝑔−1.𝑠−2 Newtonian constant of gravitation estimated by Francis M. Sanchez et al. (2019) in Back to Cosmos
• 𝐺𝑠=6.67545372.10−11 𝑚3.𝑘𝑔−1.𝑠−2 Newtonian constant of gravitation estimate by Francis M. Sanchez (Jan 2020)