https://github.com/laguer/hubble-radius

48 Methods exhibiting the Hubble radius (distance estimates) mistakenly known as the age of the universe
https://github.com/laguer/hubble-radius

age astrophysics bondi calculation calculus constant cosmology eddington gravitational jupyter kotov nambu notebook planck radius sanchez schwarzschild steady-state theory universe

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48 Methods exhibiting the Hubble radius (distance estimates) mistakenly known as the age of the universe

Host: GitHub
URL: https://github.com/laguer/hubble-radius
Owner: LaGuer
Created: 2020-01-14T14:32:55.000Z (almost 6 years ago)
Default Branch: master
Last Pushed: 2025-03-30T14:15:23.000Z (10 months ago)
Last Synced: 2025-03-30T14:28:47.400Z (10 months ago)
Topics: age, astrophysics, bondi, calculation, calculus, constant, cosmology, eddington, gravitational, jupyter, kotov, nambu, notebook, planck, radius, sanchez, schwarzschild, steady-state, theory, universe
Language: Jupyter Notebook
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Open Issues: 0
Metadata Files:
- Readme: README.md

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README

          # Hubble radius distance estimates

[![Cite ptep](https://img.shields.io/badge/PP-57-12-yellow.svg?style=flat)](http://www.ptep-online.com/2019/PP-57-12.PDF)

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[![Cite rXiv](https://img.shields.io/badge/rXiv-1904.0218-orange.svg?style=flat)](http://rxiv.org/abs/1904.0218)

[![Cite viXra](https://img.shields.io/badge/viXra-1811.0146-green.svg?style=flat)](http://vixra.org/pdf/1811.0146v8.pdf)

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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)

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