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How to determine the horizon of the visible universe in a Jupyter Notebook¶

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20 correlations establishing the theoritical value of the horizon of the visible universe see Observable_universe and Schwarzschild_radius ¶

with ± 0.0001 billion (10^9) years approximation¶

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List of physical (NIST CODATA2018) and mathematical constants used:

$\pi=3.141592653589793$
Euler natural logarithm $e=2.718281828459045$
Euler Mascheroni $\gamma=0.5772156649015329$
Apery Constant $\xi(3)=1.202056903159594$ Apery Constant
Atiyah's $\Gamma=25.178097241906$ Michael_Atiyah
Lucas Mersenne number Edouard Lucas
OM Monster Monster_group also known as Fischer–Griess monster
OB Baby Monster Baby_monster_group baby monster group B
OP Pariah Group Pariah_group six sporadic simple groups
Eddington Electric Constant $a=137.0359990836958$ also known as the inversed fine structure constant CODATA2018
$c=299792458.0$ m/s CODATA2018
$h=6.62607015.10^{-34}$ $J.Hz^{-1}$ Planck constant CODATA2018
$\hbar=1.0545718176461565.10^{-34}$ $J.s$ Planck constant over 2 pi CODATA2018
$l_P=1.616255.10^{-35}$ m Planck length CODATA2018
$m_P=2.176434.10^{-8}$ kg Planck mass CODATA2018
ƛ$_e$ $=3.861 592 6796.10^{-13}$m Reduced (Electron) Compton Wavelength CODATA2018
ƛ$_p$ $=2.103 089 103 36.10^{-16}$m Reduced (Proton) Compton Wavelength CODATA2018
$\lambda_{wi}=2.897771955.10^{-3}$ m K (exact) Wien wavelength displacement law constant CODATA2018
$k_B=1.380 649.10^{-23}$ J K^-1 (exact) Boltzmann constant CODATA2018
$t_K=9600.60(1)$s non-Doppler Kotov Periodicity (1000 ppb)
Fermi coupling constant : $1.1663787.10^{-5}$ (GeV)-2 (51 ppb)
Fermi's ratio F = Fermi's mass / m_e = 573007.3625 (25 ppb)
Fermi-Atiyah's ratio : $F_A = (2\Gamma . 137)^{3/2}$ ≈ 573007.3652 (0.22 ppb)
Mass of the electron $m_e=9.1093837015.10^{-31}$ kg CODATA2018
Mass of the proton $m_p=1.672 621 923 69.10^{-27}$ kg CODATA2018
Magnetic Moment of the electron/Bohr magneton: $d_e$ ≈ 1.00115965218128 (2 × 10-4 ppb)
Magnetic moment anomaly $1159.65218091.10^{−6}$ ± 0.00000026 Particle Data Group Leptons M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019
Boson $W=80.379 GeV$ ± 0.012 Particle Data Group Bosons M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019
Boson $Z=91.1876 GeV$ ± 0.0023 Particle Data Group Bosons M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019
Lepton $e=0.5109989461 MeV$ Particle Data Group Leptons M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019
Baryon $p=938.272081 MeV$ Particle Data Group Baryons M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019
Baryon $n=939.565413 MeV$ Particle Data Group Baryons M. Tanabashi et al. (Particle Data Group), Phys. Rev. D 98, 030001 (2018) and 2019
$G=6.6743.10^{-11}$ $m^{3}.kg^{-1}.s^{-2}$ Newtonian constant of gravitation CODATA2018
$G_q=6.6755.10^{-11}$ $m^{3}.kg^{-1}.s^{-2}$ Newtonian constant of gravitation measured by T.Quinn et al. (2013) BIPM Sevres Improved determination of G using two methods
$G_s=6.67545372.10^{-11}$ $m^{3}.kg^{-1}.s^{-2}$ Newtonian constant of gravitation estimate by Francis M. Sanchez (Jan 2020)
Galaxies Doppler radius $R=c.f /\Delta.f$=Universe Schwarzschild radius: $2G.M/c^2$
Hubble-Lemaître radius: $c/H_0$

𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔

In [190]:

from scipy import constants
import numpy as np

In [191]:

c=constants.c
print(c)
h=constants.h
print(h)
hbar=constants.hbar
print(hbar)
a=137.0359990836958
print(a)
d_e=1.00115965218128
print(d_e)
xi3=1.202056903159594
print(xi3)
xi2=np.pi**2/6
print(xi2)
#print(c,h,hbar,a,d_e,xi3,xi2)
e=np.e
print(e)
gamma=np.euler_gamma
print(gamma)
Gamma=(constants.fine_structure**-1*gamma)/np.pi
print(Gamma)
# F**(2/3)=2*137*Gamma
# Fermi's mass given by 
F=(2*137*Gamma)**1.5
print(F)
lambdabare=constants.hbar/(constants.m_e*constants.c)
print(lambdabare)
lambdabar_F=lambdabare/F
print(lambdabar_F)
lambdabar_p=(constants.hbar/(constants.m_p*constants.c))
print(lambdabar_p)
H=1837.152646   # (0.06 ppb)
p=1836.15267343 # 1836.1526734400013 #(0.06 ppb)
n=1838.68366173 # 1838.6836617324586 #(0.5 ppb)
print(H,p,n)
m_e=constants.m_e
m_p=constants.m_p
m_n=constants.m_n
print(m_e,m_p,m_n)
beta=(H-p)**(-1)
print(beta)
j=(8*np.pi**2)/np.log(2)
print(j)
k_B=constants.Boltzmann
T_cmb=2.72582
print(k_B,T_cmb)
lambdabar_cmb=(constants.hbar*constants.c)/(k_B*T_cmb)
print(lambdabar_cmb)
# Planck's length CODATA2018 (hbar.G/c^3)^1/2 = 1.61639471.10^-35
# l_P=1.616255*10**(-35) CODATA2018
G_s=6.67545372*10**-11
print(G_s)
l_P=((constants.hbar*G_s)/constants.c**3)**0.5
print(l_P)
ƛ_w=(constants.hbar/(constants.m_e*constants.c))/F**2
print(ƛ_w)
lambdabar_w=(constants.hbar/(constants.m_e*constants.c))/F**2
P=lambdabare/l_P
print(P)
#2.897 771 955... e-3 m.K - NIST CODATA 2018
b=2.897771955*10**(-3)
#lambdawien=(h*c)/(k_B*)
# b=lambda/lambdawien
print(b)
𝜔=(h*c)/(k_B*b)
print(𝜔)
l_ph=(lambdabare*(np.pi*(a**2))**2)
print(l_ph)
aprime=137.1106292
# Hydrogen Atom Radius
r_H=(lambdabare*aprime)
print(r_H)
# Hydrogen Atom Radius
r_H=(lambdabare*aprime)
print(r_H)
Lucas=(2**127)-1
print(Lucas)
OM=2**46 * 3**20 * 5**9 * 7**6 * 11**2 * 13**3 * 17 * 19 * 23 * 29 * 31 * 41 * 47 * 59 * 71 # 808017424794512875886459904961710757005754368000000000
print(OM)
OB=2**41 * 3**13 * 5**6 * 7**2 * 11 * 13 * 17 * 19 * 23 * 31 * 47 # 4154781481226426191177580544000000
print(OB)
OP=2**8 * 3**7 * 5**6 * 7 * 11 * 31 * 37 * 67 * 2**9 * 3**4 * 5 * 7**3 * 11 * 19 * 31 * 2**14 * 3**3 * 5**3 * 7 * 13 * 29 * 2**21 * 3**3 * 5 * 7 * 11**3 * 23 * 29 * 31 * 37 * 43 * 2**7 * 3**5 * 5 * 17 * 19 * 2**3 * 3 * 5 * 7 * 11 * 19
print(OP)

299792458.0
6.62607015e-34
1.0545718176461565e-34
137.0359990836958
1.00115965218128
1.202056903159594
1.6449340668482264
2.718281828459045
0.5772156649015329
25.178097241906
573007.364905975
3.8615926796089057e-13
6.739167620022749e-19
2.1030891033555923e-16
1837.152646 1836.15267343 1838.68366173
9.1093837015e-31 1.67262192369e-27 1.67492749804e-27
1.000027430752471
113.91063459990004
1.380649e-23 2.72582
0.0008400718024707187
6.67545372e-11
1.6163947113699356e-35
1.1761048867371158e-24
2.389015908333496e+22
0.002897771955
4.965114232061555
0.0013440173951173593
5.294654020152911e-11
5.294654020152911e-11
170141183460469231731687303715884105727
808017424794512875886459904961710757005754368000000000
4154781481226426191177580544000000
2663849798329448283764291471531459318169741293570162121768960000000000000

In [213]:

# URL= 'https://github.com/laguer/hubble-table/hubble-table.csv'
URL= 'https://raw.githubusercontent.com/LaGuer/hubble-table/master/hubble-table.csv'
from urllib.request import urlretrieve
urlretrieve (URL, 'hubble-table.csv')

Out[213]:

('hubble-table.csv', <http.client.HTTPMessage at 0x7f92a98faf90>)

In [193]:

URL= 'https://raw.githubusercontent.com/LaGuer/hubble-table/master/constant-table.csv'
from urllib.request import urlretrieve
urlretrieve (URL, 'constant-table.csv')

Out[193]:

('constant-table.csv', <http.client.HTTPMessage at 0x7f92b2a52ad0>)

In [218]:

import pandas as pd
#from StringIO import StringIO
#df = pd.read_csv('hubble-table.csv')
df = pd.read_csv("hubble-table.csv", dtype = {"gly_value" : "float64","m_value" : "float64"}, skiprows = [10], sep = ",")
#df = pd.read_csv("hubble-table.csv", dtype = {"gly_value" : "float64","m_value" : "float64"}, usecols=range(1), skiprows = [10], sep = ",")
#df1 = pd.read_csv("constant-table.csv", dtype = {" c_value" : "float64"}, skiprows = [1], sep = ",")
#df = pd.to_csv('hubble-table.csv', float_format='%.3f')
df.tail ()
#df.head ()

Out[218]:

	Formula in Latex	Formula in plain	gly_value	m_value
4	$$(ƛ_e^2/l_P)(j/16)^16( e^e )^2.d_e√2$$	(lambdabare2/l_P)(j/16)16( ee )2d_e...	1.381200e+10	1.306715e+26
5	$$((a/\sqrt(137)(4πF)^2)ƛe4 l_ph^3(\lambda_CMB...	((a/np.sqrt(137)(4np.piF)*2)lambdabar*4...	1.381189e+10	1.306715e+26
6	$$2\betaƛ_ej**{17}(4π)^2√137$$	2betalambdabare(j17)(4np.pi)2np.sqr...	1.381198e+10	1.306715e+26
7	$$ƛ_e(3j^j /2H)^{1/6}$$	lambdabare(3j*j /(2H))**(1/6)	1.381199e+10	1.306715e+26
8	$$(βFP^{3/2}(n/p)^{7/2}/2\pi).ƛ_e$$	betaFP*(3/2)(n/p)*(7/2)/(2np.pi)*lambda...	1.381198e+10	1.306716e+26

In [219]:

import pandas as pd
#from StringIO import StringIO
#df = pd.read_csv('hubble-table.csv')
df = pd.read_csv("hubble-table.csv", dtype = {"gly_value" : "float64","m_value" : "float64"}, skiprows = [10], sep = ",")
#df = pd.read_csv("hubble-table.csv", dtype = {"gly_value" : "float64","m_value" : "float64"}, usecols=range(1), skiprows = [10], sep = ",")
#df1 = pd.read_csv("constant-table.csv", dtype = {" c_value" : "float64"}, skiprows = [1], sep = ",")
#df = pd.to_csv('hubble-table.csv', float_format='%.3f')
#df.tail ()
df.head ()

Out[219]:

	Formula in Latex	Formula in plain	gly_value	m_value
0	$$2hbar^2/(G.m_e.m_p.m_n)$$	2hbar2/(G_sm_em_pm_n)	1.380048e+01	1.305626e+26
1	$$ƛ_e.exp((e{^4e-1/a}-ln2(P^4/a^3 ))/2)^{1/2}$$	\t(lambdabare*2/l_P)((j/16)*16)( ee )...	1.381000e+10	1.306719e+26
2	$$((a/\sqrt(137)(4\pi.F)^2)ƛ_e^4l{ph}^3(ƛ_{CMB...	((a/(np.sqrt(137)(4np.piF)2))(lambdabar...	1.381170e+10	1.306242e+26
3	$$ƛ_eF_e(Pe^e)^2/√(pn)$$	lambdabareF_e(Pee)*2/np.sqrt(pn)	1.381260e+10	1.306715e+26
4	$$(ƛ_e^2/l_P)(j/16)^16( e^e )^2.d_e√2$$	(lambdabare2/l_P)(j/16)16( ee )2d_e...	1.381200e+10	1.306715e+26

In [216]:

df1 = pd.read_csv("constant-table.csv", dtype = {"value" : "float64"}, skiprows = [2], sep = ",")
#df = pd.to_csv('hubble-table.csv', float_format='%.3f')
#df1.head ()
df1.tail ()

Out[216]:

	symbol	value
32	G_s	6.67545372e-11
33	Lucas	170141183460469231731687303715884105727
34	OM	8080174247945128758864599049617107570057543680...
35	OB	4154781481226426191177580544000000
36	OP	2663849798329448283764291471531459318169741293...

In [217]:

df.dtypes

Out[217]:

Formula in Latex      object
 Formula in plain     object
 gly_value           float64
 m_value             float64
dtype: object

In [21]:

# 3**210

In [22]:

# Using G_s the Newtonian constant of gravitation estimate by Francis M. Sanchez (Jan 2020)
G_s=6.67545372*10**-11
print(G_s)

6.67545372e-11

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Universe Horizon Radius calculus in 3 minutes (method 1)¶

$$R_U=\frac{2.\hbar^2}{G.m_e.m_p.m_n}$$

In [23]:

R_U=2*constants.hbar**2/(G_s*constants.m_e*constants.m_p*constants.m_n)
print(R_U)

1.3056258355517771e+26

Yields 13.802861522337478 Gly (G) and 13.80047597102314 Gyr (G_s)¶

In [24]:

R_U/(constants.light_year*10**9)

Out[24]:

13.80047597102314

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In [ ]:

How to determine the horizon of the visible universe in a Jupyter Notebook¶

20 correlations establishing the theoritical value of the horizon of the visible universe see Observable_universe and Schwarzschild_radius¶

with ± 0.0001 billion (10^9) years approximation¶

Universe Horizon Radius calculus in 3 minutes (method 1)¶

Yields 13.802861522337478 Gly (G) and 13.80047597102314 Gyr (G_s)¶

20 correlations establishing the theoritical value of the horizon of the visible universe see Observable_universe and Schwarzschild_radius ¶