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Analytic Solution in parametric form of Complete Friedmann Equation of the Universe RESULTS
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Hannu Poropudas
Jul 20, 2016, 3:36:19 AM7/20/16
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Analytic Solution in parametric form of Complete Friedmann Equation of the Universe
Omega_r0 = 5.4619*10^(-5)
Omega_m0 = 0.315
Omega_L0 = 0.685
(L_0 = cosmological constant)
Omega_k = 1-(Omega_m0+Omega_L0+Omega_r0)=1-Omega_0 = -5.4619*10^(-5)
H_0 = 0.67312*100*km*s^(-1)*Mpc^(-1) = 2.181777*10^(-18)*s^(-1)
t_0 = 13.81*10^9*yr = 4.35801137*10^17*s
(not rounded numbers)
References:
Olive K.A. et al. (Particle Data Group), 2014.
Chinese Physics, C38, 090001 (2014). 1676 pages, pp. 110-111.
("Astrophysical Constants and parameters").
Ryden Barbara, 2003.
Introduction to Cosmology.
Solution form of the Friedmann equation:
a = a(t) = scale factor of the Universe (Friedmann-Lemaitre-Robertson-Walker metrics of the Universe).
Integration Int is taken from 0 to a or constant of integration C_1 = -H_0*t(0), (a=0), is used here.
H_0*t = Int(a*(Omega_L0*(a^4+(1-Omega_0)/(Omega_L0)*a^2+Omega_m0/OmegaL0*a+Omega_r0/Omega_L0))^(-1/2)*da)
---------------------------------
I put here these two NEW solutions (Maple 9 program used when plotting figures) (y2=H_0*t and x2=a(t)=scale factor, 0 <= P<= Pi):
PHOTON ENERGY DENSITY ACCOUNTED
>y2 := P -> 0.6041220957*arctanh(72070.44347*(0.1999999999e11-0.1797940782e11*sin(P))/(-0.2595615132e30*(sin(P)-0.9634466212)^2-0.5001473259e30*sin(P)+0.5191104893e30)^(1/2))-0.6041220957*arctanh(72070.44347*(0.1999999999e11+0.1797940782e11*sin(P))/(-0.2595615132e30*(sin(P)+0.9634466212)^2+0.5001473259e30*sin(P)+0.5191104893e30)^(1/2))+0.3711518575e-26*((-1000000000+933077527*sin(P)^2)*(-1+sin(P)^2))^(1/2)*(1-sin(P)^2)^(1/2)*(0.9057038458e26*EllipticF(sin(P), 0.9659593817)+0.3312247835e26*EllipticPi(sin(P), 1.077319904, 0.9659593817))/((1000000000+933077527*sin(P)^4-1933077527*sin(P)^2)^(1/2)*cos(P))+0.54240*10^(-5);
>x2 := P -> (-0.5962896200*cos(P)-0.5958258106)/(2.109702780*cos(P)-0.5651901532);
PHOTON ENERGY DENSITY AND NEUTRINO ENERGY DENSITY (+68%) ACCOUNTED
>y2 := P -> -0.6041220939*arctanh(1153722.636*(312500000.0+280944204.5*sin(P))/(-0.1623314745e29*(sin(P)+0.9634558593)^2+0.3127984206e29*sin(P)+0.3246497844e29)^(1/2))+0.6041220939*arctanh(1153722.636*(312500000.0-280944204.5*sin(P))/(-0.1623314745e29*(sin(P)-0.9634558593)^2-0.3127984206e29*sin(P)+0.3246497844e29)^(1/2))+0.3816958436e-29*((-156250000+145800247*sin(P)^2)*(-1+sin(P)^2))^(1/2)*(1-1*sin(P)^2)^(1/2)*(0.8808944533e29*EllipticF(sin(P), 0.9659821848)+0.3219628214e29*EllipticPi(sin(P), 1.077299245, 0.9659821848))/((156250000+145800247*sin(P)^4-302050247*sin(P)^2)^(1/2)*cos(P))+0.118112*10^(-4);
>x2 := P -> (-0.5966419659*cos(P)-0.5958625516)/(2.110339037*cos(P)-0.5652904916);
RESULTS (both photon energy density and estimated neutrino energy density (+68%)
taken into account, not rounded numbers used):
Age of the present Universe: 13.80726587*10^9 years,
Age of the last scattering time: 371.6925892*10^3 years,
Age of the radiation-matter equality: 50.024678550*10^3 years,
Age of the half reionization time: 407.8112219*10^6 years,
Age of the matter-cosmological constant energy equality: 10.30632872*10^9 years.
For comparision Data from Particle Data Group 2014 publication:
Age of the present Universe: 13.81*10^9 years,
Age of the last scattering time: 372*10^3 years,
Redshift of the radiation-matter equality: 3360,
Age of the half reionization time: 462*10^6 years,
Age of the matter-cosmological constant energy equality: -.
---------------------------------
From the New Solution:
RESULTS of age calculations of different events of the Universe (photon energy density accounted,
neutrino energy density not accounted, not rounded numbers used):
Age of the present Universe: 13.80930072*10^9 years,
Age of the last scattering time: 403.2132060*10^3 years,
Age of the radiation-matter equality: 23.07038419*10^3 years,
Age of the half reionization time: 408.5979461*10^6 years,
Age of the matter-cosmological constant energy equality: 10.30837000*10^9 years.
These seems to be consistent with the Particle Data Group 2014 data (there is
both photon energy density and additional (+71.64%) estimated neutrino energy density accounted
this is why especially the radiation-matter time would be little larger.)
Hannu Poropudas
Kolamäentie 9E,
90900 Kiiminki / Oulu
Finland.
Omega_r0 = 5.4619*10^(-5)
Omega_m0 = 0.315
Omega_L0 = 0.685
(L_0 = cosmological constant)
Omega_k = 1-(Omega_m0+Omega_L0+Omega_r0)=1-Omega_0 = -5.4619*10^(-5)
H_0 = 0.67312*100*km*s^(-1)*Mpc^(-1) = 2.181777*10^(-18)*s^(-1)
t_0 = 13.81*10^9*yr = 4.35801137*10^17*s
(not rounded numbers)
References:
Olive K.A. et al. (Particle Data Group), 2014.
Chinese Physics, C38, 090001 (2014). 1676 pages, pp. 110-111.
("Astrophysical Constants and parameters").
Ryden Barbara, 2003.
Introduction to Cosmology.
Solution form of the Friedmann equation:
a = a(t) = scale factor of the Universe (Friedmann-Lemaitre-Robertson-Walker metrics of the Universe).
Integration Int is taken from 0 to a or constant of integration C_1 = -H_0*t(0), (a=0), is used here.
H_0*t = Int(a*(Omega_L0*(a^4+(1-Omega_0)/(Omega_L0)*a^2+Omega_m0/OmegaL0*a+Omega_r0/Omega_L0))^(-1/2)*da)
---------------------------------
I put here these two NEW solutions (Maple 9 program used when plotting figures) (y2=H_0*t and x2=a(t)=scale factor, 0 <= P<= Pi):
PHOTON ENERGY DENSITY ACCOUNTED
>y2 := P -> 0.6041220957*arctanh(72070.44347*(0.1999999999e11-0.1797940782e11*sin(P))/(-0.2595615132e30*(sin(P)-0.9634466212)^2-0.5001473259e30*sin(P)+0.5191104893e30)^(1/2))-0.6041220957*arctanh(72070.44347*(0.1999999999e11+0.1797940782e11*sin(P))/(-0.2595615132e30*(sin(P)+0.9634466212)^2+0.5001473259e30*sin(P)+0.5191104893e30)^(1/2))+0.3711518575e-26*((-1000000000+933077527*sin(P)^2)*(-1+sin(P)^2))^(1/2)*(1-sin(P)^2)^(1/2)*(0.9057038458e26*EllipticF(sin(P), 0.9659593817)+0.3312247835e26*EllipticPi(sin(P), 1.077319904, 0.9659593817))/((1000000000+933077527*sin(P)^4-1933077527*sin(P)^2)^(1/2)*cos(P))+0.54240*10^(-5);
>x2 := P -> (-0.5962896200*cos(P)-0.5958258106)/(2.109702780*cos(P)-0.5651901532);
PHOTON ENERGY DENSITY AND NEUTRINO ENERGY DENSITY (+68%) ACCOUNTED
>y2 := P -> -0.6041220939*arctanh(1153722.636*(312500000.0+280944204.5*sin(P))/(-0.1623314745e29*(sin(P)+0.9634558593)^2+0.3127984206e29*sin(P)+0.3246497844e29)^(1/2))+0.6041220939*arctanh(1153722.636*(312500000.0-280944204.5*sin(P))/(-0.1623314745e29*(sin(P)-0.9634558593)^2-0.3127984206e29*sin(P)+0.3246497844e29)^(1/2))+0.3816958436e-29*((-156250000+145800247*sin(P)^2)*(-1+sin(P)^2))^(1/2)*(1-1*sin(P)^2)^(1/2)*(0.8808944533e29*EllipticF(sin(P), 0.9659821848)+0.3219628214e29*EllipticPi(sin(P), 1.077299245, 0.9659821848))/((156250000+145800247*sin(P)^4-302050247*sin(P)^2)^(1/2)*cos(P))+0.118112*10^(-4);
>x2 := P -> (-0.5966419659*cos(P)-0.5958625516)/(2.110339037*cos(P)-0.5652904916);
RESULTS (both photon energy density and estimated neutrino energy density (+68%)
taken into account, not rounded numbers used):
Age of the present Universe: 13.80726587*10^9 years,
Age of the last scattering time: 371.6925892*10^3 years,
Age of the radiation-matter equality: 50.024678550*10^3 years,
Age of the half reionization time: 407.8112219*10^6 years,
Age of the matter-cosmological constant energy equality: 10.30632872*10^9 years.
For comparision Data from Particle Data Group 2014 publication:
Age of the present Universe: 13.81*10^9 years,
Age of the last scattering time: 372*10^3 years,
Redshift of the radiation-matter equality: 3360,
Age of the half reionization time: 462*10^6 years,
Age of the matter-cosmological constant energy equality: -.
---------------------------------
From the New Solution:
RESULTS of age calculations of different events of the Universe (photon energy density accounted,
neutrino energy density not accounted, not rounded numbers used):
Age of the present Universe: 13.80930072*10^9 years,
Age of the last scattering time: 403.2132060*10^3 years,
Age of the radiation-matter equality: 23.07038419*10^3 years,
Age of the half reionization time: 408.5979461*10^6 years,
Age of the matter-cosmological constant energy equality: 10.30837000*10^9 years.
These seems to be consistent with the Particle Data Group 2014 data (there is
both photon energy density and additional (+71.64%) estimated neutrino energy density accounted
this is why especially the radiation-matter time would be little larger.)
Hannu Poropudas
Kolamäentie 9E,
90900 Kiiminki / Oulu
Finland.
Hannu Poropudas
Jul 21, 2016, 5:23:34 AM7/21/16
to
1. Photon energy density accounted and neutrino energy density not accounted:
Zero point of the scale factor is P = 3.102148349 radians.
C = -y2(3.102148349) = -H_0*t(3.102148349) = 0.54245*10^(-5) .
2. Photon energy density and estimated neutrino energy (+68%) density accounted:
Zero point of the scale factor is P = 3.090472803 radians.
C = -y2(3.090472803) = -H_0*t(3.090472803) = 0.118116*10^(-4) .
Last digit of both cases is different than in complicated calculation with
direct substitution of formula of P(scale factor) into formula of H_0*t(P).
These differences is due Maple 9 calculation accuracy.
Hannu
Hannu Poropudas
Jul 26, 2016, 6:01:19 AM7/26/16
to
For mathematical completeness (primitive functions in parametric forms of these two cases. Plottings with Maple 9 program):
Case 1.
NEW Imaginary Solution divided by I = sqrt(-1). (y2/sqrt(-1)=H_0*t/sqrt(-1), Primitive Function)
(valid when scale factor ( x2 ) is between two negative real roots: -0.7718392260 <= scale factor <= -0.0001733936456)
Photon energy density only accounted. (0 <= P <= Pi).
>y2 := P -> -1.208244187*arctan(3797.925453*(0.1000000000e11-669224730*sin(P)^2)^(1/2)*sin(P)/(-1000000000+66922473*sin(P)^2))-0.1528025768e-26*((-1000000000+66922473*sin(P)^2)*(-1+sin(P)^2))^(1/2)*(1-sin(P)^2)^(1/2)*(0.8205327530e27*EllipticF(sin(P), 0.2586937823)-0.1120978182e28*EllipticPi(sin(P), -0.7731990447e-1, 0.2586937823))/((1000000000+66922473*sin(P)^4-1066922473*sin(P)^2)^(1/2)*cos(P));
>x2 := P -> (-0.595825809*cos(P)-0.596289618)/(-0.565190153*cos(P)+2.109702776);
Case 2.
Corrected NEW Imaginary Solution divided by I = sqrt(-1) (valid when scale factor ( x2 )
is between two negative real roots: -0.7718233324 <= scale factor <= -0.0002913013086).(0<=P<=Pi) (y2/sqrt(-1)=H_0*t/sqrt(-1), Primitive Function)
Photon energy density and estimated neutrino energy density (+68%) accounted. (0 <= P <= Pi).
>y2 := P -> -1.208244183*arctan(3797.073397*(0.1000000000e11-668784190.*sin(P)^2)^(1/2)*sin(P)/(-1000000000.+66878419.*sin(P)^2))-0.2544543940e-26*((-1000000000.+66878419.*sin(P)^2)*(-1.+sin(P)^2))^(1/2)*(1.-1.*sin(P)^2)^(1/2)*(0.4926564767e27*EllipticF(sin(P), .2586086213)-0.6730918042e27*EllipticPi(sin(P), -0.7729924476e-1, .2586086213))/((1000000000.+66878419.*sin(P)^4-1066878419.*sin(P)^2)^(1/2)*cos(P));
>x2 := P -> (-0.595862551*cos(P)-0.596641964)/(-0.565290491*cos(P)+2.110339036);
Best Regards,
Hannu Poropudas
Hannu Poropudas
Aug 18, 2016, 3:50:10 AM8/18/16
to
(redshift z = 8.8)
scale factor x2 = 1/(8.8+1)= 0.1020408163
P = 2.295344521 radians
y2(2.295344521)=H_0*t(2.295344521)=0.0385493175
age of the half reionization time = 559.9015140*10^6 years
which is approximately 560*10^6 years.
Anyhow this seems to fit to data given in article (this quite large
change from z = 11.1 (Planck 2013 results) to z = 8.8 (Planck 2015 results):
"Planck reveals first stars were born late"
05 February 2015
http://sci.esa.int/planck/55385-planck-reveals-first-stars-were-born-late/
Hannu
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