* *

**Sections**

- Surf
- Liddle, Ch. 11: The Early Universe:
- In fact, this chapter is almost all only about the temperature of the early universe from about the recombination era t = 377,770(3200) y = 1.192*10**13 s (formally the decoupling era which occurs during the recombination era: see Wikipedia: Λ CDM model: Parameters to t = 10**(-10) which is beginning of the quark era.
- z_rec = 1089.90(23) (formally for
the decoupling era.
Recall
(a_0/a)=z+1 and so at z_rec ≅ 1100, a/a_0 ≅ 1/1100.

- Before recombination radiation and matter had the same temperature. This temperature scales T = T_rec(a_rec/a) just like radiation alone. This is because radiation is the dominant thermal energy. Matter dominates the mass-energy, but most of this thermally inert rest mass.
- Note
(Ω_r/Ω_m) ≅ (a_e/a) = (t_e/t)**(2/3) (Ω_r/Ω_m)_rec ≅ (a_e/a_rec) = (t_e/t_rec)**(2/3) = (1/8)**(2/3) = 1/4

So at the recombination era radiation is ∼ 1/4 of matter in mass-energy. However,E_rad = a*T**4 = a*4000**4 = [(7.56573345 ...)*10**(-23)]*256*10**12 = 2.5*10**(-8) J/m**3 E_matter_thermal ≅ (3/2)*nkT = (3/2)*(5*10**8)*(1.380649*10**(-23) J/K)*4000 = 10**(-14)*4000 J/m**3 = 4*10**(-11) J/m**3

(see Wikipedia: Chronology of the universe: Tabular summary; Wikipedia: Ideal gas law, Wikipedia: Ideal gas law: Energy associated with a gas, Boltzmann contant k = 1.380649*10**(-23) J/K = (8.617333262 ... )*10**(-5) eV/K (exact) ≅ 10**(-4) eV/K ≅ 10**(-10) MeV/K; Wikipedia: blackbody spectrum, radiation density constant a=(4σ/c) = (7.56573345 ... )*10**(-15) J/*m**3/K**4 (exact)).So it seems that radiation energy must overwhelingly dominate the thermal energy at recombination. The matter thermal energy can only be a perturbation.

- Radiation-Matter era of the universe figure:
For exxact solution and approximate solutions:
see
Universe in problems:Solutions of Friedman equations in the Big Bang model: Problem 5,
but note they do
**NOT**explicitly evaluate the integral which can be done:a_e=(omega_r_0_p/omega_m_0_p)*a_0_cosmic ! Radiation era end a(t). omega_e=omega_r_0_p*(a_0_cosmic/a_e)**power_r Ω_e = omega_e t_e = (4/3)(sqrt(2)-1)/[H_0*sqrt(2Ω_e)] sqrt(Ω_r_0)*H_0*t =c1*x*(1+c0*x)**(1/2)+c2*((1+c0*x)**(3/2)+c3 where c0=omega_m_0_p/omega_r_0_p c1=2/c0 c2=-(4/3)/c0**2 c3=-c2 x=a/a_0 sqrt(Ω_r_0)*H_0*t for x << 1 which is needed for a < ∼ 10**(-8) for numerical accuracy =(1/2)*x**2-(1/6)*c0*x**3

- Early universe cosmic temperature figure. Note the Boltzmann contant k = 1.380649*10**(-23) J/K = (8.617333262 ... )*10**(-5) eV/K (exact) ≅ 10**(-4) eV/K ≅ 10**(-10) MeV/K implies that 1 mega-electron-volt (MeV) corresponds to about 10**(10) K.
- Wikipedia: Chronology of the universe: Tabular summary:
- Wikipedia: Graphical timeline from Big Bang to Heat Death but note that the left-hand vertical scale is tricky, for greater than about > 0, it is x=100*log(log(t_year)) and so t_year=10**(10**(x/100)))
- Wikipedia: Graphical timeline of the Big Bang: Redundant to the timeline above.
- ** | Evidence of Neutrino Enhanced Clustering in a Complete Sample of Sloan Survey Clusters, Implying ∑ m_ν = 0.11(3) eV.: KATRIN: neutrino mass experiment (sensitivity range ∼ 0.2--2 eV): out of luck.
- * | The eras of radiation, matter, and dark energy: new information
from the Planck Collaboration:
Kevin Cahill,
arXiv,
2016,
June 22,
13 pages:
Research:
On
cosmic scale factor
and the
cosmic energy eras
based on the
Planck 2015: Planck 2015 results. XIII. Cosmological parameters.

---Keywords: cold dark matter (CDM), concordance model, concordance model distance measures, cosmic energy eras (radiation era, matter era, dark-energy era) cosmic scale factor, cosmological redshift, cosmology, dark energy, dark matter, disk galaxies, expansion of the universe, IOP cosmological parameters summary: bit dated, Universe in Problems: lots of solutions, etc. - Keywords: Big Bang, Big Bang nucleosynthesis, big bang singularity, cosmic energy eras (radiation era, matter era, dark-energy era) early universe, IOP cosmological parameters summary: bit dated, Universe in Problems: lots of solutions.

For the time being, mainly what we have is just Liddle, Ch. 11: The Early Universe.

Plus some supplements/complements.