Sections
Plus some supplements/complements.
(a_0/a)=z+1 and so at z_rec ≅ 1100, a/a_0 ≅ 1/1100.
(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) = 1/4So 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.
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