Herleitung der exakten Planck Temperatur

Herleitung der exakten Planck-Temperatur (c = h = G = k = 1):

1.) Strahlungsära = Ideales Quantengas p = rho*c^2 / 3 = F_planck / (8*pi*R^2) = atilde*T^4 / 3 (F_planck = c^4/G)

atilde = 8*pi^5 / 15 => T_planck^4 = (45 / 64*pi^6) => T_planck = 1 / 6.08088337383 ~ 1 / 6

=> T_planck = 5.84022663e31 Kelvin

2.) Kinetische Energie für relativistische Teilchen Ek ~ 3*k*T_planck ~ m_planck*c^2 / 2

3.) Energiebilanz E_planck = m_planck*c^2 = h*ny = 6.081*k*T_planck

4.) Telchendichte zur Planck-Zeit: n_planck = 16*pi*zeta(3)*k^3*T_planck^3 / (h*c)^3 = 4.0411525e102 1/m^3

5.) Teilchenzahl im Planckvolumen V_planck = 4*pi*L_planck^3 / 3 [L_planck = sqrt(h*G/c^3)]

N_planck = V_planck * n_planck = 1.12559841369

6.) Innere Energie U_planck = E_planck / 2 = N_planck*2.701*k*T_planck

7.) Hauptsatz der Thermodynamik W = U + p*dV = E_planck/2 + F_planck / (8*pi*R^2) * 4*pi*R^2 * L_planck = E_planck/2 + E_planck/2 = E_planck

8.) Rotverschiebung der kosmischen Hintergrundstrahlung

E = 6.081*k*T = h*c/lambda => lambda/L_planck = sqrt(t/t_planck) = sqrt(R/L_planck)

Weltalter t = 1967 Gyr
Weltradius R = 1.861e28 Meter

9.) Photonenzahl im Universum: N_cmb = 1.126 * (t/t_planck)^3/2 = 1.108e94

Photonendichte für T_cmb = 2.725 K => n_cmb = 410.5e6 1/m^3

10.) Entropie im Universum

Mit S = -k*ln(P) und S = -k*ln*exp(-R^2/L_planck^2) = +k*R^2/L_planck^2 = zeta^4*k (Entropiekonstante zeta)