User:Platos Cave (physics)/4Dtime black-hole white-hole universe

Page 3: 4-D time and the black-hole white-hole universe

Using the following CMB ( cosmic microwave background) parameters as a guide, we find they give a best fit approximation for a black hole universe that is [| 14.624 billion] years old.

Universe mass density
Assume that for each unit of Planck time tp the universe expands by 1 unit of Planck mass mP and 1 unit of Planck (spherical) volume (Planck length = lp).
 * t{age} = the dimensionless age of the universe as measured in units of Planck time


 * t{sec} = the age of the universe as measured in seconds


 * $$t_p = \frac{2 l_p}{c} $$
 * $$mass: m_{universe} = \frac{2 t_{age}}{m_P} $$
 * $$volume: v_{universe} = \frac{4 \pi r^3}{3} \;\;\; (r=4 l_p t_{age} = 2 c t_{sec}) $$
 * $$\frac{m_{universe}}{v_{universe}} = {2 t_{age} m_P}.\;\frac{3}{4 \pi {(4 l_p t_{age})}^3} = \frac{3 m_P}{128 \pi t_{age}^2 l_p^3} \; (\frac{kg}{m^3}) $$

Gravitation constant G in Planck units;


 * $$G = \frac{c^2 l_p}{m_P} $$


 * $$\frac{m_{universe}}{v_{universe}} = \frac{3}{32 \pi t_{sec}^2 G} $$

Friedman equation


 * $$p = \frac{m_{universe}}{v_{universe}} $$


 * $$\lambda = \frac{3 c^2}{8 \pi G p} = 4 c^2 t_{sec}^2 $$
 * $$\sqrt{\lambda} = radius \; r = 2 c t_{sec} \;(m) $$

Universe temperature
In 1974, Stephen Hawking showed that black holes, which are objects that light cannot escape from and hence classically are at absolute zero, do radiate at temperature TBH where black hole mass M = n.mP, Planck temperature = TP;


 * $$T_{BH} = \frac{h c^3}{16 \pi^2 G k_B M} = \frac{T_P}{8 \pi n}\; (K) $$

(Proposed) temperature of the universe Tuniverse as a function of the sqrt of t_age


 * $$T_{universe} =  \frac{T_P}{8 \pi \sqrt{t_{age}}} $$

The mass/volume formula uses tage2, the temperature formula uses sqrt of tage. Eliminating the age variable t_age and combining both formulas gives a single constant of proportionality that resembles the Stefan Boltzmann constant σSB.
 * $$\frac{m_{universe}}{v_{universe} T_{universe}^4} = \frac{3 m_P}{128 \pi t_{age}^2 l_p^3}.\;\frac{1}{T_{universe}^4} = \frac{2^8 3 \pi^6 k_B^4}{h^3 c^5}$$


 * $$\sigma_{SB} = \frac{2 \pi^5 k_B^4}{15 h^3 c^2}$$

Radiation density

 * $$\frac{8 \pi^5 k_B^4}{15 h^3 c^5}. T_{universe}^4 = \frac{1}{15}.\frac{m_P}{2^{12} \pi^2 l_p^3 t_{age}^2}$$

Casimir formula
F = force, A = plate area, dc.lp = distance between plates in units of Planck length
 * $$\frac{-F_c}{A} = \frac{\pi h c}{480 {(d_c l_p)}^4} = \frac{1}{15}.\frac{\pi^2 m_P c^2}{16 d_c^4 l_p^3}$$

when dc = 4.pi.t_age, distance = 0.42mm ;
 * $$\frac{1}{15}.\frac{\pi^2 m_P c^2}{16 d_c^4 l_p^3} = \frac{1}{15}.\frac{m_P c^2}{2^{12} \pi^2 l_p^3 t_{age}^2}

$$

Hubble constant
1 Mpc = 3.08567758 x 10-22m
 * $$H = \frac{1 Mpc}{t_{sec}} $$

Wien's displacement law

 * $$\frac{x e^x}{e^x-1} - 5 = 0, x = 4.965114231... $$


 * $$\lambda_{peak} = \frac{2 \pi l_p T_P}{x T_{universe}}  = \frac{16 \pi^2 l_p \sqrt{t_{age}}}{x}  $$

Black body peak frequency

 * $$\frac{x e^x}{e^x-1} - 3 = 0, x = 2.821439372...$$


 * $$v_{peak} = \frac{k_B T_{universe} x}{h} = \frac{x}{8 \pi^2 t_p \sqrt{t_{age}}}$$


 * $$f_{peak} = \frac{x c}{16 \pi^2 l_p \sqrt{t_{age}}}$$

Cosmological constant
Tmax = maximum temperature. Let us set 1/Tmax = minimum temperature

Tmin = 1/Tmax
 * $$T_{min} \sim \frac{1}{T_{max}} \sim \frac{8 \pi}{T_P} \sim  0.177\;10^{-30}\; K$$

The age of the universe at Tmin


 * $$t_{end} = T_{max}^4 \sim 1.014 \;10^{123}$$

The mid way point when Tuniverse = 1K = 108.77 billion years, or in units of Planck time


 * $$t_u = T_{max}^2 \sim 3.18 \;10^{61}$$

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