Black-hole (Planck)

Programming cosmic microwave background parameters for deep-universe (Planck scale) Simulation Hypothesis modeling

The simulation hypothesis or simulation argument is an argument that proposes that the universe in its entirety, down to the smallest detail, could be an artificial simulation such as a computer simulation. Neil deGrasse Tyson put the odds at 50-50 that our entire existence is a program on someone else’s hard drive.

Referenced here is a deep-universe simulation model that operates at the Planck scale, and uses the Planck units as the scaffolding upon which particles are embedded.

Simulation clock-rate
The (dimensionless) simulation clock-rate would be defined as the minimum discrete 'time variable' (tage) increment to the simulation. It may be that Gods use analog computers, but as an example; 'begin simulation FOR tage = 1 TO the_end          //big bang = 1 conduct certain processes ........  NEXT tage                         //tage is an incrementing variable and not the dimensioned unit of time 'end simulation

For each increment to tage, a set of Planck units (MLTA as geometrical objects) are added to the simulation.

FOR tage = 1 TO the_end generate 1 unit of Planck time T = tp                generate 1 unit of Planck mass M = mP                generate 1 unit of Planck volume (radius L = Planck length lp) ........  NEXT tage

As each tage increment adds 1 unit of Planck time tp, then in a 14 billion year old universe, numerically (note tp has the units s, tage is dimensionless)


 * tage = tp = 1062

Measuring only the Planck units (in the absence of particle matter), the simulation would have the following parameters (see table 1.). These are compared with the observed CMB parameters.

Mass density
Setting bh as the sum universe and tsec as time measured in seconds;


 * $$mass:\; m_{bh} = 2 t_{age} m_P$$


 * $$volume:\; v_{bh} = \frac{4 \pi r^3}{3} \;\;\; (r=4 l_p t_{age} = 2 c t_{sec})$$


 * $$\frac{m_{bh}}{v_{bh}} = {2 t_{age} m_P}.\;\frac{3}{4 \pi {(4 l_p t_{age})}^3} = \frac{3 m_P}{128 \pi t_{age}^2 l_p^3} \; (\frac{kg}{m^3})$$

Gravitation constant G in Planck units;


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


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

From the Friedman equation; replacing p with the above mass density formula, √(λ) reduces to the radius of the universe;


 * $$\lambda = \frac{3 c^2}{8 \pi G p} = 4 c^2 t_{sec}^2 $$


 * $$\sqrt{\lambda} = radius \; r = 2 c t_{sec} \;(m) $$

Temperature
Measured in terms of Planck temperature TP;


 * $$T_{bh} =  \frac{T_P}{8 \pi \sqrt{t_{age}}}\; (K) $$

The mass/volume formula uses tage2, the temperature formula uses √(tage). We may therefore eliminate the age variable tage and combine both formulas into a single constant of proportionality that resembles the radiation density constant.


 * $$T_p = \frac{m_P c^2}{k_B} = \sqrt{\frac{h c^5}{2\pi G {k_B}^2}}$$


 * $$\frac{m_{bh}}{v_{bh} T_{bh}^4} = \frac{2^5 3 \pi^3 m_P}{l_p^3 T_P^4} = \frac{2^8 3 \pi^6 k_B^4}{h^3 c^5} $$

Radiation energy density
From Stefan Boltzmann constant σSB


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


 * $$\frac{4\sigma_{SB}}{c}.T_{bh}^4 = \frac{c^2}{1440 \pi}. \frac{ m_{bh}}{v_{bh}}$$

Casimir formula
The Casimir force per unit area for idealized, perfectly conducting plates with vacuum between them; F = force, A = plate area, dc 2 lp = distance between plates in units of Planck length


 * $${-F_c}{A} = \frac{\pi h c}{480 {(d_c 2 l_p)}^4}$$

if dc = 2 π √tage then the Casimir force equates to the radiation energy density formula.


 * $$\frac{-F_c}{A} = \frac{c^2}{1440 \pi}. \frac{m_{bh}}{v_{bh}} $$

The diagram (right) plots Casimir length dc2lp against radiation energy density pressure measured in mPa for different tage with a vertex around 1PaA. A radiation energy density pressure of 1Pa occurs around tage = 0.8743 1054 tp (2987 years), with Casimir length = 189.89nm and temperature TBH = 6034 K.

Hubble constant
1 Mpc = 3.08567758 x 1022.


 * $$H = \frac{1 Mpc}{t_{sec}} $$

Black body peak frequency

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


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

Entropy

 * $$S_{BH} = 4 \pi t_{age}^2 k_B$$

Cosmological constant
Riess and Perlmutter using Type 1a supernovae to show that the universe is accelerating. This discovery provided the first direct evidence that Ω is non-zero giving the cosmological constant as ~ 1071 years;


 * $$t_{end} \sim 1.7 x 10^{-121} \sim 0.588 x 10^{121}$$ units of Planck time;

This remarkable discovery has highlighted the question of why Ω has this unusually small value. So far, no explanations have been offered for the proximity of Ω to 1/tuniv2 ~ 1.6 x 10-122, where tuniv ~ 8 x 1060 is the present expansion age of the universe in Planck time units. Attempts to explain why Ω ~ 1/tuniv2 have relied upon ensembles of possible universes, in which all possible values of Ω are found.

The maximum temperature Tmax would be when tage = 1. What is of equal importance is the minimum possible temperature Tmin - that temperature 1 Planck unit above absolute zero, this temperature would signify the limit of expansion; tage = the_end (the 'universe' could expand no further). For example, taking the inverse of Planck temperature;
 * $$T_{min} \sim \frac{1}{T_{max}} \sim \frac{8 \pi}{T_P} \sim  0.177\;10^{-30}\; K $$

This then gives us a value for the final age in units of Planck time (about 0.35 x 1073 yrs);
 * $$t_{end} = T_{max}^4 \sim 1.014 \;10^{123} $$

The mid way point (Tuniverse = 1K) would be when (about 108.77 billion years);


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

Spiral expansion
As this is a geometrical model, by expanding according to the geometry of the Spiral of Theodorus, where each triangle refers to 1 increment to tage, we can map the mass and volume components as integral steps of tage (the spiral circumference) and the radiation domain as a sqrt progression (the spiral arm). A spiral universe can rotate with respect to itself differentiating between an L and R universe without recourse to an external reference. If mathematical constants and physical constants are also a function of tage then their precision would depend on tage, for example using this progression when tage = 1, π2 = 6;


 * $$ \frac{\pi^2}{6} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots$$