# Simulation argument (coding cosmic microwave background)

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.
A deep-universe simulation model operates at the Planck scale and uses the Planck units as the scaffolding upon which particles are embedded.

### Simulation clock-rate

Assuming a digital computer, the simulation clock-rate would be the minimum increment to the simulation. For each increment to t$_{age}$ a set of Planck units are generated. As an example;
At t$_{age}$ = 10$^{62}$, the simulation would have generated a 14 billion year old universe, in units of Planck time 10$^{62}$t$_p$. Measuring only the Planck units (in the absence of particle matter), the simulation would have the following parameters (see table). These compare with the observed CMB parameters.
cosmic microwave background parameters; calculated vs observed :
| Parameter | Calculated | Calculated | Observed |
| Age (billions of years) | 13.8 | 14.624 | 13.8 |
| Age (units of Planck time) | 0.404 10$^{61}$ | 0.428 10$^{61}$ | 0.404 10$^{61}$ |
| Mass density | 0.24 x 10$^{-26}$ kg.m-3 | 0.21 x 10$^{-26}$ kg.m-3 | 0.24 x 10$^{-26}$ kg.m-3 |
| Radiation energy density | 0.468 x 10$^{-13}$ kg.m-1.s-2 | 0.417 x 10$^{-13}$ kg.m-1.s-2 | 0.417 x 10$^{-13}$ kg.m-1.s-2 |
| Hubble constant | 70.85 km/s/Mp | 66.86 km/s/Mp | 67 (ESA's Planck satellite 2013) |
| CMB temperature | 2.807K | 2.727K | 2.7255K |
| CMB peak frequency | 164.9GHz | 160.2GHz | 160.2GHz |
| Casimir length | 0.41mm | 0.42mm | |

### Mass density

Setting bh as the sum universe and t$_{sec}$ as time measured in seconds;
    $mass:\;m_{bh}=2t_{age}m_{P}$
    $volume:\;v_{bh}={\frac {4\pi r^{3}}{3}}\;\;\;(r=4l_{p}t_{age}=2ct_{sec})$
    ${\frac {m_{bh}}{v_{bh}}}={2t_{age}m_{P}}.\;{\frac {3}{4\pi {(4l_{p}t_{age})}^{3}}}={\frac {3m_{P}}{128\pi t_{age}^{2}l_{p}^{3}}}\;({\frac {kg}{m^{3}}})$
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;
    $\lambda ={\frac {3c^{2}}{8\pi Gp}}=4c^{2}t_{sec}^{2}$
    ${\sqrt {\lambda }}=radius\;r=2ct_{sec}\;(m)$

### Temperature

Measured in terms of Planck temperature T$_P$;
    $T_{bh}={\frac {T_{P}}{8\pi {\sqrt {t_{age}}}}}\;(K)$
The mass/volume formula uses t$_{age}$$^2$, the temperature formula uses √(t$_{age}$). We may therefore eliminate the age variable t$_{age}$ and combine both formulas into a single constant of proportionality that resembles the radiation density constant.
    $T_{p}={\frac {m_{P}c^{2}}{k_{B}}}={\sqrt {\frac {hc^{5}}{2\pi G{k_{B}}^{2}}}}$
    ${\frac {m_{bh}}{v_{bh}T_{bh}^{4}}}={\frac {2^{5}3\pi ^{3}m_{P}}{l_{p}^{3}T_{P}^{4}}}={\frac {2^{8}3\pi ^{6}k_{B}^{4}}{h^{3}c^{5}}}$

### Radiation energy density

From Stefan Boltzmann constant σ$_{SB}$
    $\sigma _{SB}={\frac {2\pi ^{5}k_{B}^{4}}{15h^{3}c^{2}}}$
    ${\frac {4\sigma _{SB}}{c}}.T_{bh}^{4}={\frac {c^{2}}{1440\pi }}.{\frac {m_{bh}}{v_{bh}}}$

### Casimir formula

The Casimir force per unit area for idealized, perfectly conducting plates with vacuum between them; F = force, A = plate area, d$_c$ 2 l$_p$ = distance between plates in units of Planck length
    ${-F_{c}}{A}={\frac {\pi hc}{480{(d_{c}2l_{p})}^{4}}}$
if d$_c$ = 2 π √t$_{age}$ 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 d$_c$2l$_p$ against radiation energy density pressure measured in mPa for different t$_{age}$ with a vertex around 1PaA. A radiation energy density pressure of 1Pa occurs around t$_{age}$ = 0.8743 10$^{54}$ t$_p$ (2987 years), with Casimir length = 189.89nm and temperature T$_{BH}$ = 6034 K.

### Hubble constant

1 Mpc = 3.08567758 x 10$^{22}$.
    $H={\frac {1Mpc}{t_{sec}}}$

### Black body peak frequency

    ${\frac {xe^{x}}{e^{x}-1}}-3=0,x=2.821439372...$
    $f_{peak}={\frac {k_{B}T_{bh}x}{h}}={\frac {x}{8\pi ^{2}{\sqrt {t_{age}}}t_{p}}}$

### 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 ~ 10$^{71}$ years;
    $t_{end}\sim 1.7x10^{-121}\sim 0.588x10^{121}$ 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/t$_{univ}$$^2$ ~ 1.6 x 10$^{-122}$, where t$_{univ}$ ~ 8 x 10$^{60}$ is the present expansion age of the universe in Planck time units. Attempts to explain why Ω ~ 1/t$_{univ}$$^2$ have relied upon ensembles of possible universes, in which all possible values of Ω are found .
The maximum temperature T$_{max}$ would be when t$_{age}$ = 1. What is of equal importance is the minimum possible temperature T$_{min}$ - that temperature 1 Planck unit above absolute zero, this temperature would signify the limit of expansion; t$_{age}$ = 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 10$^{73}$ yrs);
    $t_{end}=T_{max}^{4}\sim 1.014\;10^{123}$
The mid way point (T$_{universe}$ = 1K) would be when (about 108.77 billion years);
    $t_{u}=T_{max}^{2}\sim 3.18\;10^{61}$

### Spiral expansion

By expanding according to a Spiral of Theodorus pattern where each triangle refers to 1 increment to t$_{age}$, we can map the mass and volume components as integral steps of t$_{age}$ (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 t$_{age}$ then their precision would depend on t$_{age}$, for example using this progression when t$_{age}$ = 1, π$^2$ = 6;
    ${\frac {\pi ^{2}}{6}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots$