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

Further information: v:God (programmer) § Simulation Time

The (dimensionless) simulation clock-rate would be defined as the minimum discrete 'time variable' (t_age) 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 t_age, 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 t_age increment adds 1 unit of Planck time t_p, then in a 14 billion year old universe, numerically (note t_p has the units s, t_age is dimensionless)

t_age = t_p = 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.

table 1. 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 1061	0.428 1061	0.404 1061
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_age², 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

y-axis = mPa, x-axis = d_c2l_p (nm)}

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_c2l_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 10⁵⁴ t_p (2987 years), with Casimir length = 189.89nm and temperature T_BH = 6034 K.

Hubble constant

1 Mpc = 3.08567758 x 10²².

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

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 ~ 10⁷¹ 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_univ² ~ 1.6 x 10^-122, where t_univ ~ 8 x 10⁶⁰ is the present expansion age of the universe in Planck time units. Attempts to explain why Ω ~ 1/t_univ² 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 10⁷³ 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

Planck black-hole universe; Planck units mapped onto a Theodorus spiral giving universe mass, size, temperature per value of t_age

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 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, π² = 6;

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

External links

• The mathematical electron
• Physical constant anomalies
• Planck units as geometrical objects
• Programming relativity at the Planck scale
• Programming gravity at the Planck level
• The sqrt of Planck momentum
• The Programmer God
• The Simulation hypothesis
• Programming at the Planck scale using geometrical objects