Programming cosmic microwave background parameters for Planck scale Simulation Hypothesis modeling

Universe simulation hypothesis models operating at the Planck scale can use the Planck units as the scaffolding upon which particle structures are embedded ^[1]. Data address in universe 'space' are represented by a Planck micro black-hole defined as a discrete entity that comprises the Planck units for mass m_P, length l_p and time t_p. The simulation clock-rate can therefore be measured in units of Planck time (or equally Planck mass or Planck length ...). Consequently, from 1 variable, the age of the universe for example, other variables can be calculated. In the following table, the cosmic microwave background peak frequency value = 160.2GHz was used.

Mass density

For each expansion step of universe 'space', to the sum (black-hole bh) universe is added a Planck micro black-hole which includes; a unit of Planck time t_p, Planck mass m_P and Planck (spherical) volume (Planck length = l_p), such that we can calculate the mass, volume and so density of this sum black-hole for any chosen time by setting t_age; the age of the black-hole universe as measured in units of Planck time or t_sec the age of the black-hole universe as measured in seconds.

${\displaystyle mass:\;m_{bh}=2t_{age}m_{P}}$

${\displaystyle volume:\;v_{bh}={\frac {4\pi r^{3}}{3}}\;\;\;(r=4l_{p}t_{age}=2ct_{sec})}$

${\displaystyle {\frac {m_{bh}}{v_{bh}}}={2t_{age}m_{P}}.\;{\frac {3}{4\pi {(4l_{p}t_{age})}^{3}}}={\frac {3m_{P}}{128\pi t_{age}^{2}l_{p}^{3}}}\;({\frac {kg}{m^{3}}})}$

Gravitation constant G as Planck units;

${\displaystyle G={\frac {c^{2}l_{p}}{m_{P}}}}$

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

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

${\displaystyle \lambda ={\frac {3c^{2}}{8\pi Gp}}=4c^{2}t_{sec}^{2}}$

${\displaystyle {\sqrt {\lambda }}=radius\;r=2ct_{sec}\;(m)}$

Temperature

Measured in terms of Planck temperature = T_P;

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

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.

${\displaystyle T_{p}={\frac {m_{P}c^{2}}{k_{B}}}={\sqrt {\frac {hc^{5}}{2\pi G{k_{B}}^{2}}}}}$

${\displaystyle {\frac {m_{bh}}{v_{bh}T_{bh}^{4}}}={\frac {2^{5}3\pi ^{3}m_{P}}{l_{p}^{3}T_{P}^{4}}}={\frac {2^{8}3\pi ^{6}k_{B}^{4}}{h^{3}c^{5}}}}$

Radiation energy density

From Stefan Boltzmann constant σ_SB

${\displaystyle \sigma _{SB}={\frac {2\pi ^{5}k_{B}^{4}}{15h^{3}c^{2}}}}$

${\displaystyle {\frac {4\sigma _{SB}}{c}}.T_{bh}^{4}={\frac {c^{2}}{1440\pi }}.{\frac {m_{bh}}{v_{bh}}}}$

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

${\displaystyle {-F_{c}}{A}={\frac {\pi hc}{480{(d_{c}2l_{p})}^{4}}}}$

if d_c = 2 π √t_age then the Casimir force equates to the radiation energy density.

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

A radiation energy density pressure of 1Pa gives t_age = 0.8743 10⁵⁴ t_p (2987 years), Casimir length = 189.89nm and temperature T_BH = 6034 K.

Hubble constant

1 Mpc = 3.08567758 x 10²².

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

Black body peak frequency

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

${\displaystyle f_{peak}={\frac {k_{B}T_{bh}x}{h}}={\frac {x}{8\pi ^{2}{\sqrt {t_{age}}}t_{p}}}}$

Entropy

${\displaystyle S_{BH}=4\pi t_{age}^{2}k_{B}}$

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;

${\displaystyle t_{end}\sim 1.7x10^{-121}\sim 0.588x10^{121}}$ 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 ^[3] .

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 (the Black hole could expand no further). For example, taking the inverse of Planck temperature;

${\displaystyle T_{min}\sim {\frac {1}{T_{max}}}\sim {\frac {8\pi }{T_{P}}}\sim 0.177\;10^{-30}\;K}$

This then gives us a value for the final age in units of Planck time (about 0.35 x 10⁷³ yrs);

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

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

${\displaystyle t_{u}=T_{max}^{2}\sim 3.18\;10^{61}}$

Spiral expansion

Planck black-hole universe; discrete Planck micro black-holes mapped onto a Theodorus spiral

By expanding according to a Spiral of Theodorus pattern where each triangle refers to 1 step (additional micro black-hole), 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 (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 when t_age = 1, π² = 6;

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

External links

• Simulation Argument -Nick Bostrom's website
• Our Mathematical Universe: My Quest for the Ultimate Nature of Reality -Max Tegmark
• Micro black hole
• Planck particle
• Simulation hypothesis
• Mathematical electron
• Relativity in the Planck level
• Gravity and Planck mass
• Digital time in a simulation hypothesis
• the Source Code of God; a programming approach -online resource

References