# Black-hole (Planck)

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 mP, length lp and time tp. 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, the other variables can be calculated.

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

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 tp, Planck mass mP and Planck (spherical) volume (Planck length = lp), such that we can calculate the mass, volume and so density of this sum black-hole for any chosen time by setting tage; the age of the black-hole universe as measured in units of Planck time or tsec 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 = TP;

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

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.

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

From Stefan Boltzmann constant σSB

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

#### Casimir formula

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

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

if dc = 2 π √tage 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 tage = 0.8743 1054 tp (2987 years), Casimir length = 189.89nm and temperature TBH = 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/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 [2] .

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 (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 (Tuniverse = 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 tage (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 tage then their precision would depend on tage, for example when tage = 1, π2 = 6;

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