# Black-hole (Planck)

Method for programming Planck Black-hole Cosmic Microwave Background parameters for use in Simulation Hypothesis models

A primordial Planck size micro black-hole that incorporates the Planck units (units of Planck mass, Planck time, Planck length ... ) as fundamental (non-divisible) discrete units is premised . Computer Simulation Hypothesis models that use digital time instead of analog (continuous) time where the time variable (tage) is measured in units of Planck time can calculate comparable cosmic microwave background parameters (see table) with good approximation at low computational cost by modelling the scaffolding of the universe as expanding in Planck size micro black-holes per increment in tage. .

cosmic microwave background parameters; calculated vs observed
Parameter Calculated Observed
Age (billions of years) 14.624 13.8
Age (units of Planck time) 0.428 1061
Mass density 0.21 x 10-26 kg.m-3 0.24 x 10-26 kg.m-3
Radiation energy density 0.417 x 10-13 kg.m-1.s-2 0.417 x 10-13 kg.m-1.s-2
Hubble constant 66.86 km/s/Mpc 67 (ESA's Planck satellite 2013)
CMB temperature 2.727K 2.7255K
CMB peak frequency 160.2GHz 160.2GHz
Entropy CEH 2.3 x 10122kB 2.6 x 10122kB
Casimir length 0.42mm

#### Mass density

For each increment to time tage, to the sum black-hole is added a Planck black-hole comprising a unit of Planck time tp, Planck mass mP and Planck (spherical) volume (Planck length = lp), such that we can then calculate the mass, volume and so density of the sum black-hole for any chosen unit of time, (tsec as the age of the black-hole as measured in seconds).

$t_{p}={\frac {2l_{p}}{c}}$
$mass:\;m_{universe}=2t_{age}m_{P}$
$volume:\;v_{universe}={\frac {4\pi r^{3}}{3}}\;\;\;(r=4l_{p}t_{age}=2ct_{sec})$
${\frac {m_{universe}}{v_{universe}}}={2t_{age}m_{P}}.\;{\frac {3}{4\pi {(4l_{p}t_{age})}^{3}}}={\frac {3m_{P}}{128\pi t_{age}^{2}l_{p}^{3}}}\;({\frac {kg}{m^{3}}})$

Gravitation constant G as Planck units;

$G={\frac {c^{2}l_{p}}{m_{P}}}$
${\frac {m_{universe}}{v_{universe}}}={\frac {3}{32\pi t_{sec}^{2}G}}$

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 = 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.

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

From Stefan Boltzmann constant σSB

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

#### Casimir formula

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

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

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

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

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.

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

#### 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/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 (the black hole 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);

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

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