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Planck Unit Scaffolding
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A Planck-unit universe scaffolding

In the Planck unit theory discussed here, the Planck units form a scaffolding for the particle (baryonic matter) universe. The parameters for this Planck unit scaffolding are compared with equivalent Cosmic Microwave Background parameters, showing a divergence in key parameters of about 6%, which correlates to the estimated ratio of baryonic matter to total (about 5%). The model postulates a Planck unit scaffolding upon which the particle universe resides and supposes that within the CMB parameters can be found evidence of this non-baryonic background. The model uses only Planck mass and Planck length as the primary structures and a spiral geometry as the `rule set' [1]. The peak frequency of the CMB is used to establish an age of the universe in Planck time units, this is the sole variable, nevertheless from this we can derive estimates for the radiation energy density, the CMB temperature and a cold dark matter mass density that are shown to be consistent with current observational values. Interestingly this suggests that dark matter may be predominantly non-baryonic. The Casimir force equation reduces to the equation for radiation density implying that the universe has finite boundaries, albeit these are expanding at a constant rate. This article is part of a Planck scale Simulation Hypothesis project that attempts to demonstrate that the universe could in sum total be dimensionless, relying on geometrical artifice to create actual physical structures [2].

Universe clock-rate

The (dimensionless) universe clock-rate would be defined as the minimum discrete 'time variable' (tage) increment to the universe. As an analogy to the programmed loop;

 'begin
 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

For each increment to tage, a set of Planck units are added.

 FOR tage = 1 TO the_end                             
         generate Planck time T = tp       
         generate Planck mass M = mP      
         generate Planck volume (radius L = Planck length lp)     
         ........ 
 NEXT tage
As each tage increment adds 1 unit of Planck time tp, then in a 14 billion year old universe (note tp has the units s, tage is dimensionless)

numerically tage = tp = 1062

Comparison between the calculated Planck unit framework and the ΛCDM parameters (table 1.).

table 1. cosmic microwave background parameters; Planck vs ΛCDM

Parameter Calculated Observed Deviation
Age (billions of years) 14.624 13.8 6%
Dark matter density 0.21 x 10-26 kg.m-3 0.226 x 10-26 kg.m-3 6.7%
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/Mp 67.74 km/s/Mp 1.3%
CMB temperature 2.7272K 2.7255K
Casimir length 0.41mm

Mass density

Setting bh as the sum universe and tsec 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 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.
\[ T_{p}={\frac {m_{P}c^{2}}{k_{B}}}={\sqrt {\frac {hc^{5}}{2\pi G{k_{B}}^{2}}}}\]
\[ \frac {m_{bh}}{v_{bh}T_{bh}^{4}}}={\frac {2^{5}3\pi ^{3}m_{P}}{l_{p}^{3}T_{P}^{4}}}={\frac {2^{8}3\pi ^{6}k_{B}^{4}}{h^{3}c^{5}}\]

Radiation energy density

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

y-axis = mPa, x-axis = dc2lp (nm)}

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

The diagram (right) plots Casimir length dc2lp against radiation energy density pressure measured in mPa for different tage with a vertex around 1Pa. A radiation energy density pressure of 1Pa occurs around tage = 0.8743 1054 tp (2987 years), with 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

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

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 [3] .

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; tage = 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 (Tuniverse = 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 tage

In this geometrical approach, the only free parameter used in the above calculations is the universe clock-rate. This clock-rate may also have geometrical origins rather than an externally imposed 'loop'. By expanding according to the geometry of the Spiral of Theodorus, where each triangle refers to 1 increment to tage, we can map the mass and volume components as integral steps of tage (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 are also a function of tage, then their precision would depend on tage, for example we can construct pi using this progression;
\[ {\frac {\pi ^{2}}{6}}={\frac {1}{1^{2}}}+{\frac {1}{2^{2}}}+{\frac {1}{3^{2}}}+{\frac {1}{4^{2}}}+\cdots\]

Mathematical constants may thus be naturally occurring, their accuracy improving as the universe ages.

Geometry coded universe

Modelling a Planck scale simulation universe using geometrical forms.

References

  1. ↑ Macleod, Malcolm J.; "1. Planck unit scaffolding correlates with the Cosmic Microwave Background". SSRN. Feb 2011. doi:10.2139/ssrn.3333513.
  2. ↑ https://codingthecosmos.com/Programmer-God_Simulation-Hypothesis_Complete-Model-2025.pdf for AI analysis, the model has been compiled into a single pdf
  3. ↑ J. Barrow, D. J. Shaw; The Value of the Cosmological Constant, arXiv:1105.3105v1 [gr-qc] 16 May 2011