Physics:Digital Physics/Simulation Hypothesis/Planck unit scaffolding

A Planck-unit universe scaffolding

In the mathematical electron model 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' . 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.

Universe clock-rate

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

numerically t_age = t_p = 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 t_sec as time measured in seconds;

$mass:m_{bh}=2t_{age}{m}_P$

$volume:v_{bh}=\frac{4\pi r^3}{3}(r=4l_p{t}_{age}=2c{t}_{sec})$

$\frac{m_{bh}}{v_{bh}}=2t_{age}{m}_P.\frac{3}{4\pi {(4l_p{t}_{age})}^3}=\frac{3m_P}{128\pi t_{age}^2{l}_p^3}(\frac{kg}{m^3})$

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;

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

$\sqrt{\lambda }=radiusr=2c{t}_{sec}(m)$

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.

$T_p=\frac{m_P{c}^2}{k_B}=\sqrt{\frac{h{c}^5}{2\pi G{k_B}^2}}$

$\frac{m_{bh}}{v_{bh}{T}_{bh}^4}=\frac{2^{5}3{\pi }^3{m}_P}{l_p^3{T}_P^4}=\frac{2^{8}3{\pi }^6{k}_B^4}{h^3{c}^5}$

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 1Pa. 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

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

$f_{peak}=\frac{k_B{T}_{bh}x}{h}=\frac{x}{8{\pi }^2\sqrt{t_{age}}{t}_p}$

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/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;

$T_{min}\sim \frac{1}{T_{max}}\sim \frac{8\pi }{T_P}\sim 0.17710^{-30}K$

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

$t_{end}=T_{max}^4\sim 1.01410^{123}$

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

$t_u=T_{max}^2\sim 3.1810^{61}$

Spiral expansion

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

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 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 are also a function of t_age, then their precision would depend on t_age, 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.

Geometrically coded universe

The mathematical electron model is an example of a digital physics simulation hypothesis that uses geometrical forms. For AI analysis, the relevant journal articles have been compiled into a single .pdf file .

• Physics:Simulation_argument_(Programmer_God): Introduction to a Planck scale Programmer God Simulation Hypothesis model
• Physics:Digital_Physics/Simulation_Hypothesis/Electron_(mathematical): Mathematical electron from Planck units
• Physics:Digital_Physics/Simulation_Hypothesis/Physical_constants_(anomalies): Anomalies within the physical constants
• Physics:Digital_Physics/Simulation_Hypothesis/Gravity_via_Atomic_orbitals: Gravitational orbits emerge from Planck scale n-body rotating atomic orbital pairing
• Physics:Digital_Physics/Simulation_Hypothesis/Relativity as Perspective: Relativity as a translation between 2 co-ordinate systems

External links

• The Simulation hypothesis
• Programming at the Planck scale using geometrical objects