2 Oct 2025: Grok4’s calculations and comments.
In the context of the base-2 model, the universe is conceptualized as emerging from discrete, fundamental units at the Planck scale—specifically, Planck spheres generated one per unit of Planck time (approximately 5.39 × 10-44 seconds). This raw generation rate equates to about 1.85 × 1043 spheres per second, which can be thought of as an ultra-high-frequency “thrust” or driving force for expansion at the most infinitesimal level. It’s like the universe’s primordial engine, pumping out these basic building blocks continuously from the very start (Notation-0 in the model).
However, this thrust doesn’t translate directly to the large-scale expansion we observe today. Over cosmic time, as spheres accumulate through 202 base-2 doublings (each notation representing a multiplicative step in scale, time, and structure), the total number of spheres builds up enormously—roughly 8 × 1060 by the present day, corresponding to the age of the universe divided by Planck time (t / t_p). This accumulation effectively dilutes or scales down from the initial rate.
To see how this works mathematically:
- The raw Planck-scale rate is 1 / t_p ≈ 1.85 × 10⁴³ s⁻¹.
- The scaled effective rate becomes (1 / t_p) / (t / t_p) = 1 / t ≈ 2.3 × 10^{-18} s^{-1}, where t is the current age of the universe (about 4.35 × 10^{17} seconds).
- Converting this to standard Hubble constant units (km/s/Mpc) involves multiplying by the distance conversion factor (1 Mpc ≈ 3.086 × 10^{19} km), yielding approximately 71 km/s/Mpc—remarkably close to observed values from supernovae measurements (~73 km/s/Mpc).
In essence, the model posits that the universe’s expansion isn’t a smooth, continuous velocity but an emergent property from these discrete Planck-scale additions. The “thrust” remains constant at the fundamental level, but its relative impact on the overall scale decreases as the universe grows larger/older, much like how the growth rate in an exponentially expanding system (via doublings) slows in relative terms over time. This scaling could help explain tensions in H_0 measurements, as different observational methods might probe averages over these discrete steps differently, without needing exotic physics like inflation.
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To convert 18.5 tredecillion Planckspheres per second to Hubble km/s/Mpc
The rate of 18.5 tredecillion (1.85 × 10⁴³) spheres per second can be interpreted as a frequency of 1.85 × 10⁴³ s⁻¹.
To convert a rate in s⁻¹ to km/s/Mpc (the units of the Hubble constant H₀), multiply by the conversion factor derived from the definition of 1 Mpc = 3.086 × 10¹⁹ km:
H = (1.85 × 10⁴³ s⁻¹) × (3.086 × 10¹⁹ km/Mpc) ≈ 5.71 × 10⁶² km/s/Mpc.
How to arrive at the solution:
- Recognize the sphere rate as equivalent to 1 / tₚ, where tₚ (Planck time) ≈ 5.39 × 10⁻⁴⁴ s, so 1 / tₚ ≈ 1.85 × 10⁴³ s⁻¹.
- The Hubble constant in s⁻¹ (H_s) relates to its value in km/s/Mpc (H_km) by H_s = H_km / (3.086 × 10¹⁹).
- Rearranging, H_km = H_s × (3.086 × 10¹⁹).
- Substitute the rate for H_s and compute the product.
Note that this direct conversion yields an enormous value due to the vastly different scaling at the Planck level versus the macroscopic universe. In the base-2 model context, the effective current expansion rate is this raw rate divided by the total accumulated spheres (≈ 8 × 10⁶⁰, or t / tₚ), yielding ≈ 2.3 × 10⁻¹⁸ s⁻¹, which converts to ≈ 71 km/s/Mpc—aligning with observed H₀ values. The Planck-scale rate thus represents a fundamental “thrust” scaled down over cosmic time.