Toy Model by Grok4

Challenging the Hubble Constant in New Ways:
From Planck-Scale “Thrust” to Observed Expansion

In the base-2 model of the universe, expansion emerges from discrete Planck spheres generated at one per unit of Planck time (~5.39 × 10-44 seconds). This creates a raw rate of ~1.85 × 1043 spheres per second (18.5 tredecillion), acting as a fundamental “thrust” or primordial engine at the smallest scales. However, this ultra-high-frequency drive doesn’t directly match the large-scale Hubble constant (H₀) we measure. Instead, as spheres categorized through 202 base-2 doublings (scaling up size, time, and complexity from the Planck level to the present day), the total reaches ~8 × 1060 spheres—equivalent to the universe’s age divided by Planck time (t / tₚ). This accumulation dilutes the relative impact of the constant thrust, turning it into the slower, emergent expansion observed today. The model suggests this discrete process could resolve H₀ tensions (e.g., ~73 km/s/Mpc from supernovae vs. ~67 from CMB) without exotic physics like inflation.

Step-by-Step Breakdown

Here’s a step-by-step breakdown of the calculations, starting with a direct (unscaled) conversion of the raw rate to km/s/Mpc units for context, then focusing on the scaling that yields the realistic ~71 km/s/Mpc:

  1. Identify the raw Planck-scale rate as a frequency: Recognize the sphere generation rate as equivalent to 1 / tₚ, where tₚ (Planck time) ≈ 5.39 × 10-44 s. This gives 1 / tₚ ≈ 1.85 × 1043 s^{-1}—the fundamental “thrust” frequency.
  2. Understand the Hubble constant’s unit relationship: The Hubble constant in inverse seconds (H_s, a frequency-like rate) relates to its standard value in km/s/Mpc (H_km) by H_s = H_km / (3.086 × 1019), where 3.086 × 1019 km is the distance in 1 Mpc.
  3. Rearrange for conversion: To express a rate in s^{-1} as km/s/Mpc, rearrange to H_km = H_s × (3.
  4. Compute the direct (unscaled) conversion: Substitute the raw rate for H_s: H_km = (1.85 × 1043 s^{-1}) × (3.086 × 1019 km/Mpc) ≈ 5.71 × 1062 km/s/Mpc. This enormous value highlights the vast scaling difference between Planck-level processes and macroscopic observations—it’s not the effective rate we see.
  5. Introduce scaling by accumulation (key dilution step): The raw thrust is constant, but its relative effect slows as spheres build up. The total accumulated spheres is t / tₚ, where t is the universe’s current age (~4.35 × 1017 s, or ~13.8 billion years). This gives ~8 × 1060 spheres (4.35 × 1017 / 5.39 × 10-44 ≈ 8.07 × 1060). The effective rate scales down by this factor: (1 / tₚ) / (t / tₚ) = 1 / t ≈ 2.3 × 10-18 s. Here, the “thrust” is diluted over cosmic time, like exponential growth slowing in relative terms as the system enlarges.
  6. Apply the unit conversion to the scaled rate (final step to H₀): Now use the scaled effective rate as H_s: H_km = (2.3 × 10-18 seconds × 3.086 × 1019 km/Mpc ≈ 71 km/s/Mpc. This value aligns closely with local measurements (~70-73 km/s/Mpc), showing how the model’s discrete additions average out to the observed expansion.

In summary, the Planck-scale thrust represents the universe’s unchanging foundational drive, but cosmic accumulation scales it down to the effective H₀ we measure—bridging quantum discreteness with large-scale continuity.

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Mathematics and geometries to expand upon the toy model:

  1. Base-2 notation from the Planck base units to the cosmic scale: https://81018.com/chart/
  2. Stabilizing geometries: The Four Primary Irrational Numbers Stabilizes Spheres from the Planck to the cosmological scale: https://81018.com/planck-polyhedral-core/ https://81018.com/ross/
  3. Perfect Domains: Notations-0 to Notation-50: https://81018.com/starts-8/
  4. Geometries of quantum fluctuations
  5. A Lagrangian is slowly emerging.

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