Elaboration on Distance Scales in De Sitter Space Formation

The core conflict is between the immense scale of the cosmos and the microscopic scale of binding forces. The numbers involved are so extreme they defy everyday intuition.

The Governing Scale: The Cosmological Horizon

In a De Sitter universe (like our distant future or an idealized inflationary epoch), the most important length scale is the de Sitter radius or cosmological horizon length (L_dS). This is related to the cosmological constant (Λ).

Order of Magnitude: L_dS ~ 1 / H, where H is the Hubble constant. For our future de Sitter state, H is tiny, making L_dS astronomically large.

Approximate Value: L_dS ≈ 10⁶¹ in Planck units, or about 10²⁶ meters (over 10 billion light-years).

The Disruptive Effect: This horizon defines the maximum distance two particles can ever communicate. The expansion is characterized by this scale, and its "strength" is given by the Gibbons-Hawking temperature (T_GH), which is inversely proportional to L_dS.

Step 1: Forming Hadrons (Success)

Binding Force Scale (Strong Nuclear):

Distance: Operates at ~ 1 femtometer (10⁻¹⁵ m).

Energy: The characteristic energy scale (e.g., the pion mass) is ~100 MeV.

Disruptive Cosmological Scale:

Gibbons-Hawking Temperature: T_GH ~ ħH / k_B ≈ 10^-30 K, which is ~10⁻⁴³ eV.

Analysis: The ratio of binding energy to disruptive thermal energy is: (100 MeV) / (10^-43 eV) = 10⁵¹. The cosmological expansion provides an energy perturbation fifty-one orders of magnitude smaller than the strong force's binding energy. It's like comparing the energy of a flying mosquito to the energy required to shatter a planet. The strong force is completely oblivious to the expansion on these scales. Protons and neutrons are eternally stable.

Step 2: Forming Atomic Nuclei (Failure)

Here, the challenge is not the stability of the nucleus itself, but getting the ingredients (protons and neutrons) together in the first place.

Binding Distance Scale (Nuclear):

Nucleons must come within ~1 femtometer (10⁻¹⁵ m) to be captured by the strong force.

Disruptive Cosmological Scale:

The expansion acts over the initial separation distance of particles. In an empty De Sitter space, the average distance between any two hadrons is of the order of the horizon scale, ~10²⁶ m.

Analysis: This is the crux of the problem. To form a nucleus, two particles starting even a modest distance apart (say, 1 meter) must fight against an expansion that wants to push them to a final separation of (1 meter) × e^Ht. The Coulomb Barrier and the fact that the mean free path for a collision is vastly larger than the horizon size L_dS means particles will never find each other. The distance scale mismatch is fatal: You are trying to orchestrate a meeting on a scale of 10⁻¹⁵ m in a universe where the natural distance between objects is 10²⁶ m. It's like trying to get two specific ants to meet on opposite sides of the observable universe.

Step 3: Forming Atoms (Failure)

This step highlights a more subtle but equally decisive failure. Even if you were given a pre-formed, stable nucleus (like a proton), you cannot bind an electron to it.

Binding Distance Scale (Atomic):

The Bohr radius, the characteristic size of a hydrogen atom, is ~5 × 10⁻¹¹ m.

Binding Energy: 13.6 eV.

Disruptive Cosmological Scale:

Again, the expansion acts over cosmological timescales. The quantum mechanical wavefunction of the electron is spread out over the Bohr radius. The expansion of space "stretches" this wavefunction, effectively increasing the average electron-proton separation over time.

Analysis: This is a quantum mechanical tunneling/ionization process driven by cosmology. The electron is bound in a potential well with a depth of 13.6 eV. The De Sitter expansion provides a persistent, time-dependent perturbation. Calculations show that the electron's wavefunction evolves into a continuum state over a sufficiently long time. The ionization timescale for a hydrogen atom in De Sitter space is proportional to the horizon scale 1/H (~10¹⁷ seconds). This means that on a timescale comparable to the current age of the universe, a lone hydrogen atom will be ionized by the cosmic expansion. The bond, while strong on human timescales, is metastable on cosmological timescales.

Summary: The Scale Mismatch

Formation Step	Binding / Interaction Scale (meters)	Disruptive Cosmological Scale (meters)	Result
1. Hadrons	10−15 m (Strong Force)	Horizon: 1026 m	SUCCESS. Binding energy is incomparably stronger.
2. Nuclei	10−15 m (Interaction Range)	Initial Separation: ~1026 m	FAILURE. Particles cannot find each other across the cosmic void.
3. Atoms	10−11 m (Bohr Radius)	Horizon/Ionization Time: 1026 m / 1017 s	FAILURE. Quantum bonds are ionized by persistent cosmic expansion.

Conclusion: The analysis demonstrates that in a pure De Sitter universe, the hierarchy of structure formation is not just slowed down but is fundamentally and permanently arrested after the formation of individual hadrons. The exponential expansion severs the causal connections necessary for larger structures to ever assemble, making the universe a forever-empty, ever-cooling void dominated by the energy of the vacuum itself.