A Framework for a Theory of Everything

Test 29 — Recombination Visibility Function Test

Do neutral hydrogen atoms and freely-streaming photons — the pairing that becomes the cosmic microwave background — form at the redshift, over the duration, and with the optical-depth profile that standard recombination physics and the Planck 2018 last-scattering measurement say they must?

For about 380,000 years the young universe was an opaque fog — a plasma so hot that light could not travel a single step without being knocked sideways by a free electron. Then it cooled just enough for protons and electrons to lock together into the first neutral hydrogen atoms, the fog lifted in one cosmic instant, and the light that had been trapped since the beginning finally streamed free. That flash is the oldest light we can ever see — the cosmic microwave background — and it carries a single, sharp timestamp: the redshift at which it was released.

Two completely different ways of reading that moment arrive at the very same number. Standard cosmology runs the century-old ionization physics — the Saha and Peebles equations — and finds the fog clears at redshift z* = 1089.92. This framework asks a different question entirely: when does a hydrogen atom and its escaping photon first become a stable, public record — something the universe can carry forward and we can later read? It lands on the identical redshift, with the same measured inputs and the same thickness to the last-scattering surface. Planck’s own measurement pins that number to z* = 1089.92 ± 0.25. Two roads, one moment, one number.

Here is the honest fine print, stated plainly: this is not the framework predicting a new number out of the geometry. Both roads use the same measured atomic physics, so the agreement is a consistency reading — the record-cost picture reproduces the standard answer rather than forcing it independently. That still matters. It shows the framework’s way of counting “when does a distinction become recordable” lines up exactly with the textbook physics of when the universe went transparent — no seam, no contradiction, at the very edge of the observable past.

1. Verdict

Agrees with existing models. Ordinary recombination physics — hydrogen atoms forming and photons breaking free of the plasma, run through the standard Saha-then-Peebles ionization calculation — puts this transition at \(z_*=1089.92\), about 380,000 years after the Big Bang, matching the Planck 2018 measurement of the last-scattering surface to high precision. We use the same measured inputs everyone else does (\(\Omega_b h^2\), \(T_{\rm CMB,0}\)) and reach the same number. Our framework adds one extra reading on top — that this is the moment the atom-and-photon record becomes stable and public — and that reading fits comfortably alongside the standard result without changing any of the numbers.

2. Tested Claim

The precise granularity claim under test: as the universe cools, hydrogen recombination (\(e^- + p \rightarrow \mathrm{H} + \gamma\)) must occur, and the photon mean free path must grow past the Hubble radius (photon decoupling), at a redshift \(z_*\) and over a finite-thickness "last-scattering surface" set by ordinary atomic physics plus the measured baryon-to-photon ratio and expansion history — not at some other, framework-tuned epoch. The framework interprets this transition as the point where two new stable, recordable distinctions are minted simultaneously: (a) a neutral, bound hydrogen atom as opposed to a free electron+proton pair, and (b) a freely propagating photon that no longer Thomson-scatters and can therefore carry an undisturbed, encodable record (the CMB) forward for the observer to read billions of years later. A qualitative match to "atoms and light decouple somewhere in the early universe" is not sufficient; the redshift, duration, and optical-depth profile must match the measured visibility function.

3. Data Used

• Photon-decoupling redshift and last-scattering thickness: Planck Collaboration 2018, "Planck 2018 results. VI. Cosmological parameters," A&A 641, A6 (arXiv:1807.06209) — TT,TE,EE+lowE+lensing base-ΛCDM chains give \(z_* = 1089.92 \pm 0.25\) (redshift at which the photon optical depth \(\tau(z_*)=1\)) and \(z_{\rm drag} = 1059.94 \pm 0.30\) (baryon-drag epoch); the angular size of the sound horizon at last scattering, \(\theta_* = (1.04109 \pm 0.00030)\times10^{-2}\) rad, is measured to 0.03% precision and is the single most constraining number in the whole Planck fit.
• Baryon density (recombination input): Planck 2018, \(\Omega_b h^2 = 0.02237 \pm 0.00015\) (A&A 641, A6, Table 2) — sets the baryon-to-photon ratio \(\eta_b \approx 6.1\times10^{-10}\) that controls the Saha-suppression delay of recombination relative to the naive \(kT \sim 13.6\) eV expectation.
• CMB monopole temperature: Fixsen (2009), ApJ 707, 916, COBE/FIRAS, \(T_{\rm CMB,0} = 2.7255 \pm 0.0006\) K — used to convert \(z_*\) into the temperature at last scattering, \(T(z_*) = T_{\rm CMB,0}(1+z_*)\).
• Last-scattering surface thickness: Planck 2018 and standard recombination-code (RECFAST/ CosmoRec/HyRec, e.g. Chluba & Thomas 2011, MNRAS 412, 748) results give a visibility-function full width of \(\Delta z \approx 195\) (roughly \(\Delta z/z_* \approx 0.18\)), corresponding to a finite-duration "surface" rather than an instantaneous transition — standard result, not distinctive.
• Reionization optical depth (downstream, distinguished from \(z_*\)): Planck 2018, \(\tau_{\rm reio} = 0.0544 \pm 0.0073\) (A&A 641, A6) — the much later (\(z\sim7\)–\(10\)) partial re-ionization by first stars/galaxies, kept explicitly separate here from the \(z_*\approx1090\) recombination optical-depth-unity surface that this test concerns.
• Hydrogen binding energy: \(E_1 = 13.6057\) eV (CODATA), standard atomic physics input to the Saha equation, not a framework output.

4. Calculation Summary

Window definition (per README Minimum Calculation 1): \(W = \){ age \(\approx 372{,}000\) yr (Planck 2018 best-fit; often rounded to "~380,000 years" in popular accounts), redshift \(z_*\approx1090\), temperature \(T(z_*) \approx 2973\) K, density/expansion regime: matter-dominated (matter-radiation equality already passed, per Test 28, at \(z_{\rm eq}\approx3400\)), relevant interaction: Thomson scattering of photons off free electrons, competing against Hubble expansion and hydrogen recombination removing free electrons }.

Why recombination happens at \(kT \ll 13.6\) eV, not \(kT\sim13.6\) eV (Saha suppression): the naive expectation is that hydrogen recombines when the thermal photon energy drops to the binding energy, \(kT\sim13.6\) eV. The actual measured temperature at \(z_*\) is:

This is a factor of \(13.6/0.256 \approx 53\) below the hydrogen binding energy — recombination is delayed to much lower temperature than the naive binding-energy estimate because of the enormous photon-to-baryon ratio (\(\eta_b^{-1}\sim1.6\times10^9\) photons per baryon): even at \(kT\ll13.6\) eV, the exponential tail of the photon Planck distribution above 13.6 eV still contains enough photons per baryon to keep re-ionizing newly-formed atoms, via the Saha equation

This factor-of-~50 suppression is a textbook, well-established result (e.g. Peebles 1968, ApJ 153, 1; standard cosmology texts such as Dodelson & Schmidt, "Modern Cosmology," 2nd ed., ch. 3) — it is exactly the kind of "rate/threshold check against a naive expectation" the README's Minimum Calculation 3 asks for, and it passes: the measured \(z_*\) is consistent with the Saha-suppressed calculation, not with the naive \(kT\sim13.6\) eV estimate (which would predict a much higher, wrong redshift \(z\sim6000\) if the photon-to-baryon ratio were ignored).

Decoupling condition (photon mean free path vs. Hubble): photons decouple once the Thomson scattering rate \(\Gamma = n_e \sigma_T c\) drops below the Hubble expansion rate \(H(z)\). Because \(n_e \propto x_e(z)\,(1+z)^3\) falls steeply as \(x_e\) collapses from \(\sim1\) toward \(\sim10^{-4}\) over a narrow redshift range (the Peebles-equation "recombination" itself, following the initial Saha collapse), \(\Gamma/H\) crosses unity at \(z_*\approx1090\) — consistent, within the precision of the full Boltzmann-code treatment (RECFAST/CosmoRec/HyRec), with the Planck-measured \(z_*=1089.92\pm0.25\).

Finite-thickness check (visibility function width): because \(x_e(z)\) does not drop instantaneously, the "surface" of last scattering has finite width. Standard recombination codes give \(\Delta z \approx 195\), i.e. \(\Delta z/z_*\approx0.18\) — a ~18% fractional thickness, not a razor-thin surface. This finite width is itself a measured/derived feature of the visibility function (its peak location is \(z_*\); its width is \(\Delta z\)) and is what sets the CMB's photon diffusion (Silk) damping scale, independently cross-checked against the Planck TT damping-tail data (used in Test 31, CMB Acoustic Peak Geometry). No inconsistency is found between the recombination-code width and the Planck-fit damping tail.

Cross-epoch consistency (README Minimum Calculation 5): the same \(\Omega_b h^2=0.02237\) used here is the identical value used in the BBN light-element test (Test 23) and the baryon-to-photon-ratio test (Test 21) — no separate, inconsistent baryon density is introduced for this test. Matter-radiation equality (Test 28, \(z_{\rm eq}\approx3400\)) occurs safely before \(z_*\approx1090\), as required for the standard matter-dominated growth of perturbations assumed in the recombination/last-scattering calculation and in the subsequent acoustic-peak fit (Test 31).

5. Granularity Interpretation

On the framework's reading, recombination/last-scattering is the epoch at which two new distinctions become simultaneously stable and publicly recordable: (a) the neutral hydrogen atom, as a bound, stable configuration distinguishable from the free electron-proton plasma that preceded it; and (b) the free-streaming photon, no longer erased by Thomson rescattering, which becomes a high-fidelity compressed record — not a full copy of the plasma, but its statistical imprint (temperature, polarization, and their correlations) frozen in at the moment of decoupling and carried forward largely undisturbed for 13.8 billion years to the observer. This matches the framework's validity rule: prior to \(z_*\), a "distinction" between individual photon and electron trajectories could not be stabilized or encoded (each photon was rescattered \(\mathcal{O}(10^{3}\text{-}10^{4})\) times before decoupling); after \(z_*\), the distinction between free-streaming-photon and neutral-atom is stable at the window's physical resolution and becomes the cosmic microwave background — the compressed record actually used by every subsequent CMB observation.

6. Gate Routing

This test informs the atom-and-photon record decoupling gate, part of the broader question of how and when the universe starts keeping stable, readable records (Encoding / Observation). It depends on and is consistent with Test 21 (baryon-to-photon ratio), Test 23 (BBN light elements, same \(\Omega_b h^2\)), and Test 28 (matter-radiation equality, which must precede \(z_*\)); it feeds forward into Test 30 (CMB blackbody/spectral distortions), Test 31 (CMB acoustic peak geometry, which uses the same \(z_*\) and sound horizon), and Test 32 (CMB polarization and the separate, later reionization optical depth \(\tau_{\rm reio}\), which this test explicitly does not conflate with the \(z_*\) optical-depth-unity surface).

7. Failure Mode

Not applicable in the strict sense — the recombination timing and visibility-function width match the Planck 2018 constraints using only standard atomic physics and the measured baryon density as inputs. For completeness, the specific failure modes this test screened for (per the test suite's guardrails), and confirmed absent:

• No premature-appearance error: the calculation confirms recombination does not occur at the naive binding-energy temperature (\(kT\sim13.6\) eV, which would place it far too early, near \(z\sim6000\)); the Saha-suppression factor of ~53 correctly delays it to \(z_*\approx1090\), matching Planck.
• No field-versus-particle confusion: the "atom" and "free-streaming photon" here are genuinely stable, long-lived, individually trackable objects (mean atomic lifetime against re-ionization at \(z\ll z_*\) is cosmologically long; photon mean free path exceeds the Hubble radius after decoupling) — not a field-theoretic degree of freedom mistaken for a stable particle.
• No measured-value-as-derived confusion: \(z_*\), \(\Omega_b h^2\), \(T_{\rm CMB,0}\), and the hydrogen binding energy are all explicitly flagged as imported measured/atomic-physics inputs, not framework outputs.
• No conflation of \(z_*\) with reionization: the much later, partial reionization optical depth \(\tau_{\rm reio}=0.0544\pm0.0073\) (Test 32's concern) is kept separate from the optical-depth-unity surface \(z_*\) that defines recombination/last-scattering.

8. Next Action

Data lookup / cross-check, not derivation: (a) carry \(z_*=1089.92\), \(\Delta z\approx195\), and \(\theta_*=1.04109\times10^{-2}\) rad forward as fixed inputs to Test 31 (CMB acoustic peak geometry) rather than re-deriving them there; (b) if the framework's technical program ever proposes a modified baryon density, extra relativistic species, or a nonstandard recombination history, re-run a full recombination code (RECFAST/CosmoRec/HyRec) against the Planck likelihood to confirm the proposal does not shift \(z_*\) or \(\theta_*\) outside the measured uncertainty; (c) no dead-end routing is needed for this test — it resolved as a consistency check against firmly established recombination physics and the Planck 2018 measurement, not as an unresolved distinction/encoding question.

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