Test 28 — Matter–Radiation Equality Timing — A Framework for a Theory of Everything

Test 28 — Matter–Radiation Equality Timing Test

Does the redshift at which matter density overtakes radiation density, computed from the measured density parameters, land where the cosmic microwave background and large-scale-structure data independently say it must — and does the framework's "structure can finally grow" granularity reading add anything beyond restating that standard result?

Agrees Matter–radiation equality

What we're checking

Observable: Redshift of matter–radiation equality, z_eq — the moment the density of matter first overtakes the density of radiation.
Standard cosmology: z_eq ≈ 3402 ± 26 — a firmly settled, directly measured number.
Granularity: z_eq ≈ 3422 — computed from the matter-to-radiation density ratio, 1 + z_eq = Ω_m/Ω_r.
Measured: z_eq = 3402 ± 26 (Planck's own direct fit to the cosmic microwave background).
The epoch: Roughly 50,000 years after the beginning — deep in the radiation era, just as matter takes the lead.
Verdict: AGREES 3422 vs 3402 ± 26 — a match to 0.6%, well inside the measured uncertainty.

For the first fifty thousand years, the universe belongs to light. Radiation presses outward, sets the tempo, and holds matter in check. Then, quietly, the balance tips. As everything expands and cools, radiation thins faster than matter does — and at a redshift near 3400, matter finally outweighs the light that once ruled. That crossover has a precise address in the sky, and it is one of the best-pinned numbers in all of cosmology.

There are two honest roads to that address. Take the textbook ratio of matter density to radiation density, feed it the measured amounts of each, and the equality lands at z_eq ≈ 3422. Then set that aside and ask the cosmic microwave background directly — read the crossover straight off the pattern of hot and cold spots — and Planck answers 3402 ± 26. Two entirely different ways of looking at the same instant, and they land within 0.6% of each other.

This is not an open question with no settled answer — it is the opposite. The moment matter's gravity begins to steer the cosmos is measured, agreed upon, and reproduced here to better than a percent. Both roads arrive at the same redshift, the same fifty-thousand-year mark, the same turning point. The number matches. The verdict is agreement.

What we did: two independent ways of pinning down when matter took over from radiation as the universe's dominant stuff agree with each other. Plug Planck's measured density numbers (\(\Omega_m h^2\), \(\Omega_r h^2\)) into the textbook ratio and you get \(z_\text{eq}\approx3422\); Planck's own direct fit to the CMB gives \(z_\text{eq}=3402\pm26\) independently. Those two numbers land within about 0.6% of each other — well inside the expected uncertainty. On top of that arithmetic, this framework reads the moment as the point where matter density differences stop being washed out and become permanent, growable structure. That reading fits the data; it doesn't add a new number, and it isn't a derivation of \(\Omega_m\) or \(\Omega_r\) from first principles — those remain measured inputs, same as they are for every other cosmology. Agreement here builds confidence in the picture; it doesn't prove it.

1. Verdict

The number, both ways

Number we’re testing: Redshift of matter–radiation equality, z_eq — the moment matter density first overtakes radiation density
Standard cosmology & measured: Standard cosmology here is the measurement itself — z_eq ≈ 3402 ± 26 — a firmly settled, directly measured number (Planck's own direct fit to the CMB) · z_eq = 3402 ± 26 (Planck 2018 direct fit); cross-checked by the matter power spectrum turnover scale (eBOSS DR16)
This framework (granularity): The independent read: z_eq ≈ 3422 — computed from the density ratio 1 + z_eq = Ω_m/Ω_r using Planck 2018 parameters (Ω_m h² = 0.1430, Ω_r h² = 4.177 × 10^-5 with N_eff = 3.044)
Agreement: 3422 vs 3402 ± 26 — a match to 0.6%, well inside the measured uncertainty (consistency check — shared inputs)

Check the source → the calculation shown on this page (Data Used · Calculation Summary)

Agrees. The matter-radiation equality redshift computed from Planck 2018 density parameters, \(z_\text{eq} \approx 3422\), agrees with the Planck-reported directly-fit value \(z_\text{eq} = 3402 \pm 26\) to within about 0.6%, well inside the combined uncertainty of the two independent ways of extracting it (density-parameter ratio vs. direct CMB-peak fit). This is a consistency check of measured cosmological parameters against each other, not a derivation of \(\Omega_m\), \(\Omega_r\), or \(z_\text{eq}\) from the framework's own machinery — all inputs are Planck-collaboration measured quantities, same as any other cosmology would use.

2. Tested Claim

The precise granularity claim under test: the transition from radiation-dominated to matter-dominated expansion — defined by \(1+z_\text{eq} = \Omega_m/\Omega_r\), i.e. the redshift at which the matter and radiation energy densities are equal — must occur at the redshift/time independently inferred from the cosmic microwave background acoustic-peak structure and the turnover scale of the matter power spectrum. The framework reads this transition as a granularity event: below \(z_\text{eq}\), matter density perturbations are no longer suppressed by radiation pressure and can grow via gravitational instability, so density contrasts become "stable, recordable distinctions" (the seeds of large-scale structure) for the first time, rather than transient radiation-pressure-supported oscillations.

3. Data Used

Planck 2018 cosmological parameters (Planck Collaboration VI, 2020, "Planck 2018 results. VI. Cosmological parameters," A&A 641, A6, arXiv:1807.06209), base-\(\Lambda\)CDM, TT,TE,EE+lowE+lensing+BAO column (Table 2):
- \(\Omega_m h^2 = 0.1430 \pm 0.0011\)
- \(h = 0.6766 \pm 0.0042\) (so \(\Omega_m = 0.3127 \pm 0.0060\))
- directly-fit equality redshift: \(z_\text{eq} = 3402 \pm 26\)
CMB temperature (photon density input): \(T_{\text{CMB},0} = 2.7255 \pm 0.0006\) K, Fixsen (2009), "The Temperature of the Cosmic Microwave Background," ApJ 707, 916 (COBE/FIRAS), giving the standard \(\Omega_\gamma h^2 = 2.47\times10^{-5}\) used in all subsequent \(\Omega_r h^2\) calculations.
Effective neutrino number: \(N_\text{eff} = 3.044\), Bennett et al. (2021), "Towards a precision calculation of \(N_\text{eff}\) in the Standard Model," JCAP 04 (2021) 073 (arXiv:2012.02726) — refines the historical \(N_\text{eff}=3.046\), used here in the standard \(\Omega_r h^2 = \Omega_\gamma h^2(1+0.2271\,N_\text{eff})\) relation for the neutrino contribution to radiation density.
Large-scale-structure cross-check (independent of CMB): the matter power spectrum turnover scale near the equality wavenumber \(k_\text{eq}\), as measured by galaxy-clustering and BAO surveys (e.g. eBOSS DR16, Alam et al. 2021, Phys. Rev. D 103, 083533, arXiv:2007.08991), is consistent with the same \(z_\text{eq}\) inferred from the CMB — used as the "compare to CMB and large-scale-structure inferred equality" step the test-suite procedure requires.

4. Calculation Summary

Governing relation (standard \(\Lambda\)CDM background cosmology):

\[ 1+z_\text{eq} \;=\; \frac{\Omega_m}{\Omega_r}\;=\;\frac{\Omega_m h^2}{\Omega_r h^2}. \]

Step 1 — radiation density parameter:

\[ \Omega_r h^2 \;=\; \Omega_\gamma h^2\,\bigl(1 + 0.2271\,N_\text{eff}\bigr) \;=\; 2.47\times10^{-5}\times\bigl(1 + 0.2271\times3.044\bigr) \;=\; 4.177\times10^{-5}. \]

Step 2 — equality redshift from the parameter ratio:

\[ 1+z_\text{eq} \;=\; \frac{0.1430}{4.177\times10^{-5}} \;=\; 3423, \qquad z_\text{eq}^{\text{(computed)}} \;\approx\; 3422. \]

Step 3 — compare to the Planck-reported direct fit: \(z_\text{eq}^{\text{(Planck, direct)}} = 3402\pm26\). The fractional difference is

\[ \frac{3422-3402}{3402} \;\approx\; 0.6\%, \]

which is within the combined uncertainty budget once the small approximations in Step 1 (e.g. treating \(N_\text{eff}\) neutrinos as exactly massless and the photon-to-neutrino temperature ratio as the instantaneous-decoupling value) are accounted for — this is the expected, well-documented level of agreement between the parameter-ratio method and Planck's own direct multi-parameter fit, not a discrepancy.

Cross-check against large-scale structure: the equality wavenumber implied by \(z_\text{eq}\approx3400\) sets the turnover scale of the linear matter power spectrum; galaxy-survey and BAO measurements of this turnover (eBOSS DR16 and earlier SDSS/BOSS results) are consistent with the CMB-inferred value within their (larger) survey uncertainties, satisfying the "compare to CMB and large-scale-structure inferred equality" step independently of the CMB-only number above.

Sensitivity check (Step 3 of the minimum calculations): extra relativistic species (a higher \(N_\text{eff}\)) or an early-dark-energy component would each shift \(\Omega_r h^2\) or \(\Omega_m h^2\) and hence \(z_\text{eq}\); Planck's own joint fit already marginalizes over \(N_\text{eff}\) and finds no significant deviation from the Standard-Model value 3.044, so no such shift is required or favored by the current data (Planck 2018 VI, §6.3; Bennett et al. 2021).

5. Granularity Interpretation

On the framework's reading, matter-radiation equality is the point at which matter density-contrast distinctions become stable and recordable in the sense the validity rule requires: before \(z_\text{eq}\), radiation pressure suppresses the growth of matter overdensities (they oscillate rather than collapse); after \(z_\text{eq}\), matter perturbations grow via gravitational instability essentially unimpeded, laying down the density field that later CMB decoupling (Test 29) and structure-formation surveys (Tests 34–38) record. The fossil record this test relies on is twofold and mutually consistent: the location of the equality "turnover" imprinted in the linear matter power spectrum, and the amplitude/phase relationships of the CMB acoustic peaks (which depend on the ratio of radiation to matter density at recombination). Nothing new is created at this epoch that is not already tracked by \(\Omega_m\) and \(\Omega_r\) — the transition is a change in which distinctions can grow, not the appearance of a new object requiring its own accounting.

6. Gate Routing

This test informs the structure-growth regime gate. It feeds forward into Test 29 (Recombination Visibility Function), Test 31 (CMB Acoustic Peak Geometry, which is directly sensitive to the matter/radiation ratio at recombination), and Tests 34–36 (primordial power spectrum to large-scale structure, BAO, and \(\sigma_8\)/\(S_8\)), all of which depend on this equality epoch having occurred at the value fixed here.

7. Failure Mode

Not applicable in the strict sense — the two independent extractions of \(z_\text{eq}\) (parameter-ratio calculation and Planck's direct multi-parameter fit) agree to within 0.6%, and the large-scale-structure cross-check is consistent within its (larger) uncertainties. For completeness, the specific failure modes this test screened for (per the test suite's guardrails), and confirmed absent:

No hidden-sector or early-dark-energy shift: Planck's joint fit finds no significant deviation of \(N_\text{eff}\) from the Standard-Model value, so no additional relativistic or early-dark- energy component is required to explain the data, and none is smuggled in here.
No measured-value-as-derived confusion: \(\Omega_m h^2\), \(\Omega_r h^2\), \(h\), and \(N_\text{eff}\) are all explicitly flagged as Planck-collaboration observed givens, not outputs of the framework's own geometric or granularity machinery.
No single-method reliance: the CMB parameter-ratio value and the independent large-scale-structure turnover-scale value are cross-checked against each other rather than one number being taken on faith.

8. Next Action

Data lookup / cross-check, not derivation: (a) carry \(z_\text{eq}\approx3400\) forward as a fixed input to Tests 29, 31, and 34–36 rather than re-deriving it there; (b) if the framework's technical program ever proposes an extra relativistic species, early dark energy, or a hidden sector with cosmologically relevant density, re-run this same ratio-vs-direct-fit comparison to confirm the proposal does not shift \(z_\text{eq}\) outside the Planck-allowed range; (c) no further routing is needed for this test — it resolved as a consistency check between two independent extractions of an already-measured quantity.

Bottom line

\(\Omega_m\), \(\Omega_r\), and the resulting \(z_\text{eq}\) all belong to standard \(\Lambda\)CDM cosmology as measured by Planck, and this framework uses the same numbers. Our reading of matter-radiation equality — the moment matter-density differences become stable enough to grow into structure — fits comfortably alongside that data. It's agreement between two ways of looking at the same measurement, not an independent confirmation from new physics.

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