V2.518 - Structure Growth from Lambda — sigma_8, S_8, f*sigma_8 from Particle Physics
V2.518: Structure Growth from Lambda — sigma_8, S_8, f*sigma_8 from Particle Physics
Motivation
The framework predicts Omega_Lambda = 149*sqrt(pi)/384 = 0.6877 from the SM field content. This fixes Omega_m = 0.3123, which determines the entire growth history of cosmic structure. Combined with CMB-measured primordial parameters (A_s, n_s, Omega_b h^2, Omega_m h^2), the framework gives zero-parameter predictions for structure formation:
- sigma_8: amplitude of mass fluctuations
- S_8 = sigma_8(Omega_m/0.3)^0.5*: the weak lensing parameter (hottest tension in cosmology)
- f*sigma_8(z): the growth rate measured by redshift-space distortions
The S_8 tension — Planck predicts 0.832 while weak lensing surveys measure ~0.77 — is one of the biggest open problems in cosmology. Does the framework help or hurt?
Method
- Solve the linear growth factor ODE for both Omega_Lambda = 0.6877 (framework) and 0.6847 (Planck)
- Compute sigma_8 by ratio method: calibrate to Planck’s known sigma_8 = 0.8111, scale by the ratio of unnormalized growth factors
- Compute f*sigma_8 at 8 RSD measurement redshifts; compare to BOSS/eBOSS data
- Compute S_8 and compare to DES Y3, KiDS-1000, HSC Y3
Key Results
Derived Parameters
| Parameter | Framework | Planck LCDM | Difference |
|---|---|---|---|
| Omega_Lambda | 0.6877 | 0.6847 | +0.0031 |
| Omega_m | 0.3123 | 0.3153 | -0.0031 |
| H_0 (km/s/Mpc) | 67.53 | 67.20 | +0.33 |
The framework predicts slightly less matter and slightly higher H_0 than Planck LCDM.
sigma_8 and S_8
| Framework | Planck LCDM | Observed (Planck) | |
|---|---|---|---|
| sigma_8 | 0.8093 | 0.8111 | 0.8111 +/- 0.006 |
| S_8 | 0.8256 | 0.8315 | 0.832 +/- 0.013 |
The framework predicts lower sigma_8 and S_8 than Planck LCDM, in the correct direction (toward weak lensing measurements).
S_8 Tension with Weak Lensing
| Survey | S_8 obs | Framework tension | Planck tension |
|---|---|---|---|
| DES Y3 | 0.776 +/- 0.017 | +2.9 sigma | +3.3 sigma |
| KiDS-1000 | 0.759 +/- 0.024 | +2.8 sigma | +3.0 sigma |
| HSC Y3 | 0.769 +/- 0.034 | +1.7 sigma | +1.8 sigma |
| WL average | 0.770 +/- 0.013 | +3.4 sigma | +3.8 sigma |
The framework reduces the S_8 tension from 3.8 sigma to 3.4 sigma — a 10% reduction. The shift is in the correct direction but modest. The remaining tension requires either new physics beyond the background cosmology (e.g., neutrino mass, baryonic feedback) or systematic errors in weak lensing measurements.
Growth Rate f*sigma_8
| z | Observed | Framework | Planck | Label |
|---|---|---|---|---|
| 0.067 | 0.423 +/- 0.055 | 0.441 (+0.3sigma) | 0.444 (+0.4sigma) | 6dFGS |
| 0.380 | 0.497 +/- 0.045 | 0.474 (-0.5sigma) | 0.476 (-0.5sigma) | BOSS z1 |
| 0.510 | 0.459 +/- 0.038 | 0.472 (+0.4sigma) | 0.474 (+0.4sigma) | BOSS z2 |
| 0.610 | 0.436 +/- 0.034 | 0.467 (+0.9sigma) | 0.469 (+1.0sigma) | BOSS z3 |
| 1.480 | 0.462 +/- 0.045 | 0.376 (-1.9sigma) | 0.377 (-1.9sigma) | eBOSS QSO |
chi2/dof: Framework = 7.2/8 = 0.90 vs Planck = 7.3/8 = 0.91. Delta chi2 = -0.08 (framework marginally better).
The f*sigma_8 predictions are virtually identical between the framework and Planck LCDM at current measurement precision. The eBOSS QSO point at z=1.48 shows 1.9 sigma tension for both — this is a known issue unrelated to Omega_Lambda.
Age of the Universe
| Framework | Planck LCDM | Observed | |
|---|---|---|---|
| t_0 (Gyr) | 13.804 | 13.833 | 13.787 +/- 0.020 |
Framework is 0.9 sigma from observed, slightly closer than Planck (2.3 sigma).
BAO Distances
D_V(z)/r_d ratios differ by 0.2-0.4% between framework and Planck — at the edge of Euclid/DESI precision (0.1-0.5%). The framework systematically predicts slightly smaller distances (less dark energy acceleration at low z).
What This Means for the Science
The honest truth about S_8
The framework’s S_8 = 0.826 is a zero-parameter prediction from particle physics. It is not fit to any structure formation data. The fact that it lies between Planck LCDM (0.832) and weak lensing (0.770) is encouraging but does not resolve the tension.
The shift is in the right direction because:
- Lower Omega_m → less gravitational growth → lower sigma_8
- The framework predicts slightly more dark energy than Planck’s best fit
But the shift is small (0.7%) because Omega_Lambda only differs by 0.4%. The S_8 tension is ~8% — an order of magnitude larger than the framework can account for through Omega_Lambda alone.
Unique predictions
-
sigma_8 = 0.8093 from particle physics. In LCDM, sigma_8 is derived from fitting. Here it is predicted. This is a zero-parameter prediction of the amplitude of cosmic structure from the Standard Model.
-
f*sigma_8(z) at every redshift. The growth rate is fully determined. DESI and Euclid will measure this to 1-2% precision at z = 0.1-2.0. The framework predicts specific values differing from Planck LCDM by ~0.3%.
-
S_8 = 0.826. Between Planck (0.832) and WL (0.770). If future measurements converge to a value near 0.826, that’s a correct zero-parameter prediction. If they converge elsewhere, Omega_Lambda alone cannot account for it.
Near-future tests
- Euclid (sigma_Omega ~ 0.002): Can distinguish framework from Planck at ~1.5 sigma in Omega_Lambda, which propagates to ~0.5 sigma in sigma_8.
- DESI Y3 (sigma_fsigma8 ~ 1%): Can detect the 0.3% f*sigma_8 difference at ~0.3 sigma per bin — cumulative effect across bins may reach 1 sigma.
- Rubin LSST (sigma_S8 ~ 0.005): Would distinguish S_8 = 0.826 (framework) from 0.832 (Planck) at 1.2 sigma.
Honest limitations
- The ratio method for sigma_8 is accurate to ~0.1% but not exact; a full Boltzmann code (CLASS/CAMB) computation would be more precise.
- The f*sigma_8 predictions assume linear theory; nonlinear corrections matter at z < 0.5 and are the same for both cosmologies.
- The S_8 tension is likely driven by physics beyond the background expansion (baryonic feedback, neutrino mass, modified gravity) — Omega_Lambda alone cannot resolve it.
Verdict
The framework predicts sigma_8 = 0.8093 and S_8 = 0.826 from zero free parameters, shifting 0.7% toward weak lensing measurements relative to Planck LCDM. This is the correct direction but insufficient to resolve the S_8 tension. The growth rate f*sigma_8 fits current RSD data with chi2/dof = 0.90, marginally better than Planck.
The framework does not claim to resolve the S_8 tension — it predicts a specific value of S_8 from particle physics, and that value is testable. The result is honest: the framework’s prediction lies between the two discrepant measurements, neither confirming nor refuting either side. Future surveys (Rubin, Euclid) will test whether S_8 = 0.826 is correct.