Slope Law Beyond Equilibrium
Experiment V2.05: Slope Law Beyond Equilibrium
Status: COMPLETE
Goal
Characterize how the slope law breaks down away from exact KMS (thermal equilibrium). Specifically: for an adiabatic FRW background with slowly varying Hubble parameter H(t), measure the deviation of the slope-law temperature from the instantaneous Gibbons-Hawking temperature T_GH = H/(2pi).
Model
Hubble profile: H(t) = H_0 exp(-eps_H H_0 t)
- |Hdot(0)|/H(0)^2 = eps_H (the adiabatic parameter at the measurement center)
- H(t) > 0 for all t (avoids sign issues)
Adiabatic detector response: The detector response is a Gaussian-weighted average of instantaneous KMS responses:
F_adi(Omega) = integral ds w(s) F_KMS(Omega; T(s))where w(s) = exp(-s^2/sigma^2) / (sqrt(pi) sigma) and T(s) = H(s)/(2pi).
This follows from the adiabatic decomposition of the two-point function: G+(tau, tau’) ~ G+_KMS(tau-tau’; H((tau+tau’)/2)) in the Jacobian change to center-of-mass and relative time coordinates.
Key Findings
1. Exact de Sitter Limit (Phase 1 — 12 H_0 values)
At eps_H = 0, the slope law holds to numerical precision:
| H_0 | T_GH | T_extracted | Deviation |
|---|---|---|---|
| 0.50 | 0.0796 | 0.0796 | 2.1 x 10^-5 |
| 2.55 | 0.4051 | 0.4051 | 2.1 x 10^-5 |
| 5.00 | 0.7958 | 0.7958 | 2.1 x 10^-5 |
Power law: Q_t ~ H_0^1.000000, residual < 10^-15. The slope law is exact for exact KMS states (confirming V2.04).
2. Deviation Scales Quadratically (Phase 2 — Key Result)
At fixed H_0 = 2.0, sigma = 4.0:
| eps_H | eta = eps_H H_0 sigma | deviation |
|---|---|---|
| 0.000 | 0.00 | 2.1 x 10^-5 |
| 0.001 | 0.008 | 1.4 x 10^-5 |
| 0.005 | 0.040 | 8.6 x 10^-4 |
| 0.010 | 0.080 | 3.5 x 10^-3 |
| 0.020 | 0.160 | 1.4 x 10^-2 |
| 0.050 | 0.400 | 9.4 x 10^-2 |
| 0.100 | 0.800 | 4.7 x 10^-1 |
Best fit: deviation ~ eta^2.15 (essentially quadratic).
The research plan predicted deviation ~ eps_H (first-order). The actual result is second-order: deviation ~ eta^2 = (eps_H H_0 sigma)^2.
Why second-order? The symmetric Gaussian switching function w(s) = w(-s) kills the first-order correction. The linear term in T(s) = T_0(1 - eps_H H_0 s + …) averages to zero under the symmetric weight. Only the quadratic variance term survives:
T_eff ~ T_0 (1 + O(eta^2))This makes the slope law a better thermometer than expected — the errors are second-order in the adiabatic parameter, not first-order.
3. Slope Law Degradation (Phase 3)
The power law Q_t ~ H_0^alpha degrades as eps_H increases:
| eps_H | alpha | |alpha - 1| | Residual | |-------|-------|-------------|----------| | 0.00 | 1.000 | ~10^-15 | ~10^-16 | | 0.01 | 1.006 | 6.3 x 10^-3 | 2.1 x 10^-3 | | 0.02 | 1.026 | 2.5 x 10^-2 | 8.7 x 10^-3 | | 0.05 | 1.171 | 1.7 x 10^-1 | 6.0 x 10^-2 | | 0.10 | 1.621 | 6.2 x 10^-1 | 1.8 x 10^-1 | | 0.20 | 2.142 | 1.14 | 1.5 x 10^-1 |
Two effects compete:
- alpha shifts from 1: The power law still approximately holds but with wrong exponent (temperature extraction biased).
- Residual grows: The power law itself breaks down (the relationship between Q_t and H_0 is no longer a simple power law).
4. Universal Scaling Collapse (Phase 4 — Most Important)
The deviation is a universal function of eta = eps_H x H_0 x sigma:
| H_0 | eps_H | eta | deviation |
|---|---|---|---|
| 0.5 | 0.02 | 0.04 | 8.61 x 10^-4 |
| 1.0 | 0.01 | 0.04 | 8.61 x 10^-4 |
| 2.0 | 0.005 | 0.04 | 8.61 x 10^-4 |
| 0.5 | 0.10 | 0.20 | 2.22 x 10^-2 |
| 1.0 | 0.05 | 0.20 | 2.22 x 10^-2 |
| 1.0 | 0.10 | 0.40 | 9.44 x 10^-2 |
| 2.0 | 0.05 | 0.40 | 9.44 x 10^-2 |
Points with the same eta give identical deviations (to 4+ significant figures), confirming that eta is the unique control parameter.
Approximate formula:
deviation ~ 0.55 eta^2 = 0.55 (eps_H H_0 sigma)^2The Slope Law as a Thermometer
The slope law Q_t = C_Q T is a valid thermometer when the adiabatic condition eta << 1 is satisfied:
| eta = eps_H H_0 sigma | Deviation | Thermometer quality |
|---|---|---|
| < 0.04 | < 0.1% | Excellent |
| 0.04 - 0.15 | 0.1% - 1% | Good |
| 0.15 - 0.4 | 1% - 10% | Fair |
| > 0.4 | > 10% | Poor |
Physical interpretation of eta:
- eps_H = |Hdot|/H^2 controls how fast H changes
- H_0 sigma is the measurement duration in Hubble units
- eta = eps_H H_0 sigma = how much H changes during the measurement
The thermometer works when H doesn’t change much during the measurement.
What This Proves
-
The slope law is a genuine thermometer, not just a mathematical identity. It extracts the correct temperature from QFI alone, with no reference to the metric or temperature.
-
The errors are second-order in the adiabatic parameter. This is BETTER than the research plan expected (which predicted first-order). The symmetric detector switching function provides natural error cancellation.
-
Universal scaling: The deviation depends only on eta = eps_H H_0 sigma, not on eps_H or H_0 independently. This is a testable prediction.
-
Applicability criterion: The slope law is accurate to 1% when eta = eps_H H_0 sigma < 0.15, i.e., when the Hubble parameter changes by less than ~15% during the measurement.
Correction to Research Plan
The research plan (V2.05 spec) predicted: “epsilon ~ |Hdot|/H^2”
The actual result is: epsilon ~ (|Hdot|/H^2 x H sigma)^2 = eta^2
This is second-order rather than first-order, because:
- The Gaussian switching function is symmetric: w(t) = w(-t)
- The first-order correction <T(s) - T_0>_w ~ integral w(s) Hdot s ds = 0
- Only the quadratic variance term survives
Files
| File | Purpose | Tests |
|---|---|---|
src/adiabatic_frw.py | Adiabatic FRW model, deviation measurement, scaling | |
tests/test_adiabatic.py | Validation tests | 15/15 |
run_experiment.py | Full 4-phase experiment |
Next Steps -> V2.06
V2.06 will test cross-family universality: the slope law should give the same temperature regardless of which UDW detector variant is used (qudit, Gaussian, etc.), provided all use the same Q_t = F_timing^(1/3) definition.