Discovery Engine Guide

Overview

The Extreme-Field QED Simulator has evolved into a comprehensive discovery engine for systematically searching for new physics in gravitational coupling, vacuum structure, and spacetime interactions. This guide explains how to use the framework to:

Compute precise predictions from known physics (QED + GR)
Parameterize ignorance via κ-ansätze for unknown couplings
Run systematic parameter sweeps across source configurations
Derive experimental constraints from null results
Publish discovery reach curves and κ-upper-bounds

Core Methodology: κ-Constraint Framework

The Big Idea

Most experiments searching for new physics yield null results. Instead of discarding these, we turn them into meaningful constraints:

"We didn't see anything, therefore new physics coupling κ must be smaller than X."

How It Works

Baseline Prediction: Compute gravitational wave strain h_EM from pure electromagnetic stress-energy using Einstein's equations:
```
h_ij(t, R) ≈ (2G/c⁴R) Q̈_ij(t)
```
where Q_ij is the mass-energy quadrupole moment.
Anomalous Coupling: Model unknown physics as an effective modification to stress-energy:
```
T^μν_eff = T^μν_EM + κ F^μν[fields, ...]
```
where:
- κ = dimensioned coupling strength (to be constrained)
- F^μν = "functional" chosen from physics-motivated ansätze
Detection Threshold: Compare h_EM to detector sensitivity (LIGO, LISA, quantum sensors):
- If h_EM < threshold → signal is undetectable with known physics alone
- Anomalous coupling must boost strain to threshold for detection
κ-Constraint: Solve for required κ:
```
h_total ≥ h_threshold  ⟹  κ ≥ κ_required
```
Null result (no detection) implies: κ < κ_required

Ansätze Catalog

The framework provides 7 physics-motivated ansätze:

1. Vector Potential Squared (`vector_potential_squared`)

F ∝ |A|² η^μν

Motivation: Beyond-SM theories with direct gravity–vector-potential coupling
Units: [κ] = J·s²/(kg·m⁴)
Use Case: Test gauge-field–gravity interactions

2. Field Invariant F² (`field_invariant_F2`)

F ∝ (B² - E²/c²)² / (16ε₀²)

Motivation: Lorentz-invariant coupling to first EM field invariant
Units: Dimensionless or [κ] = J·m³
Use Case: Standard QFT-like couplings

3. Photon Number (`photon_number`)

F ∝ n_γ = u/(ħω)

Motivation: Quantum gravity proposals scaling with coherent photon number
Units: [κ] = J·s
Use Case: High-Q cavity experiments with ~10¹⁵ photons

4. Axion-Like (Parity-Odd) (`axion_like`)

F ∝ E·B

Motivation: Axion–photon coupling g_aγγ a F F̃ where F̃^μν = ε^μνρσ F_ρσ/2
Units: [κ] = J·m/(V·T) maps to g_aγγ × a₀
Use Case: ALP searches; compare to CAST/ADMX constraints g_aγγ < 10⁻¹⁰ GeV⁻¹
Parity: Violating (pseudoscalar)

5. Dilaton-Like (Scalar-Tensor) (`dilaton_like`)

F ∝ T^μ_μ  (with QED corrections: α/π × F² / E_s²)

Motivation: Brans-Dicke, dilaton gravity coupling scalar φ to trace
Units: Dimensionless or [κ] = ⟨φ⟩ (scalar VEV)
Use Case: Tests of scalar-tensor modifications; conformal anomaly effects

6. Chern-Simons-Like (Lorentz-Violating) (`chern_simons_like`)

F ∝ A·B  (proxy for k^μ ε_μνρσ A^ν F^ρσ)

Motivation: Standard Model Extension (SME) Lorentz violation
Units: Depends on k^μ; typically dimensionless or 1/GeV
Use Case: SME bounds |k^μ| < 10⁻¹⁵ – 10⁻³²
Parity: Violating

7. Spatial Gradient (`spatial_gradient`)

F ∝ ∇·(|A|²)

Motivation: Localized coupling near sources/boundaries
Units: Similar to vector_potential_squared
Use Case: Geometry-dependent signatures

Detector Sensitivities

The framework includes realistic noise models for major gravitational wave detectors:

Detector	Frequency Band	Strain ASD @ 100 Hz	Integration Time
LIGO O1	10 Hz – 5 kHz	~3×10⁻²² Hz⁻¹/²	1 hour
aLIGO Design	10 Hz – 5 kHz	~1×10⁻²² Hz⁻¹/²	1 hour
LISA	0.1 mHz – 1 Hz	~1×10⁻²⁰ Hz⁻¹/²	1 year
Einstein Telescope	1 Hz – 10 kHz	~1×10⁻²⁴ Hz⁻¹/²	1 hour
Quantum Sensor (Aspirational)	1 Hz – 1 MHz	~1×10⁻³⁰ Hz⁻¹/²	1 second
Tabletop Interferometer	0.1 Hz – 10 kHz	~1×10⁻¹⁸ Hz⁻¹/²	100 seconds

SNR Calculation: Matched-filter SNR via band integration:

SNR² = 4 ∫ |h̃(f)|² / S_n(f) df

where S_n(f) = [ASD(f)]² is the power spectral density.

Detection Criterion: SNR ≥ 5 (5-sigma) for confident detection.

Running Experiments

Single Experiment

python scripts/run_experiments.py --config configs/experiments.yaml --experiment experiment_1

Output:

results/experiment_1.h5: HDF5 file with strain timeseries, power, quadrupole, metrics
Terminal summary: h_rms, P_avg, κ_required for each detector

Parameter Sweep

python scripts/run_experiments.py --config configs/sweeps.yaml --sweep sweep_E0_colliding_pulses

Output:

results/sweeps/E0_colliding_pulses/: Directory with individual HDF5 per sweep point
sweep_E0_colliding_pulses_summary.csv: Consolidated table with columns:
- sweep_value: Parameter value (e.g., E₀ = 10¹⁴ V/m)
- R_10.0m_h_rms, R_10.0m_P_avg: Strain and power at 10 m
- kappa_<ansatz>_<detector>: Required κ for each ansatz–detector pair
sweep_E0_colliding_pulses_plots.png: Auto-generated 2×2 plot grid

Example Sweep Configurations (in configs/sweeps.yaml):

sweep_E0_colliding_pulses: Peak field 10¹³ → 10¹⁶ V/m (7 points)
sweep_waist_colliding_pulses: Focal spot 0.5 → 20 μm (6 points)
sweep_Q_cavity: Cavity Q-factor 10⁵ → 10⁹ (5 points)

Creating Custom Sweeps

YAML Template

my_custom_sweep:
  description: "Brief description of sweep purpose"
  
  geometry:
    type: "colliding_pulses"  # or "cavity_mode", "plasma_toroid", etc.
    parameters:
      E0: 1.0e15  # Base value; will be overridden by sweep
      waist: 5.0e-6
      wavelength: 800.0e-9
      pulse_duration: 10.0e-15
      collision_delay: 0.0
      polarization: "linear"
  
  grid:
    x_min: -20.0e-6
    x_max: 20.0e-6
    nx: 31  # Balance resolution vs. compute time
    # ... y, z similarly
  
  time_evolution:
    t_start: -15.0e-15
    t_end: 15.0e-15
    nt: 51
  
  physics:
    heisenberg_euler: true
    pair_production: true
    qed_stress_energy: true
  
  gravitational:
    observer_distances: [1.0, 10.0, 100.0]  # Multiple R for 1/R scaling check
    use_spectral_derivatives: true  # FFT-based (more stable for noisy Q̈)
    apply_TT_projection: true  # Transverse-traceless gauge
  
  anomalous_coupling:
    ansatze: ["axion_like", "dilaton_like", "field_invariant_F2"]
    kappa_scan: false
  
  detection_thresholds:
    LIGO: 1.0e-21
    aLIGO: 1.0e-22
    Einstein_Telescope: 1.0e-23
    quantum_sensor: 1.0e-30
  
  # SWEEP SPECIFICATION
  sweep_parameter: "geometry.parameters.E0"  # Dot-separated path to nested param
  sweep_values: [1.0e14, 3.0e14, 1.0e15, 3.0e15, 1.0e16]  # List of values
  
  output:
    sweep_dir: "results/sweeps/my_custom_sweep"
    save_fields: false  # Set true to save snapshots (large files!)
    save_quadrupole: true
    save_strain: true
    save_metrics: true

Sweep Parameter Paths:

"geometry.parameters.E0": Peak electric field
"geometry.parameters.waist": Gaussian beam waist
"geometry.parameters.Q_factor": Cavity quality factor
"grid.nx": Grid resolution (use cautiously; affects all dimensions)
"time_evolution.nt": Time resolution

Interpreting Results

Expected Scaling Laws

Strain vs. Energy Density:
```
h ∝ Q̈ ∝ T₀₀ ∝ E²  ⟹  h ∝ E₀²
```
Doubling E₀ → 4× strain
Radiated Power:
```
P_GW ∝ (d³Q/dt³)² ∝ E₀⁴
```
Doubling E₀ → 16× power
Distance Scaling:
```
h(R) = h(R₀) × (R₀/R)
```
10× distance → 1/10 strain
κ-Constraint Scaling:
```
κ_required ∝ (h_threshold / h_EM) ∝ 1/E₀²
```
Higher field → smaller required κ → stronger constraint

Physical Benchmarks

Schwinger Field: E_s = m²c³/(eℏ) ≈ 1.3×10¹⁸ V/m

Below E_s: Perturbative QED (Heisenberg-Euler)
Near E_s: Non-perturbative pair production dominates

Planck Energy Density: ρ_Pl = c⁷/(Gℏ) ≈ 10¹¹³ J/m³

Quantum gravity effects expected near this scale

PVLAS Limit (vacuum birefringence):

Observed: null at Δn ~ 10⁻²⁰
QED prediction (1-loop HE): Δn ~ 10⁻²³ at E ~ 10⁵ V/m, B ~ 5 T
Framework can reproduce and extend to higher fields

Publication-Quality Outputs

From Sweep CSV

import pandas as pd
import matplotlib.pyplot as plt

# Load sweep results
df = pd.read_csv('results/sweeps/E0_colliding_pulses/sweep_E0_colliding_pulses_summary.csv')

# Discovery reach plot
fig, ax = plt.subplots(figsize=(10, 7))
ax.loglog(df['sweep_value'], df['kappa_axion_like_LIGO'], 'o-', label='Axion-like (LIGO)')
ax.loglog(df['sweep_value'], df['kappa_dilaton_like_aLIGO'], 's-', label='Dilaton-like (aLIGO)')
ax.set_xlabel('Peak Field E₀ [V/m]', fontsize=14)
ax.set_ylabel('κ_required for Detection', fontsize=14)
ax.set_title('Discovery Reach: New Physics Coupling Constraints', fontsize=16)
ax.legend()
ax.grid(True, alpha=0.3, which='both')
ax.axhline(1e30, color='red', linestyle='--', alpha=0.5, label='Naturalness scale')
plt.savefig('discovery_reach.png', dpi=300, bbox_inches='tight')

From HDF5 (Single Experiment)

import h5py
import numpy as np

with h5py.File('results/colliding_pulses_qed.h5', 'r') as f:
    h_t = f['gravitational/R_10.0m/h_timeseries'][:]  # (T, 3, 3)
    P_t = f['gravitational/R_10.0m/P_timeseries'][:]  # (T,)
    
    # Extract h_+ polarization (assuming TT gauge, +z propagation)
    h_plus = h_t[:, 0, 0]  # xx component
    
    # Plot
    fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(12, 8))
    ax1.plot(h_plus, linewidth=0.8)
    ax1.set_ylabel('Strain h_+')
    ax1.grid(True, alpha=0.3)
    
    ax2.semilogy(P_t, linewidth=1.0, color='darkred')
    ax2.set_xlabel('Time Step')
    ax2.set_ylabel('Radiated Power [W]')
    ax2.grid(True, alpha=0.3)
    
    plt.tight_layout()
    plt.savefig('strain_power_timeseries.png', dpi=300)

Best Practices

1. Grid Resolution

Start coarse (nx=15) for parameter space exploration
Refine (nx=31–51) for publication-quality runs
Check convergence: does doubling nx change h_rms by <1%?

2. Time Evolution

Cover at least 2–3 pulse durations for transients to pass
Use nt ≥ 50 for clean FFT frequency spectra
Enable use_spectral_derivatives=true for stability

3. Ansatz Selection

Use axion_like for parity-odd searches
Use dilaton_like for scalar-tensor tests
Use field_invariant_F2 as baseline (Lorentz-invariant, parity-even)
Run all ansätze in sweeps to build comprehensive constraint matrix

4. Detector Choice

LIGO/aLIGO: Near-term realistic (10–5000 Hz)
LISA: Space-based (mHz range), ideal for low-frequency sources
Quantum sensors: Aspirational (h~10⁻³⁰), sets ultimate limits
Include multiple detectors to span frequency space

5. Computational Efficiency

Disable save_fields: true unless debugging (huge files)
Use pair_production: false if E₀ ≪ E_s (saves time)
Run sweeps on clusters: each point is independent (embarrassingly parallel)

Troubleshooting

Q: Strain values are ~10⁻⁶⁰, way below any detector. Is this wrong?

A: No! Extreme-field QED produces tiny quadrupole moments due to:

Small spatial scale (μm focal spots)
Cancellation in symmetric field configurations
Radiation distance R (strain ∝ 1/R)

Solutions:

Increase E₀ (strain ∝ E₀²)
Increase source volume (larger waist or cavity)
Bring detector closer (R = 1 m vs. 10 m)
Use asymmetric geometry (rotating capacitor, counter-rotating rings)

Q: κ_required is enormous (10⁵⁰). What does this mean?

A: The anomalous coupling would need to be unnaturally large to produce a detectable signal. This is good—it means:

Known physics (EM + GR) predicts nearly zero signal
Any detection would indicate strong new physics
Null result constrains κ < 10⁵⁰ (useful if theory predicts κ ~ 10⁴⁰)

Q: Why do axion-like constraints differ from field_invariant_F2?

A: Different ansätze have different:

Units (κ dimensions vary)
Symmetries (axion is parity-odd; F² is parity-even)
Field dependence (E·B vs. E²)

Compare κ-constraints within the same ansatz across different experiments.

Q: Sweep plots show non-monotonic κ vs. E₀. Bug?

A: Likely due to:

Frequency spectrum shifting out of detector band as E₀ changes
Destructive interference in certain field configurations
Numerical noise in low-signal regime

Check:

Plot h_rms vs. E₀ (should be monotonic)
Plot peak frequency vs. E₀
Increase grid resolution or integration time

Next Steps: Advanced Topics

1. Bayesian κ Inference

Instead of point estimates, compute posterior distributions:

P(κ | data, model) ∝ P(data | κ, model) × P(κ)

Use MCMC (emcee, PyMC3) to sample posteriors and plot credible intervals.

2. Multi-Geometry Comparisons

Run same ansatz across:

Colliding pulses (pulsed, high-field)
Cavity modes (CW, high-Q)
Plasma toroid (rotating, low-frequency)

Build "discovery matrix": best geometry × ansatz pair for each physics scenario.

3. Experimental Proposals

Use κ-constraint plots to guide:

Laser facility designs (target E₀, focus)
Cavity specifications (Q, finesse)
Detector requirements (bandwidth, integration time)

Optimize cost–benefit: which upgrade path improves κ-reach most?

4. SME Coefficient Mapping

For Lorentz-violating ansätze, map κ → SME coefficients:

c^μν: CPT-even, parity-even
k^μν: CPT-odd, parity-odd

Compare to existing bounds from:

Astrophysical birefringence (radio polarization)
Laboratory tests (spin-polarized torsion balance)
Collider limits (photon sector)

References

Theoretical Background

Heisenberg-Euler QED:
- W. Heisenberg & H. Euler, Z. Phys. 98, 714 (1936)
- J. Schwinger, Phys. Rev. 82, 664 (1951)
- Two-loop: Gies & Karbstein, JHEP 1703, 108 (2017)
Gravitational Wave Production:
- Landau & Lifshitz, Classical Theory of Fields (Ch. 11)
- Misner, Thorne & Wheeler, Gravitation (Ch. 36)
- Rothman & Boughn, "Can Gravitons Be Detected?" Found. Phys. 36, 1801 (2006)
Axion-Photon Coupling:
- CAST: Nature Phys. 13, 584 (2017)
- ADMX: Phys. Rev. Lett. 120, 151301 (2018)
Standard Model Extension:
- Kostelecký & Mewes, Phys. Rev. D 80, 015020 (2009)

Experimental Limits

PVLAS: Phys. Rev. D 77, 032006 (2008)
LIGO O1: Phys. Rev. Lett. 116, 061102 (2016)
Quantum Sensing: Tobar et al., Phys. Rev. D 104, 064054 (2021)

Support

Issues: https://github.com/arcticoder/extreme-field-qed-simulator/issues
Examples: configs/experiments.yaml, configs/sweeps.yaml
Tests: pytest tests/ (validate installation)
Notebooks: notebooks/experiment_analysis.ipynb (visualization templates)

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