Exploratory Phenomenology of Post-Merger Gravitational Wave Signatures through the Monad-Field Lens

Authors: Monad-Field Research Collaboration (CFD Framework)
Date: 2026-05-08
Status: Methodological demonstration – not a detection claim

Important: This work is exploratory and educational. It does not claim detection of gravitational wave echoes or validation of the Monad-Field ontology.

Abstract

We present an exploratory phenomenological pipeline designed to detect periodic or quasi-periodic structure in gravitational wave post-merger signals without assuming the standard General Relativistic (GR) ringdown model as the only possible ontology.

The pipeline is motivated by the Monad-Field (S/Ψ) framework, which replaces geometric curvature with a nonlinear, saturable substrate and predicts that the post-merger phase may exhibit echo-like features, power-law relaxation tails, or beat-frequency modulations.

Using a coherent stacking cross-correlation method, we analyze synthetic data generated from three distinct waveform families:

1. Standard GR exponential ringdown
2. Power-law amplitude decay (substrate relaxation)
3. Nonlinear beats from two close frequencies

All three families produce peaks in the delay-scan cross-correlation, but at different characteristic delays. This demonstrates that the pipeline is sensitive to generic oscillatory coherence, not uniquely to delayed echo copies.

We conclude that while the Monad-Field framework offers a coherent ontological alternative, current correlation-based methods require Bayesian model comparison, multi-detector coherence, and PSD-weighted matched filtering to discriminate between competing physical interpretations.

No claim of actual echo detection is made.

1. Introduction

The observation of gravitational waves from merging black holes (LIGO/Virgo, 2015–present) has spectacularly confirmed the exterior predictions of general relativity (GR). However, the GR interpretation of the interior of a black hole — a curvature singularity — remains an extrapolation beyond empirical reach.

The Monad-Field (CFD) framework proposes a different ontology: spacetime is not a geometric manifold but a nonlinear, finite-capacity substrate field S with excitation field Ψ.

In this view, what GR calls “ringdown” is the relaxation of substrate tension after a merger. The framework naturally suggests that the post-merger signal might contain:

• Scalar echo modes
• Power-law tail deviations from exponential decay
• Beat patterns arising from nonlinear coupling

This white paper summarises exploratory numerical experiments designed to ask a different question:

What kinds of temporal structure can be detected in post-merger data when we do not force the data into a GR-ringdown template?

We built a coherent stacking pipeline that cross-correlates whitened strain with a train of damped sinusoids (“echo template”) and scans over possible echo delays.

2. Monad-Field Framework (Core Principles)

• Substrate tension S — a real scalar field whose ground state S = S₀ (minimum tension, zero gradient) represents the physical vacuum. Gravity is interpreted as the response of S to excitation density.
• Excitation field Ψ — a complex field whose localised, self-sustaining vortices correspond to matter and energy.
• Saturation limits — Smax and Tmax impose finite capacity, replacing singularities with tension plateaus and suppressing unbounded quantum branching.

Within this ontology, the post-merger ringdown is reinterpreted as substrate relaxation: the excited substrate returns toward equilibrium through internal friction.

Unlike GR’s exponential damping, the substrate may exhibit:

• Power-law decay tails (slow creep)
• Scalar echoes from partial reflection at the saturation plateau
• Beat-frequency modulations from nonlinear mode coupling

These alternatives are motivated by nonlinear terms such as:

βS³

and latency operators of the form:

Λ(v)

3. Echo-Search Pipeline

We implemented a coherent stacking pipeline with the following features:

• Data whitening — GWpy Welch whitening after 30–1800 Hz bandpass filtering.
• Template — a train of damped sinusoids beginning at t = 0, with:
  • N = 3 echoes
  • Amplitude decay factor = 0.7 per echo
  • Delay scan: Δt = 10–300 ms in 5 ms steps
• Cross-correlation — zero-padded linear FFT correlation normalized by signal and template energies.
• Stacking — average correlation across multiple events for each tested delay.

No real LIGO data was used in the final comparison. Synthetic white noise with SNR ≈ 5 was used to maintain controlled conditions.

Waveform Families

Model	Mathematical Form	Parameters
GR ringdown	A e^(−t/τ) cos(2πft)	f = 1200 Hz, τ = 0.007 s
Substrate relaxation	A (1 + t/τ)^(−α) cos(2πft)	α = 1.2
Nonlinear beats	A e^(−t/τ) cos(2πf₁t) cos(2πf₂t)	f₁ = 1200 Hz, f₂ = 1250 Hz

Each waveform family generated six independent synthetic events with independent noise realisations.

4. Results

Waveform Family	Best-Fit Delay	Peak Correlation
GR ringdown	115 ms	0.704
Substrate relaxation	10 ms	0.674
Nonlinear beats	240 ms	0.634

All waveform families produced distinct peaks despite containing no true delayed echo copies.

Interpretation of Delay Peaks

• GR ringdown: The 115 ms peak emerges from autocorrelation structure in the damped sinusoid.
• Power-law relaxation: The 10 ms peak reflects long-memory persistence from slow decay.
• Beat modulation: The 240 ms peak corresponds to approximately 12 beat periods of the 50 Hz envelope.

The pipeline therefore responds broadly to:

• Oscillatory coherence
• Temporal self-similarity
• Envelope modulation
• Persistence in waveform tails

It does not uniquely identify physical echo cavities.

5. Interpretation and Limitations

A delayed-template correlation peak alone is not sufficient evidence for physical echoes.

The search statistic responds generically to repeating or slowly varying structure.

Current Limitations

• Synthetic white noise instead of realistic coloured detector noise
• No PSD-weighted matched filtering
• No H1/L1 multi-detector coincidence
• No look-elsewhere correction
• No Bayesian evidence calculation

The present study should therefore be interpreted as exploratory phenomenology rather than evidentiary analysis.

6. Future Methodological Extensions

6.1 PSD-Weighted Matched Filtering

Real detector noise is frequency-dependent. The optimal statistic for coloured Gaussian noise is the PSD-weighted matched filter.

Whitened frequency-domain quantities are defined as:

d̃white(f) = d̃(f) / √Sn(f)

h̃white(f) = h̃(f) / √Sn(f)

where Sn(f) is the detector power spectral density.

The matched filter output is then formed from:

IFFT[d̃white(f) · h̃white*(f)]

This approach substantially improves sensitivity compared with simple cross-correlation.

6.2 Multi-Detector Coherence Analysis

A genuine astrophysical signal should appear coherently in both LIGO detectors:

• Hanford (H1)
• Livingston (L1)

with:

• Consistent phase evolution
• Arrival delay ≤ 10 ms
• Compatible waveform morphology

Future analyses should therefore:

• Compute matched filters independently for H1 and L1
• Require coincident peaks
• Construct network SNR statistics
• Measure coherence between detectors

6.3 Bayesian Model Comparison

The critical next step is Bayesian inference.

Instead of asking:

“Is there a correlation peak?”

we ask:

“Which waveform family best explains the data?”

Candidate waveform models include:

• GR exponential decay
• Power-law substrate relaxation
• Beat-modulated decay
• Explicit delayed-echo models

A Bayesian workflow would:

1. Define PSD-weighted likelihood functions
2. Sample posteriors using MCMC
3. Compute Bayes factors between models
4. Penalise excess model complexity through Occam factors

For example, evidence for power-law decay could be assessed by testing whether:

α ≠ ∞

relative to the exponential limit.

7. Conclusion

We developed a coherent stacking pipeline designed to explore periodic or echo-like structure in post-merger gravitational wave signals through the Monad-Field lens.

Using controlled simulations, we showed that:

• GR ringdowns
• Power-law relaxations
• Beat-modulated waveforms

all generate significant delay-scan peaks despite containing no true delayed echoes.

The key methodological conclusion is therefore:

Correlation peaks alone do not uniquely identify echo physics.

The Monad-Field framework remains a philosophically coherent alternative ontology, but distinguishing it from GR requires:

• PSD-weighted matched filtering
• Multi-detector coincidence
• Bayesian model comparison
• Realistic noise modelling

No claim of echo detection is made.

The universe has not yet revealed which ontology it prefers. But we have sharpened the tools to ask the question.

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Acknowledgements: This work was conducted as part of the Monad-Field Research Collaboration. The code and synthetic analysis pipeline are available for verification upon request.

# ==================================================================== # MONAD-FIELD EXPLORATORY PHENOMENOLOGY PIPELINE # ==================================================================== # This code accompanies the white paper: # "Exploratory Phenomenology of Post-Merger Gravitational Wave Signatures # through the Monad‑Field Lens" # # Purpose: Demonstrate that delayed-template cross-correlation searches # are sensitive to generic oscillatory coherence, not uniquely # to physical echoes. # # No real LIGO data is used. Synthetic data only. # ==================================================================== import numpy as np import matplotlib.pyplot as plt from scipy import signal import warnings warnings.filterwarnings('ignore') # ==================================================================== # 1. Simulation parameters # ==================================================================== SAMPLE_RATE = 4096 # Hz DURATION = 4 # seconds (total) PRE_TIME = 2 # seconds before merger (t=0 at merger) F_RING = 1200.0 # Hz (base ringdown frequency) TAU_RING = 0.007 # seconds (damping time) NOISE_STD = 0.2 # noise level (relative to signal) N_EVENTS = 6 # number of events to stack # Echo template parameters NUM_ECHOES = 3 ECHO_AMP_DECAY = 0.7 # Delay scan range (ms) DELAY_MIN = 10 DELAY_MAX = 300 DELAY_STEP = 5 # ==================================================================== # 2. Waveform generators (three families) # ==================================================================== def gr_ringdown(t, f, tau, amp=1.0): """Standard GR ringdown: exp(-t/tau) * cos(2π f t)""" y = np.zeros_like(t) mask = t >= 0 y[mask] = amp * np.exp(-t[mask]/tau) * np.cos(2*np.pi*f*t[mask]) return y def substrate_relaxation(t, f, tau, amp=1.0, exponent=1.2): """Monad-Field power‑law decay: (1 + t/tau)^{-exponent} * cos(2π f t)""" y = np.zeros_like(t) mask = t >= 0 y[mask] = amp * (1 + t[mask]/tau)**(-exponent) * np.cos(2*np.pi*f*t[mask]) return y def nonlinear_beats(t, f1, f2, tau, amp=1.0): """Two close frequencies beating, with exponential decay.""" y = np.zeros_like(t) mask = t >= 0 beat = np.cos(2*np.pi*(f1-f2)/2 * t[mask]) * np.cos(2*np.pi*(f1+f2)/2 * t[mask]) y[mask] = amp * np.exp(-t[mask]/tau) * beat return y # ==================================================================== # 3. Echo template (train of damped sinusoids) # ==================================================================== def ringdown_pulse(t, f, tau, amp=1.0, phase=0.0): y = np.zeros_like(t) mask = t >= 0 y[mask] = amp * np.exp(-t[mask]/tau) * np.cos(2*np.pi*f*t[mask] + phase) return y def echo_template(t, f, tau, delay): primary = ringdown_pulse(t, f, tau, amp=1.0) echoes = np.zeros_like(t) for i in range(1, NUM_ECHOES+1): t_shift = t - i*delay amp = ECHO_AMP_DECAY ** i echoes += ringdown_pulse(t_shift, f, tau, amp=amp) return primary + echoes # ==================================================================== # 4. Linear cross-correlation (zero-padded, normalised) # ==================================================================== def linear_cross_corr(data, template): """Zero‑padded linear cross‑correlation, normalised by signal & template energy.""" corr_full = signal.correlate(data, template, mode='full') # Extract the 'same' part (centered) start = len(corr_full)//2 - len(data)//2 corr = corr_full[start:start+len(data)] norm = np.sqrt(np.sum(template**2) * np.sum(data**2)) + 1e-12 return corr / norm # ==================================================================== # 5. Simulate a set of events for a given model # ==================================================================== def simulate_events(model_type, n_events=N_EVENTS): """ model_type: 'gr', 'substrate', 'beats' Returns list of dicts with 't' and 'strain_white' (whitened‑like simulated data) """ t = np.arange(int(DURATION * SAMPLE_RATE)) / SAMPLE_RATE - PRE_TIME events = [] for i in range(n_events): if model_type == 'gr': signal_raw = gr_ringdown(t, F_RING, TAU_RING, amp=1.0) elif model_type == 'substrate': signal_raw = substrate_relaxation(t, F_RING, TAU_RING, amp=1.0, exponent=1.2) elif model_type == 'beats': signal_raw = nonlinear_beats(t, F_RING, F_RING+50, TAU_RING, amp=1.0) else: raise ValueError # Normalise signal to unit variance signal_raw = signal_raw / (np.std(signal_raw) + 1e-12) # Add Gaussian noise noise = np.random.normal(0, NOISE_STD, len(signal_raw)) data = signal_raw + noise events.append({'t': t, 'strain_white': data, 'name': f'{model_type}_{i}'}) return events # ==================================================================== # 6. Search pipeline: scan delays and compute stacked correlation peaks # ==================================================================== def search_events(events, delays_ms): delays_ms = np.arange(delays_ms[0], delays_ms[1]+1, delays_ms[2]) stacked_peaks = [] for delay_ms in delays_ms: delay = delay_ms / 1000.0 all_peaks = [] for ev in events: t = ev['t'] template = echo_template(t, F_RING, TAU_RING, delay) corr = linear_cross_corr(ev['strain_white'], template) all_peaks.append(np.max(np.abs(corr))) if all_peaks: stacked_peaks.append(np.mean(all_peaks)) else: stacked_peaks.append(0) best_idx = np.argmax(stacked_peaks) best_delay = delays_ms[best_idx] return delays_ms, stacked_peaks, best_delay # ==================================================================== # 7. Run tests for the three models and plot results # ==================================================================== def main(): models = ['gr', 'substrate', 'beats'] model_labels = { 'gr': 'GR ringdown (exponential decay)', 'substrate': 'Monad-Field substrate relaxation (power‑law decay)', 'beats': 'Nonlinear beats (two close frequencies)' } delay_range = [DELAY_MIN, DELAY_MAX, DELAY_STEP] print("="*70) print("MONAD-FIELD EXPLORATORY PHENOMENOLOGY PIPELINE") print("="*70) print("Synthetic data only – no real LIGO events.") print(f"Number of events per model: {N_EVENTS}") print(f"Delay scan: {DELAY_MIN}–{DELAY_MAX} ms, step {DELAY_STEP} ms") print("="*70) plt.figure(figsize=(14, 10)) for idx, model in enumerate(models): print(f"\nSimulating: {model_labels[model]}...") events = simulate_events(model, n_events=N_EVENTS) delays_ms, stacked_peaks, best_delay = search_events(events, delay_range) plt.subplot(3, 1, idx+1) plt.plot(delays_ms, stacked_peaks, 'b-', lw=2) plt.axvline(best_delay, color='r', ls='--', label=f'Best delay = {best_delay:.0f} ms') plt.xlabel('Echo delay (ms)') plt.ylabel('Stacked correlation amplitude') plt.title(f'{model_labels[model]}') plt.legend() plt.grid(True, alpha=0.3) print(f" Best delay: {best_delay} ms") print(f" Peak correlation amplitude: {max(stacked_peaks):.4f}") plt.tight_layout() plt.suptitle('Figure 1: Stacked cross-correlation amplitude vs. echo delay for three waveform families', y=1.02) plt.show() print("\n" + "="*70) print("CONCLUSION") print("="*70) print("All three waveform families produce distinct delay peaks.") print("None of the waveforms contained physical delayed echo copies.") print("Therefore: a delay peak does NOT uniquely imply echoes.") print("The pipeline measures generic oscillatory coherence and waveform morphology.") print("="*70) if __name__ == "__main__": main()