Part IV

Gravitational Waves

The LIGO 2015 detection of GW150914 opened gravitational-wave astronomy. Binary black-hole mergers produce characteristic chirps (frequency rising through the detector band), followed by ringdown to a final Kerr BH. 100+ coalescences catalogued to date reveal a rich astrophysical population.

Chirp Signal

During inspiral, orbital energy is radiated as GWs; frequency and amplitude rise:

\[ \dot f \;\propto\; f^{11/3}\,\mathcal{M}^{5/3},\qquad \mathcal{M} = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}} \]

ℳ is the chirp mass, the single parameter that dominates the inspiral signal. Post-Newtonian expansion provides the full waveform to 4-PN order; numerical-relativity simulations provide the merger; quasinormal-mode expansion provides the ringdown. Matched-filter analysis detects signals buried below the noise.

Ringdown & No-Hair Theorem

After merger, the remnant Kerr BH “rings down” via quasi-normal modes at frequencies determined entirely by final mass and spin. Multiple observed modes provide a direct test of the no-hair theorem: ringdown spectroscopy constrains deviations from Kerr. GW230529 Kerr-plus-deviations analyses consistent with GR to within current sensitivity.

Simulation: Chirp Waveform & Sensitivity

Python

script.py48 lines

import numpy as np, matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# Chirp waveform: inspiral + merger + ringdown
t = np.linspace(-0.5, 0.1, 6000)
# Inspiral: f(t) ~ (-t)^(-3/8)
f_inspiral = 35 * (np.maximum(-t, 1e-4))**(-3/8)
f_inspiral = np.clip(f_inspiral, 35, 250)
# Phase
phase = 2*np.pi * np.cumsum(f_inspiral) * (t[1] - t[0])
# Amplitude grows as -t^{-1/4} then rings down
A = np.where(t < 0, np.minimum(1, 0.3 / np.maximum(-t, 0.01)**0.25),
             np.exp(-t*25))
h = A * np.sin(phase)

fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(12, 7), facecolor='#0a0a1a')
for ax in (ax1, ax2):
    ax.set_facecolor('#111827'); ax.tick_params(colors='#cbd5e1')
    for s in ax.spines.values(): s.set_color('#334155')
    ax.grid(True, color='#334155', alpha=0.3)

ax1.plot(t, h, color='#fbbf24', lw=1.5)
ax1.set_xlabel('Time (s)', color='#cbd5e1')
ax1.set_ylabel('Strain h (arb.)', color='#cbd5e1')
ax1.set_title('Chirp waveform: inspiral + merger + ringdown',
              color='#fde68a', fontweight='bold')
ax1.axvline(0, color='#f87171', ls='--', alpha=0.7, label='Merger')
ax1.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

# Strain spectrum vs frequency (detector sensitivity)
f = np.logspace(0, 4, 300)
# LIGO design ASD
asd = np.where(f < 10, 1e-20 * (f/10)**-4,
        np.where(f < 100, 3e-23,
          np.where(f < 1000, 3e-23*(f/100)**0.5, 3e-22*(f/1000)**2)))
ax2.loglog(f, asd, color='#2dd4bf', lw=2.5, label='LIGO design ASD')
ax2.axvspan(35, 300, color='#fbbf24', alpha=0.15, label='BH binary band')
ax2.set_xlabel('Frequency (Hz)', color='#cbd5e1')
ax2.set_ylabel('Strain (1/sqrt(Hz))', color='#cbd5e1')
ax2.set_title('LIGO detector sensitivity curve',
              color='#fde68a', fontweight='bold')
ax2.legend(facecolor='#1e293b', edgecolor='#334155', labelcolor='#cbd5e1')

plt.tight_layout()
plt.savefig('output.png', dpi=120, bbox_inches='tight', facecolor='#0a0a1a')
print('GW150914: first direct BH-BH merger detection (36 + 29 -> 62 Msun, 3 Msun in GWs)')
print('LIGO strain sensitivity ~10^-23 / sqrt(Hz) in 50-300 Hz band')

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Population & Future Detectors

LIGO-Virgo-KAGRA O4 run 2023–2024 has catalogued ~200 coalescences. Third-generation detectors (Einstein Telescope underground, Cosmic Explorer 40-km) will detect every stellar-BH merger in the observable universe. LISA (launch ~2037) will observe supermassive BH mergers and extreme mass-ratio inspirals at mHz frequencies. Pulsar Timing Arrays (NANOGrav, EPTA) reported 2023 evidence for the nanohertz GW background from supermassive BH binaries.

Key References

• Abbott, B. P. et al. (2016). “Observation of gravitational waves from a binary black hole merger.” Phys. Rev. Lett., 116, 061102.

• LIGO-Virgo-KAGRA (2023). “GWTC-3.” Phys. Rev. X, 13, 041039.

• Berti, E. et al. (2009). “Quasinormal modes of black holes and black branes.” Class. Quantum Grav., 26, 163001.

• NANOGrav Collaboration (2023). “The NANOGrav 15-yr data set: evidence for a gravitational-wave background.” Astrophys. J. Lett., 951, L8.

Share:X Reddit LinkedIn

← EHT Overview