Astrophysical Sources of Gravitational Waves

1. Introduction: The Gravitational Wave Spectrum

Just as the electromagnetic spectrum spans from radio waves to gamma rays, the gravitational wave spectrum encompasses an enormous frequency range. Each band corresponds to fundamentally different source classes and requires distinct detection strategies:

The characteristic strain $h_c(f)$ provides a unified framework for comparing sources across the spectrum. For a source at luminosity distance $d_L$ emitting gravitational wave energy $dE_{\rm GW}/df$ per unit frequency, we have:

A source is detectable when its characteristic strain exceeds the detector's sensitivity curve$h_n(f) = \sqrt{f\, S_n(f)}$, where $S_n(f)$ is the one-sided noise power spectral density. The signal-to-noise ratio for an optimally oriented and located source is$\rho^2 = 4\int_0^\infty \frac{|\tilde{h}(f)|^2}{S_n(f)}\,df$.

2. Compact Binary Inspiral Waveform

The inspiral phase of a compact binary — two objects spiraling together under gravitational radiation reaction — is the most analytically tractable and observationally important source. We derive the waveform in the restricted post-Newtonian (PN) approximation, where the amplitude is kept at leading (Newtonian) order while the phase includes higher-order corrections.

2.1 Keplerian Setup and the Chirp Mass

Consider two compact objects of masses $m_1$ and $m_2$ in a quasi-circular orbit with separation $a$ and orbital frequency $f_{\rm orb}$. Kepler's third law gives:

The gravitational wave frequency is twice the orbital frequency, $f = 2 f_{\rm orb}$, because the quadrupole moment has a 2-fold symmetry. The energy of the circular orbit is:

where $M = m_1 + m_2$ is the total mass, $\mu = m_1 m_2/M$ the reduced mass, and $\omega = 2\pi f_{\rm orb}$. The chirp mass, which uniquely determines the leading-order inspiral dynamics, is defined as:

2.2 Frequency Evolution

Balancing the orbital energy loss with the gravitational wave luminosity from the quadrupole formula,$dE/dt = -P_{\rm GW}$, we derived in the previous chapter:

This can be integrated to find the time remaining until coalescence. Starting from a frequency $f$and integrating $dt = df/\dot{f}$ to $f \to \infty$:

For a $1.4\,M_\odot$–$1.4\,M_\odot$ BNS at $f = 20$ Hz (LIGO's lower cutoff), $\tau \approx 17$ minutes. For a $30\,M_\odot$–$30\,M_\odot$ BBH at the same frequency, $\tau \approx 0.4$ seconds.

2.3 The Inspiral Waveform

The two GW polarizations from a circular binary at luminosity distance $d_L$ with inclination angle $\iota$ are:

For the detector response $h(t) = F_+ h_+ + F_\times h_\times$, we write the compact form:

The phase evolution is obtained by integrating the instantaneous frequency:

where $\Phi_0$ is the initial phase. Using the frequency evolution, the phase in the Fourier domain (stationary phase approximation) becomes:

2.4 Frequency at the Innermost Stable Circular Orbit

The inspiral approximation breaks down at the ISCO. For a test particle around a Schwarzschild black hole of mass $M$, the ISCO radius is $r_{\rm ISCO} = 6GM/c^2$. The corresponding orbital frequency is:

Since $f_{\rm GW} = 2f_{\rm orb}$, the gravitational wave frequency at ISCO is:

For a $60\,M_\odot$ total-mass BBH system, $f_{\rm ISCO} \approx 73$ Hz — well within LIGO's sensitive band. For a $10^6\,M_\odot$ SMBH binary,$f_{\rm ISCO} \approx 4.4$ mHz, falling in LISA's band.

3. Binary Black Hole Merger Phases

A BBH coalescence proceeds through three distinct phases: inspiral, merger, and ringdown. Each requires different theoretical tools, and their seamless combination produces the complete inspiral-merger-ringdown (IMR) waveform used in LIGO/Virgo analyses.

3.1 Inspiral Phase (Post-Newtonian Approximation)

During inspiral, the orbital velocity satisfies $v/c \ll 1$ and the gravitational field is weak. The post-Newtonian expansion expresses the equations of motion as a series in$\epsilon \sim v^2/c^2 \sim GM/(rc^2)$. The phase is known to 3.5PN order:

where $\eta = m_1 m_2/M^2$ is the symmetric mass ratio. The coefficients$\alpha_k$ and $\beta_k$ depend on the component masses and spins. The leading term (0PN, $k=0$) reproduces the Newtonian result derived above.

3.2 Merger Phase (Numerical Relativity)

When $v/c \sim 0.3$–$0.5$, the PN expansion diverges and only full numerical solutions of Einstein's equations can track the dynamics. The 2005 breakthrough by Pretorius, and independently by Campanelli et al. and Baker et al., achieved stable BBH merger simulations. Key results include:

• ●The peak gravitational wave luminosity reaches $L_{\rm peak} \sim 3.6 \times 10^{56}$ erg/s ($\sim 200\,M_\odot c^2/\text{s}$), exceeding the combined luminosity of all stars in the observable universe
• ●For equal-mass non-spinning BBH, approximately 5% of the total mass-energy is radiated as gravitational waves
• ●Asymmetric emission produces a recoil (gravitational wave kick) up to $\sim 5000$ km/s for spinning BHs

3.3 Ringdown Phase (Quasinormal Modes)

After merger, the remnant BH settles to a Kerr state by radiating quasinormal modes (QNMs). These are damped sinusoids characterized by complex frequencies $\omega_{\ell m n}$ that depend only on the remnant's mass $M_f$ and spin $a_f$ (the no-hair theorem).

For a Schwarzschild black hole ($a_f = 0$), the fundamental $\ell=2, m=2, n=0$quasinormal mode frequency and damping time are:

where the quality factor $Q$ determines the number of oscillation cycles before decay. For the Schwarzschild $\ell=2$ mode, numerical calculations give:

The real part gives the oscillation frequency and the imaginary part gives the damping rate$\tau_{\rm damp} = Q/(\pi f_{\rm QNM})$. The ringdown waveform takes the form:

3.4 The Complete IMR Waveform

Phenomenological waveform models (e.g., IMRPhenom, SEOBNR families) stitch together the three phases. In the frequency domain, the IMR waveform amplitude schematically follows:

where $\mathcal{L}$ is a Lorentzian centered on the ringdown frequency with width$\sigma = f_{\rm QNM}/Q$. The transition frequencies $f_{\rm merg}$ and$f_{\rm ring}$ are calibrated to numerical relativity simulations.

4. Binary Neutron Stars and Tidal Effects

Binary neutron star (BNS) inspirals are distinguished from BBH by the finite size of the neutron stars. Near merger, tidal interactions modify the orbital dynamics and leave a measurable imprint on the gravitational waveform. This provides a unique probe of the nuclear equation of state (EOS) at supranuclear densities.

4.1 Tidal Deformability

In an external tidal field $\mathcal{E}_{ij}$, a neutron star develops a quadrupole moment $Q_{ij} = -\lambda\,\mathcal{E}_{ij}$, where $\lambda$ is the tidal deformability parameter. The dimensionless tidal deformability is:

where $k_2$ is the $\ell=2$ tidal Love number (typically 0.05–0.15 for neutron stars), $R$ is the stellar radius, and $M$ is its mass. The Love number $k_2$ depends on the internal structure and must be computed by integrating the perturbation equations for a given EOS:

where $C = GM/(Rc^2)$ is the compactness and $y = R\,H'(R)/H(R)$ is determined by integrating the perturbation function $H(r)$ through the stellar interior.

4.2 Tidal Phase Correction

Tidal effects enter the gravitational wave phase at 5PN order (relative to the point-particle baseline). The leading tidal correction to the Fourier-domain phase is:

where $\tilde{\Lambda}$ is the mass-weighted combination of tidal deformabilities:

The negative sign means tidal interactions accelerate the inspiral relative to the point-particle case — the tidal bulge transfers energy from the orbit to the NS internal modes, causing the system to coalesce faster.

4.3 Constraints from GW170817

GW170817, the first BNS merger detected by LIGO and Virgo on August 17, 2017, provided the first measurement of tidal effects in gravitational waves. Key results:

• ●Component masses: $m_1 \in [1.36, 1.60]\,M_\odot$, $m_2 \in [1.17, 1.36]\,M_\odot$ (90% credible intervals)
• ●Combined tidal deformability: $\tilde{\Lambda} = 300^{+420}_{-230}$ (90% highest density interval)
• ●This rules out very stiff EOS models with $\tilde{\Lambda} > 800$ and constrains the NS radius to $R_{1.4} \approx 11.9 \pm 1.4$ km
• ●Combined with the electromagnetic counterpart, confirmed that BNS mergers are a primary site of r-process nucleosynthesis (heavy element formation)

The $f^{5/3}$ scaling of the tidal phase means these effects are most significant at high frequencies ($f \gtrsim 400$ Hz), near the merger. Next-generation detectors (Einstein Telescope, Cosmic Explorer) will measure $\tilde{\Lambda}$ to within a few percent.

5. Continuous Waves from Rotating Neutron Stars

A rotating neutron star with a non-axisymmetric mass distribution emits continuous, nearly monochromatic gravitational waves. Although individually weak, these signals can be coherently integrated over months to years, building up signal-to-noise ratio.

5.1 Ellipticity and GW Emission

Define the ellipticity of the neutron star as the fractional difference in its principal moments of inertia:

where the rotation axis is along $z$. For a biaxial star rotating at angular frequency$\Omega = 2\pi f_{\rm rot}$, the time-varying quadrupole moment has a component oscillating at $2\Omega$. Hence the gravitational wave frequency is:

The quadrupole formula gives the strain amplitude for an optimally oriented source at distance $d$:

To derive this, we compute $\ddot{Q}_{ij}$ for a biaxial body with $I_{xx} \neq I_{yy}$rotating about the $z$-axis. The non-vanishing components oscillate at frequency $2\Omega$with amplitude $\ddot{Q} \sim \varepsilon I_{zz}\Omega^2$, and the quadrupole formula$h \sim 2G\ddot{Q}/(c^4 d)$ yields the result.

5.2 Spin-Down Limit

If a pulsar's spin-down is entirely due to gravitational wave emission, conservation of energy gives$P_{\rm GW} = -dE_{\rm rot}/dt = I_{zz}\Omega\dot{\Omega}$. Using the GW luminosity for a triaxial rotator:

Setting $P_{\rm GW} = I_{zz}\Omega|\dot{\Omega}|$ and solving for $h_0$gives the spin-down upper limit:

For the Crab pulsar ($f_{\rm rot} = 29.7$ Hz, $\dot{f}_{\rm rot} = -3.7\times 10^{-10}$ Hz/s,$d = 2$ kpc), the spin-down limit is $h_0^{\rm sd} \approx 1.4 \times 10^{-24}$. LIGO has beaten this limit, constraining $\varepsilon < 10^{-4}$, proving that less than$\sim 1\%$ of the Crab's spin-down is due to gravitational wave emission.

5.3 Sources of Asymmetry

• ●Crustal mountains: Elastic strain in the NS crust supports deformations up to$\varepsilon \sim 10^{-7}$–$10^{-5}$ depending on the breaking strain
• ●Magnetic deformation: Strong internal toroidal fields ($B \sim 10^{12}$–$10^{15}$ G) create $\varepsilon \propto B^2$
• ●R-mode instabilities: Rossby waves in rotating NSs can be unstable to GW emission via the CFS mechanism, with $f_{\rm GW} = 4f_{\rm rot}/3$
• ●Accretion torques: In low-mass X-ray binaries, accretion can build up and sustain asymmetries (torque balance scenario)

6. Stochastic Gravitational Wave Background

The stochastic gravitational wave background (SGWB) is a random superposition of unresolved signals from both astrophysical and cosmological sources. It is characterized by the dimensionless energy density spectrum.

6.1 Energy Density Spectrum

The fractional energy density in gravitational waves per logarithmic frequency interval is defined as:

where $\rho_c = 3H_0^2 c^2/(8\pi G)$ is the critical energy density needed to close the universe. To relate $\Omega_{\rm GW}$ to the characteristic strain, we start from the energy density in gravitational waves expressed via the metric perturbation:

where $S_h(f)$ is the one-sided power spectral density of the GW strain. Computing$d\rho_{\rm GW}/d\ln f = f\,d\rho_{\rm GW}/df$ and dividing by $\rho_c$:

Since the characteristic strain is $h_c(f) = \sqrt{f\,S_h(f)}$ (for a stochastic background), we can write:

6.2 Astrophysical Contributions

The astrophysical SGWB arises from the incoherent superposition of unresolved sources throughout cosmic history. For compact binary mergers:

where $R(z)$ is the merger rate density and $dE_{\rm GW}/df_s$ is the source-frame energy spectrum. In the inspiral regime, $dE_{\rm GW}/df_s \propto f_s^{-1/3}$, giving $\Omega_{\rm GW} \propto f^{2/3}$.

6.3 Cosmological Sources

• ●Inflation: Tensor perturbations amplified during cosmic inflation produce a nearly scale-invariant background with $\Omega_{\rm GW} \propto f^{n_T}$ where$n_T \approx -r/8$ and $r$ is the tensor-to-scalar ratio
• ●Phase transitions: First-order phase transitions in the early universe (e.g., electroweak at $T \sim 100$ GeV) generate GWs from bubble collisions, sound waves, and turbulence, peaking at $f \sim$ mHz
• ●Cosmic strings: Topological defects from symmetry breaking produce a broad spectrum $\Omega_{\rm GW} \sim G\mu/c^2$ where $\mu$ is the string tension

6.4 NANOGrav and the Nanohertz Background

In 2023, NANOGrav (along with EPTA, PPTA, and CPTA) reported strong evidence for a stochastic process in pulsar timing data consistent with a gravitational wave background at nanohertz frequencies. The measured spectrum is consistent with $\Omega_{\rm GW} \propto f^{2/3}$, as expected from a population of supermassive black hole binaries, though contributions from cosmological sources cannot be excluded. The characteristic strain amplitude at $f = 1\,\text{yr}^{-1}$ is$A \approx 2.4 \times 10^{-15}$.

7. Applications

8. Historical Context

9. Python Simulation

The following simulation plots the gravitational wave strain spectrum with sensitivity curves for LIGO, LISA, and pulsar timing arrays, overlaid with source tracks for binary black holes, binary neutron stars, supermassive black hole binaries, and the stochastic background. It also generates a BBH chirp waveform showing the inspiral, merger, and ringdown phases.