Gravitational-wave lensing beyond rays: a disordered-system approach

1. Problem Statement

The paper addresses the propagation of gravitational waves (GWs) through a stochastic distribution of weak gravitational lenses (matter inhomogeneities).

• Limitations of Current Models: Existing studies of GW lensing generally focus on:
  • Geometric Optics: Valid when the GW wavelength is much smaller than the lens scale (ray-based).
  • Single/Finite Lenses: Wave-optics treatments usually consider isolated lenses or a finite number of them.
• The Gap: There is no general framework for wave-optics lensing in a random, continuous medium where diffraction and interference are significant. Standard trajectory-based techniques (like Boltzmann equations) fail in the wave-optics regime because the concept of a "ray" is lost due to diffraction.
• Goal: To develop a unified framework describing GW propagation through a static, random background of weak lenses, treating the problem as a system with quenched disorder.

2. Methodology

The authors employ techniques from statistical physics and quantum field theory (QFT), specifically adapting the Schwinger-Keldysh (in-in) formalism and the path-integral representation of the density matrix.

A. Physical Setup

• Metric: A perturbed Minkowski metric with a static, stochastic gravitational potential $\Phi(\mathbf{x})$.
• Wave Equation: The GW is modeled as a real scalar field $\zeta$ (neglecting polarization for clarity) satisfying the wave equation in the weak-field limit:
  $$[\partial_x^2 - (1 - 4\alpha\Phi)\partial_t^2]\zeta = 0$$
  where $\alpha$ is a bookkeeping parameter for the perturbative expansion.
• Disorder Statistics: The potential $\Phi$ is treated as a Gaussian random field with zero mean and a specified power spectrum $\tilde{C}(k)$.

B. Theoretical Framework

1. Density Matrix Approach: Instead of solving for a single realization, the authors define the disorder-averaged density matrix $\rho_{\text{av}}$. This object captures both occupation probabilities (diagonal elements) and phase correlations (off-diagonal elements), which are crucial for interference effects.
2. Schwinger-Keldysh Formalism: The evolution is formulated on a closed time contour with two branches ($+$ and $-$). The density matrix elements are represented as a path integral over field configurations $\zeta_+$ and $\zeta_-$:
   $$\rho_{\text{av}} \sim \int \mathcal{D}\Phi \, P[\Phi] \int \mathcal{D}\zeta_+ \mathcal{D}\zeta_- \, e^{i\Omega(S[\zeta_+] - S[\zeta_-])}$$
   Here, $\Omega$ is a large semiclassical parameter identified with the wave frequency ($\omega$).
3. Perturbative Expansion:
   • Background Field Method: The field is split into a classical solution ($\zeta_{\text{cl}}$) satisfying the equation of motion and quantum fluctuations ($\eta$).
   • Classical Action: Expanded to second order in $\alpha$ to capture the interaction with the potential.
   • Loop Corrections: The Gaussian integration over fluctuations yields a functional determinant (one-loop correction), which renormalizes the statistical weight of the disorder.

C. The Averaging Procedure

The core calculation involves performing the Gaussian functional integral over the stochastic potential $\Phi$. This couples the forward ($+$) and backward ($-$) branches of the path integral, leading to an effective non-unitary evolution for the averaged system.

3. Key Contributions and Results

A. Structure of the Averaged Density Matrix

The authors derive an explicit perturbative form for the averaged density matrix, which separates naturally into two distinct factors:

1. Pure Phase Factor (Oscillatory):
   • Arises from the mismatch in free propagation between the two branches, corrected by a disorder-induced term ($\delta A_2$).
   • Interpreted as a disorder-induced modification of the propagation kernel (analogous to a self-energy).
   • Represents elastic scattering: it modifies coherent propagation without energy exchange but introduces phase shifts.
2. Damping Factor (Real Exponential):
   • Takes the form $\exp\left( -\frac{\alpha^2 \Omega^2}{2} \int \delta A_1 \tilde{C} \delta A_1 \right)$.
   • This is a quadratic exponential that suppresses off-diagonal elements of the density matrix.
   • Physical Meaning: This term represents decoherence. It quantifies the loss of phase coherence due to the random accumulation of phase shifts across different realizations of the lens distribution.

B. Mechanism of Decoherence

• Branch Mismatch: The suppression depends on $\delta A_1$, which measures the difference between the field configurations on the forward and backward branches.
• Scale Dependence: Decoherence is most efficient when the wavelength of the GW is comparable to the correlation length of the disorder ($\ell_c$).
  • If $\lambda \gg \ell_c$: The wave averages over many fluctuations, and effects cancel out.
  • If $\lambda \ll \ell_c$: The background appears locally smooth.
  • If $\lambda \sim \ell_c$: The wave "resolves" the inhomogeneities, leading to maximal dephasing.
• Time Dependence: The decoherence effect is cumulative. For short times, the suppression scales as $T^6$, indicating a slow onset, but grows significantly for longer propagation times.

C. Application to Gaussian Wave Packets

The authors explicitly calculate the decoherence rate for Gaussian wave packets.

• They show that for a real scalar field (required for GWs), the wave packet must be a superposition of two momentum lobes ($\pm p_0$) to satisfy reality conditions.
• The disorder can cause decoherence not just via small momentum transfers, but also via transfers connecting the two lobes ($|k| \sim 2p_0$).
• This contrasts with complex scalar fields where a single-sided packet might remain coherent if the disorder spectrum lacks support at the specific momentum transfer.

D. Influence Functional

The paper reformulates the result using the Feynman-Vernon influence functional. Averaging over the static disorder induces an effective non-local, quartic self-interaction between the forward and backward branches of the field. This provides a complementary view where the random medium acts as an environment generating a non-unitary effective action.

4. Significance and Implications

• Unified Framework: The work provides the first general framework for GW lensing in the wave-optics regime that handles continuous, stochastic distributions of lenses, bridging the gap between geometric optics and single-lens wave optics.
• Beyond Geometric Optics: It demonstrates that even for weak lenses, the cumulative effect of a random distribution can lead to significant decoherence and diffraction, effects invisible to ray-tracing methods.
• Observational Prospects: The formalism allows for the calculation of observable quantities (like the GW strain power spectrum) directly from the statistical properties of the matter distribution (power spectrum $\tilde{C}(k)$). This opens a new avenue to probe the matter power spectrum and small-scale structure of the universe using GWs.
• General Applicability: While framed in GW lensing, the derivation applies to any wave propagation in a static disordered medium, including electromagnetic, acoustic, and seismic waves.
• Theoretical Bridge: It successfully adapts "quenched disorder" techniques from condensed matter physics (typically used for electrons in impure metals) to cosmological GW propagation, treating the static matter distribution as a frozen environment.

Conclusion

The paper establishes that the propagation of GWs through a stochastic matter distribution is fundamentally a problem of decoherence in a disordered system. The authors derive a path-integral representation of the averaged density matrix, showing that disorder induces both a modification of the propagation kernel (self-energy) and an exponential damping of coherence. This damping is controlled by the overlap between the wave's momentum spectrum and the disorder's power spectrum, offering a new theoretical tool to interpret future GW observations in the context of large-scale structure.