Detecting gravitational wave background with equivalent configurations in the network of space based optical lattice clocks

The network of space based optical lattice clocks (OLCs) has been proposed to detect the stochastic gravitational wave background. We investigate the overlap reduction function (ORF) of the OLC detector network and analytically derive a transformation that leaves the ORF invariant. This transformation is applicable to configurations with two OLC detectors, each equipped with a one-way link. It can map a configuration with small separation and high noise correlation to another configuration with larger separation and reduced noise correlation. Using this transformation, we obtain a favourable OLC detector network configuration with high cross-correlation response, and compare its sensitivity to that of space-based laser interferometer gravitational wave detectors.

💡 Research Summary

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The paper proposes and analyzes a novel method for detecting the stochastic gravitational‑wave background (SGWB) using a network of space‑based optical lattice clocks (OLCs). Each OLC detector consists of two satellites linked by a single one‑way laser beam; the clock on each satellite provides an ultra‑stable optical frequency that can be compared across the link to sense spacetime strain. The key figure of merit for a pair of detectors is the overlap reduction function (ORF), which quantifies how the geometric configuration (separation, arm length, orientation) modulates the correlated response to an isotropic SGWB.

The authors first review the standard expression for the ORF, derived from the product of the individual detector response functions and an exponential phase factor that depends on the baseline vector. They then ask a fundamental design question: can two distinct detector configurations produce identical ORFs, thereby yielding the same GW‑signal response while possibly differing in instrumental noise correlation? By reformulating the ORF in a form that separates real amplitude factors from complex phase factors, they identify a set of conditions under which the magnitude of the ORF is invariant under a non‑trivial transformation.

Two trivial families of solutions are recognized: (i) a global translation of the whole network, which leaves all relative distances unchanged, and (ii) an orthogonal transformation that simultaneously rotates both baselines and preserves their lengths. These do not change the geometry in any meaningful way. The novel contribution is the discovery of a non‑trivial mapping that swaps the endpoints of the laser links. Mathematically, this is expressed by equating the real prefactors (leading to conditions (6) and (7) in the paper) and then matching the complex exponentials. The result is a transformation that takes an original configuration {A} = {x₁, L₁ û₁; x₂, L₂ û₂} and produces an equivalent configuration {B} = {x₁ + L₁ û₁, L₂ û₂; x₂ + L₂ û₂, L₁ û₁}. Geometrically, the end‑point of each link is shifted by the length of the opposite link, effectively moving the detectors farther apart while preserving the ORF magnitude.

When the two arms have equal length (L₁ = L₂ = L), the β term in the phase factor vanishes, and the condition α₁₂ = α₃₄ guarantees that the full complex ORF, not just its magnitude, is identical. This is crucial because data analysis typically treats the real and imaginary parts of the cross‑correlation separately. Consequently, the transformation can be used to convert a compact, high‑signal‑to‑noise configuration (where the two OLCs are close together and thus have a large ORF but also strong local‑noise correlation) into a more widely separated configuration with the same GW response but reduced noise correlation.

The authors illustrate the transformation with several concrete geometries. In Figure 3, configuration (a) shows two links sharing a common endpoint with an included angle θ; configuration (b) is its equivalent where the endpoints are separated by d = 2L sin(θ/2); configuration (c) has the two links parallel at the same separation d. Numerical evaluation of the ORF shows that a small angle (θ ≈ 20°) yields a higher response at low frequencies, whereas the parallel configuration performs slightly better at higher frequencies. This behavior mirrors the Hellings‑Downs curve familiar from pulsar‑timing array analyses, indicating that the optimal angle depends on the target frequency band.

To assess practical sensitivity, the paper derives the effective strain noise for the OLC network: \