Robustness-based selection of laser-locking architectures for time-delay interferometry in space-based gravitational-wave detectors

Author(s)

Chen, Hao-Kang, Wang, Pan-Pan, Shao, Cheng-Gang

Abstract

Space-based gravitational-wave detectors compare laser phases exchanged between distant spacecraft. Offset phase locking keeps beatnotes in the phasemeter band, while time-delay interferometry suppresses laser frequency noise. A disturbance of one reference can degrade the functions that depend on it, making robustness an additional criterion for choosing a phase-reference architecture. We introduce a quantitative comparison based on established cascade equations and percolation measures, with directed reference graphs and Erdős--Rényi support layers. Under uniform disturbances, the robustness score is determined by the reference-depth distribution. Under nonuniform disturbances and partial dependence, the connected-support criterion can select a different configuration from both the published depth and local-lock criteria and a reference-path baseline. The phase-locking comparisons favor configurations that limit accumulated degradation and supply downstream lasers from references retaining greater connected support. Paired finite-network calculations support these differences in preference. In a separate comparison of optical-input support for time-delay interferometry, requiring connected support on all spacecraft gives a different preference from requiring it on at least one spacecraft. These findings identify disturbance location, reference dependence and the required extent of support as factors that can change architecture selection. Within the stated statistical model, robustness rankings can complement phase-locking feasibility and interferometric performance when prioritizing configurations for space-based gravitational-wave detectors.

Figures

Caption

Physical reference architectures and ER comparisons. Panels a and b show swap-S1 and swap-S6, with a star marking $L_{12}$. Blue and red arrows identify shared and changed reference edges, respectively. Panel c shows their common evaluation through dependency and connectivity pruning. Panel d compares uniform-disturbance scores at $q=1$. Panel e gives a separate six-candidate non-swap comparison under the disturbance of Fig.~\ref{fig:choices}. Its ER and path scores use separate scales. Gray diamonds mark catalogue choice N1. Filled blue and red markers identify the respective maxima N3 and N6. Every layer has $k_i=8$.

Caption

Uniform-disturbance curves for the representative reference trees at $q=1$ and $k_i=8$. The final score averages the six layer fractions. S1 and N5 belong to different optical layouts and illustrate the dependence on reference depth. Selection tests compare configurations within each layout.

Caption

Integrated uniform-disturbance scores. Orange and blue dots give swap and non-swap means at all nine dependency fractions. Capped intervals give the 15th to 85th configuration percentiles. Triangles and squares identify S1 and N5. Family positions are horizontally offset for legibility. Integration uses the same 41-point $p$ grid.

Caption

Selection among all six non-swap candidates at $p_{12}=0.25$, $p_{31}=0.55$, unit fractions elsewhere, $q=0.9$, and $k_i=8$. Panels a and b give ER and path scores on separate horizontal scales. Filled blue and red markers identify the respective maxima N3 and N6. The gray diamond locates catalogue choice N1 in each panel. Table~\ref{tab:criteria} provides the scores.

Caption

Synchronous pruning for S1 and S6 with $p_{13}=p_{23}=0.15$, unit fractions elsewhere, $q=0.9$, $k_i=8$, and 80000 nodes per layer. Colored bars give means across 24 realizations with error bars spanning 1.96 standard errors. Gray portions include initial removal and subsequent pruning. Round zero contains the initial qualified sets. Later rounds follow Eq.~\eqref{eq:finite_stages}. Stable realizations are padded at their final values. Final means are 0.1982 for S1 and 0.1900 for S6.

Caption

Conditional ER scores at $q=0.95$ for the five patterns in Table~\ref{tab:motifs}, labeled 6L and A to D. The ordinate averages three functional giant-component fractions. Vertical segments connect one-sided limits at discontinuities, including the secondary receiving-pair onset of C. The downstream onset of B is continuous. Section~\ref{sec:tdi_mapping} defines the optical-input support policy.

Caption

Support-pattern landscapes on a common color scale. Colors give $\bar P$. White lines locate the display threshold $p_*(q)$ at $\bar P\ge10^{-3}$. Values at the twenty displayed dependency fractions follow the continuous crossing, with the fold used when a jump crosses the level. Adjacent points are joined for display. Appendix~\ref{app:critical} specifies the threshold calculations.