Time delay interferometry with minimal null frequencies and shortened time span

TL;DR

This work presents PD4L, a 4L-time-span time-delay interferometry configuration that minimizes null frequencies by merging two first-generation schemes. PD4L provides a more compact data combination window, reducing margins and aliasing, while delivering stable null streams and competitive GW response in high-frequency bands. Through noise-budget analysis, GW waveform simulations, and frequency-domain Bayesian parameter inference, PD4L demonstrates superior high-frequency parameter estimation and reliable noise characterization for up to four months, albeit with some interpolation-related challenges in the low-frequency regime. Overall, PD4L emerges as a promising, space-mensor-friendly alternative for high-frequency GW data analysis in LISA-like missions, with ongoing work to mitigate its low-frequency limitations.

Abstract

Time-delay interferometry (TDI) is essential for suppressing laser frequency noise in space-based gravitational wave (GW) observatories such as LISA. However, current second-generation TDI schemes often exhibit undesirable null frequencies and require long delay spans, which can impair data analysis performance. In this work, we introduce an alternative TDI configuration PD4L designed to minimize null frequencies and operate with a shorter effective time span. Constructed by synthesizing two distinct first-generation TDI schemes, PD4L achieves a delay span of 4$L$ (where $L$ is the arm length), half that of the standard Michelson and hybrid Relay configurations. We assess PD4L's performance by evaluating the spectral stability of instrumental noise via arm-length derivatives, simulating chirping GW signals from coalescing massive black hole binaries, and comparing waveform responses. Parameter estimation is performed in the frequency domain, and noise characterization is examined under realistic orbital dynamics. As demonstrated by the comparisons, the compact structure of PD4L offers several advantages: (1) reduced data margins at segment boundaries, (2) mitigated aliasing effects in the high-frequency regime, and (3) shortened signal tails arising from extended delay spans. Additionally, PD4L's null channels exhibit the same minimal null frequencies as its science channels, while maintaining greater spectral stability than other null streams. Overall, PD4L improves parameter estimation accuracy at high frequencies and supports reliable noise characterization over observation periods of up to four months. These results highlight PD4L as a compact and effective alternative for future TDI implementations, especially in high-frequency GW data analysis for LISA-like missions.

Figures (11)

• Figure 1: The geometric diagrams of X1 and $\mathrm{U\overline{U}}$Wang:2011Wang:2020pkk. The vertical lines represent the spacecraft trajectories over time, with $i$ denoting S/C$i$ ($i = 1, 2, 3$). Additional trajectories for S/C2 and S/C3 are plotted to avoid the crossings at noninteger delay time points. The blue solid lines depict the path of one virtual laser beam, while the magenta lines represent the path of another beam. (Diagrams reused from Wang:2024alm.)
• Figure 2: Geometric diagrams of the TDI channels Beacon-P and Monitor-D, and PD4L-1 Wang:2011. The Beacon-P and Monitor-D are the first-generation TDI channels, shown in the upper panel. The lower plot illustrates PD4L-1, the first channel of the second-generation TDI PD4L configuration, which formed by combining P($t$) + D($t$).
• Figure 3: Noise PSDs (first row) and their derivatives with respect to three arm lengths (second to fourth rows) for the orthogonal observables from three TDI schemes. Columns correspond to the A, E, and T channels from left to right. The specific null stream $C^{12}_3$ is also shown in the third column for comparison with other T channels. As indicated in the first row, the A and E channels are identical within each TDI configuration. The Michelson observables exhibit null frequencies at $u=m/4 \ (m=1,2,3...)$, while the science channels of hybrid Relay and PD4L, as well as T$_\mathrm{PD4L}$ and $C^{12}_3$, show nulls at $u=m$. The second to fourth rows show the derivatives with respect to the $L_{12}$, $L_{13}$, and $L_{23}$, respectively. Despite identical PSDs, the A and E channels respond differently to changes in arm lengths. The largest variations occur near the null frequencies due to the vanishing PSD values. Consequently, the Michelson configuration (dashed blue curves) exhibits the highest instability at high frequencies. The science channels of hybrid Relay (dashed orange curves) show improved stability, benefiting from fewer null frequencies. The PD4L observables (solid green curves) display frequency- and arm-dependent stability. At low frequencies, its science channels are relatively unstable compared to the other two configurations. At high frequencies, their fluctuations could also be higher than those of hybrid Relay near null frequencies; however, A$_\mathrm{PD4L}$ and E$_\mathrm{PD4L}$ can achieve greater stability at certain frequencies. Notably, E$_\mathrm{PD4L}$ is particularly stable under changes in $L_{12}$ and $L_{23}$. Among the null streams (right column), T$_\mathrm{PD4L}$ is the most robust, while $C^{12}_3$ is generally stable but slightly less so. T$_\mathrm{X1}$, in contrast, is highly unstable and susceptible to performance degradation across the band.
• Figure 4: Sky-averaged GW responses (upper) and sensitivities (lower) of selected TDI channels. The left column presents results for the science A channels, while the right column depicts the null streams. The spikes in the sensitivity curves are caused by numerical error near the null frequencies. In the bottom-right plot, the curves for T$_\mathrm{U\overline{U}8L}$, T$_\mathrm{PD4L}$ and $C^{12}_3$ largely overlap.
• Figure 5: A merging GW signal from a MBBH ($m_1=3 \times 10^4 M_\odot$, $m_2=1 \times 10^4 M_\odot$ at $z=0.2$) and the corresponding waveforms in different TDI channels. The top row shows the original waveform. Blue and orange frames indicate the TDI operation time span for $8L$ (Michelson and hybrid Relay) and $4L$ (PD4L) configurations, respectively. From left to right, three frame pairs illustrate: 1) margin effects at data boundaries, 2) frequency aliasing at high frequencies, and 3) tails effects at the signal's end. (Note: realistic TDI involves GW responses across multiple inter-S/C links.) The middle and bottom rows display the responded waveforms in selected TDI channels. The merger time arriving at the solar-system barycenter is set to be $t=0$, and the signal arrives later at the detector in current simulation setup. The middle left plot depicts the waveforms in the Michelson optimal channels (A$_\mathrm{X1}$, E$_\mathrm{X1}$, T$_\mathrm{X1}$), the middle right panel displays those from hybrid Relay (A$_\mathrm{U\overline{U}8L}$, E$_\mathrm{U\overline{U}8L}$, T$_\mathrm{U\overline{U}8L}$). The bottom left plot illustrates the waveforms in PD4L observables, and the bottom right shows the null stream $C^{12}_3$. These waveforms demonstrate that shorter-span TDI observables produce smoother waveforms with reduced signal tails.