## Author(s)

Wang, Gang## Abstract

Time delay interferometry (TDI) is essential for suppressing laser frequency noise and achieving the targeted sensitivity for space-borne gravitational wave (GW) missions. In Paper I, we examined the performance of the fiducial second-generation TDI Michelson configuration versus an alternative, the hybrid Relay, in noise suppression and data analysis. The results showed that both TDI schemes have comparable performances in mitigating laser and clock noises. However, when analyzing chirp signal from the coalescence of massive binary black holes, the Michelson configuration becomes inferior due to its vulnerable T channel and numerous null frequencies. In contrast, the hybrid Relay is more robust in dynamic unequal-arm scenarios. In this work, we further investigate the noise characterization capabilities of these two TDI configurations. Our investigations demonstrate that hybrid Relay achieves more robust noise parameter inference than the Michelson configuration. Moreover, the performance could be enhanced by replacing the T channel of hybrid Relay with the null stream from Sagnac configuration. The combined data streams, two science observables from the hybrid Relay and a null observable from the Sagnac, could form an optimal dataset for characterizing noises.

## Figures

The geometric diagrams for X1 and $\mathrm{U\overline{U}}$ \cite{Wang:2011,Wang:2020pkk} (diagrams reused from \cite{Wang:2024alm}).

The geometric diagrams for X1 and $\mathrm{U\overline{U}}$ \cite{Wang:2011,Wang:2020pkk} (diagrams reused from \cite{Wang:2024alm}).

The real and imaginary components of CSDs for hybrid Relay observables. The real parts of CSDs dominate over imaginary components, enabling orthogonal transform from (U$\mathrm{\bar{U}}$, V$\mathrm{\overline{V}}$, W$\mathrm{\overline{W}}$) to (A$_\mathrm{U\overline{U}}$, E$_\mathrm{U\overline{U}}$, T$_\mathrm{U\overline{U}}$). (The real components from three pairs are overlapped.)

The cross-correlations between noises and GW responses among optimal TDI channels for Michelson and hybrid Relay. The upper panel depicts correlations for Michelson, while the lower panel shows those for hybrid Relay with the additional T channel of Sagnac, $T_\mathrm{\alpha 1}$. The left and right columns display noise and GW response correlations, respectively. In the upper panel, T$_\mathrm{X1}$ is fully correlated with E$_\mathrm{X1}$ at frequencies lower than 1 mHz. The correlations across three optimal channels of hybrid Relay are lower than those of Michelson. T$_\mathrm{U\overline{U}}$ shows relative independence from A$_\mathrm{U\overline{U}}$/E$_\mathrm{U\overline{U}}$ at frequencies lower than 15 mHz. While T$_\mathrm{U\overline{U}}$ and T$_\mathrm{\alpha 1}$ are highly correlated except at few particular null frequencies, T$_\mathrm{\alpha 1}$ remains independent from A$_\mathrm{U\overline{U}}$/E$_\mathrm{U\overline{U}}$ until 40 mHz.

The cross-correlations between noises and GW responses among optimal TDI channels for Michelson and hybrid Relay. The upper panel depicts correlations for Michelson, while the lower panel shows those for hybrid Relay with the additional T channel of Sagnac, $T_\mathrm{\alpha 1}$. The left and right columns display noise and GW response correlations, respectively. In the upper panel, T$_\mathrm{X1}$ is fully correlated with E$_\mathrm{X1}$ at frequencies lower than 1 mHz. The correlations across three optimal channels of hybrid Relay are lower than those of Michelson. T$_\mathrm{U\overline{U}}$ shows relative independence from A$_\mathrm{U\overline{U}}$/E$_\mathrm{U\overline{U}}$ at frequencies lower than 15 mHz. While T$_\mathrm{U\overline{U}}$ and T$_\mathrm{\alpha 1}$ are highly correlated except at few particular null frequencies, T$_\mathrm{\alpha 1}$ remains independent from A$_\mathrm{U\overline{U}}$/E$_\mathrm{U\overline{U}}$ until 40 mHz.

The cross-correlations between noises and GW responses among optimal TDI channels for Michelson and hybrid Relay. The upper panel depicts correlations for Michelson, while the lower panel shows those for hybrid Relay with the additional T channel of Sagnac, $T_\mathrm{\alpha 1}$. The left and right columns display noise and GW response correlations, respectively. In the upper panel, T$_\mathrm{X1}$ is fully correlated with E$_\mathrm{X1}$ at frequencies lower than 1 mHz. The correlations across three optimal channels of hybrid Relay are lower than those of Michelson. T$_\mathrm{U\overline{U}}$ shows relative independence from A$_\mathrm{U\overline{U}}$/E$_\mathrm{U\overline{U}}$ at frequencies lower than 15 mHz. While T$_\mathrm{U\overline{U}}$ and T$_\mathrm{\alpha 1}$ are highly correlated except at few particular null frequencies, T$_\mathrm{\alpha 1}$ remains independent from A$_\mathrm{U\overline{U}}$/E$_\mathrm{U\overline{U}}$ until 40 mHz.

Ratios between time-varying PSDs and their averages for optimal TDI channels of Michelson, Sagnac, and hybrid Relay over 30 days. The spectra vary significantly around their null/characteristic frequencies. The characteristic frequencies of Michelson and Sagnac are $\frac{i}{4 L} \simeq 0.03 i$ Hz $(i=1, 2, 3...)$ and $\frac{i}{ 3 L} \simeq 0.04 i$ Hz for LISA mission, respectively. The null frequencies of A/E from hybrid Relay are $\frac{i}{L} \simeq 0.12 i$ Hz, with additional characteristic frequencies appearing in the T$_\mathrm{U\overline{U}}$ which the lowest is $0.015$ Hz as shown in lower panel. For the T$_\mathrm{X1}$, besides its spectrum is unstable at null frequencies, it also exhibits large variance at frequencies lower than $\sim$2 mHz.

Ratios between time-varying PSDs and their averages for optimal TDI channels of Michelson, Sagnac, and hybrid Relay over 30 days. The spectra vary significantly around their null/characteristic frequencies. The characteristic frequencies of Michelson and Sagnac are $\frac{i}{4 L} \simeq 0.03 i$ Hz $(i=1, 2, 3...)$ and $\frac{i}{ 3 L} \simeq 0.04 i$ Hz for LISA mission, respectively. The null frequencies of A/E from hybrid Relay are $\frac{i}{L} \simeq 0.12 i$ Hz, with additional characteristic frequencies appearing in the T$_\mathrm{U\overline{U}}$ which the lowest is $0.015$ Hz as shown in lower panel. For the T$_\mathrm{X1}$, besides its spectrum is unstable at null frequencies, it also exhibits large variance at frequencies lower than $\sim$2 mHz.

Ratios between time-varying PSDs and their averages for optimal TDI channels of Michelson, Sagnac, and hybrid Relay over 30 days. The spectra vary significantly around their null/characteristic frequencies. The characteristic frequencies of Michelson and Sagnac are $\frac{i}{4 L} \simeq 0.03 i$ Hz $(i=1, 2, 3...)$ and $\frac{i}{ 3 L} \simeq 0.04 i$ Hz for LISA mission, respectively. The null frequencies of A/E from hybrid Relay are $\frac{i}{L} \simeq 0.12 i$ Hz, with additional characteristic frequencies appearing in the T$_\mathrm{U\overline{U}}$ which the lowest is $0.015$ Hz as shown in lower panel. For the T$_\mathrm{X1}$, besides its spectrum is unstable at null frequencies, it also exhibits large variance at frequencies lower than $\sim$2 mHz.

The inferred parameter distributions from 30-day data (A, E, T) for a static unequal-arm constellation with high frequency cutoffs of 0.01 Hz and 0.1 Hz. Acceleration noises dominates the spectra at frequencies lower than $\sim$4 mHz, and increasing high frequency cutoff from 0.01 Hz to 0.1 Hz introduces more OMS noise data rather than acceleration noise. Therefore, the amplitudes of acceleration noise are estimated with same constraint in the left plot, and the constraints on OMS noise are improved with a higher high frequency cutoff in the right plot. Results from Michelson and hybrid Relay are overlapped for the same setup. ($A_{ij}$ represents the noise component on S/C$i$ facing S/C$j$; the three color gradients show the 1$\sigma$ to 3$\sigma$ credible regions).

The inferred parameter distributions from 30-day data (A, E, T) for a static unequal-arm constellation with high frequency cutoffs of 0.01 Hz and 0.1 Hz. Acceleration noises dominates the spectra at frequencies lower than $\sim$4 mHz, and increasing high frequency cutoff from 0.01 Hz to 0.1 Hz introduces more OMS noise data rather than acceleration noise. Therefore, the amplitudes of acceleration noise are estimated with same constraint in the left plot, and the constraints on OMS noise are improved with a higher high frequency cutoff in the right plot. Results from Michelson and hybrid Relay are overlapped for the same setup. ($A_{ij}$ represents the noise component on S/C$i$ facing S/C$j$; the three color gradients show the 1$\sigma$ to 3$\sigma$ credible regions).

The inferred parameter distributions from 30-day and 90-day dataset (A, E, T) based on dynamic constellation motion with a high frequency cutoff of 0.01 Hz. Michelson's results diverge from the true values due to its time-varying PSD/CSD with T channel. In contrast, the inference results from hybrid Relay are more sensible and robust. A longer data duration improves data constraints on both types of noise parameters.

The inferred parameter distributions from 30-day and 90-day dataset (A, E, T) based on dynamic constellation motion with a high frequency cutoff of 0.01 Hz. Michelson's results diverge from the true values due to its time-varying PSD/CSD with T channel. In contrast, the inference results from hybrid Relay are more sensible and robust. A longer data duration improves data constraints on both types of noise parameters.

The inferred parameter distributions from 90-day data streams (A, E) based on static and dynamic unequal-arm constellation with a high frequency cutoff 0.1 Hz. The determinations on acceleration noises are identical for Michelson and hybrid Relay with same data setups. In the right plot, Michelson achieves more precise distributions for OMS noise compared to hybrid Relay. The degeneracy between different OMS noise components could not be effective broken using (A, E) compared to results from (A, E, T). Compared to the static case, results from dynamic case exhibit larger uncertainties because the unstable spectra around null frequencies are not fully gated.

The inferred parameter distributions from 90-day data streams (A, E) based on static and dynamic unequal-arm constellation with a high frequency cutoff 0.1 Hz. The determinations on acceleration noises are identical for Michelson and hybrid Relay with same data setups. In the right plot, Michelson achieves more precise distributions for OMS noise compared to hybrid Relay. The degeneracy between different OMS noise components could not be effective broken using (A, E) compared to results from (A, E, T). Compared to the static case, results from dynamic case exhibit larger uncertainties because the unstable spectra around null frequencies are not fully gated.

The inference results from 120-day and 180-day datasets using $(\mathrm{A_{U\overline{U}}}, \mathrm{E_{U\overline{U}}}, \mathrm{T_{\alpha1}})$ in dynamic case with a high frequency cutoff of 0.03 Hz. After substituting $\mathrm{T_{U\overline{U}}}$ with $\mathrm{T_{\alpha1}}$, the new data combination can determine the noise parameters more accurately and precisely compared to the results from other case. However, a duration of 180 days would be too long to maintain sufficiently stable spectra, leading to some parameters of OMS noises falling outside the $3\sigma$ credible regions.

The inference results from 120-day and 180-day datasets using $(\mathrm{A_{U\overline{U}}}, \mathrm{E_{U\overline{U}}}, \mathrm{T_{\alpha1}})$ in dynamic case with a high frequency cutoff of 0.03 Hz. After substituting $\mathrm{T_{U\overline{U}}}$ with $\mathrm{T_{\alpha1}}$, the new data combination can determine the noise parameters more accurately and precisely compared to the results from other case. However, a duration of 180 days would be too long to maintain sufficiently stable spectra, leading to some parameters of OMS noises falling outside the $3\sigma$ credible regions.

The correlations of spectra between optimal TDI channels for Michelson and hybrid Relay with non-identical noises amplitudes.

The correlations of spectra between optimal TDI channels for Michelson and hybrid Relay with non-identical noises amplitudes.

The inferred parameter distributions from 120-day ordinary and optimal datasets based on dynamic constellation motion with a 0.03 Hz high frequency cutoff. However, due to the instabilities of T$_{U\overline{U}}$'s spectrum in 120 days duration, the distributions could not infer the noise parameter accurately.

The inferred parameter distributions from 120-day ordinary and optimal datasets based on dynamic constellation motion with a 0.03 Hz high frequency cutoff. However, due to the instabilities of T$_{U\overline{U}}$'s spectrum in 120 days duration, the distributions could not infer the noise parameter accurately.

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