Time-delay interferometric ranging for space-borne gravitational-wave detectors

Space-borne interferometric gravitational-wave detectors, sensitive in the low-frequency (mHz) band, will fly in the next decade. In these detectors, the spacecraft-to-spacecraft light-travel times will necessarily be unequal and time varying, and (because of aberration) will have different values on up- and down-links. In such unequal-armlength interferometers, laser-phase noise will be canceled by taking linear combinations of the laser-phase observables measured between pairs of spacecraft, appropriately time shifted by the light propagation times along the corresponding arms. This procedure, known as time-delay interferometry (TDI), requires an accurate knowledge of the light-time delays as functions of time. Here we propose a high-accuracy technique to estimate these time delays, and we study its use in the context of the Laser Interferometer Space Antenna (LISA) mission. We refer to this ranging technique, which relies on the TDI combinations themselves, as time-delay interferometric ranging (TDIR). For every TDI combination, we show that, by minimizing the rms power in that combination (averaged over integration times $\sim {10}^4\mathrm{s}$) with respect to the time-delay parameters, we obtain estimates of the time delays accurate enough to cancel laser noise to a level well below the secondary noises. Thus TDIR allows the implementation of TDI without the use of dedicated interspacecraft ranging systems, with a potential simplification of the LISA design. In this paper we define the TDIR procedure formally, and we characterize its expected performance via simulations with the Synthetic LISA software package.

Figure 1

Distribution of errors $\Delta L$ [see Eq. (9) and the main text above it] in the determination of light-travel times, using $X_1$-based TDIR with chunk durations of 8192 s (for the LM and ODM models), and 16 384 and 32 768 s (for the ODM model only). As expected, the errors are lower for longer integration times $T$; for the LM model, the larger errors are due to the unmodeled curvature in the time dependence of the light-travel times. The distributions shown correspond to samples of 512, 256, and 128 chunks for $T=8192$, 16 384, and 32 768 s, respectively, spread across a year.

Figure 2

Spectra of frequency laser noise and GWs plus secondary noises before and after TDIR. Beginning from the top, the left-panel curves plot: (“uncanceled laser”) laser noise for one of the $s_{ij}$ measurements, before cancellation in the TDI observables; (“residual noise”) residual laser noise in $X_1$ at the beginning of TDIR minimization, with imposed armlength errors $\simeq 40\mathrm{k}\mathrm{m}$; (“$X_1^{(n)}$”) GWs plus secondary noises in $X_1$; (“$X_1^{(0)}$”) residual laser noise in $X_1$ at the end of TDIR minimization, as performed using chunk durations of 8192 s (for the LM and ODM models), and 16 384 and 32 768 s (for the ODM model only). We show averages of the spectra computed separately for each chunk using triangle-windowed periodograms; the averages are taken over populations of 512, 256, and 128 chunks for $T=8192$, 16 384, and 32 768 s, respectively, spread across a year. In all cases, the initially overwhelming laser noise is suppressed to levels several orders of magnitude below the secondary noises: the cutout graph on the right shows that the typical laser-noise suppression factor with respect to secondary noise is $\sim 5\times {10}^3$ for the worst case considered (8192-s LM); it improves by a factor $\sim 2$ for 8192-s ODM, and by factors of $\sim \sqrt{2}$ for each successive doubling of $T$. The GWs from the two circular binaries stand clearly above the noise at 1 and 3 mHz.