Bayesian time delay interferometry

Laser frequency noise (LFN) is the dominant source of noise expected in the Laser Interferometer Space Antenna (LISA) mission, at approximately 7 orders of magnitude greater than the typical signal expected from gravitational waves (GWs). Time-delay interferometry (TDI) suppresses LFN to an acceptable level by linearly combining measurements from individual spacecraft delayed by durations that correspond to their relative separations. Knowledge of the delay durations is crucial for TDI effectiveness. The work reported here extends upon previous studies using data-driven methods for inferring the delays during the postprocessing of raw phase meter data, also known as TDI ranging (TDIR). Our TDIR analysis uses Bayesian methods designed to ultimately be included in the LISA data model as part of a “Global Fit” analysis pipeline. Including TDIR as part of the Global Fit produces GW inferences which are marginalized over uncertainty in the spacecraft separations and allows for independent estimation of the spacecraft orbits. We demonstrate Markov chain Monte Carlo (MCMC) inferences of the six time-independent delays required in the rigidly rotating approximation of the spacecraft configuration (TDI 1.5) using simulated data. The MCMC uses fractional delay interpolation to digitally delay the raw phase meter data, and we study the sensitivity of the analysis to the filter length. Varying levels of complexity in the noise covariance matrix are also examined. Delay estimations are found to result in LFN suppression well below the level of secondary noises and constraints on the arm lengths to $\mathcal{O}(30)\mathrm{cm}$ over the approximately $2.5\times {10}^9\mathrm{m}$ baseline.

Figure 1

(a) TDI $X$ channel residual of the LFN using maximum likelihood parameters from the MCMC posterior (purple) compared to analytic optical metrology (OMS) and test mass (TM) noise PSD (green), which is an approximate representation of the GW sensitivity. The LFN component to the science measurement ${\mathrm{s}}_{32}$ (black) is also shown to demonstrate LFN input to spacecraft #1 measurements before TDI suppression. (b) Marginalized posterior distribution functions of the six independently measured delay estimates in nanoseconds for data containing LFN only. The distributions are centered on the true value for the arm length.

Figure 2

Schematic showing direction of primed and unprimed delays. Primed delays are clockwise, and unprimed delays are counterclockwise.

Figure 3

(a) Comparison of MCMC posteriors for the two data generation methods. Corner plots are arm-length posteriors subtracted by the true values used in the data generation. Shaded regions are 90% credible interval. Comparing oversampled plus interpolated data using an initial sampling rate of $1\times {10}^3\mathrm{Hz}$ and $N=101$ downsampled to 4 Hz (green) to interpolated-only data using the LaGrange filter with $N=49$ (blue). (b) Comparison of the ${\chi }_i^2$ sum over frequency bins distributions for the two data generation methods.

Figure 4

Delay posteriors at varying LaGrange window filter lengths with their corresponding median ${\chi }^2$ values. Posterior estimates are given as the difference from truth values used in the data simulation in nanoseconds. Data were sampled at 4 Hz using LaGrange filter length $N=49$. By restricting the frequency band in the likelihood to below 0.1 Hz, all filter lengths tested above $N=21$ achieve similar delay estimates. $N=29$ marks the beginning of the asymptotic portion of the filter length vs ${\chi }^2$ distributions.

Figure 5

(a) Posterior delay time estimates using the AET, equal-, and unequal-arm versions of the noise covariance matrix. Estimates are given as the difference from truth values used in the data simulation in nanoseconds. No significant differences in estimates were found. (b) Posterior delay time point estimates from the maximum of the likelihood distribution as input to the $X$ channel. $s_{31}$ (black) is shown as an example of the data. The resulting residual of the LFN portion of the data is shown for the AET (purple), equal-arm (blue), and unequal-arm (green) versions of the noise covariance matrix. Placement of the (0,0) and (0,1) elements of each matrix is shown and is representative of the entire matrix.

Figure 6

(a) Comparison of arm-length posteriors using data that included TM and OMS noises with LFN (orange) vs data containing LFN only (purple). Estimates are given by sample values subtracted by the truth values that were used as input in the data simulation in nanoseconds. (b) $X$-channel residual of full data containing $\mathrm{LFN}+\mathrm{TM}+\mathrm{OMS}$ noise using maximum likelihood parameters from MCMC posterior (orange) compared to analytic OMS and TM noise PSD (green). $X$-channel residual of LFN data using maximum likelihood parameters from MCMC posterior (purple). The science measurement ${\mathrm{s}}_{32}$ (black) is also shown to demonstrate LFN input to spacecraft #1 measurements before TDI suppression.