Calculating the precision of tilt-to-length coupling estimation and noise subtraction in LISA using Fisher information

Tilt-to-length (TTL) noise from angular jitter in LISA is projected to be the dominant noise source in the milli-Hertz band unless corrected in post-processing. The correction is only possible after removing the overwhelming laser phase noise using time-delay interferometry (TDI). We present here a frequency domain model that describes the effect of angular motion of all three spacecraft on the interferometric signals after propagating through TDI. We then apply a Fisher information matrix analysis to this model to calculate the minimum uncertainty with which TTL coupling coefficients may be estimated. Furthermore, we show the impact of these uncertainties on the residual TTL noise in the gravitational wave readout channel, and compare it to the impact of the angular witness sensors’ readout noise. We show that the residual TTL noise post-subtraction in the TDI variables for a case using the LISA angular jitter requirement and integration time of one day is limited to the $8\mathrm{pm}/\sqrt{\mathrm{Hz}}$ level by angular sensing noise. However, using a more realistic model for the angular jitter we find that the TTL coupling uncertainties are 70 times larger, and the noise subtraction is limited by these uncertainties to the $14\mathrm{pm}/\sqrt{\mathrm{Hz}}$ level.

Figure 1

Simplified schematic of the movable optical subassemblies (MOSAs) on a single SC. TM: test mass; OB: optical bench, connected through the backlink fiber; Tel: telescope. The telescope transmits the local beam with magnification of around 300 (134 telescope magnification and about 2.2 from the OB), and simultaneously receives light from the far SCs.

Figure 2

LISA constellation and reference frames. The blue rectangles represent the MOSAs with the primed and nonprimed nomenclature as in the literature [6], with their respective coordinate axes. Yaw ($\varphi$) is positive for counterclockwise rotation around the z-axis; pitch ($\eta$) is positive for counterclockwise rotation around the y-axis; roll ($\theta$) is positive for counterclockwise rotation around the x-axis. Every SC has two MOSAs with an opening angle of roughly 60°, forming a roughly equilateral triangle constellation. Each primed MOSA forms a science interferometer with a non-primed MOSA. The $i$th delay, ${\tau }_i$, along an arm corresponds to the opposite ${\mathrm{SC}}_i$, with counterclockwise (clockwise) light travel paths being denoted by primed (nonprimed) delays.

Figure 3

Total angular jitter (left: yaw, right: pitch) profiles considered for the LISA requirement (case (A) and total, SC, and MOSA jitter profiles from LISASim (case B). We do not display the SC and MOSA curves of case A as they are all similar.

Figure 4

Noise contributions for case A after going through second-generation TDI X. TM: test-mass acceleration; IFO: IFO noise; DWS: DWS readout noise. Case B’s noise curve is given by LISASim total. Delays used: ${\tau }_{2,2^{\prime }}=8.29\mathrm{s};{\tau }_{3,3^{\prime }}=8.40\mathrm{s}$.

Figure 5

TDI X residual computed from the error covariance matrix $\mathrm{\Sigma }$. $S_{\mathrm{n}}$: All noise sources as seen in Fig. 4 for case A, excluding the gravitational background; $S_{\mathrm{A}}$, $S_{\mathrm{B}}$: Noise contributions from TTL assuming $\mathrm{k}=3\mathrm{mm}/\mathrm{rad}$; $S_{\text{residual}}$: Expected residual $S_{{\mathrm{res}}_{\mathrm{A}}}$ ($S_{{\mathrm{res}}_{\mathrm{B}}}$) for case A (case B) using expected jitter and an integration time of $T\sim 1\mathrm{day}$ as computed in Eq. (24); case A has a roughly $6\times$ lower residual. DWS: readout noise assuming $\mathrm{k}=3\mathrm{mm}/\mathrm{rad}$.