Michele's research

Time-Delay Interferometry and the LISA measurement concept

Understanding TDI

Developed by J. W. Armstrong, F. B. Estabrook, M. Tinto, and others, Time Delay Interferometry (TDI) is the LISA-specific technique used to combine the one-way phase measurements performed between the three spacecraft into virtual interferometric observables that cancel the otherwise overwhelming laser phase noise. TDI is what makes LISA a true interferometer.

Geometric TDI is my take on time-delay interferometry: instead of deriving the expressions for laser-noise-free observables using algebraic manipulations, it relies on directed graphs (not unlike Feynman diagrams), whose ``closure'' identifies laser-noise cancellation. These graphs are a very intuitive way to understand how TDI works; they also allow the exhaustive enumeration of all possible TDI observables of a given length, identifying useful glitch- and gap-resistant variants of the standard expressions.

Synthetic LISA, my high-fidelity simulator of the LISA science process, can produce synthetic data streams for all the TDI observables, according to a full model of TDI.

  • Geometric Time Delay Interferometry
    M. Vallisneri
    Phys. Rev. D 72, 042003 (2005) [+]
  • To understand the different notations and conventions in the TDI literature, use my Rosetta Stone (PDF, v. 1/5/2005).
  • See also eLISA, a visual demonstration of TDI (requires Java).

Improvements to the LISA concept: TDI ranging and post-processed TDI

Thinking computationally about LISA (which I began to do while simulating its response) helped me devise a series of improvements to TDI, and to the LISA measurement concept.

TDI observables are assembled by summing delayed one-way phase measurements; the delays must correspond very accurately to the evolving distances between the spacecraft. Tinto, Armstrong, and I were able to show that these distances can be determined in post-processing from the phase-measurement data itself, so that LISA does not a dedicated ranging system (or at least would still be able to operate were the ranging system to fail).

Until 2004, it was thought that the TDI observables needed to be assembled in real time aboard the spacecraft, which requires complicated onboard logic and could lead to irreversible data corruption. Shaddock, Ware, Spero, and I proved instead that the observables can be reconstructed in post-processing from the time series of the LISA phase measurements, sampled at low rates that can be transmitted economically to Earth.

  • TDIR: Time-Delay Interferometric Ranging for space-borne gravitational-wave detectors
    M. Tinto, M. Vallisneri, and J. W. Armstrong
    Phys. Rev. D 71, 041101(R) (2005) [+]
  • Post-processed time-delay interferometry for LISA
    D. A. Shaddock, B. Ware, R. E. Spero, and M. Vallisneri
    Phys. Rev. D 70, 081101(R) (2004) [+]

A general theory of LISA sensitivity in alternate configurations

In the case where not all of the LISA laser links are working, one can still build TDI observables, although their space (usually three-dimensional) is reduced to two or one dimensions. In this paper, Crowder, Tinto and I develop a general formalism to predict the LISA scientific performance (its sensitivity and expected parameter-estimation precision) in those configurations.


From the exhaustive survey described in my Geometric TDI paper, a list of all modified TDI observables of lengths 8 to 16, and of all second-generation TDI observables of lengths 16 to 24. These are ASCII text files, so the notation is a bit different from the paper: link indices (1, 2, 3 CCW, 1', 2', 3' CW) are in the forward time direction unless prefixed by a minus.

  • Modified TDI: 8, 10, 12, 14, 16 (0.5Mb).
    Each row corresponds to a unique TDI combination (modulus cyclic string shift and reversal symmetries), and displays an ordinal label, the link string, the number of beams (contiguous substrings with the same time direction), and the type of the combination.
  • Second-generation TDI: 16, 18, 20, 22 (0.3Mb), 24 (1.9Mb).
    Each row corresponds to a unique TDI combination (modulus cyclic string shift and reversal symmetries), and displays an ordinal label, the link string, the temporal footprint, the number of beams (contiguous substrings with the same time direction), the type of the combination, and its splicing decomposition in shorter modified TDI combinations, identified by their ordinal label and their type.

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© M. Vallisneri 2014 — last modified on 2012/05/15