Time-delay interferometry for Gravitational Wave Detectors in Space: The Underlying Mathematical Structure

ABSTRACT:

Several ground-based detectors of gravitational waves (GW) around the world are in various stages of development including LIGO-India. However, there are astrophysically interesting GW sources, such as supermassive black-holes, white dwarf binaries, etc., which emit GW at low frequencies and lie below the band-width of ground-based detectors.

A detector in space overcomes this difficulty. However, a major source of noise in space detectors is the laser frequency noise which arises due to phase fluctuations in the laser. Amongst the important noise sources, laser phase noise is expected to be several orders of magnitude larger than other noises in the instruments. Thus, cancelling the laser frequency noise is vital for reaching the requisite sensitivity.

This talk will be in the context of the Laser Interferometric Space Antenna (LISA). The LISA configuration forms an unequal arm interferometer in the shape of a giant triangle which is almost equilateral. Time-delay interferometry (TDI) is a technique in which the data streams are combined with appropriate time delays so that the laser frequency noise is cancelled or suppressed.

Interestingly this scheme can be translated to a problem in algebraic geometry posed by David Hilbert in 1890 and solved later ∼ 1970 using Gr¨obner basis methods. The laser noise cancelling data combinations form a module over a polynomial ring, well known in algebraic geometry, as the first module of syzygies. Gr¨obner basis methods lie at the heart of this approach which are used to find the generators of the module. The talk will describe this approach and also touch upon the recent approaches involving matrices and Taylor expansions of the timedelay operators for time-dependent arm-lengths.