2 Physical and Historical Motivations of TDI
Equal-arm interferometer detectors of gravitational waves can observe gravitational radiation by
cancelling the laser frequency fluctuations affecting the light injected into their arms. This is done by
comparing phases of split beams propagated along the equal (but non-parallel) arms of the
detector. The laser frequency fluctuations affecting the two beams experience the same delay
within the two equal-length arms and cancel out at the photodetector where relative phases are
measured. This way gravitational wave signals of dimensionless amplitude less than
can be
observed when using lasers whose frequency stability can be as large as roughly a few parts in
.
If the arms of the interferometer have different lengths, however, the exact cancellation of the laser
frequency fluctuations, say
, will no longer take place at the photodetector. In fact, the larger the
difference between the two arms, the larger will be the magnitude of the laser frequency fluctuations
affecting the detector response. If
and
are the lengths of the two arms, it is easy to see that the
amount of laser relative frequency fluctuations remaining in the response is equal to (units in which the
speed of light
)
In the case of a space-based interferometer such as LISA, whose lasers are expected to display relative
frequency fluctuations equal to about
in the mHz band, and whose arms will differ by a few
percent [3
], Equation (1) implies the following expression for the amplitude of the Fourier components of
the uncancelled laser frequency fluctuations (an over-imposed tilde denotes the operation of Fourier
transform):
At
, for instance, and assuming
, the uncancelled fluctuations from the
laser are equal to
. Since the LISA sensitivity goal is about
in this part of
the frequency band, it is clear that an alternative experimental approach for canceling the laser frequency
fluctuations is needed.
A first attempt to solve this problem was presented by Faller et al. [9
, 11, 10], and the scheme
proposed there can be understood through Figure 1. In this idealized model the two beams
exiting the two arms are not made to interfere at a common photodetector. Rather, each is
made to interfere with the incoming light from the laser at a photodetector, decoupling in this
way the phase fluctuations experienced by the two beams in the two arms. Now two Doppler
measurements are available in digital form, and the problem now becomes one of identifying an
algorithm for digitally cancelling the laser frequency fluctuations from a resulting new data
combination.
The algorithm they first proposed, and refined subsequently in [14], required processing
the two Doppler measurements, say
and
, in the Fourier domain. If we denote
with
,
the gravitational wave signals entering into the Doppler data
,
,
respectively, and with
,
any other remaining noise affecting
and
, respectively, then
the expressions for the Doppler observables
,
can be written in the following form:
From Equations (3, 4) it is important to note the characteristic time signature of the random process
in the Doppler responses
,
. The time signature of the noise
in
, for instance,
can be understood by observing that the frequency of the signal received at time
contains laser
frequency fluctuations transmitted
earlier. By subtracting from the frequency of the received signal
the frequency of the signal transmitted at time
, we also subtract the frequency fluctuations
with
the net result shown in Equation (3).
The algorithm for cancelling the laser noise in the Fourier domain suggested in [9] works as follows. If
we take an infinitely long Fourier transform of the data
, the resulting expression of
in the Fourier
domain becomes (see Equation (3))
If the arm length
is known exactly, we can use the
data to estimate the laser frequency
fluctuations
. This can be done by dividing
by the transfer function of the laser noise
into the observable
itself. By then further multiplying
by the transfer
function of the laser noise into the other observable
, i.e.
, and then subtract
the resulting expression from
one accomplishes the cancellation of the laser frequency
fluctuations.
The problem with this procedure is the underlying assumption of being able to take an infinitely long
Fourier transform of the data. Even if one neglects the variation in time of the LISA arms, by taking a finite
length Fourier transform of, say,
over a time interval
, the resulting transfer function
of the laser noise
into
no longer will be equal to
. This can be seen
by writing the expression of the finite length Fourier transform of
in the following way:
where we have denoted with
the function that is equal to 1 in the interval
, and zero
everywhere else. Equation (6) implies that the finite-length Fourier transform
of
is equal to the
convolution in the Fourier domain of the infinitely long Fourier transform of
,
, with the Fourier
transform of
[15] (i.e. the “Sinc Function” of width
). The key point here is that we can
no longer use the transfer function
,
, for estimating the laser noise
fluctuations from one of the measured Doppler data, without retaining residual laser noise into the
combination of the two Doppler data
,
valid in the case of infinite integration time. The
amount of residual laser noise remaining in the Fourier-based combination described above, as a
function of the integration time
and type of “window function” used, was derived in the
appendix of [29
]. There it was shown that, in order to suppress the residual laser noise below
the LISA sensitivity level identified by secondary noises (such as proof-mass and optical path
noises) with the use of the Fourier-based algorithm an integration time of about six months was
needed.
A solution to this problem was suggested in [29
], which works entirely in the time-domain. From
Equations (3, 4) we may notice that, by taking the difference of the two Doppler data
,
,
the frequency fluctuations of the laser now enter into this new data set in the following way:
If we now compare how the laser frequency fluctuations enter into Equation (7) against how they appear in
Equations (3, 4), we can further make the following observation. If we time-shift the data
by the
round trip light time in arm 2,
, and subtract from it the data
after it has been
time-shifted by the round trip light time in arm 1,
, we obtain the following data set:
In other words, the laser frequency fluctuations enter into
and
with the same time structure. This implies that, by subtracting Equation (8) from Equation (7) we
can generate a new data set that does not contain the laser frequency fluctuations
,
The expression above of the
combination shows that it is possible to cancel the laser frequency noise in
the time domain by properly time-shifting and linearly combining Doppler measurements recorded by
different Doppler readouts. This in essence is what TDI amounts to. In the following sections we will further
elaborate and generalize TDI to the realistic LISA configuration.