Pre-Phase A Report - Lisa - Nasa

84 Chapter 4 Measurement Sensitivity
Adding all these contributions quadratically, one arrives at the total path length variation
of 40×10−12 m/ √ Hz in the bottom line. It is this estimate of 40 pm/ √ Hz that formed
the basis of the sensitivity curves in the figures of Section 1.2 .
The shot noise contribution in Table 4.1 was calculated assuming a 1 W laser and a
diameter of 0.30 m for the optics (see Section 3.1).
Some of the major noise effects are discussed in more detail in the remainder of this
section or in earlier sections.
Laser phase noise. Another optical noise source creating spurious optical-path signals
is the phase fluctuations of the master laser. The four lasers on the four spacecraft are
phase-locked with each other, but with about 17 s time delays for two of them because of
the length of the interferometer arms. The requirements for measuring and correcting for
the laser phase noise are discussed in Section 4.3.2 .
Thermal vibrations. The proof masses and the optical structures have their resonant
frequencies orders of magnitude above the frequency range of the gravitational wave signals.
Nevertheless, thermal vibrations (Brownian noise) can produce spurious signals. We
are searching for signals whose frequencies are well below the lowest resonant frequencies
of the optical structure. The (kT) thermal vibration noise is composed of the sub-resonant
tails of the (various) structural resonances. These vibrations, per proof mass, have linear
spectral densities δL(f) of apparent arm-length fluctuations of a general form
δL(f) =
4 kT
MQω2 0 ω
1/2 (4.5)
if we assume the (noisier) displacement-dependent ‘structural damping’ (imaginary spring
constant) [96]. These noise sources will be very small for LISA, so they are not included
in Table 4.1.
4.2.3 Acceleration noise budget
Table 4.2 gives the error allocation for spurious accelerations, mainly of the individual
proof masses, the allowed value (in units of 10−15 ms−2 / √ Hz at 10−4 Hz), and the number
of such effects entering in one inertial sensor.
The allowed total effect of acceleration noise for one inertial sensor is 3×10−15 ms−2 / √ Hz .
The effect on the difference in geometrical arm lengths is then 6×10−15 ms−2 / √ Hz . It is
with these values, after multiplication with factors 5 (SNR) and √ 5 (for orientation and
polarisation), that the sensitivity curves of Figures 1.3 and 1.4 were drawn.
To ease comparison of these acceleration errors with the allowed errors in optical path,
the total was multiplied by two in the final line, to give 12×10−15 ms−2 / √ Hz . A multiplication
with ω−2 will give the optical-path error due to the contributions in Table 4.2 .
3-3-1999 9:33 Corrected version 2.08