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Comment: noise power spectra for ``advanced" LIGO & the Cutler-Flanagan model

0 During the years 1992-96, the upcoming LIGO experiment was planned in two stages, an ``initial" and an ``advanced" stage. These words were taken from an important article in Science [38] which described the LIGO plans. (The ``advanced" stage has now been supplanted by a series of seven enhancements and is now described as ``enhanced" LIGO).

During this period 1992-96, research on data analysis algorithms made use of detector noise curves taken from the Science article. Unfortunately two of the figures in the article (Figs. 7 and 10) which gave the noise curve were inconsistent, and also inconsistent with the parameters given in the article. The GRASP parameters/ directory contains a noise curve corresponding to Fig. 7, and another noise curve, generally called the ``Cutler and Flanagan" approximation, which is an approximation to the curve used in Fig. 10.

Note that the noise level in Fig. 7 in the Science article [38] is a factor of 3 in $h_{\rm rms}$ [or a factor of $\sim
10$ in $S_n(f)$] lower than that of Fig. 10 in between $\sim
10$ Hz and $\sim 70$ Hz. Kip Thorne has informed us that Fig. 10 is the correct figure and Fig. 7 is in error, and that the error does not appear in the corresponding figure V.4 of the 1989 LIGO proposal. The error can be seen by inserting the parameter values $m = 1000 \, {\rm kg}$, $f_0
= 1 \, {\rm Hz}$, and $Q_0 = 10^9$ given in [38] into the standard equation for suspension thermal noise due to viscous damping, as given in, e.g., Eq. (4.3) of Reference [15]. The resulting noise level is a factor of 3 higher than the noise level shown in Fig. 7, and agrees with the noise level of Fig. 10. Note however that the noise curve of Fig. 7 has been adopted and used by several researchers as the ``advanced ligo noise curve", and that the GRASP ``advanced" advanced noise curve is that of Fig. 7.

The Cutler and Flanagan approximation to the advanced ligo noise curve is

\begin{displaymath}
S_n(f) = {1 \over 5} S_0 \left[{f_0^4 \over f^4} +2 \left(1 + {f^2
\over f_0^2} \right) \right]
\end{displaymath} (11.3.224)

for $f \ge 10$ Hz, and $S_n(f) = \infty$ for $f < 10$ Hz, where $S_0
= 3 \times 10^{-48} \, {\rm Hz}^{-1}$ and $f_0 = 70 \,{\rm Hz}$. This is Eq. (2.1) of Reference [21] with $S_0$ replaced by $S_0/5$ to correct a typo in the published paper. The noise curve ([*]) is an approximate analytic fit to the advanced noise curve shown in Fig. 10 (not Fig. 7) of the LIGO Science article [38]. It is the GRASP noise curve noise_cutler_flanagan.dat. The accuracy of the fit is fairly good but not a perfect fit - in particular the noise curve ([*]) is slightly larger than the noise curve in [38] at high frequencies. A slightly more accurate fit has been obtained by Scott Hughes and Kip Thorne (quoted in Reference [6]), which uses the same functional form but the slightly different parameter values $f_0 = 75$ Hz and $S_0 = 2.3 \times 10^{-48}$ with a lower shutoff frequency of $12$ Hz.


next up previous contents
Next: Function: noise_power() Up: GRASP Routines: Stochastic background Previous: Function: detector_site()   Contents
Bruce Allen 2000-11-19