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Example: omega_min program

0

The example program described in the previous section calculates the theoretical signal-to-noise ratio after a total observation time $T$, for a given pair of detectors, and for a given strength $\Omega_0$ of the stochastic background. A related--and equally important--question is the inverse: ``What is the minimum value of $\Omega_0$ required to produce a given SNR after a given observation time $T$?'' For example, if ${\rm SNR}=1.65$, then the answer to the above question is the minimum value of $\Omega_0$ for a stochastic background that is detectable with 95% confidence after an observation time $T$. The following example program calculates and displays this 95% confidence value of $\Omega_0$ for the inital Hanford, WA and Livingston, LA LIGO detectors, for approximately 4 months ( $T=1.0\times 10^7$ seconds) of observation time. (The answer is $\Omega_0=2.87\times 10^{-6}$.) Again, we are assuming in this example program that the stochastic background has a constant frequency spectrum: $\Omega_{\rm gw}(f)=\Omega_0$ for $5\ {\rm Hz}\le f\le 5000\ {\rm Hz}$. By modifying the parameters in the #define statements listed at the beginning of the program, one can calculate and display the minimum required $\Omega_0$'s for different detector pairs, for different signal-to-noise ratios, and for different observation times $T$.

Note: As shown in Sec. [*], the squared signal-to-noise ratio can be written in the following form:

\begin{displaymath}
\left({\rm SNR}\right)^2={T\ \Omega_0^2\over
A + B\ \Omega_0 + C\ \Omega_0^2}\ ,
\end{displaymath} (11.22.264)

where $A$, $B$, and $C$ are complicated expressions involving integrals of the the overlap reduction function and the noise power spectra of the detectors, but are independent of $T$ and $\Omega_0$. Thus, given SNR and $T$, Eq. ([*]) becomes a quadratic for $\Omega_0$:
\begin{displaymath}
a\ \Omega_0^2 + b\ \Omega_0 + c = 0\ ,
\end{displaymath} (11.22.265)

which we can easily solve. It is this procedure that we implement in the following program.

The omega_min example program can be run in two ways. Without any arguments:
machine-prompt> omega_min
uses the detectors defined by SITE1_CHOICE and SITE2_CHOICE. The program can also be run with two command line arguments which specify alternative detector site choices, for example:
machine-prompt> omega_min 23 31
which produces the output:

Detector site 1 = LIGO-WA_enh7
Detector site 2 = LIGO-LA_enh7
S/N ratio = 1.650000e+00
f_low  = 0.000000e+00 Hz
f_high = 1.000000e+04 Hz
Observation time T = 1.000000e+07 sec
Minumum Omega_0 = 5.290809e-09

Includes/omega_min.tex

Authors: Bruce Allen, ballen@dirac.phys.uwm.edu, and Joseph Romano, romano@csd.uwm.edu
Comments: None.


next up previous contents
Next: Function: analyze() Up: GRASP Routines: Stochastic background Previous: Example: snr program   Contents
Bruce Allen 2000-11-19