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## Example: omega_min program

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The example program described in the previous section calculates the theoretical signal-to-noise ratio after a total observation time , for a given pair of detectors, and for a given strength of the stochastic background. A related--and equally important--question is the inverse: What is the minimum value of required to produce a given SNR after a given observation time ?'' For example, if , then the answer to the above question is the minimum value of for a stochastic background that is detectable with 95% confidence after an observation time . The following example program calculates and displays this 95% confidence value of for the inital Hanford, WA and Livingston, LA LIGO detectors, for approximately 4 months ( seconds) of observation time. (The answer is .) Again, we are assuming in this example program that the stochastic background has a constant frequency spectrum: for . By modifying the parameters in the #define statements listed at the beginning of the program, one can calculate and display the minimum required 's for different detector pairs, for different signal-to-noise ratios, and for different observation times .

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

 (11.22.264)

where , , and are complicated expressions involving integrals of the the overlap reduction function and the noise power spectra of the detectors, but are independent of and . Thus, given SNR and , Eq. () becomes a quadratic for :
 (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