10. What would it look like if there were a pulsar near enough to detect?

If there were a pulsar close enough to confidently detect, we expect that it would exceed the 2F=25 threshold in most or all of the sixty different 10-hour data segments. Thus it would lead to a large number count for some sky position in the coincidence plots shown in the next sections. In fact, we will see in the following sections that Gaussian noise alone produces a number count of only around six by chance.

How can we know if this would really work? To verify this, we employ two different checks that our data-analysis procedures are working correctly and could detect a source.

HARDWARE INJECTIONS
When the S3 data was taken, eleven simulated pulsar signals were 'injected' in real time into the detector hardware. To be sure that they would not cause unanticipated problems in subsequent data analysis, these hardware injections were not ``on'' during the entire S3 run. So they do not completely duplicate what would happen if a real signal were present. These fake pulsar signals are present in about one-third (roughly 20) of the ten-hour data segments analyzed by Einstein@Home.

SOFTWARE INJECTIONS
In preparing the (final, second time through) S3 data for distribution by Einstein@Home, we added an additional set of six 'software simulated' pulsars into the data set. The signals were present in all of the data, but their amplitudes were chosen near the expected threshold of sensitivity. Hence these simulated signals exceed the 2F=25 threshold in approximately half (roughly 30) of the ten-hour data segments analyzed by Einstein@Home.

Shown in the figures below are sky-maps for the frequency bands containing these hardware and software injected signals. Note the large number of segments for which F exceeds the threshold value.

The following two pairs of figures show a part of the results of the hardware and software injections. These (putative) pulsars have parameters ^10.1 given by:

	$h_0$	$f$ [Hz]	$\cos\iota$	$\psi$	$\Phi_0$	$\alpha$	$\delta$	$<N>$
hardware injected pulsar	6.16125E-23	108.8571594	-0.0806666107	0.444280306	5.5318207	3.11318871	-0.583578803	20
software injected pulsar	1.56171E-23	574.1	-0.9281	-0.2218	4.0345	3.7569	0.0601	29-31

Many of the figures that appear in the following pages are color maps of the celestial sphere. The coordinates are conventional astronomical coordinates: right ascension [40] and declination [41]. These are roughly analogous to longitude and latitude on the surface of the Earth. The color maps show the number of 10-hour data segments which exceed the 2F=25 threshold in each different sky and frequency region. In the figure captions we summarize this by saying ``Color map of the number of coincidences among sixty 10-hour data segments''. A strong pulsar signal would exceed the 2F=25 threshold in many or all of the sixty different data segments, and would lead to red regions on these color maps. Regions with no detectable pulsar signals only exceed the threshold by random accident (chance!) in a handful (about six) of data segments and lead to dark blue regions on these color maps.

Figure 10.1: Hardware injection. Color map of the number of coincidences among sixty 10-hour data segments. (a) All-sky map. (b) Frequency-declination plane.

(a) All-sky map	(b) declination - frequency map

Figure 10.2: Software injection. Color map of the number of coincidences among sixty 10-hour data segments. (a) All-sky map. (b) frequency-declination plane.

(a) All-sky map	(b) declination - frequency map

The most important single quantity in the above table is the gravitational-wave strain amplitude $h_0$. This strain amplitude $h_0$ depends on the distance to the pulsar, the gravitational-wave frequency, the ellipticity $\epsilon$ of the source and the moment of inertia of the source. (It is proportional to the ellipticity, moment of inertia, and square of the frequency, and inversely proportional to the distance.)

The figure below shows the relationship between $h_0$ and the distance in parsecs at a fixed ellipticity $\epsilon =10^{-6}$ and at the same frequencies as the hardware injection and the software injection pulsars mentioned above. The ellipticity is a measure of how 'out of round' or lumpy the star is. This value of $\epsilon =10^{-6}$ is about the upper limit of values considered reasonable for a normal neutron star, although some of the more exotic possibilities such as quark stars might have ellipticities 10 to 100 times larger [42] and thus would produce gravitational waves of 10 to 100 times higher amplitude $h_0$.

Figure 10.3: The gravitational-wave strain amplitude $h_0$ as a function of distance in parsecs. Here the ellipticity $\epsilon$ of the pulsar is assumed to be $\epsilon =10^{-6}$ (and a conventional value for the moment of inertia of $I\sim10^{38}~\textrm{kg\,m}^2$). The red circle indicates the software injection described above. A pulsar with $\epsilon =10^{-6}$ and the same properties as the fake software injected pulsar would be about $25$ parsecs away. The fact that Einstein@Home sees this software-injected pulsar thus means that it has a reasonable chance to detect pulsars at distances of tens of parsecs. These curves are drawn based on the equation (25) of [38]

You can see that with the S3 data set, we should be able to detect pulsars at distances of up to tens of parsecs (about one hundred light-years). This distance depends upon the frequency of the source. We'll also show later that this distance depends upon the source's location on the sky.

Contents[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]Bibliography

Einstein@Home S3 Analysis Summary	Last Revised: 2007.03.28 08:59:23 UTC
Copyright © 2005 Bruce Allen for the LIGO Scientific Collaboration	Document version: 1.132