Image: A schematic pulsar timing array. (Credit: NASA/DOE/Fermi LAT Collaboration via Nature)

• NANOGrav observes the best astrophysical clocks in the Universe, millisecond pulsars. The pulsars spin rapidly and we see a beam of emission pass our line of sight many times per second. Since the emission is very stable, we can use each observed pulse as the "tick" of a clock, allowing us to measure time very precisely and accurately.
• As the distance between the pulsar and the Earth changes, the time it takes pulses to arrive at the Earth will also change. We develop models of radio-pulse arrival times which take into account the pulsar's spin and spindown rate, the position and motion on the sky relative to the Earth, and for many pulsars the orbital motions around a binary companion. With this information, we can predict when pulses will arrive at the Earth.
• Deviations from these timing models can be due to a wide range of effects. If a gravitational wave passes by the Earth or the pulsar, the distance will change ever so slightly. While pulsars have radii of about 10,000 meters, we expect a change of perhaps 10 to 1,000 meters depending on the type of passing gravitational wave. The distances to pulsars are of the order of 30,000,000,000,000,000,000 meters, which is why we need such precise clocks to measure distances so precisely.
• For gravitational waves that pass by the Earth, we can look for the correlation of arrival times between sets of pulsars either early or late. For example, if space stretches then two pulsars observed in the same direction will have their pulses both arrive later than we expect from our timing models. The pulses will arrive early if space contracts. Our demonstration illustrates a number of the techniques used in the pulsar timing and in the search for gravitational waves from a correlated signal without the need for a Galactic-scale detector but with one that can be built cheaply and can easily fit in a small room.

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• Just as we can observe radio pulses with telescopes and measure the times of their arrival, we can "observe" sound pulses with a microphone. While they do not spin like pulsars, the sound pulses from metronomes behaves like what we would detect.
• With a quality microphone and metronome, we were able to measure the arrival times of the sound pulses to well within one microsecond with even the most basic of data analyses. Since the speed of sound is approximately 340 meters per second (760 miles per hour), we are able to localize the relative position of the microphone to within a fraction of a millimeter. As with a true pulsar timing array, the absolute distance does not matter but only how it changes relative to the observer.
• With a single metronome on a circular track, one can observe the motion of the metronome analogous to a pulsar in a binary orbit as long as the size of the orbit is of order many centimeters (it can be smaller, depending on how quickly the pulses are being generated). If the microphone is on a circular track, the size of the orbit is analogous to measurement of parallax from the Earth orbiting around the Sun.
• With two metronomes, one is able to move the microphone around analogous to a gravitational wave passing by the Earth. Just as we do with pulsars, we are able to correlate the arrival times of the microphone pulses in order to detect the motion of the microphone.

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Image: Cartoon of a pulsar orbiting around its binary companion (left) and the Earth moving around the Sun (right). Both of these motions will change the distance that pulses must travel and therefore their arrival times.

Image: The Hellings-Downs curve, showing the correlation of a gravitational wave signal for pulsars separated by some angle. Unlike the asymmetric shape due to gravitational waves, our acoustic correlation curve will be related only by the cosine of the angle.

• We chose Seiko model SQ50-V quartz metronomes in our demonstration, which allow for a range of beats per minute (analogous to the pulsar spin period), volume (analogous the brightness of the pulsar), and pitches (analogous to the unique pulse shape of the pulsar). By having two different available pitches, we were able to record two metronomes simultaneously while distinguishing which pulses came from which metronome.
• For each metronome separately, we record a series of pulses that we can then fold (average) together in order to build a high signal-to-noise pulse profile. Then when we record the set of pulses simultaneously from both metronomes, by cross-correlating the profiles with the data, we are able to extract high-precision arrival times.
• The precision of our measurements allows us to find deviations from the expected beats per minute for the metronomes. Once we know the "period" of the metronomes, we are able to compare the observed arrival times with the expected arrival times from just the metronome period and create timing residuals, the difference between the two. Any variations in the residuals outside of the errors should be due to the motion of the microphone.
• To simpify the mathematics, we move the metronome in a circle at a uniform speed. The correlation between the residuals is simply related to the cosine of the angle between the metronomes. When the angle is 0 degrees, the residuals should be fully correlated and when it is 180 degrees, the residuals should be anti-correlated. This is analogous to finding the Hellings-Downs curve correlation for a pulsar timing array caused by gravitational waves.

• We developed a graphical user interface (GUI) to help perform the analyses. There are two versions: a single-metronome GUI for analyzing each metronome's period and profile shape separately and a double-metronome GUI for doing the correlations.
• For each 45 degree interval from 0 to 180 degrees, we performed the experiment and calculated the observed correlation coefficient. We used the errors on the fit to our residuals to simulate results rather than run the experiment thousands of times. Our results matched the expected dependence of the correlation coefficient on the angular separation of the metronomes.
• We show the GUIs in action in the paper and also provide documentation to their uses. We also provide sample data files that one can use to more easily perform the experiments on previously recorded data.
• Our software is freely available. The cost of the setup is very cheap and the space required is small, only a few meters by a few meters at the most. One can use freely available software-based metronomes though the errors on the arrival times, and thus the correlation coefficients, will be larger due to lower quality. One can also use a built-in laptop microphone rather than purchasing a separate one though moving the laptop may be more difficult, depending on the experimental setup.

Image: Setup of the two metronomes laid out at 45 degree intervals with the microphone position in the center.

Image: An example of the graphical user interface to perform the correlation analysis. The correlation of the arrival times in the bottom right is clearly seen with the two red curves.

• To avoid the installation of the Python packages and their requirements, we are developing a web-based interface for running the data analysis. While the data taking will be done locally, the data analysis will be done remotely on the web server after uploading of the appropriate files.
• We are working on the completion of a smartphone app called TableTopPTA which will allow smartphones to act as the metronomes or the microphone. The app is built in Javascript and runs on Android smartphones, and the code is available on github.
• As part of broader NANOGrav outreach efforts, we are developing a Raspberry-Pi-based hardware package that will come with instructions on how to construct a setup that will be compatible with our software and easy to install. While it will include a number of possible demonstrations that one can construct, the metronome demonstration will be the primary focus.
• To illustrate the pulsar spin more accurately, we are developing a similar demonstration using a rotating flashlight on a turntable and a camera. The sweep of the flashlight beam is analogous to the pulsar radiation moving across our line of sight. Since the speed of light is so much faster than the speed of sound, measuring arrival times used in a correlation analysis will not be possible, and this will be designed as a single-flashlight demonstration only.

• Members of the NANOGrav Collaboration: M. T. Lam, J. D. Romano, J. S. Key, M. Normandin, J. S. Hazboun
• Contact: Dr. Michael T. Lam (corresponding author), Dr. Scott Ransom (NANOGrav chair).

The NANOGrav Collaboration at the 2017 Fall meeting in Lafayette College, PA