# Frequency and amplitude for the three gravitational wave events at LIGO

I was reading LIGO/VIRGO's article Binary Black Hole Mergers in the First Advanced LIGO Observing Run, and have two questions concerning the frequencies and the amplitudes of the three gravitational wave events.

The question arises from figure 1 in the article:

As far as I understand, the y-axis shows the strain (which corresponds to the amplitude of the GW). The x-axis left is the frequency of the modulation, and right it is the time.

1. On the right image (and explained in the GW articles), one sees that the gravitational wave increases both in frequency and amplitude before the merger. Therefore I would expect in the left image, that the indicated amplitude-frequency bands for the three events increase with increasing frequency. However, it shows that the strain of the bands decreases with increasing frequency. Why is that so?

2. All three events seem to be significantly above noise even below 30Hz, as one can see from the left plot. In my naive picture, for GW150914 that could lead to a much longer signal. Why do they analyse the waves only above 30Hz (right)?

Good question, had me wondering also.

But it's relatively straightforward. On your question 1

1. the left graph is total power (or really square root of that to simply plot amplitude, and on a strain scale) at each frequency. It includes all the power of the signal at that frequency, even if it was at different times. The waveforms spent more time (more cycles), at lower frequencies, that's why it looks that way -- the power at the lower energies integrated power for a longer period of time. It That's the problem with freq/ampl graphs, a freq may include longer periods of time. For instance, in the last detection for GW 151226 the lower frequencies around 30 Hz went on (yes, slowly increasing) for about a second or so. They explain it, in the quote below from the pdf of the paper, in the next paragraph after Fig. 1 (bold italization is mine for the relevant sentence). You really would like to see a 3D plot, with freq and time in two axis, and amplitude the z axis, you could then see the chirp as a ridge int he time freq plane as in the figure below from Matlab at https://www.mathworks.com/help/signal/examples/practical-introduction-to-time-frequency-analysis.html?requestedDomain=www.mathworks.com

From the pdf you referred us to, at http://journals.aps.org/prx/pdf/10.1103/PhysRevX.6.041015 "The amplitude of the signal is maximum at the merger, after which it decays rapidly as the final black hole rings down to equilibrium. In the frequency domain, the amplitude decreases with frequency during inspiral, as the signal spends a greater number of cycles at lower frequencies. This is followed by a slower falloff during merger and then a steep decrease during the ringdown.

1. On your second question, no GW150914 has about an SNR of 0 dB at 20 Hz, and better at 30 Hz (at 30 it looks like about 3x$10^{-22}$ at 30, just sort of reading off the graph), so SNR of about $3^2$ or about 10 dB, so just detectable. At 20 Hz it was too low. They will extend the lower limit in later runs I understand.
• Thanks Bob, OK its the integrated power. That clearly explains both questions - great. The last thing I wonder, why that metric is interesting, in contrast to frequency/amplitude. Do you have any idea? – Mario Krenn Jan 9 '17 at 8:48
• I am not sure what you mean. But if you mean the graph on the left it still allows you to get snr at each freq. It is the power at that freq. i suppose they could have set it up so vertical axis is per unit freq, maybe they could have seen more detail at the higher freq but am not sure if one would see more, if you had matlab you could try.. I'd have to get into analysis details – Bob Bee Jan 9 '17 at 14:43
• To add to it, they do matched filtering for some (pattern based) detection, but I don't know their integration time or BW. The other detection is blind, and the needed snr will depend on parameters also. For the 150914 detection they published in their paper a lot of the specific details. – Bob Bee Jan 9 '17 at 18:22
• Btw, in their. Papaer they also show a time freq plot with color for the amplitude, and you can see the chirp, in the GW case it is an up chirp – Bob Bee Jan 12 '17 at 4:25