# How to interprete the gravitational wave measurement?

I saw this picture: the LIGO measurements of a gravitational wave.

I have a few questions about the graphs. First the graph with residual, what does that mean? If the data of Hanford is projected on the Livingstone graph, it is shifted and turned around. Why is the latter, is it because the two detectors aren't in the same direction?

Can you know the mass of the objects that collided and the mass of the resulting object with only this graph, if yes, how, if no, how did LIGO do it?

This figure comes from the paper describing LIGO's initial detection of gravitational waves. This GW event was named GW150914. The LIGO Virgo Collaboration writes a "science summary" for each major paper they write. The summary for this paper is here.

## Observed GW Strain

The top panels show LIGO's strain measurement, or fractional change in length $$\Delta L/L$$, for each detector after it has been "whitened". To whiten the data it is essentially divided by the average noise. This is kind of like a bandpass filter which suppresses frequencies where the instrument is not sensitive. The detectors' noise spectra are shown in Figure 3b of the same paper (below).

To compare the data between the two interferometers one must account for the fact that the detectors are not colocated, nor are they oriented exactly the same.

Figure 3a of the same paper, shows the location of each detector in the USA and the orientation of the detector arms. The time shift accounts for the finite signal travel time for the GW to pass from one detector to the other. The inversion is a phase shift of $$\pi = 180^\circ$$ in the GW waveform, which results from the flipped orientation of the two detectors. The orientation isn't perfectly flipped, so the actual phase shift isn't quite $$\pi$$, but it's pretty close.

## Data residual

The residual is the difference between the data and the reconstructed signal. If the reconstructed signal is good fit to the data, then the residual should be white noise (white, because of the data whitening step).

This particular plot shows non-white noise because the numerical relativity waveform subtracted from the data to make this plot was not the maximum likelihood signal. The actual data analysis used waveforms calculated in a different manner. Some people questioned the LIGO detection due to confusion surrounding this plot, but their claims have been thoroughly debunked. For example see this paper by Nielsen, et al (preprint, arXiv:1811.04071).

## measuring masses and other parameters

The shape of the waveform carries information about the motion of the sources. Assuming the motion of the two black holes is well described by general relativity, the motion is determined by the masses, and spin angular momentum of the two black holes. For instance, the peak frequency of the signal encodes the total mass $$M = m_1 + m_2$$ of the system.

You can make a crude estimate of the peak frequency from the original plot, by measuring the time, $$\Delta t$$ between successive peaks of the waveform. The peak frequency is $$f_\mathrm{max} = 1/\min(\Delta t)$$.

Other questions on this stack address this in detail, for example: How were the solar masses and distance of the GW150914 merger event calculated from the signal?