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Danish "LIGO discovery is noise" paper is hogwash

All of them (and maybe lots or even all of LIGO folks) are missing that the two LIGO detectors aren't predicted to see signals that are exactly proportional to each other

At a Forbes blog, Sabine Hossenfelder uncritically promotes a combative Danish paper in her article

Was It All Just Noise? Independent Analysis Casts Doubt On LIGO's Detections
It is all about a fresh paper
On the time lags of the LIGO signals
by Andrew Jackson and pals (Creswell, von Hausegger, Liu, and Naselsky). They say that the LIGO discoveries could be noise or based on completely fake data (as Bulgarian crank Pentcho Valev helpfully says under Hossenfelder's article). ATdotDE and Telescoper claim to be agnostic.



Has everyone lost his or her mind? Please give me a break.




When the first LIGO discovery was released in February 2016, I downloaded the raw data from both LIGO detectors sampled at 4096 Hz, wrote and ran all the required codes in Mathematica (that did especially the filtering of frequencies and the whitening), and discovered my own LIGO gravitational wave. I am absolutely certain that the probability that such clear signal-like events occur by chance is negligible. Also, the gravitational waves are the only conceivable signal that can make the delay between the two detectors this small. Any seismic or similar process would almost certainly lead to a much longer delay, basically because the vibrations would spread through the Earth by the speed of sound etc.




Andrew Jackson et al. is saying that "something is wrong" because "the residual noise from LIGO-LA and LIGO-WA detectors are correlated and have the same delay" but this correlation between these two "noises" shouldn't exist.

Except that this statement is wrong. The residuals aren't just noise. They're the difference between the best fit and the actual observation. But the best fit isn't the same thing as the actual gravitational wave.

In particular, as I have emphasized from the very beginning, the two detectors – LIGO-LA and LIGO-WA – are directed in somewhat different directions, so they basically observe different polarizations of the gravitational wave. Because that gravitational wave is rather general – it is neither circularly polarized, nor linearly polarized – the two detectors that basically measure the linearly polarized components in two different directions must yield somewhat different signals.



If the incoming gravitational wave were linearly polarized, the two signals could be proportional to each other with no extra phase delay, as on the animations above. If the incoming gravitational wave were circularly polarized, the two detectors' signals would look like delayed by a phase and in principle, one should say that the two functions are about as independent as the sine and cosine (times a modulating function). In reality, the actual incoming gravitational wave is something in between. It is rather well approximated by a linearly polarized wave, but it is not quite linearly polarized, so the two signals aren't proportional to each other.

Just an example: Imagine that you have some simple wave modulated by a Gaussian that is "circularly polarized" and it's reflected by the complex exponential\[

\exp(i\omega t - ct^2).

\] Well, if you measure this wave in the two detectors, assuming that they're rotated by 45 degrees for the sake of simplicity, they will produce the real and imaginary parts of the function above, respectively. Those two functions – i.e. the Washington and Louisiana signals – will be\[

e^{-ct^2} \cos\omega t, \quad e^{-ct^2} \sin\omega t

\] and these two functions aren't proportional to each other, not even if you shift them by an arbitrary time delay. It's easy to see why: the first function is even around its "center of mass" while the second one is odd around its "center of mass". Any nontrivial linear combinations of these two functions will be signal-like! That's true even if you shift them by a time delay to "make them look proportional to each other". Some time shift may approximately and temporarily "mimic the phase delay", which is \(\pi/2\) in my example, but the fixed time delay can't replace the phase delay exactly because the frequency of the gravitational wave is changing with time or is modulated (i.e. because the wave isn't monochromatic and periodic). The replacement is only approximately good if the modulating frequency of the signal and the rate of the frequency change are much slower than the peak frequencies of the signal. To put it mathematically:\[

\forall (A,B)\in \RR^2 \backslash \{(0,0)\}, \,\,\forall \Delta t\in \RR:\\
A e^{-ct^2} \cos\omega t + B e^{-c(t-\Delta t)^2} \sin [\omega (t-\Delta t)] \neq 0

\] I have absolutely no doubt that the signal is real and comes from a black hole merger. But I am not so sure whether LIGO members actually understand that the two LIGO detectors' signals aren't supposed to be proportional to each other because the incoming gravitational wave isn't linearly polarized – and for a circularly polarized wave, the non-parallel detectors detect the phase-shifted parts of the incoming signals, like the real and imaginary part of a complex wave, and the actual polarization is "somewhat elliptic and general", so everything that can be nonzero is nonzero.

I have been always worried that LIGO – and maybe every single person who has ever written a paper predicting the waves in two non-parallel detectors – makes this trivial mistake because they talked about things like the "inverted signal" which seems to assume that the LIGO-LA signal should be equal to the LIGO-WA signal times a positive number, or a negative number. But none of these choices is true in the real-world, generic case. Instead, they detect two parts of an elliptically, generally polarized gravitational wave.

So it's completely obvious that the signals aren't supposed to be proportional to each other and if you just compute the optimal linear combination of the two signals that should be noise-like, it simply won't be noiselike but will contain a part of the signal in the same period of time – basically the linear polarization of the signal with respect to a different axis than one that maximizes the signal's strength.

Dear LIGO folks, be sure that you have discovered the gravitational waves from a black hole merger but if I have pointed out something that you really did in a sloppy way by neglecting the general, non-linear polarization of the incoming wave, I urge you to redo the theoretical predictions and comparisons with the experiments, do it before your gurus are named the physics Nobel prize in Fall 2017, and thank me in the Nobel prize lecture.

Thank you very much, Ladies and Gentlemen.

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