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I actually have two questions:

  1. Since only a gravitational wave that is strong enough can be detected and the GW passes by the earth at light speed, can I say that we are lucky to detect one this time. It may take hundreds or even thousands years to detect the next one?

  2. Is there any chance that the detected gravitational wave is the result of the interactions of two or even more similarly strong sources? If yes, how should it affect the calculation of the location of where the two black holes merged?

user107189
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1 Answers1

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The probability of detecting an even is not known a priori, and that includes detection of a gravitational wave.

However, once you have observed the event once, you can set a reasonable bound on the probability that it will happen again in time x.

Here is how the argument might work. Let's say that it took a time $t$ to detect just one event; and you are asking

"what is the chance of not detecting an event in time $nt$ for some (large) value of $n$?"

The probability of detection is the product of "number of events occurring" times "probability of detecting an event, given that it occurred". Now we can safely assume that the number is large, and the probability is small- meaning we can approximate the phenomenon with a Poisson distribution.

Let's assume that we were lucky, and that the chance of detecting the event when we did was only 5%; if the mean expected number of observations per unit time was $\mu$, and we observed "no fewer than zero", then the probability of that occurrence is

$$p = 1-e^{-\mu t}$$

Setting that probability to 5%, we find $\mu t = 0.051$. If, given that probability, we want a 95% certainty of detecting an event, then we need to wait for a time $n t$ such that

$$1 - e^{-n \mu t} = 0.95$$

from which it follows that $n = 58$.

A slightly less conservative approach would use a p=0.22 ($\sqrt{0.05}$); then the probability of the first event having occurred so soon, and the second event taking this long, would be 95%. In that case, we find n = 3.4

Now LIGO went into "engineering mode" in February 2015, and detected an event in September - 7 months later.

It would be highly unlikely that it will take more than 58 * 7 months ~ 34 years before the next event is detected; it is quite unlikely that it will take more than 3.4 * 7 months ~ 2 years; and more likely, it will take another 7 months.

And that would be assuming that the system was "fully functional" in February, when in fact it was of course undergoing all kinds of testing and wasn't yet fully operational. In fact, according to the Wiki link above, it didn't start formal science observations until four days after the famous event... the event was detected on September 14, and the instruments started formal observations on September 18th.

Now there's a bit of relativity to ponder...

Floris
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