gravitational waves - Could (old) LIGO have detected GW150914?

The merging black hole binary system GW150914 was detected in only 16 days of aLIGO data at a signal level that appears to be well above the detection threshold at around 5 sigma. There are no further events between 4 and 5 sigma in the same data.

Could this event have been detected by previous LIGO/VIRGO incarnations which observed for much longer, albeit with lower sensitivity? If so, does this indicate that the aLIGO team have struck lucky and that this is a comparatively rare event that may not be repeated for many years?

EDIT: The answers I have agree that LIGO couldn't have seen it, but don't yet completely explain why. GW150914 had a strain that rose from a few $10^{-22}$ to $10^{-21}$ over about 0.2 seconds. This appears to make the characteristic strain maybe a few $10^{-22}$ Hz$^{-1/2}$ and thus would appear from the published sensitivity curves to lie above original LIGO's detection sensitivity at frequencies of $\sim 100$ Hz. Is my estimate of the characteristic strain way off?

Answer

To expand on HDE's answer, initial LIGO indeed wouldn't have detected GW150914, but it's not quite as simple as the peak strain being below the curve in the sensitivity plot: the integration time also matters.

These plots can be misleading; the curves they show don't represent a minimum detectable strain. Indeed, the units on the y-axis of these plots are $\mathrm{Hz}^{-1/2}$, while the GW strain is dimensionless, so you can't actually compare them! It's entirely possible to detect a signal that peaks well below the noise curve, as long as it's in-band for sufficiently long.

The curves that you see describing LIGO detector sensitivities conventionally show the amplitude spectral density of the detector noise. Meanwhile, the threshold for a detection is determined by the signal-to-noise ratio (SNR) from matched (Wiener) filtering. Assuming we know the form of the signal $h$ in advance (see caveats below), this is defined in terms of the noise-weighted inner product of $h$ with itself: $$ \mathrm{SNR}^2 = \left \equiv \int_0^\infty \frac{4|\tilde{h}(f)|^2}{S_n(f)}\,\mathrm{d}f $$ where $S_n(f)$ is the noise power spectral density (i.e., the square of what's shown in the sensitivity plots). The SNR therefore depends on the spectral composition of the signal and its overlap with the detector bandwidth.

If you imagine this in the time domain (Parseval's theorem), the (squared) SNR actually accumulates in proportion to the number of cycles the waveform spends in-band. For a monochromatic source, this is proportional to the integration time. For example, if $\tilde{h}(f) = \delta(f-f_0)h_0$ and, without loss, the noise PSD is a constant $S_n(f_0)$, then the SNR is given by: $$ \mathrm{SNR}^2 = \frac{2}{S_n}\int_{-\infty}^\infty|\tilde{h}(f)|^2\,\mathrm{d}f = \frac{2}{S_n}\int_{-\infty}^\infty|h(t)|^2\,\mathrm{d}t $$ Therefore, since $|h(t)| = h_0$, for a finite observation window $T$, the SNR scales with $\sqrt{T}$: $$ \mathrm{SNR} = \sqrt{\frac{2T}{S_n}}h_0 $$

So, let's approximate GW150914 as a monochromatic source. Reading off the plots in the detection paper, let's say it has a average frequency of $f_0 \approx 60 \,\mathrm{Hz}$, an amplitude of $h_0\approx 5\times10^{-22}$, and a duration of $T \approx 0.2\,\mathrm{sec}$. Then, reading off a strain ASD of $\sqrt{S_n(f_0)} \approx 10^{-22}$ for initial LIGO, we'd get an SNR of around 3, which doesn't meet the standard detection threshold of 8 (also, see the caveats below).

There's a much more complete discussion of detector sensitivity curves in this paper; it's worth a read! A more useful quantity, described in this paper, is the characteristic strain, which attempts to account for the frequency evolution of an inspiral signal such as GW150914, to ease comparison between detector sensitivity and strain amplitude.

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Caveats: in practice, it's more complicated than the matched filter model, since the detector noise is annoyingly non-stationary and non-Gaussian. There are more sophisticated search algorithms that use things like signal quality vetoes and $\chi^2$ discriminants that reject spurious responses of the matched filter. There are also search algorithms that don't require a priori knowledge of the signal waveform and can detect unmodelled bursts. It was actually this sort of generic search that detected GW150914; references are available in the detection paper.

Also note that the SNR defined above is the optimal SNR that you get if:

1. you filter the data stream with the exact signal that you're looking for, and


2. the noise realisation is zero.

Since the mean of the noise is zero, number 2 above is equivalent to taking the expectation of the SNR over all noise realisations.

In practice, we don't know the precise signal a priori, and some SNR is lost in the approximation. For a candidate waveform $u$, the expected SNR (over all noise realisations) is then given by $$ \mathrm{SNR} = \frac{\left}{\|u\|} $$