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The principle behind the detection of gravitational waves is that stellar-mass sources can act as coherent oscillators, producing a gravitational wave with a well defined, and relatively slowly varying, amplitude and frequency, whose effects can be detected. As a result the response of a gravitational wave detector scales as the amplitude of the wave - as $r^{-1}$.

When detecting electromagnetic waves, astronomers are almost exclusively dealing with incoherent sources and must rely on detecting the summed intensity, which scales as $r^{-2}$.

However, in the laboratory, surely a dipole transmitter can be driven such as to produce a very stable, monochromatic and coherent electromagnetic wave that could then be received by a detecting aerial such that the current in the aerial would then be directly proportional to the electric field of the propagating electromagnetic wave. In which case would the detection response of that aerial also scale as $r^{-1}$, or am I missing something fundamental?

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  • $\begingroup$ for detectebility SNR is what matters and that involves a detection bandwidth; placing an sqrt after the power measurement of the incoherent source will also get you an amplitude of 1/r behavior. $\endgroup$
    – hyportnex
    Jan 22, 2020 at 0:50
  • $\begingroup$ @hypnotex I don't understand the comment. SNR is what matters for GW detection too. If your $S$ depends on $r^{-2}$ then of course $SNR$ depends on $r^{-2}$ for a fixed noise? $\endgroup$
    – ProfRob
    Jan 22, 2020 at 7:58
  • $\begingroup$ the detection of a noncoherent signal, i.e., detecting noise signal in noise, has a threshold behavior (the output SNR is not proportional to the input SNR) that coherent detection does not, this has nothing to do with 1/r or 1/r^2, it has to do with the noncoherent detection; see Eq 22 in Mills, Prescott "A Comparison of Various Radiometer Detection Models" IEEE T-AES 1996, pp467-473 $\endgroup$
    – hyportnex
    Jan 22, 2020 at 11:09
  • $\begingroup$ @hypnotex The question is about detecting coherent signals. $\endgroup$
    – ProfRob
    Jan 22, 2020 at 12:39
  • $\begingroup$ I was trying to make the point that the supposed "1/r^2" behavior you are comparing to the coherent signal detection is due to the way the radiometer detects random rayleigh distributed envelope (noise signal) in an independent gaussian noise and is not because the way the antenna senses the electric field (that has 1/r dependence in free space far field, always). If this does not answer your question then I admit that I do not understand what you are asking. $\endgroup$
    – hyportnex
    Jan 22, 2020 at 12:47

1 Answer 1

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Coherent versus incoherent is a red herring here. There isn't any fundamental difference between gravitational and electromagnetic waves, either. The facts are the following:

  • For both coherent and incoherent waves, whether electromagnetic or gravitational, the typical intensity falls off as $1/r^2$, and the typical amplitude falls off as $1/r$.
  • For a white noise spectrum and a reasonably sharply peaked signal, the signal to noise ratio in both cases is given by the Dicke radiometer equation, $$\text{SNR} \sim \frac{P_s}{S_n} \sqrt{\frac{t_{\text{int}}}{\Delta \omega}}$$ where $P_s$ is the signal power, $S_n$ is the noise power spectral density (in units of $\text{W}/\text{Hz}$), $t_{\text{int}}$ is the total integration time, and $\Delta \omega$ is the frequency bandwidth of the measurement. The signal is detectable if $\text{SNR} \gtrsim 1$.
  • Of these quantities, $S_n$, $t_{\text{int}}$ and $\Delta \omega$ don't depend on $r$, while for both electromagnetic and gravitational waves, $P_s \propto 1/r^2$.
  • For an incoherent signal, $\Delta \omega$ above is the frequency width of the whole signal. For a coherent signal (i.e. one that appears monochromatic on the timescale of the experiment), the frequency width is the minimum frequency resolution achievable during the experiment, $\Delta \omega \sim 1/t_{\text{int}}$. The difference between coherent and incoherent is thus that the SNRs increase as $t_{\text{int}}$ or $\sqrt{t_{\text{int}}}$, respectively. This has nothing to do with the scaling with $r$.
  • The above analysis is phrased in terms of powers, but applies to both coherent and incoherent signals. You could also equivalently phrase everything in terms of amplitudes, by "taking the square root" of everything. In this case, you would have a signal amplitude, and the signal noise would be characterized by something with units of $\text{Ampl.}/\sqrt{\text{Hz}}$, and the SNR would increase as $\sqrt{t_{\text{int}}}$ or $t_{\text{int}}^{1/4}$, for a coherent or incoherent signal respectively.
  • These two ways of thinking about it are equivalent in power as long as the signal doesn't have any kind of special substructure (which the transient events in LIGO do). In any case, your desired example doesn't, and whether or not this substructure exists doesn't depend on $r$.
  • In the example you propose, we can phrase the result in terms of either power or amplitude, with equivalent results. For example, in terms of power, we could cool down the receiver by a factor of $4$, reducing the thermal noise power spectral density by a factor of $4$. So since power goes as $1/r^2$, we can see twice as far. This is completely equivalent to saying that the cooling reduces the voltage fluctuations (in units of $\text{V}/\sqrt{\text{Hz}}$) by a factor of $2$, so since amplitude goes as $1/r$, we can see twice as far.
  • For things like radiometers, you want to phrase the analysis in terms of power, because it's easier to think of the dominant noise sources (thermal noise, shot noise, etc.) in this way. For things like LIGO, you want to phrase the analysis in terms of amplitude, for the same reason -- the amount the mirror wobbles or the ground shakes is naturally an amplitude. But there's no fundamental difference, you could describe radiometers in terms of amplitudes if you wanted to.
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  • $\begingroup$ So you are saying that when LIGO say we are increasing our sensitivity by a factor of two, so we can see twice as far; that is is merely an artefact of them expressing their sensitivity in terms of amplitude? The fact that the signal is coherent then just means that you can detect GWs at all, not that it changes how detectability varies with distance. $\endgroup$
    – ProfRob
    Mar 6, 2020 at 9:38
  • $\begingroup$ @RobJeffries "So you are saying that when LIGO say we are increasing our sensitivity by a factor of two, so we can see twice as far; that is is merely an artefact of them expressing their sensitivity in terms of amplitude?" Yes, exactly! $\endgroup$
    – knzhou
    Mar 7, 2020 at 9:22
  • $\begingroup$ @RobJeffries "The fact that the signal is coherent then just means that you can detect GWs at all, not that it changes how detectability varies with distance." Sure. One can quibble a bit here over exactly what the word "coherent" means (LIGO's chirps are nowhere near monochromatic, for instance), but yes. $\endgroup$
    – knzhou
    Mar 7, 2020 at 9:23
  • $\begingroup$ By coherent, I mean a single oscillating source. When you look at electromagnetic sources in the universe you are looking at the superposition of countless little dipoles that are not coherent in time. $\endgroup$
    – ProfRob
    Mar 7, 2020 at 10:01
  • $\begingroup$ I'm still confused. eg. Equation 48 of this review on GW detection ncbi.nlm.nih.gov/pmc/articles/PMC5255901 posts a similar formula to the one in your answer, but the LHS is actually the square of the SNR. But that's exactly my point. The SNR seems to scale as $h$ not as $h^2$. $\endgroup$
    – ProfRob
    Mar 7, 2020 at 10:58

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