Here's a picture showing how different sources of noise affect the sensitivity of LIGO

enter image description here

I'm trying to understand the frequency dependence of each curve. I'll specifically focus on seismic noise, suspension thermal noise and shot noise since they are the ones that determine the bandwitdh of the experiment. I think I understand seismic and shot noise but I'm not sure about thermal noise.

  • Seismic noise: The earth shakes constantly with an rms amplitude that approximately scales as

\begin{equation} \Delta l_{rms}=10^{-8}cm \big(\frac{100\text{Hz}}{f}\big)^{3/2} \end{equation}

where $f$ is the frequency of the vibrations. This immediately explains the shape of the seismic curve in the plot, the amplitude is bigger at smaller frequencies.

  • Shot noise: Shot noise affects the sensitivity of phase measurements at the photodiode. The noise scales as $1/\sqrt{N}$ where $N$ is the number of photons measured over a time interval $\Delta t$. The reason why this increases with the frequency of gravitational waves is that the time light spends trapped in the cavities should be roughly half a cycle of wave we are trying to detect so aiming for higher frequency gravitational waves means that the light will spend less time in the cavity. A smaller $\Delta t$ with a fixed laser power means that there will be less photons reaching the photodiode, which increases the error in phase measurements.
  • Suspension thermal noise: I understand that this source of noise comes from vibrational modes of the four masses used in the mirrors. the rms amplitude of one of this modes can be estimated as

\begin{equation} \Delta l_{rms}=\sqrt{\frac{kT}{m\omega^2}} \end{equation}

where $m$ is is the total mass of the mirror and $\omega$ the frequency of one of its modes. I also understand that this type of noise is reduced using isolation systems for the masses and using high Q factor materials such that vibrations are steady and they average out. However, I don't understand why this source of noise increases at lower frequencies.

  • $\begingroup$ What exactly is your question? Are you asking why $\Delta l_\text{rms} \propto 1 / \omega$? $\endgroup$
    – DanielSank
    Nov 29, 2021 at 22:39

1 Answer 1


The thermal force is approximately white, that is, frequency independent. Let's call the frequency domain force $\tilde{F}_t$.

If we write $F=ma$ in the frequency domain, we have \begin{equation} m \omega^2 \tilde{x} = \tilde{F}_t \end{equation} where $\tilde{x}$ is the frequency domain displacement, where $\omega=2\pi f$ is the angular frequency and $f$ is the frequency.

Solving this for the displacement, in the frequency domain, will grow at small frequencies \begin{equation} \tilde{x} = \frac{\tilde{F}_t}{m \omega^2} \end{equation} In physical terms, a low frequency force has a positive sign, before switching signs, for a longer time than a high frequency force. Since the amplitude of the forces at different frequencies is the same (since we have a white noise force), this means the displacement at low frequencies is larger (more time with the same acceleration).

  • $\begingroup$ Hi! Thanks for your reply! So what you are saying is that low frequency modes have higher amplitude and therefore interfere with low frequency gravitational waves more, is that correct? $\endgroup$ Nov 29, 2021 at 19:02
  • 1
    $\begingroup$ @P.C.Spaniel The noise power spectrum describes motion in the detector that occurs in the absence of gravitational waves; it's harder to see GWs at frequencies where there is a large noise power. Thermal noise causes fluctuations in the surface of the mirror, which leads to changes in the phase of the light at the output port of the interferometer, which is (mis)interpreted as motion of the mirror. If the amplitude of the thermal noise force is independent of frequency, then the amplitude of the displacement due to that force will be larger at lower frequencies. $\endgroup$
    – Andrew
    Nov 29, 2021 at 19:07
  • $\begingroup$ Does that make sense? Basically there is more noise at low frequencies due to thermal noise. Any noise will be superposed with any signals present (you could say the noise and signal "interfere"), more noise will make it harder to pull out a signal. To explain why thermal noise grows at low frequencies, you need this argument about how "the acceleration in $ma$ causes a $F(\omega)/\omega^2$ growth in displacement in the displacement" where $F(\omega)$ is the force in the frequency domain. $\endgroup$
    – Andrew
    Nov 29, 2021 at 19:09
  • $\begingroup$ Yes! Thank you. The thing that still confuses me a bit is that the curve in the plot is a rather continuous function that increases at low frequencies, but I assume thermal vibrations occur for the normal modes of the setup, which occur at discrete frequencies. In other words, there must be a lowest frequency mode of the hanging masses, you can't have thermal noise below that lowest frequency normal mode, can you? Do you happen to know what these frequencies are, approximately? $\endgroup$ Nov 29, 2021 at 19:15
  • $\begingroup$ @P.C.Spaniel Ah yeah, good point. There actually are suspension thermal noise resonances, called "violin modes", that appear in real noise spectra (what you're showing is a simplification based on simulations). You can see some real spectra here: gw-openscience.org/detector_status/day/20200105 in the LIGO detectors, there is a violin mode resonance at about 500 Hz and harmonics of that (around roughly 1000 Hz, around roughly 1500 Hz, etc). The "smooth" part of the thermal noise is a kind of residual "white noise" term. $\endgroup$
    – Andrew
    Nov 29, 2021 at 19:23

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