Frequency sensitivity of gravitational wave interferometer with its armlength and with mass of the mergers

To the best of my knowledge, LIGO is capable of observing gravitational waves (GW) from stellar mass black hole (BH) mergers but not mergers of supermassive black holes. In order to detect the latter mergers, we need GW interferometers (or detectors) with longer arms such a LISA. So it seems that with the increase of the mass of the merging BHs, the range of frequencies of the emitted GWs falls outside the frequency window that LIGO is sensitive to, which is in turn related to its armlengths.

Question $$1$$ What is the formula to find out the range of frequencies of the GWs that could be produced in a merger with the masses of the merging BHs?

Question $$2$$ What is the formula that relates the range of frequencies that LIGO or any other GW detector can observe with the length of the interferometer arms?

• These really are two separate questions, that should be asked separately. Commented Feb 2, 2021 at 13:37
• LIGO is seismic noise limited at low frequencies, not arm length limited. The noise properties of GW detectors like LIGO, LISA, and PTAs determine their frequency ranges. See, for instance, this answer: physics.stackexchange.com/a/543844/47594 Commented Feb 3, 2021 at 1:16
• If you stay on the ground increasing the arm length $L$ gets you more strain sensitivity for the same frequency range you already have. It's pretty much impossible to build a ground based detector that can go below 1 Hz because of seismic noise. You may be interested in reading about the next generation of ground based detectors like the Einstein Telescope or LIGO Cosmic Explorer. Commented Feb 3, 2021 at 2:26
• In addition to seismic noise there is also Newtonian noise produced by local fluctuations in the gravitational field due to seismic or atmospheric waves, which is another major limiting factor at low frequencies. Commented Feb 3, 2021 at 4:04
• @SRS There will literally be zero overlap between the answers to the two questions. Commented Feb 3, 2021 at 8:21

To answer your second question lets look at the sensitivity curve for LISA. (There are two reasons to look at LISA instead of LIGO. First, the arm length of LISA was a significant topic of discussion during the recent mission design phase. Consequently, the is plenty of sources discussing the impact of the arm length on the sensitivity. Second, LIGO for the most part observes gravitational waves whose wavelength is much longer than the arm length, whereas LISA will also see sources with wavelengths shorter than the arm length, making the arm length more relevant in the sensitivity curve.)

According to a recent paper by Robson, Cornish, and Liu, a good approximation to the LISA sensitivity curve (lower is higher sensitivity) is given by

$$S_n(f) = \frac{10}{3L^2}\left(P_{OMS}(f)+2(1+\cos^2(f/f_{*})\frac{P_{acc}(f)}{(2\pi f)^4}\right)\left(1+\frac{6}{10}\left(\frac{f}{f_{*}} \right)^2 \right),$$

where $$P_{OMS}$$ characterizes the noise introduced by the optical measurement system, $$P_{acc}$$ is the acceleration noise (i.e. how well the space craft can keep the test masses in freefall), $$L$$ is the arm length, and $$f_{*} = c/(2\pi L)$$ (the characteristic frequency if light traveling around the detector). We thus see that the sensitivity of LISA depends on the arm length in two ways.

• There is an overall suppression of the noise by a factor $$L^2$$. That is, making the arms longer, improves sensitivity over the entire frequency range.

• The second is through $$f_{*}$$, which accounts for a penalty to the sensitivity due to the wave length of the gravitational waves being comparable our smaller than the arm length. At higher, frequencies this penalty pretty much negates any advantage of making the arms longer.

The combined effect of these two effects is that increasing arm length shift the minimum of the sensitivity curve to lower frequencies.

For LIGO, the second effect is less relevant, and the location of the minimum is determined mostly by the competition by other sources of noise that do not depend sensitively to the arm length. (Mostly seismic noise at low frequencies, and shot noise at large frequencies.)

To give a quick, approximate, answer to the first question (see below for the significant caveats), the relevant figure of merit is the detectable chirp mass for a given configuration defined as $$M_\textrm{chirp} = (1+z) \left(\frac{M_1M_2}{M_1 + M_2}\right)^{3/5} (M_1 + M_2)^{2/5},$$ where $$M_1$$ and $$M_2$$ are the masses of the merging black holes an $$z$$ is the redshift of the black hole. The sensitivity of a given experiment to a given chirp mass will depend a lot on the noise properties of an experiment. For an example for Advanced LIGO, see here: https://core.ac.uk/download/pdf/267293216.pdf

We can do a very rough approximation for the inspiraling phase of a merger fully classically, assuming no relativistic corrections and no black hole spin. The specific derivation can be found here: https://arxiv.org/abs/gr-qc/9402014 but in the end we get a differential equation relating the chirp mass to the frequency evolution of the merger, $$\frac{df}{dt} = \frac{96}{5} \pi^{8/3} \left(\frac{G M_\textrm{chirp}}{c^3}\right)^{5/3}f^{11/3}.$$ We can use an extension to newtonian gravity to understand the merger better as we start going into the relativistic regime (so called post-newtonian formalism). Derivations related to using post-newtonian formalism applied to black hole mergers can be found here: https://arxiv.org/abs/1310.1528

The general solution for the actual observed frequency is a very complicated formula it depend on a lot of factors of the merger.

1. The mass of the black holes
2. The redshift the merger occurs
3. The angular momenta (i.e. spin) of the incoming black holes
4. The orbital eccentricity of the system
5. The orientation of the merger
6. The relative angle between the arms of interferometer and the incoming gravitational wave

In addition to this, there are multiple regimes of the black hole merger; inspiraling, merger, ringdown. In the case of current black hole detection, we don't look for single individual spikes but try to find an entire waveform. There are multiple templates for various events that are generated from numerical simulations which can be used as matched filters to try to find merger signatures.

• This seems unnecessarily complicated and avoids giving an answer. The mass-dependence of the chirp frequency just before merger is reasonably well defined. Commented Feb 3, 2021 at 8:34
• I edited it to highlight the chirp mass analysis first. Commented Feb 3, 2021 at 17:03