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This answer to Why can't supermassive black holes merge? (or can they?) describes the barrier to merging that two supermassive black holes face when two galaxies are in the process of merging or have "successfully" merged.

Here is a bit of the answer, but its worth taking a moment to go there and read the whole thing:

When two galaxies merge, their supermassive black holes both have angular momentum. Through a phenomenon known as "dynamical friction," gravitational interactions with other stars sap the black holes of much of their angular momentum, until they are brought within a few parsecs or so of each other. At this point, the black holes have flung out all of the stars that were in the region and there is (presumably) nothing left for dynamical friction to sap their angular momentum.

  1. What kind of timescale are we talking about to reach "the last (few) parsec(s)? Is it similar to the time it takes for the merged galaxy to reach some kind of centralized shape (things rotating around a common center rather than having two identifiable centers of rotation)? Much longer or shorter?
  2. What kinds of processes are involved that allow the two black holes to reach the last few parsecs of separation? What is "dynamical friction" and "star-flinging"?
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    $\begingroup$ Possible useful answer here: astronomy.stackexchange.com/a/14521/7411 $\endgroup$ Commented Jul 15, 2019 at 8:45
  • $\begingroup$ @PeterErwin aha, yep that's the longer version of your comment as an answer, but it doesn't yet address question #1 about the timescales. Something about the timescales plus a reference to that answer together probably answers this. Thanks! $\endgroup$
    – uhoh
    Commented Jul 15, 2019 at 9:21
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    $\begingroup$ Section 4.1 of the paper you linked to in your original question (Goulding et al. 2019) has some discussion of the timescales. $\endgroup$ Commented Jul 15, 2019 at 16:03
  • $\begingroup$ @PeterErwin that links to Begelman, Blandford & Rees 1980 in Nature Massive black hole binaries in active galactic nuclei which contains a concise and readable explanation and links to further sources. This is really interesting and I'll dig in. Thanks! $\endgroup$
    – uhoh
    Commented Jul 21, 2019 at 5:38

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There's a paper that gives exactly the formulae you are looking for by Sobolenko et al. (2021).

They describe the merger of supermassive black holes as a three-phase process. These phases may overlap or even not occur depending on the timescales involved, which depend on the masses of the black holes and other parameters.

The first part of a "dry" merging process (one in which is gas-free) assumes that the central BHs are surrounded by stars and experience the process of dynamical friction - the gravitational drag experienced by a massive body as it moves through a cloud of less massive objects. This transfers energy from the black holes to the surrounding stars, the black holes lose kinetic energy and move closer together. The timescale is $$\tau_{\rm df} \sim 200\left(\frac{1}{\ln N}\right) \left( \frac{10^6M_\odot}{M_{\rm BH}}\right)\left(\frac{r_c}{100\ {\rm pc}}\right)^2 \left(\frac{\sigma}{100\ {\rm km/s}}\right)\ {\rm Myr}\ , $$ where the central black holes, each of mass $M_{\rm BH}$ are sinking towards the centre of a stellar distribution with $N$ stars with a core radius of $r_c$ and a velocity dispersion of $\sigma$. The timescale is that to reach a separation roughly given by the "sphere of influence" of the black holes, which is roughly where their combined mass equals the stellar mass within this separation.

After that, dynamical friction becomes much less effective but three-body scattering interactions with stars can become important when the separation reaches $a_h \sim G\mu/4\sigma^2$, where $\mu = m_1m_2/(m_1+m_2)$ is the reduced mass of the black holes. Below this separation the timescale for hardening the binary (energy is extracted from the black holes by transferring it to the scattered stars and the black hole separation decreases) is $$\tau_{\rm hard} \sim 70\left(\frac{\sigma}{100\ {\rm km/s}}\right)\left(\frac{10^4 M_\odot\ {\rm pc}^{-3}}{\rho}\right)\left(\frac{10^{-3}\ {\rm pc}}{a}\right)\ {\rm Myr}\ , $$ where $\rho$ is the stellar density and $a$ is the black hole separation.

The presence of gas would usually shorten the merger timescales - offering another dissipative route to lose orbital energy.

The third phase is then the gravitational wave energy loss.

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