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Reference: the rotation speed of galaxies in a galaxy cluster:

Is the dark matter associated with each galaxy in a cluster, sufficient to explain the rotation speed of galaxies in the cluster, or extra additional dark matter in the cluster in the intergalactic space is required to explain the galaxy rotation speed in the cluster?

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Assuming that by "rotation speed of galaxies in a galaxy cluster" you mean something like orbital speeds (since most galaxies in a cluster have rather radial orbits and don't "rotate" within the cluster):

Yes, you absolutely need extra dark matter in the cluster, outside of the dark matter in the individual galaxies. The mass in a cluster comes primarily in three forms:

  • Ordinary (baryonic) matter in galaxies, mostly in stars (about 1-2% of the total mass);
  • Baryonic matter in the form of thin, extremely hot gas in between the galaxies, filling the cluster (about 10% of the total mass);
  • Dark matter (about 85+% of the total mass).

If you assume that each individual galaxy has dark matter amounting to about 10 times the baryonic matter (as is the case for most galaxies, including the Milky Way), then the per-galaxy dark matter is only about 10-20% of the total mass. Given that the total dark matter content is about 85% or more, that means $\sim 70$% of the total mass is in the form of dark matter filling the cluster in between the galaxies.

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Dark matter is clustered together with galaxies rather than spread across intergalactic space. Larger, more dense galaxies have more dark matter than smaller or more diffuse galaxies. And smaller galaxies that pass through larger galaxies can have their dark matter stripped away. As well, dark matter tends to be out front of moving galaxies, while the normal matter lags behind.

If you look at this map of the bullet cluster of galaxies, you can see the visible and dark matter together and separately. Take care to look at the 4th image where you can clearly see the purple dark matter out in front of the normal matter in pink.

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    $\begingroup$ Dark matter is clustered with galaxies and spread across intergalactic space. Also, dark matter does not tend to be "out front of moving galaxies". (The Bullet Cluster is a peculiar situation where the hot, X-ray-emitting intergalactic gas (the pink stuff) has become separated from the dark matter, which is still associated with the galaxies.) $\endgroup$
    – Peter Erwin
    Apr 14 at 19:41
  • $\begingroup$ @PeterErwin The bullet cluster is the best model we have so far, so I'll take it as strong evidence. It would be wrong to imply that dark matter is spread evenly. "a clear connection between the distribution of stars withing massive galaxies and halo mass. At fixed total stellar mass, massive galaxies with more extended distributions tend to live in more massive dark matter haloes." From "Weak lensing reveals a tight connection between dark matter halo mass and the distribution of stellar mass in massive galaxies." And the study on Ultra Diffuse Galaxies found a lack of dark matter. $\endgroup$
    – foolishmuse
    Apr 14 at 21:07
  • $\begingroup$ @PeterErwin and if you look at this video from NASA youtube.com/watch?v=0rGz862VPRo starting at 2:30, you'll note that the bow shock wave in the bullet cluster matches with the dark matter as shown here viewspace.org/interactives/unveiling_invisible_universe/…. Darned good evidence that they are associated. I would say that the bow shock wave causes the dark matter, but that's just my speculative idea. $\endgroup$
    – foolishmuse
    Apr 14 at 21:16
  • $\begingroup$ The Bullet Cluster is not "the best model we have" for clusters; it's a very peculiar system (which is useful for thinking about certain things), which is the result of a high-velocity, head-on collision between two smaller clusters. Also, I did not say the dark matter is spread "evenly". It has a clumpy distribution throughout a cluster, with a strong central concentration. $\endgroup$
    – Peter Erwin
    Apr 15 at 10:12
  • $\begingroup$ Dwarf galaxies in general have relative more dark matter (i.e., ratio of dark matter to normal matter) than large galaxies like the Milky Way. Some ultra-diffuse galaxies (UDGs) appear to have abnormally low amounts of dark matter; other have normal or even excessive amounts of dark matter (e.g., Dragonfly 44). $\endgroup$
    – Peter Erwin
    Apr 15 at 10:16

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