Why doesn't the centrifugal force cause rotating dark halos to flatten? 
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*Consider a spherical cloud of dark matter like the spherical halo around our galaxy. Please see the diagram below  

*Assuming that the halo is purely made up of dark matter which has only gravitational interaction and nothing else. Therefore, it cannot dissipate energy with time. 

*Now imagine the system is given a uniform rotation at $t=0$ about the axis shown in the figure. 
Wouldn't such a motion cause the dark matter distribution to flatten out and take up the shape of a disc? After all, these dark matter particles will experience a centrifugal force. But this is not observed for the galactic halo. The halo retains its spherical shape even though the galaxy is rotating. What is wrong with my reasoning? Isn't the halo also expected to rotate like the visible matter in the galaxy? 
 A: The premises of your question are false, according to current theory and measurements. Halos do rotate and their shapes are not spherical. Some references:


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*Current constraints on the Milky Way dark halo shape from the orbits of globular clusters suggest an axis ratio of $\sim 1.3$: Posti & Helmi (2019).

*The halo angular momentum has been theoretically well motivated for a long time: Hoyle (1949), Efstathiou & Jones (1979).

*The dimensionless halo spin parameter is $\lambda\sim0.035$, indicating that dark halos are mostly supported by dispersion (random motions) rather than rotation, so flattening into a disc is not expected, but a slight elongation is : Mo, van den Bosch & White (2010), p. 358-359 - this is a textbook reference, there are also many articles on this well-studied topic.

*There are also theoretically derived constraints on discs of dark matter that could in principle form by collecting within the stellar discs of galaxies. This seems not to be a very prevalent phenomenon, though: Schaller et al. (2016) - disclaimer, I am one of the authors.

