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Conformal gravity is an "alternative" theory of gravity, where instead of using the Einstein-Hilbert action composed of the Ricci scalar, the square of the conformal Weyl tensor is used. It was originally designed to arrive at the inflationary cosmological models without the use of dark energy.

However, it was later noticed that the galactic rotation curves of a certain matter distribution commonly seen in galaxies can also be accurately predicted using the conformal gravity and without the use of dark matter, but where in addition to total luminosity of the galaxy and its mass used in the matter distribution, two new constants appear. However, both of the constants turn out to be universal and are found to be equal for all galaxies (within the errors permitted by the deviation from the assumed baryonic matter distribution).

A while ago it was widely reported that the Bullet Cluster lensing effects rule out the alternative theories of gravity and provide evidence for dark matter. Does conformal gravity sufficiently explain the lensing effects observed in Bullet Cluster or is it similarly ruled out?

Kyle Oman
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SMeznaric
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Based on what I've heard during a few talks, and from skimming what I can from papers like this, I think the answer is probably not. In essence, my understanding is that conformal gravity leads to a different form of gravitational potential from mass distributions---but it can't create potential gradients where there isn't mass.

If you look at a picture of the bullet cluster (e.g., from here), you can clearly see that the primary source of field is in an entirely different place than the baryonic matter distribution.

Bullet Cluster

DilithiumMatrix
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    So does that mean it is ruled out? From what I have heard, one of the reasons why conformal gravity seems to reproduce the dark matter rotation curve profiles is that the iron sphere theorem does not apply and that matter outside can contribute to the potential. Given that, is it obvious how this misalignment means conformal gravity is ruled out? – SMeznaric Jan 11 '13 at 18:02
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    @SMeznaric I would say the Bullet Cluster is very damaging to all theories that claim there is no dark matter, precisely for the reason zhermes stated: you can change how potentials drop off with distance, but you can't create massive regions of attraction in the middle of empty space. Of course, you can always come up with some scenario, however implausible, to say the Bullet Cluster data is being misinterpreted. –  Jan 11 '13 at 18:10
  • @ChrisWhite That is true. I can see how this can rule out MOND theories, where you change the potential in a classical (Newtonian) way. I think conformal gravity is a bit different in this regard, as here you modify action to be the square of the conformal tensor from the Ricci scalar, which changes space-time around a matter distribution in a way potentially dependent on matter outside the distribution of interest. It's not obvious to me that this could not effectively change the lensing effects to reproduce the apparent shift in the Newtonian potential. Can you explain more on this? – SMeznaric Jan 11 '13 at 19:06
  • @SMeznaric, you need a way to establish the two centers of mass in the bullet cluster as preferred locations -- separate from that of the normal matter. That would seem to require a conspiracy of extended matter to form these highly symmetric, spheriodal mass distributions in just the right place -- which is quite improbable. – DilithiumMatrix Jan 11 '13 at 21:54
  • @SMeznaric, It also seems to be the case that if potentials depended on distant objects in such a convoluted way, it would make the normal dynamics of galaxies and galaxy clusters very different. – DilithiumMatrix Jan 11 '13 at 21:56
  • @zhermes But does it really require that much of a conspiracy? The question here is really whether there is some fundamental reason why the preferred locations are special and we expect those exact locations to be preferred. You can equally consider the dark matter distribution to have conspired to produce those points, just as you can consider the universe to have conspired. On the other hand, perhaps we can consider it simply as a coincidence of either the dark matter distribution or the universe. Or did you have another reason in mind why this could be a conspiracy? – SMeznaric Jan 14 '13 at 13:19
  • @SMeznaric sorry, I definitely didn't explain that well. With dark matter, the preferred locations are natural - two spherical galaxy 'halos' which simply pass through each other. The baryonic matter interacts, forming the conical distributions. If, instead, you try to construct these potentials based on a distribution of matter outside the local region - that distribution would have to be just right to create the same type of potential, but seemingly for no physical reason (i.e. just coincidence). – DilithiumMatrix Jan 14 '13 at 14:27
  • @zhermes Yeah, that sounds very convincing. I can see how those locations would be natural if you just assume nice spherical halos. Consider the following question as a result of my lack of understanding of the dark matter models: Is there a standard dark matter halo distribution that applies to every galaxy or do you need to find it in each individual case based on the rotation curves and/or lensing data? – SMeznaric Jan 14 '13 at 15:00
  • @zhermes Another question in this regard is whether it could be that the natural mass-energy distribution in the universe might not somehow lead to the same results anyway. Of course for consistency it must be the exact same distribution required to arrive at the rotation curves for galaxies, otherwise you have falsified the theory and you need to discard it. Is it inherently obvious why this scenario could not lead to the same results? (Sorry had to put it into two comments due to the space limit). – SMeznaric Jan 14 '13 at 15:00
  • @SMeznaric the distribution of DM in halos is a very active area of study and there are numerous unresolved questions. Overall, they are more homogenous than normal matter, and usually their profiles are described by (something like) the NFW profile: http://en.wikipedia.org/wiki/Navarro%E2%80%93Frenk%E2%80%93White_profile – DilithiumMatrix Jan 14 '13 at 19:06
  • @SMeznaric, regarding your second question, you can always devise a complex and elaborate physical theory that manages to explain individual scenarios (e.g. bullet cluster). Simpler models are generally better, and models which can explain more situations... such a 'conspiratory' model seems highly unlikely – DilithiumMatrix Jan 14 '13 at 19:07