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As I understand, the evidence for dark matter is the observation of the gravitational dynamics of objects around the size of galaxies and above. They move in a way such that the equations imply there is more mass than what we can see. There has never been any direct observation of dark matter, and our theories tell us it cannot be composed of any elementary particles from the standard model. Furthermore, there is no observation which suggests that dark matter is unevenly distributed, that some galaxies consisting of roughly the same amount of normal matter have varying amounts of dark matter.

I wonder if therefore an alternative explanation for the phenomena is that we have the wrong dynamical equations for gravity. Just as Newtonian gravitation is a good approximation to general relativity at low masses/energies, perhaps general relativity is a good approximation to the true gravitational theory at sub-galactic distances. Maybe when distances become very large we start seeing that gravatational force is described by slightly different curves than what general relativity predicts.

Is there any good evidence speaking against this alternative approach?

The reason this question is different from the "possible duplicate" is that the other question asks if the source of the effect attributed to dark matter is parallel universes, rather than different dynamical laws. However, the answer there does mention these alternative gravitational theories.

I realize now that this is not a new idea, and I thank the commenters and answerers for pointing me to existing research. However, this part of StackExchange does not bill itself as an "experts only" forum, so I don't think the downvotes are fair. In mathematics for example, there is one forum for experts, and another for everyone. I'm an expert in something, but not astronomy. Thanks.

Multiple theories and hypotheses have been proposed as an alternative to dark matter (DM). The most popular are, arguably,

• 
  MOND (MOdified Newtonian Dynamics):
  A term for various theories where the gravitational force falls off less steeply than $1/r^2$ at large distances.


• 
  TeVeS (Tensor–vector–scalar):
  A relativistic generalization of MOND.


• 
  f(R) gravity:
  A generalization of general relativity, where the so-called Ricci scalar — which describes the curvature of space(time) — may take different forms than just a number.


• 
  Entropic gravity:
  A string-theory-based theory describing gravity not as a fundamental interaction, but as a so-called entropic force.

Note, however, that DM is not just "a fudge factor to explain weird dynamics". What makes most (but not all) physicists believe that DM exists is, I think, that several independent observations yield consistent results, not only qualitatively, but quantitatively. In addition to rotation curves of spiral galaxies, or velocity dispersions in stellar clusters, elliptical galaxies, and galaxy clusters, you have for instance

• Gravitational lensing:
  Massive clusters of galaxies "bend" space around them, causing the light from background sources to diverge from a straight path. Measuring this divergence tells us the mass of the foreground object, which can then be compared to the visible mass..




• Cosmic microwave background:
  The patterns in the inhomogeneities of the background radiation is sensitive to the amounts of both dark and "normal" matter.




• Structure formation:
  DM explains how matter may collapse and form galaxies a few 100 million years after Big Bang, before the Universe expanded too much.




• Supernova observations:
  The way that the brightness of distant supernovae decreases with redshift depends on the expansion history of the Universe, which in turn depends on the densities of the various components of the Universe. A certain amount of matter is needed to explain the expansion, and this amount is larger than the observed amount by the same factor that the other experiments yield.

Some of the above phenomena are explained by the alternative theories as well, but in general several ad hoc hypotheses have had to be made. In my view (but I am no expert on this!), DM is simply the simplest explanation, and so, by Occam's Razor, is favored by most physicists. I think the theory, unfairly, has a reputation for being "some magic that astronomers pulled out of their hat to explain what they don't understand, and which we made sure can't be observed so you can't prove us wrong". But firstly, postulating an unknown object to explain some phenomenon is quite normal in physics — this is how e.g. Neptune and several elementary particles were found. And secondly, there are several candidates for what DM might be, and at least some of these make falsifiable predictions. If DM is a particle, then that particle might produce a signal that can be observed. There are many efforts to search for a DM annihilation signal, which is expected to be in the (tens of) GeV region (e.g. H.E.S.S Collaboration 2011 and Cui et al. 2017). There are also efforts for direct detections — see e.g. Liu et al. (2017) for a review.

One of the most compelling pieces of evidence yet for DM came, I think, with the observation of the Bullet Cluster (and subsequently many other clusters), where lensing clearly shows how unaccounted-for mass follows the collisionless galaxies, but not the collisional gas, as seen in this beautiful image:

^{The Bullet Cluster. Pink shows X-rays from the gas, whereas blue shows where the mass is concentrated. Credit: NASA/CHANDRA.}