Do we have any proof for the existence of dark matter?

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It depends on what you require for something to exist. Dark matter does exist, in the sense that we have experimental proof from which we can infer its existence and characteristics.

All the observed effects are of gravitational nature so far. One evidence is the observation of gravitational lensing (the bending of light by massive objects) caused by "invisible" structures. We can use this method to pinpoint areas of the universe where there is dark matter. 

The main problem is that we do not know what dark matter really is. All we know is that there is something with the properties given by the data we observed (all the gravitational lensing, for example).

So given the experimental data, we believe is that dark matter has mass but does not interact at all with other matter, nor with electromagnetic forces ( the reason why dark matter is invisible). One of the most accepted hypothetical explanation for dark matter is that they are made up of particles that interact only via gravitational forces and the weak force (responsible for the decay of atoms), called weakly interacting massive particles (WIMPs). But these particles have not been detected so far, by direct or indirect detection. All experiments have been inconclusive so far, and thus, its existence has not been verified to this time.

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