I know that the expansion of the universe is speeding up, but is there an equation to measure this? Also, what is the cause of this?

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    $\begingroup$ Hi Daosof, welcome to StackExchange. This is a great question with interesting and not entirely well-established answers, but both the expansion, the equations, and the possible causes are extensively covered on Wikipedia. I suggest you read those articles, or others like this, and then get back with more specific questions if you still have any :) $\endgroup$ – pela Jan 29 at 19:50

The fact that the distance between galaxies is increasing is found by Redshift of light, where, the electromagnetic radiation from galaxies, as they travel large distances, get their wavelengths increased, and turn towards the red side of the spectrum. Upon working on these measurements, Astronomers concluded that the rate of increase of separation between two galaxies is directly proportional to the distance between those two galaxies.

So, consider two galaxies A and B, the distance between them being r

Then, dr/dt α r i.e, the speed at which they are separating is proportional to r.

The cause of this is not completely and surely known as of today, so it has been named a mysterious force called dark energy.

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    $\begingroup$ This does not address why the "expansion of the universe is speeding up", it is a statement of Hubble's law. $\endgroup$ – Rob Jeffries Feb 5 at 6:56
  • $\begingroup$ We do not know the reason for that yet. $\endgroup$ – Rikhil Gupta Feb 5 at 8:05

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