What is the Role of the Cosmological Constant in the Expansion of the Universe?

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E proposed cosmological constant to prove the universe is stationary. Some others suggested Universe is contracting, while some others suggested Universe is exanding and for either of them cosmological constant was considered responsible. Today when this is discovered universe is expanding with acceleration, cosmological constant is again thought to be the reason.

So, if universe was stationary, reason cosmological constatn.
If it was expanding at constant rate, reason cosmological constant.
If it was contracting, reason cosmological constant.
It is expanding with increasing rate, reason cosmological constant.

CC is the same thing as zero point energy, or dark energy and has its own 'mass'. It permeates the entire universe and every point in space. As the universe expands, the amount of dark energy also increases. And this leaves a question what is the origin of dark energy? When there was no space, there was no dark energy, so where does it come from?
 
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You seem somewhat confused in your historical statements here.

At the time when GR was formulated, the general consensus was that we were living in a static universe. However, Einstein realized that his theory did not allow for a static solution and so introduced a cosmological constant to force a contracting universe to be static. However, observations which were made in the following years by Hubble showed that the universe was indeed expanding -- an observation which was consistent with Einstein's theory without a cosmological constant -- and so Einstein dropped this constant from his field equations.

Then, around 1998, new observations indicated that the universe was not expanding uniformly, but that it is in fact accelerating in its expansion. One way of accounting for this acceleration is to reintroduce the cosmological constant though we don't know whether this is the correct solution, or whether there is in fact some other matter in the universe driving this accelerated expansion, which we term 'dark energy'.
 
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