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I think just about anyone who has studied Einstein or relativity has heard of the cosmological constant. He added it so that his equation would return an universe that is static. Of course, the universe is expanding but the cosmological constant is actually correct. So, my question is, what does the cosmological constant actually do in the equation? How does it affect the result? And lastly, what does this mean or represent in the universe?

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  • $\begingroup$ This is a rather broad question that is answered (in various gardes of detail) in many introductions and textbooks. You coould look at Carroll's lecture notes (arxiv.org/abs/gr-qc/9712019) in Chapter 8, in particular from p 222. $\endgroup$ – Toffomat Feb 10 at 14:44
  • $\begingroup$ Does this answer your question? Field equations with cosmological constant $\endgroup$ – ohneVal Feb 10 at 15:02
  • $\begingroup$ The question is not why there seems to be a small cosmological constant, but rather why there is not one 120 orders of magnitude larger. The cosmological constant is simply the lowest-order term when expanding the gravitational action in powers of the curvature tensor. $\endgroup$ – G. Smith Feb 10 at 16:51
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A cosmological constant on the left side of Einstein’s field equations has exactly the same effect as a so-called perfect fluid with the unusual equation of state $p=-\rho$ on the right. (I’m using units where $c$ is $1$.)

Because the Friedmann equation for the acceleration of the scale factor for a homogeneous and isotopic universe depends on the combination $\rho+3p$, the large negative pressure acts like a kind of antigravity, making the expansion of the universe get faster and faster. This acceleration has been observed.

Anything with the unusual equation of state $p=-\rho$ is often called “dark energy”. Whether it has anything to do with the vacuum expectation value of the energy-momentum tensor of quantum fields (which must have this equation of state to be Lorentz-invariant) is an open question. It may be due to a new scalar field that we don’t know about, with a nonzero vacuum value, perhaps changing on cosmological time scales.

It may also have nothing at all to do with fields and their energy-momentum tensors, and simply be a true constant in a term on the “geometry” side of Einstein’s equations. This would not be surprising, because such a cosmological constant would just be the leading (i.e., zeroth-order) term in the gravitational action when expanded in powers of the curvature. The puzzle is why this constant seems to have an unnaturally small but nonzero value.

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