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Physics LibreTexts

16. The Cosmological Constant (INCOMPLETE)

It is a logical possibility, consistent with Einstein's equations, that there is an energy density associated with space itself; i.e., a certain amount of energy in every cubic centimeter, an amount that does not change with time. In fact, as we will see, there is evidence supporting this possibility. Einstein considered this possibility early on as a means to explain why the universe is static (as he thought it was), rather than expanding or contracting, when he introduced the ``cosmological constant'' as an additional term in his field equations.

 A static universe means \(\dot a = 0\), which can be arranged by balancing the two terms in the Friedmann equation. For a given density, choose the curvature appropriately and the expansion rate is zero. However, will it remain zero? From the Friedmann equation one can solve for \(\dot a\) and then differentiate with respect to time and use \(ad\rho/da = -3(P+\rho)\) to get


 \[ \frac{\ddot a}{a} = -\frac{4\pi G}{3}(3P+\rho).\] 


So in order for \(\dot a\) to remain zero, we need \(\ddot a = 0\) which means there has to be at least some contribution to the pressure that is negative. If it is to offset the positive pressure of other components we must have \(P < -\rho/3\) for the cosmological constant. As it turns out, this is not a problem. 
We know \(dE = -PdV\), and we know that the energy density of the cosmological constant remains unchanged by expansion, so we have \(dE = \rho dV = -PdV\), so indeed, \(P=-\rho\)! 

Einstein's reasons for introducing the cosmological constant turned out to be unfounded. In 1929 Edwin Hubble reported his inferences of recession velocity and distance for a set of (relatively nearby) galaxies, that showed a roughly linear trend of increasing velocity with distance, just as one would expect from a uniform expansion. Einstein missed the opportunity to predict the discovery of the expansion of the universe, a missed opportunity he referred to as his greatest blunder.

The cosmological constant though has refused to die. There are two reasons for this. The first is due to the fact that if one tries to calculate, using quantum field theory, the energy density that is in every cubic centimeter of space from the zero-point enenrgy of all the quantum fields, one gets an enormously large energy density, larger than observational limits by about \(10^{120}\). This huge embarrassment of modern physics is called ``the cosmological constant problem.'' 

The second is that over the past twenty years strong evidence has emerged that the dominant contribution to the mean energy density of the universe in the current epoch is something that is behaving a lot like a cosmological constant. As we will see shortly, the observational evidence comes from inferences of the relationship between distance and redshift that indicated the expansion rate is accelerating. The first claims of acceleration from redshift-distance inferences were published in 1998, and were based on  observations of Type 1a supernovae. This work led to a Nobel Prize in Physics in 2011.