Vacuum energy as an effective cosmological constant

2.5 Vacuum energy as an effective cosmological constant

So far we discussed the cosmological constant introduced in the l.h.s. of the Einstein equation. Formally one can move the $Λ$ -term to the r.h.s. by assigning

This effective matter field, however, should satisfy an equation of state of $p = − ρ$ . Actually the following example presents a specific example for an effective cosmological constant. Consider a real scalar field whose Lagrangian density is given by

Its energy-momentum tensor is

and if the field is spatially homogeneous, its energy density and pressure are

Clearly if the evolution of the field is negligible, i.e., $˙ϕ2 ≪ V(ϕ )$ , $pϕ ≈ − ρϕ$ and the field acts as a cosmological constant. Of course this model is one of the simplest examples, and one may play with much more complicated models if needed.

If the $Λ$ -term is introduced in the l.h.s., it should be constant to satisfy the energy-momentum conservation $Tμν;ν = 0$ . Once it is regarded as a sort of matter field in the r.h.s., however, it does not have to be constant. In fact, the above example shows that the equation of state for the field has $w = − 1$ only in special cases. This is why recent literature refers to the field as dark energy instead of the cosmological constant.