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
ρΛ = -Λ--, pΛ = − -Λ--. (35 ) 8πG 8πG
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
1 ℒ = --gμν∂μϕ ∂νϕ − V (ϕ). (36 ) 2
Its energy-momentum tensor is
T = ∂ ϕ∂ ϕ − ℒg , (37 ) μν μ ν μν
and if the field is spatially homogeneous, its energy density and pressure are
ρϕ = 1ϕ˙2 + V (ϕ), p ϕ = 1ϕ˙2 − V (ϕ). (38 ) 2 2
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.


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