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
-term to the r.h.s. by assigning
This effective matter field, however,
should satisfy an equation of state of
. 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.,
,
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
-term is introduced in the
l.h.s., it should be constant to satisfy the energy-momentum
conservation
. 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
only in special cases. This
is why recent literature refers to the field as dark energy instead of the cosmological
constant.