2.4 Einstein’s static model and
Lemaître’s model
So far we have shown that solutions of the Einstein equation are
dynamical in general, i.e., the scale
factor
is time-dependent. As a digression, let us examine
why Einstein once introduced the
-term to obtain a static
cosmological solution. This is mainly important for historical
reasons, but is also interesting to observe how the operationally
identical parameter (the
-term, the cosmological
constant, the vacuum energy, the dark energy) shows up in
completely different contexts in the course of the development of
cosmological physics.
Consider first the case of
in Equations (4) and (5). Clearly the
necessary and sufficient condition that the equations admit the
solution of
is given by
Namely, any static model requires that the Universe is dominated by
matter with either negative pressure or negative density. This is
physically unacceptable as long as one considers normal matter in the standard model of
particle physics. If
on the other hand, the
condition for the static solution is
yielding
Thus both
and
can be positive if
In particular, if
,
This represents the closed Universe (with positive spatial
curvature), and corresponds to Einstein’s static model.
The above static model is a special case of
Lemaître’s Universe model with
and
. For simplicity, let us assume that the Universe is
dominated by non-relativistic matter with negligible pressure, and
consider the behavior of Lemaître’s model. First we define the
values of the density and the scale factor corresponding to
Einstein’s static model:
In order to study the stability of the model around the static
model, consider a model in which the density at
is a factor of
larger than
. Then
and Equations (4) and (5) reduce to
For the period of
, Equation (33) indicates that
and the Universe is decelerating (
). When
reaches
,
takes the minimum value
and
the Universe becomes accelerating (
). Finally
the Universe approaches the exponential expansion or de Sitter
model:
. If
becomes closer to unity, the minimum value reaches zero and the
expansion of the Universe is effectively frozen. This phase is
called the coasting period, and the case with
corresponds to Einstein’s static model in which the
coasting period continues forever. A similar consideration for
indicates that the Universe starts collapsing (
) before
. Thus the behavior of
Lemaître’s model is crucially different if
is larger or smaller than unity. This suggests that Einstein’s
static model (
) is unstable.