Annu. Rev. Astron. Astrophys. 1984. 22:
157-184
Copyright © 1984 by . All rights reserved |

**2.4. The Cosmological Constant**

When the cosmological constant is taken into account, the field equations (Equation 7) are altered to

(12) |

which can be rewritten in the form

(13a) |

where

(13b) |

that is, the field equations can be regarded as the same as before but
with an altered stress-energy tensor. Writing the effective
stress-energy tensor *T'*_{ab} in the form of Equation 4a, one
finds that

(14a) |

and so Equation 10 becomes

(14b) |

If the -term is
large enough, the argument by which Equation 10 leads
to Equation 11 no longer applies, for the effective active gravitational
density (*µ* + 3*p*)' can violate the energy condition
Equation 9b. If it
does so at all times, a singularity in the past can be avoided (cf.
Equation 14b), and indeed it has long been known
(86) that there are
FLRW universes with a large positive cosmological constant that (*a*)
start expanding from a state asymptotically like the Einstein static
universe [Eddington (23)
suggested that this was preferable to a
"big bang" origin]; or (*b*) collapse from infinity to a minimum
radius value *R*_{min} > 0 and then reexpand to infinity.

There are two problems with these models. Firstly, the deceleration
parameter
*q*_{0}
- (1 / *R*)_{0}(*d*^{2}*R* /
*dt*^{2})_{0} / (*H*_{0}^{2})
must be negative at all times in
these universes (for if it ever becomes positive, the universes must
come from a singular origin), but observations suggest that the present
value of *q*_{0} is positive
(114). Secondly, an
alternative explanation for
the CMBR must be found in these cases, for it is not possible for this
radiation to arise from a hot early stage in such a universe. The reason
is that if such a hot early stage were to explain the CMBR, the
asymptotic phase or turnaround caused by
would have to occur
during the early plasma phase before decoupling when *T* > 3000
K; thus
inequality Equation 9b would have to be violated by the effective matter
(see Equation 14) for values of *R* less than
(*R*_{0} / 1000). But at that time,
(*µ* + 3*p*) would be at least 10^{9} times
larger than at present,
while
would have the same value as at present (because it is constant);
violation of Equation 9b at that time therefore implies a value of
so large that we
would undoubtedly have detected it from *q*_{0}. It is
difficult to find alternative explanations for the CMBR
(54). In
addition, these models require element abundances to be arbitrarily set
by initial conditions in the Universe (for the argument above shows the
turnaround would occur long before nucleosynthesis temperatures were
achieved). Thus, these models leave so many questions open as to not be
presently viable.

One should note here that it follows from Equations 5, 9, and 14b that
a static FLRW universe model will necessarily be unstable.
[Perturbing *R* from a static value *R*_{c} will lead
either to collapse or expansion
(23,
25).]
This implies that even if we could find some other explanation for
redshift (abandoning Equations 1 and 2) we would not expect a FLRW
universe to be static unless we abandon either Einstein's field
equations or the energy conditions.