**7.1. A Decaying Cosmological Constant?**

One method of resolving the dilemma between a very large cosmological
constant (predicted by field theory) and an extremely small one
(suggested by observations) with obvious cosmological advantages
is to make the cosmological term time-dependent. An initially large
cosmological term would give rise to Inflation, ameliorating
the horizon and flatness problems and (possibly) seeding
galaxy formation. The subsequent slow decay of
(*t*) would
enable a small present value (*t*_{0}) to be reconciled with observations
suggesting _{} ~ 0.7.
(A time dependent cosmological term of course arises in Inflationary models
and during cosmological phase transitions, but in such cases the
post-inflationary decay of the cosmological term is very rapid.)

The first proposal for dynamically reducing the cosmological constant was made by Dolgov (1983) who considered a massless non-minimally coupled scalar field having the Lagrangian density

(74) |

and the resulting equation of motion

(75) |

where *R* is the scalar curvature and
the coupling to
gravity. Considering the special case of a homogeneous scalar field,
the Einstein equations become

(76) |

where

(77) |

is the scalar field energy density. The scalar field equation (75) reduces to

(78) |

Dolgov made the discovery that, for negative values of
,
the scalar field is unstable: its energy density
_{} becomes large
and negative compensating for the
cosmological constant in (76), so that the resulting *effective*
cosmological constant
rapidly decays to zero. Let us demonstrate this by examining
the Einstein equation (76) which together with the scalar field
equation (78)
defines a pair of nonlinear differential equations determining
the behaviour of the scale factor *a*(*t*) and the scalar
field (*t*).
The term 3
*H*^{2}
^{2} in
_{} can be
carried over into the left hand side of (76) resulting in

(79) |

As Dolgov demonstrated,
(*t*) grows with time
if < 0, so
that the effective cosmological constant
_{eff}
= /(1 +
8
*G*||
^{2}) decreases.
The late time behaviour of *a*(*t*),
(*t*) obtained
by solving (76 - 78) with
_{m}
<< _{} has the asymptotic form
[46,
66]

As a result,
lim_{t}
_{eff}
0
*i.e.* the cosmological term vanishes at late times.

Unfortunately this mechanism cannot be used in real universe. The first problem with this approach is that the very mechanism which decreases the cosmological constant also quenches the effective gravitational constant, since from (76),

(80) |

As a result, the effective gravitational constant becomes noticeably
time-dependent:
_{eff} /
*G*_{eff} = - 2/*t* ~ - 10^{-10} yr^{-1},
which strongly contradicts upper limits from Viking radar ranging
[87]
and lunar laser ranging experiments
[211].
Another problem is that such screening of
is still not
sufficient. The remaining part of
remains of the
order of the Ricci tensor
all the time, while we need it to be much less than the Ricci tensor
during the matter dominated epoch to obtain sufficient growth of
scalar perturbations. Finally,
_{m}
<< 1 during this regime.

An extension of this method to higher spin fields (massless vector and tensor) can remove the first drawback by making a cancellation of the cosmological constant possible while keeping the gravitational coupling constant time-independent [47]. However, the other difficulties (especially, the second one) remain. This shows that it is not easy to explain the observed -term by a cancellation mechanism. Still this aesthetically attactive possibility should be investigated further (some variants of the early Dolgov mechanism are discussed in [10, 205]).