The cosmological constant is the energy density associated to the vacuum. Its presence modifies the property of the space time and the matter. When we consider a homogenous universe, we can put the cosmological equation in the form
where
, 
, 
, 
denote respectively the rate of the nonradiative energy, radiative energy, the curvature contribution, the cosmological constant contribution, 
is the scale factor and
, the present Hubble constant. From this equation, one can deduce some remarks:
• The lenght scale associated to the cosmological scale [
. This value is too small comparatively to fondamental interaction scale.
• By Equation (1), 
and 
vary in the same way; so if 
is large, 
is big too. But at the very earlier epoch of the universe, when 
how does 
behave?
• From different mesures, we can write 
.
• It follows from the previous remarks that 
is perhaps a dynamical quantity.
The first theory in this domain is the standard hot universe. According to this theory, the universe has been expanding and gradually cooling from a state with infinite temperature and density. In this standard scenario, it is usually assumed, in the very early stages of evolution of the universe, that was very flat and the evolution law is given by
Despite the great phenomegical success of the standard hot universe scenario, this scenario was still somewhat incomplete. We give here some problems arising from this scenario.
• The flatness problem: The universe would be closed and it would have collapsed millions of years ago or the universe would be opened and the present energy density of the universe would be negligible;
• The singularity problem: From Equation (2) it follows that the scale factor of the universe 
vanishes at 
whereas the energy density becomes infinitely large;
• The homogeneity and isotropy problems: It was assumed that the universe was initially absolutely homogenous and isotropic. Meanwhile, even at present, the universe is not totally homogenous and isotropic, at least at a sufficiently small lenght scale;
• The galaxy formation problem: It was not quite clear what was the source which generates galaxies;
• The inflationnary universe: According to this, in the very earlier stages, the expansion of the universe was exponential from an instable vacuum state. At the end of this state, the energy of this state transforms itself in energy of hot universe. This theory suppose that, there was a time when the pressure was negative and the negative pressure happens due to a single new real scalar field
. In this case the energy and the pressure densities can be written
If the potential energy V is a slowly varying function of the field 
and if the initial value of the time derivation of 
is not too large, the kinetic energy 
can be small compared to
. If in addition 
is large enough to make a significant contribution to the stressenergy tensor, the pression can satisfy the inflation condition.
Quintessence has been proposed as the missing energy component that must be added to the baryonic and the matter density in order to reach the critical density [
, lies between 0 et –1. A key problem with quintessence proposal is explaining why 
and the matter energy density should be comparable today. One of the aspect to this problem is the coincidence problem [
is nearly constant and lies between 
and
.
, 
, 
denote the background matter lagrangian and F the contribution of the scalar field. From the action (4), we get the Einstein equation
and 
repectively, one can combine the cosmological equations to obtain
. They observed that the energy density 
grows below the line 
and decreases above this line. They conclude that all the point lying on the line 
are attractors. These points correspond to exponentially inflation points:
; thus inflation appears by the only kinetical term in the lagrangian; this motivates the term kinetically driven inflation. With this method, we analyse some model where the function 
is not a scalar field but a F-harmonic map. The F-harmonic maps are the critical points of l functional energy defined on the space of the regular maps enter riemanian varieties. Ara [
.
negligeable, the cosmological constant equal to 0 and
; then we otain
,
. Note that we use the metric
very small, we can do a limited developpement of the harmonic function, the physical quantities known, the energy and the pressure can be written
negative, the two physical quantities are negative; It is not physically.
positive, they are all positive and we obtain
, acceptable only near the origin
.
does’t allow developpment and is positive then the two physical quantities are positive and grows as an exponentially function when the Hubble constant grows.
does’t allow developpment and is negative, then we can pose 
to obtain
, 
grows when 
and 
decrease to zero.
, if 
grows, 
decreases to 
when 
to
. Eliminating 
in the expression of 
and 
we get the equation of state
corresponding to 
and the point 
corresponding to very large values of
, but now this model have no point of exponential inflation model because the point 
is never attained.
negligeable, the cosmological constant equal to 0 and
gives:
and
, then this differential equation can be written as the following system
and 
and
.
must be even; and the egeinvalues can be written as: 
These points are instable fixed points. At these points, 
and the cosmological Equations (20) and (21) become:
is negative, and 
decreases. Let us look now at the behaviour near the points 
where
, 
very small. The system (23) becomes
, we find that the acceptable values of 
are old. The egeinvalues read
.These points are stable. Near these points:
very small, the system (23) leads to
; which implies 
and
. The points 
correspond then to exponential inflation points. At these points
. By the same, from the equation of state (24) when
, we find
. The points 
crrespond to a negative energy density, so it is not useful. The points 
corresponding to 
are acceptable only near the origin. The points 
correspond to the points of exponential inflation.
, we look at the solutions of the cosmological which gives 
constant. These are the tracking solution [
which can be written in the form
, one poses 
an look at the solutions for which 
because in this case,
and 
and 
and 
is a generalisation of exponentially harmonic function.
and 
can be written
as constant function of
.
constant imply
. With the assumption 
. Putting of this value in the conserved equation of the energymomentum tensor we get a function 
and the field equation becomes:
where 
is the solution of
constant and 
constant, we can think at the tracking solutions. Here
is positive. Two cases are possible:
,
. We deduce that there are tracking solutions. Let us note that here 
is exclude.
, but 
does not verify the necessary condition. There is no tracking solutions. (34). Which the change of variables this can be read
).
constant, imply 
and 
are all constant. More precisely 
and
. With the assumption
, the conserved equation of matter leads to
if 
Hence we can not use this condition here. But we can search the function 
for which the equation of state 
varie weakly. For that, we need a relation between 
and its derivative; what we do not find yet. However if we known the potentiel we can look at the behaviour for other quantities by the study of the following dynamical system
, 
and 
, determining for the expected results. The problem of coincidence, however house. With the usual scalar fields, P. Steinhardt and his associates [
which allows without an indepth study of knowledge if a model can allow to avoid the problem of coincidence by means of fields “trackers”. T. Chiba made the same thing (matter) with the lagrangians of the field of the shape
. With the lagrangians of field exponential of the shape 
an independent relation was not able to be still found. We continue to look for a relation with the lagrangian of the shape
.