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Chaos in FRW cosmology with gently sloping scalar field potentials

The chaotic behavior in FRW cosmology with a scalar field is studied for scalar field potentials less steep than quadratic. We describe a transition to much stronger chaos for appropriate parameters of such potentials. The range of parameters which allows this transition is specified. The influence of ordinary matter on the chaotic properties of this model is also discussed.


Introduction
In the last few years the chaotic regime in dynamics of closed FRW universe filled with a scalar field becomes the issue of investigations. Initially, the model with a massive scalar field (with the scalar field potential V (ϕ) = (m 2 ϕ 2 )/2, where m is the mass of the scalar field) was studied [1,2]. Before summarizing main results obtained, we present the equation of motion (for further using we will not specify the potential V (ϕ)). The system has two dynamical variablesthe scale factor a and the scalar field ϕ: with the first integral − 3 8π m 2 P (ȧ 2 + 1) + a 2 2 φ 2 + 2V (ϕ) = 0.
Here m P is the Planck mass.
The points of maximal expansion and those of minimal contraction, i.e. the points, whereȧ = 0 can exist only in the region where Sometimes, the region defined by inequalities (4) is called the Euclidean one.
One can easily see also that the possible points of maximal expansion (wherė a = 0,ä < 0) are localized inside the region while the possible points of minimal contraction (whereȧ = 0,ä > 0) lie outside this region (5) being at the same time inside the Euclidean region (4).
The main idea of the further analysis [3] consists in the fact that in the closed isotropical model with a minimally coupled scalar field satisfying the energodominance condition all the trajectories have the point of maximal expansion. Then the trajectories can be classified according to localization of their points of maximal expansion. The area of such points is specified by (5). A numerical investigation shows that this area has a quasi-periodical structure, wide zones corresponding to the falling to singularity being intermingled with narrow those in which the points of maximal expansion of trajectories having the so called "bounce" or point of minimal contraction are placed. Then studying the substructure of these zones from the point of view of possibility to have two bounces one can see that this substructure reproduce on the qualitative level the structure of the whole region of possible points of maximal expansion. Continuing this procedure ad infinitum yields the fractal set of infinitely bouncing trajectories.
It should be noticed that even the 1-st order bounce intervals (containing maximum expansion points for trajectories having at least one bounce) are very narrow. Analytical approximation for large initial a indicates that the width of intervals is roughly inversely proportional to a [4]. The opposite case of small initial a was investigated numerically, and the ratio of the first such interval width to the distance between intervals appear to be of the order of 10 −2 , if we do not take into account zigzag-like trajectories. So, the chaotic regime, though being interesting from the mathematical point of view, may be treated as not important enough.
For steeper potentials the chaos is even less significant. The chaotic behavior may disappear completely for exponentially steep potentials [5] .
The goal of the present paper is to describe the opposite case -the potentials which is less steep than the quadratic one. We will see that in this case the transition to a qualitatively stronger chaos may occur.
The structure of the paper is as follows. It Sec.2 we consider asymptotically flat potential and explain new features of the chaos which give rise in this case.
In Sec.3 a more wide class of potentials less steep than quadratic is studied. In Sec.4 we discuss the transition to regular dynamics in the presence of ordinary matter in addition to the scalar field for potentials under consideration.
2 Asymptotically flat potentials and the merging of bounce intervals We will use the units in which m P / √ 16π = 1 for presenting our numerical results, because in these units most of the interesting events occur for the range of parameters of the order of unity.
We start with the potential where M 0 and ϕ 0 are parameters. M 0 determines the asymptotical value of the potential for ϕ → ±∞.
It can be easily checked from the equations of motion that multiplying the potential to a constant (i.e. changing the M 0 ) leads only to rescaling a. So, this procedure do not change the chaotic properties of our dynamical system.
On the contrary, this system appear to be very sensitive to the value of ϕ 0 . We The initial value of a vary in the range of the first two intervals before and after merging with the step 0.002. Before merging, the measure of so chaotic trajectories is extremely low and they are undistinguisheble on our grid. When ϕ 0 becomes slightly low than the value of the first merging, this number begin to grow rather rapidly and for ϕ 0 ∼ 0.6 near 10% of trajectories from the 1-st interval experience at least 50 oscillation before falling into a singularity.
We recall that for a simple massive scalar field potential only ∼ 10 −2 trajectories in the same range of the initial scale factors have at least one bounce. In contrast to this, the chaos for the potential (6) is really significant. Detail of intervals merging including the description out of ϕ = 0 cross-section require further analysis.
For large initial a the configuration of bounce intervals for potential (6) looks like the configuration for a massive scalar field potential with the effective mass easily derived from (6): The periods of corresponding structures coincides with a good accuracy though the widths of the intervals for the potential (6) is bigger then for V = (m 2 ef f ϕ 2 )/2.

Damour-Mukhanov potentials
The very chaotic regime described above is possible also for potentials, which are not asymptotically flat, if the potential growth is slow enough. We will illustrate this point describing a particular (but rather wide) family of potentials having power-low behavior -Damour -Mukhanov potentials [6]. They was originally introduced to show a possibility to have an inflation behaviour without slowroll regime. After, various issues on inflationary dynamics [7] and growth of perturbation [8,9] for this kind of scalar field potential was studied.
The explicit form of Damour-Mukhanov potential is with three parameters -M 0 , q and ϕ 0 .
For ϕ ≪ ϕ 0 the potential looks like the massive one with the effective mass m ef f = M 2 0 /ϕ 0 . In the opposite case of large ϕ it grows like ϕ q . As in the previous section, the chaotic behavior does not depend on M 0 . So, we have a two-parameter family of potentials with different chaotic properties.
Numerical studies with respect to possibility of bounce intervals merging shows the following picture (see Fig.3): for a rather wide range of q there exists a corresponding critical value of ϕ 0 such that for ϕ 0 less than critical, the very chaotic regime exists. Increasing q corresponds to decreasing critical ϕ 0 .
Surely, since this regime is absent for quadratic and more steep potentials, q must at least be less than 2. We can see clearly the very chaotic regime for q < 1.24. The case q = 1.24 lead to strong chaos for ϕ 0 < 1.4 × 10 −5 and the critical ϕ 0 decreases with increasing q very sharply at this point. We did not investigated further these extremely small values of ϕ 0 , because the physical meaning of such kind of potential is very doubtful.
with the constraint Here p = 1 + 3γ, Q is a constant from the equation of motion for matter which can be integrated in the form In Ref. [10] it was shown that addition of a hydrodynamical matter to the scalar field with a potential V (ϕ) = m 2 ϕ 2 /2 can kill the chaos. Here we extend this analysis to less steep potentials. Some of our results are illustrated in Fig.4.
It is interesting that increasing Q acts as increasing ϕ 0 . In Fig.4 three intervals merged at Q = 0. When Q increases, the 3-d and 2-nd interval consecutively separate and we return to the chaos typical for V (ϕ) = m 2 ϕ 2 /2. With the further increasing of Q the chaos disappear in a way discussed in [10].
The value of Q corresponding to the chaos disappearing is in general bigger for the less steep potentials with the same effective mass . In We recall that for the case V (ϕ) = m 2 ϕ 2 /2 (which is equivalent to the Damour-Mukhanov potential with q = 2, the corresponding mass is equal to the effective one m ef f = M 2 0 /ϕ 0 ) the chaos disappear for Qm > 0.023m P [10]. To compare with this value, we plotted in Fig.5(a) the values of Qm ef f leading to ceasing of the chaos with respect to ϕ 0 for several q. In units we used the case q = 2 corresponds to horizontal line Qm ef f = 1.15. All other curves have this value as an asymptotic one for large ϕ 0 . With decreasing ϕ 0 the value Qm ef f increases with the rate which is bigger for less steep potentials.
In Fig.5(b) the analogous curve is plotted for asymptotically flat potential (6). The value Qm ef f for large ϕ 0 is the same. For small ϕ 0 we can estimate Q killing the chaos as the value corresponding to disappearance of the Euclidean region for a flat potential V (ϕ) = M 4 0 . It can be easily obtained from (4) that  and for bigger Q any bounce become impossible. As the potential (6) differs significantly from the flat one only for ϕ less then ϕ 0 , this approximation appear to be good enough for small ϕ 0 . In the our units it correspond to the curve Qm ef f = 8/ϕ 0 . Intermediate values of ϕ 0 represent smooth transition between these two asymptotic behaviors.