This observation is the corner-stones of the inflationary scenario.

Imagine that, at some very ancient instant of time ti , a small region of the spacetime manifold, containing matter and energy of various types, begins to inflate since, over that region, the scalar field ϕ is approximately constant and equal to the critical value ϕ0. The approximate behavior of this expanding space-time bubble will be the same as that of a region of de Sitter space. At the beginning ti , the spatial curvature of the bubble might have been positive, negative or null, yet at a sufficiently later time tend, the effective metric of the bubble will be that of a spatially flat deSitter manifold dsdS2 0 . Suppose now that at t = tend the scalar field, which was so far approximately constant, begins to evolve more rapidly and the kinetic energy ϕ˙2 starts dominating over the potential energy V (ϕ) which has in the meantime decreased. According to Friedman equations the acceleration parameter changes sign, the expansion rate decreases and, in a finite time, the behavior of the scale factor approaches that of a Universe filled with ordinary baryonic matter and radiation. Yet, the considered space-time bubble is spatially-flat as a result of the inflation which has occurred in the previous phase. This fundamental property of an inflationary phase in the history of the Universe is the most relevant feature of the inflationary scenario. It explains in a natural way and without ad hoc fine tuning why our Universe is spatially flat as it happens to reveal itself through experimental observations.

As we plan to analyze more explicitly in the sequel, the inflationary phase explains also why the observable Universe is homogeneous to very high accuracy over regions that are causally disconnected at the present time.

5.8.2 Slow-Rolling Approximate Solutions

In view of the qualitative considerations discussed above, let us now consider approximate solutions of (5.8.7), (5.8.8), (5.8.10) corresponding to a slow-roll phase. This latter is defined by the following two conditions:

whose physical interpretation is quite transparent, the former requiring that the kinetic energy of the scalar field should be negligible with respect to the potential energy, the latter requiring that the acceleration in the evolution of ϕ should also be small with respect to the velocity.

In order to discuss the existence and the properties of such a regime, it is convenient to manipulate a little bit the evolution equations (5.8.7), (5.8.8), (5.8.10) casting them in a more manageable form. First of all it is convenient to introduce rescaled variables so as to get rid of all the constants. Introducing the Planck mass:

scenarios. We shall not enter the fine structure of inflation modeling, which is still in lively evolution, since the most relevant and appealing feature of the inflationary scenario is precisely its robustness. The main physical mechanism and the essence of its predictions are largely independent from the detailed structure of the utilized potential, provided it is sufficiently smooth and reasonable. What this concretely means we shall presently show. Therefore, for the sake of simplicity we focus on the most widely utilized type of potentials that are the polynomial ones, in particular the following one

which for various values of the parameters λ, μ, ν captures the main features of three distinct well-known modelings of inflation.

(a)For μ = ν = 0 we have the so named large field modeling of inflation. The potential has just a minimum at φ = 0 and slow roll, as we shall prove below is obtained by starting at some sufficiently high value of φi .

(b)For μ = 0 we have the so named small field modeling. The potential has just a

maximum at φ = 0 and a minimum at φ = ( 4νλ2 )1/2α . Slow rolling is obtained by starting at small values φi of the field near the unstable maximum.

(c)For ν = μ2 the potential becomes:

which again has a maximum and a minimum and corresponds to the sort of potentials appearing in the symmetry breaking scenarios. In this case, as in the previous ones, slow rolling is obtained by starting at small values φi of the field, near the maximum.

Independently from the explicit form of the potential let us consider the consistency conditions imposed on it by the assuming the existence of a slow rolling phase. From the exact result (5.8.23) we know that if the scalar field has a slow roll, even slower will be the change of the Hubble function. Consider next the slow-roll condition (5.8.17) from which we work out

and insert this result into the first condition (5.8.16). We obtain:

We see that a necessary condition for slow roll corresponds to the smallness of a certain function εW (φ) of the scalar field, defined in (5.8.27) and completely determined by the potential. This condition is not the only one. Taking a further derivative

182 5 Cosmology and General Relativity

Fig. 5.26 In this picture we present the plot of the slow roll indices εW (φ) and ηW (φ) for the example of symmetry breaking potential presented in Fig. 5.25 corresponding to (5.8.24) with parameters:

λ = 1, μ = 16, ν = 256. In this case we obtain:

respectively sit before and after the minimum of the potential where the indices develop a pole

to the potential energy that drives an exponential growth of the scale factor. This is the inflation phase during which, dominated by vacuum energy, the Universe is very cold. Notwithstanding the slow rolling the ball goes down and it reaches the point where the slow rolling conditions cease to be valid. This is the end of inflation. After that the ball continues its descent, accelerating its motion while the growth of the scale factor experiences a rapid change of gear from an exponential to a power law one. Reaching the minimum the scalar field oscillates around it rapidly but with damped amplitude until it definitely sits down in the stable position. In order to verify the above qualitative description numerically it is convenient to rewrite the evolution equations (5.8.22) as a pair of second order coupled differential equations which we achieve as follows:

Starting from (5.8.25), if we choose the power α = 2 and the parameters λ = 1, μ = 16, ν = 256, which is the choice graphically displayed in Fig. 5.25 and already

Fig. 5.27 The initial condition for cosmic evolution corresponds to a situation where the scalar field is at some value not too far from a maximum of the potential. Then the scalar starts rolling down from the hill towards the minimum. At the beginning the condition of slow rolling are satisfied. The scalar changes slowly while the scale factor increases almost exponentially. Going downhill the scalar reaches such a value where the slow-roll parameter εW becomes of order unity. There inflation ends since the kinetic energy of φ becomes comparable to its potential energy. The exponential growth of the scalar field ceases while the field φ accelerates towards the minimum. Around the minimum the scalar oscillates rapidly and its energy is dispersed by reheating the Universe

used also in the other plots, then the potential becomes:

Replacing such explicit functions in (5.8.40), we can solve them numerically if we provide initial conditions at the initial time ti which we can conventionally fix at zero ti = 0. This means that we have to give the following data:

It is important to stress that inflation will take place if and only if such initial conditions are given in such a way as to be consistent with the slow roll. This means first of all that φ0 must be chosen in the region where εW (φ0) ' 1 and ηW (φ0) ' 1.

In this way we begin with slow roll and the slow roll phase hopefully will last for a sufficiently long interval of time as to allow the expansion of the scale factor by several order of magnitudes. In Fig. 5.28 we present the numerical solution of

Fig. 5.28 Overall plot of the solution of the coupled differential equations (5.8.40) for the scale factor a(t) and the scalar field φ(t) clearly displaying an early phase of slow rolling. The potential is that given in (5.8.41) and the initial conditions are those displayed in (5.8.44). In the inflation phase, lasting approximately in the interval of time from t = 0 to t = 2.5, the scale factor increases of almost 14 orders of magnitudes. As discussed in the text, this is not yet the sufficient amount of inflation to solve the horizon problem of cosmology. For that we need about 60 order of magnitudes (e-foldings)

(5.8.40) with the potential (5.8.41) and the following initial conditions:

which saturate the bounds (5.8.43). Clearly we have an inflation phase which lasts up to φend and inflates the Universe by about 14 orders of magnitudes. In Fig. 5.29 we present an enlargement of the same plot for later times, namely for t > 3. It is evident that the exponential expansion of the scale factor is now replaced by a power law growth which is weaker than linear as in a matter dominated Universe where a t2/3. At the same time the scalar field displays a damped rapid oscillation around the value corresponding to a minimum of the potential.

The existence of a good bona-fide slow roll phase in the numerical solution corresponding to the chosen initial conditions is best certified by the plot of the function

Fig. 5.29 After the end of inflation the growth of the scalar factor becomes of power type a tp with

p < 1. At the same time the scalar field undergoes rapid damped oscillations around the value φmin corresponding to a minimum of the potential

evaluated on the solution. This plot is presented in the first picture of Fig. 5.30. As we see, f (t) is essentially constant and equal to one in the whole interval of time from t = 0 to t = 2.3. This is the epoch of inflation. After that f (t) rapidly diverges signaling the end of the slow roll.

The important question which so far has no analytic answer is the following one.

How large is the domain of initial conditions around the critical values ˙ and ˙ for a0 φ0

which there is a slow roll phase? The relevance of this question can be appreciated a posteriori considering numerical solutions with initial conditions:

ilar to the solution in Fig. 5.28. Yet for somewhat larger deviations in the initial conditions the qualitative picture of slow roll and inflation is completely destroyed.

there is essentially no period of exponential growth of the scale factor (the Hubble function is never approximately constant) and the scalar field almost immediately oscillates. The essential absence of a slow rolling phase is certified by the plot of the function f (t) which for such a solution is displayed in the second picture of Fig. 5.30.

It is evident that a decisive progress in the mathematical foundations of the inflationary scenario and on its robustness will come only from a proper definition of the domain of attraction of the slow rolling regime. Indeed it is important to establish how generic are the initial conditions that lead to inflation and therefore produce a flat, homogeneous universe like ours.