RESUMEN DE OPERACIONES LLEVADAS A CABO EN EL POZO AMS

In this section we study what happens when the inflaton rolls uphill. We will use our results to address the issue of overshooting a possible hill in the potential (see e.g. figure 4.1). Obviously the field will roll just for a short distance ∆φ which depends on the initial speed ˙φ(0) that we take from the beginning to be ˙φmin. Let us study the simple potential

V = Λ +cφ , (4.64)

where c is the positive slope. Under the simplifying assumption Λ cφ, the solution to the equation of motion is

φ(t)−φ(0) = ∆φ=−cM√ P l 3Λt+ ˙ φminMP l √ 3Λ + cM2 P l 3Λ ! 1−e− √ 3Λt/MP l . (4.65)

The term linear in t describes the constant speed rolling down that eventually dominates over the exponentially decreasing term. If the field starts with a positive ˙φmin, it will

climb up the hill for a distance

∆φ' φ˙min√MP l 3Λ − cM_{P l}2 3Λ " 1 + log √ 3Λ ˙φmin cMP l !# , (4.66) in a time ∆t' √MP l 3Λlog 1 + √ 3Λ ˙φmin cMP l ! (4.67)

before it stops and starts rolling down again. The number of e-foldings therefore is generically short unless the slope is exponentially small (if instead ˙φmin is very large, then

H is no more well approximated by a constant).

We would like to emphasize that an uphill motion is never slow roll. Even if '0 = η, when moving uphill ¨φ is always very large (the motion is hampered both by the slope and by the Hubble friction) and can not be neglected. In fact the equation of motion is genuinely of second order and the uphill phase depends critically on the initial condition

˙

φ0. On the other hand, the downhill slow-roll motion is an attractor and the solution

eventually reaches it independently of ˙φmin (of course if the potential is of the slow-roll

type).

We now have all the ingredients to address the question whether the inflaton will overshoot the maximum. We describe the part of the potential before the minimum φ > φmin with the damped oscillator of section 4.6.3 (see figure 4.4). The key result is (4.63), the speed of the inflaton ˙φmin when it reaches φmin. We then describe the uphill phase between maximum and minimum as a straight line. The estimate may seem very rough, so let

us comment on it: if we take the steepness of our straight line (c in (4.64)) to be the maximum steepness reached by the potential V(φ) betweenφmax and φmin then we have an upper bound. We show in the following that overshooting is possible in this extremel case; we conclude therefore that this is true also for the less steep non-straight uphill phase in the potential V(φ).

We use the result (4.63) as initial condition in (4.66). One can see that the time tmin in

(4.63) is always smaller than √3Λ/2M2

P l so that, to get an order of magnitude estimate, we can neglect the exponential in that formula. Neglecting also numerical factors we take

˙

φmin∼φ0η

√

3Λ/MP l, (4.68)

where φ0 was the distance from the minimum of the initial position in the damped oscil-

latory phase of section 4.6.3. Substituting it into (4.66), we obtain ∆φ ∼ φ˙min√MP l

3Λ ∼φ0η , (4.69)

where we have used cM2

P l/3Λ 1 which is generic for our potential. We conclude that overshooting can happen, with η _& 1 and a comfortably natural choice φ0 & ∆φ (also

an initial ˙φ 6= 0 at the beginning of the underdamped oscillatory phase will help to overshoot). Typically, one does not obtain a large number of e-foldings, see (4.67). We have to remember though that (4.67) is valid just when the uphill path is a straight line, in our case instead there is maximum, where the slope vanishes.

As an aside we comment on the intriguing correlation between a small cosmological con- stant and the underdamped oscillatory regime. A graceful exit from inflation typically requires that the inflaton reaches a minimum and starts oscillating and decaying (brane inflation is an interesting exception). In section 4.6.3 we have seen that the underdamped regime, leading to oscillation around the minimum, requires η _& 1. Equivalently, it re- quires that the cosmological constant Λ is smaller than the inflaton massm. Consider now an inflaton protected by some symmetry that therefore acquires an extremely small mass only from nonperturbative effects. Then an anthropic selection principle would apply: all universes with Λ_&m2_M2

P lwould not have a graceful exit from inflation and would hence be empty.

Summarizing, if the inflaton potential is just m2_φ2_{, then anthropic arguments lead to}

an upper bound for the cosmological constant of order Λ_. m2M_{P l}2 . An extremely small inflaton mass might then explain the presence of a comparably small cosmological constant which may be responsible for today’s measured cosmic accelerated expansion.

It would be interesting to study the features of a potential like (4.58). A preliminary observation is that, if an uphill phase is present, a largely non scale-invariant spectrum is produced. The spectral index during the uphill motion is given by

ns−1≡ dlogδφ dlogk ' − ˙ φ H2 − 1 H∂t log ˙ φ H2 , (4.70)