In this section we propose a new DBI inflation model. The throat geometry is the Witten’s geometry [28] and we probe a D4 brane to drive inflation. Then we discuss the property of the model and study the attractor behavior of the model.
9.1 DBI Inflation in the Witten’s geometry
Warped geometry provides a successful framework for building inflation models. If some branes in the compact manifold become very heavy, they will warp the compact manifold and form a throat geometry. We consider D4 branes to condensate in a 5D internal manifold and use this background for DBI inflation. The setup follows. D4 branes fill five dimensional noncompact spacetime and warp the internal manifold and compactify one spatial direction x4. The metric is given by [28]
ds2 = (r
The metric describes a throat geometry with the x4 circle smoothly shrinking to a tip at the IR endpoint r = rKK by requiring the period of x4 to be 4πR3/2/3r1/2KK ≡ 2π/MKK to avoid a conical singularity.
The action of a D4 brane oriented along the compact space is given as SD4 = − 1
Put all these together with the Einstein-Hilbert action and use R3 = πgsNcl3s. and Transform the coordinate variable r into the field theory variable φ = r/l2s and the IR cutoff is φKK = rKK/l2s. The ’t Hooft coupling is λ = πgslsNc. For the zeroth order evolution, we can assume the inflaton φ is spatially homogeneous and gµν∂µφ∂νφ = − ˙φ2. Then the action becomes
The feature of this lagrange is that it automatically imposes a restriction on ˙φ, the speed limit
f h3 ≥ ˙φ2 (9.3)
The speed of the inflaton field φ will become slow as the brane approaches the tip of the throat. The equation of motion for inflaton field is
d equation and obtain the equation of motion for U
φ −¨ φ3KK For a general potential V , the Friedmann equation reads
3H2 = Te3
MKK and the energy density ρ and pressure P is [29]
ρ = h3f
pf − 2h−3X − h3+ V (φ) (9.5)
P = −h3p
f − 2h−3X + h3− V (φ) (9.6) where φ is the inflaton field and X = φ˙22.
The inflationary model can be successful if the model satisfy the slow roll condi-tion. We can define a quantity which is the pressure 9.6 of the model The slow roll parameter for the DBI type inflationary model is given as
² = −H˙ the inflaton field is large enough. In addition, the inflationary model can solve the cosmological problem of the Big Bang theory if the model can produce a large amount of inflation. The amount of inflation is characterized by the ratio of the scale factors at time t and at the end of inflation. The ratio is the number of e-folding N(t) defined by
We set the tip of the throat at φKK = 0.1. Using the numerical calculation, we can show that the initial condition for the inflaton which can produce 60 e-folding is φi = 100. This means that inflation occurs when the brane starts to move far from the tip. During the inflation, the factor f is near 1
f = 1 − φ3KK
φ3 ' 1 − 0.13 1003 ' 1
The action of the D4 DBI inflationary model reduces to the form MP l2
Even though the inflaton rolls relativistically, the speed limit requires the positivity of the argument of the square root in the action. We can approximate the inflaton speed during inflation as cs ¿ 1
˙φ ' ±φ3/2 λ1/2
The requirement ² ¿ 1 implies that the potential energy dominates during inflation.
The Friedmann equation reduces to
H2 = Te3
3Mpl2V (φ)
This model is similar to the D3 DBI inflation model. The warp factor of the D4 DBI inflation model is cubic.
9.2 Attractor behavior of the D4 DBI inflation model
Inflation can be predictive only if the solution has the attractor behavior. The difference between solutions of different initial condition will vanish quickly. The Hamilton-Jacobi formulation is an effective method for analyzing the inflationary attractor property. In this formulation, we can regard inflaton field φ as the time variable φ = φ(t) and require that the inflaton field φ does not change sigh during inflation. We also consider H = H(φ) so the inflaton field U is monotonic and H will not have oscillatory behavior in φ. we can take the time derivative of the first Friedmann equation and use the equation of motion of the inflaton field φ, we can find
Then, we can substitute the ˙φ into the first Friedmann equation and use the ex-pression for ρ to get an exex-pression for the potential V (φ) in terms of the Hubble parameter H(φ)
Suppose that H(φ)0 is a solution of the above equation and consider a small pertur-bation about H(φ) = H0(φ) + δH(φ). The linearized equation for δH becomes,
2H00δH0 = 3H0δH sign, if H0 is an inflationary solution, all linear perturbations are damped at least exponentially. Thus we conclude that the inflationary attractor property is satisfied for the inflaton field φ.
9.3 Phase diagram of attractor behavior
We will study numerically the dynamics of the φ for the simplest potential that may lead to a power law acceleration
V = 1
2m2φ2 (9.12)
This potential is simple and the higher power terms in the effective potential will be small compared to the quadratic term without tuning as discussed in [26] [32]. We will fix the parameter λ = 100 in the following discussion.
To study the attractor behavior [27], we can rewrite the equation of motion as a set of two independent variables x ≡ φ and y ≡ ˙φ. In terms of x and y, the speed limit translate into
x3
R3(1 − φ3KK
x3 ) ≥ y2 (9.13)
The evolution equation of the inflaton field φ can be written as a set of two simulta-neous equations from (9.4):
Figure 11: Attractor curves of D4 DBI inflation model.
with the Friedmann equation
3H2 = Te3 MP l2
x3
λ(1 − φ3KK x3 )(
r
1 − φ3KK x3 − λ
x3y2+ V −x3
λ) (9.14)
In the following we start the numerical analysis of the dynamical system. We will set Te3 = 108 and m2 = 10−3 and the tip of the throat φKK = 0.1. The diagram is shown in figure 11. We can see that there is a curve that attracts all he trajectories which correspond to the slow-roll curve. This confirms our conclusion in the Hamilton-Jacobi formulation analysis. For all allowed initial condition, φ tends to the slow-roll curve quickly after it begins to evolve. The dynamics of φ can be divided into three stages. In the first stage, y is damped to the slow roll curve quickly with x kept almost constant. In the second stage, x decreases while y kept almost constant. In the final stage, the attractor curve coincides with the limit curve that satisfies the speed limit, (x3/R3)(1 − φ3KK/x3) = y2. The inflaton field φ will behave in this way before it reaches the critical point. The result shows that the D4 DBI inflation also exhibits the attractor behavior.