Figure 1

(a) An inflection point potential $V(\varphi )=V_0(1+{\lambda }_1(\varphi -{\varphi }_0)+\frac{1}{6}{\lambda }_2(\varphi -{\varphi }_0{)}^3)$, with $V_0={10}^{-12}$, $({\lambda }_1,{\lambda }_2)=(7\times {10}^{-5},60)$, ${\varphi }_0=0.01$ in Planck units. (b) The phase space diagram for the potential in (a) for a few sample initial condition choices (solid red). The inflationary attractor solution is shown in dashed (blue). The attractor solution only exists in a finite region corresponding to when the potential allows slow roll inflation. Notice that the solutions flow towards the inflationary attractor solution.Reuse & Permissions

Figure 2

(a) The inflection point potential of (3.39). (b) The pseudophase space diagram for a DBI kinetic term for the potential in (a) for a few sample initial condition choices (solid blue). The inflationary attractor solution is shown in dashed (red). Notice that the solutions flow towards the inflationary attractor solution, which interpolates between the DBI and slow roll regimes. The crossed out region is inaccessible due to the DBI speed limit, $\chi \le 1$.Reuse & Permissions

Figure 3

The number of e-folds as a function of the starting value of the field, ${\varphi }_i$, along the DBI attractor (solid blue) is slightly greater than for the canonical attractor (dashed red) because of the speed limit restriction $\chi \le 1$. Also note that the majority of the e-folds are generated in the standard slow roll region near the inflection point, even for the brane-inflation model with DBI kinetic term.Reuse & Permissions

Figure 4

The available initial conditions phase space is shown for the canonical kinetic term (red outline, vertical gray lined area) and brane-inflation/DBI kinetic term (black outline, gray cross-hatched area). Notice that the available phase space allowed for brane inflation is much smaller in volume than for the standard field theoretic model due to effective field theory, compactification, and stringy effects, by a factor of $2{\Pi }_{\max ,\mathrm{brane}}/\sqrt{N_{\mathrm{D}3}}\ll 1$.Reuse & Permissions

Figure 5

The shaded region shown is the region of phase space that leads to 60 e-folds of inflation for the case of a scalar field with a canonical kinetic term [(a), red] and for brane inflation with a DBI kinetic term [(b), blue] for the same choice of potential and warp factor as described in (4.9, 4.10). The region which leads to 60 e-folds of inflation is a significantly larger fraction of the total for the D-brane scalar field, implying that the initial conditions fine-tuning for the brane-inflation model is significantly reduced.Reuse & Permissions

Figure 6

(a) The overshoot problem occurs when the inflaton initial conditions are such that the solution (solid red) does not reach the inflationary attractor solution (dashed red) until after the field value where 60 e-folds can be obtained along the attractor, denoted here by ${\varphi }_{60}$. (b) Brane inflation generically avoids the overshoot problem because (i) the initial conditions which lead to overshoot solutions are excluded by backreaction and effective theory constraints, and (ii) more e-folds are typically generated, thus the critical field value ${\varphi }_{60}$ decreases somewhat. The dashed (red) line is the DBI attractor solution, and the solid (blue) line is a solution for some randomly chosen initial condition consistent with the speed limit and backreaction constraints. The crossed out area corresponds to the initial conditions inaccessible to brane inflation due backreaction.Reuse & Permissions