Universality in D-brane Inflation

TL;DR

This paper shows that six-field D3-brane inflation on a warped conifold exhibits universal, robust inflationary behavior despite a complex potential with hundreds of terms. Using an extensive Monte Carlo ensemble, the authors find that inflationary histories concentrate around an inflection-point trajectory, with the likelihood of achieving N_e e-folds following P(N_e) ∝ N_e^{-3}. An analytic inflection-point model explains the exponent, and angular attractor dynamics reduce sensitivity to initial conditions. When perturbations are constrained to effectively single-field slow-roll regimes, viable spectra consistent with observations require N_e ≥ 120 and yield an extremely small tensor-to-scalar ratio r, while 60 e-folds occur only infrequently. Overall, the results imply emergent simplicity in high-dimensional multifield inflation and motivate analytic, high-dimensional approaches to such systems.

Abstract

We study the six-field dynamics of D3-brane inflation for a general scalar potential on the conifold, finding simple, universal behavior. We numerically evolve the equations of motion for an ensemble of more than 7 \times 10^7 realizations, drawing the coefficients in the scalar potential from statistical distributions whose detailed properties have demonstrably small effects on our results. When prolonged inflation occurs, it has a characteristic form: the D3-brane initially moves rapidly in the angular directions, spirals down to an inflection point in the potential, and settles into single-field inflation. The probability of N_{e} e-folds of inflation is a power law, P(N_{e}) \propto N_{e}^{-3}, and we derive the same exponent from a simple analytical model. The success of inflation is relatively insensitive to the initial conditions: we find attractor behavior in the angular directions, and the D3-brane can begin far above the inflection point without overshooting. In favorable regions of the parameter space, models yielding 60 e-folds of expansion arise approximately once in 10^3 trials. Realizations that are effectively single-field and give rise to a primordial spectrum of fluctuations consistent with WMAP, for which at least 120 e-folds are required, arise approximately once in 10^5 trials. The emergence of robust predictions from a six-field potential with hundreds of terms invites an analytic approach to multifield inflation.

Figures (9)

• Figure 1: Examples of downward-spiraling trajectories for a particular realization of the potential. The black dots mark 60 and 120 e-folds before the end of inflation (7 of the 8 curves shown achieve $N_e>120$); inflation occurs along an inflection point that is not necessarily parallel to the radial direction. Red curves have nonvanishing initial angular velocities $\dot{\Psi}_{0}$, while blue curves have $\dot{\Psi}_{0}=0$.
• Figure 2: The rms value, $Q$, of the coefficients $c_{LM}$ has a significant role in determining whether inflation can occur. [Left panel] The success probability $P(N_e>60)$ for two different numbers of fields, $N_f=1$ and $N_f=6$, with $\Delta_{\rm max}=7.8$. [Right panel] The success probability for two different degrees of truncation, $\Delta_{\rm max}=6$ and $\Delta_{\rm max}=7.8$, with $N_f=6$.
• Figure 3: The likelihood of $N_{e}$ e-folds of inflation as a function of $N_{e}$, for $\Delta_{\rm max} = 7.8$ and $N_{f} = 6$. We find $P(N_e)\propto N_e^{-\alpha}$, with $\alpha=3.22\pm 0.07$ at the 68% confidence level. The left panel shows the power law fit to the histogram and the right panel shows the same fit on a log-log plot.
• Figure 4: Trajectories in the $\theta_{1}-\theta_{2}$ plane, with $\rm{log}(x)$ vertical, for a fixed potential. Notice the attractor behavior in the angular directions. Green trajectories correspond to $\approx 5$ e-folds of expansion, while the remaining colors correspond to trajectories with $\approx 150$ e-folds.
• Figure 5: The ratio ${\rm{KE}}_{\Psi}/{\rm{KE}}_{r}$ of angular to radial kinetic energies for trials with $\Delta_{\rm max} = 7.8$ and $N_{f} = 6$. [Left panel] Evolution of ${\rm{KE}}_{\Psi}/{\rm{KE}}_{r}$ for trials yielding $60 \le N_e \lesssim 120$ e-folds of inflation (red lines), and $120 \le N_{e} \lesssim 180$ e-folds (black lines). Notice that in some cases the angular kinetic energy is non-negligible, and nearly constant, in the final 60 e-folds, corresponding to an inflection point trajectory that is not purely radial. [Right panel] Histogram of ${\rm{KE}}_{\Psi}/{\rm{KE}}_{r}$ 60 e-folds before the end of inflation for potentials that yield more than 60 (light blue) or more than 120 (dark blue) e-folds of inflation.