Roulette Inflation with Kähler Moduli and their Axions

TL;DR

This paper investigates inflation within the large-volume regime of type IIB string theory, modeling the inflaton as the last unsettled Kähler modulus T2 and its axion θ2. By deriving the two-field potential V(τ, θ) and its noncanonical kinetics, the authors reveal a rich set of roulette-like trajectories in which axion motion can enhance the number of e-folds while preserving the observed scalar tilt and a suppressed tensor amplitude. They show that three or more Kähler moduli are necessary to stabilize the volume during inflation and that a self-reproducing, stochastic regime is possible in regions of near-flat potential. Using both analytic approximations and numerical explorations, they demonstrate that a wide class of trajectories can reproduce CMB/LSS observations, while highlighting the need to account for isocurvature effects and higher-order corrections in future work.

Abstract

We study 2-field inflation models based on the ``large-volume'' flux compactification of type IIB string theory. The role of the inflaton is played by a Kähler modulus τcorresponding to a 4-cycle volume and its axionic partner θ. The freedom associated with the choice of Calabi Yau manifold and the non-perturbative effects defining the potential V(τ, θ) and kinetic parameters of the moduli bring an unavoidable statistical element to theory prior probabilities within the low energy landscape. The further randomness of (τ, θ) initial conditions allows for a large ensemble of trajectories. Features in the ensemble of histories include ``roulette tractories'', with long-lasting inflations in the direction of the rolling axion, enhanced in number of e-foldings over those restricted to lie in the τ-trough. Asymptotic flatness of the potential makes possible an eternal stochastic self-reproducing inflation. A wide variety of potentials and inflaton trajectories agree with the cosmic microwave background and large scale structure data. In particular, the observed scalar tilt with weak or no running can be achieved in spite of a nearly critical de Sitter deceleration parameter and consequently a low gravity wave power relative to the scalar curvature power.

Figures (12)

• Figure 1: Schematic illustration of the ingredients in Kähler moduli inflation. The four-cycles of the CY are the Kähler moduli $T_i$ which govern the sizes of different holes in the manifold. We assume $T_3$ and the overall scale $T_1$ are already stabilized, while the last modulus to stabilize, $T_2$, drives inflation while settling down to its minimum. The imaginary parts of $T_i$ have to be left to the imagination. The outer $3+1$ observable dimensions are also not shown.
• Figure 2: (a) The potential surface $V(\tau_1, \tau_2)$ in a two-Kähler model, with the axionic components $\theta_1$ and $\theta_2$ fixed at their minima. (b) shows the related contour plot of the volume $\mathcal{V}$ against $\tau_2$. Although it may be possible to find a local (very shallow) minimum for the volume in this model (marked by a star), the generic situation is that both $\tau_1$ and $\tau_2$ will be dynamical, and indeed, the evolving $\tau_2$ could force $\tau_1$ out of a local minimum, thereby destabilizing what may have once been stabilized. For this reason, we have focused on models with three or more Kähler moduli, with all but one mutually enforcing their respective stabilizations, and in particular that of the volume. This large volume multi-Kähler approach to stabilization differs from the KKLT stabilization mechanism.
• Figure 3: One dimensional sections of the uplifted potential for parameter set 1. We perform a proper uplift procedure by explicitly introducing the additional field $\tau_3$ which is responsible for stabilizing the volume during inflation. The parameters for $\tau_3$ are chosen in such a way that the stability condition is fulfilled and at the same time we recover the desired value of $\mathcal{V}_{min}\approx10^6$. The minimum in $\tau_3$ direction is clearly visible at $\tau_{3,min}\approx 49$.
• Figure 4: (a) The $T_2$-potential surface $V(\tau ,\theta)$ for the parameter set 1 of Table 1. Equipotential contour lines are superposed. Both $\tau$ and $\theta$ are multiplied by the characteristic scale $a_2$. The surfaces for the other parameter sets in the Table are very similar when $a_2$-scaled, even sets 5 and 6 with high $W_0$. In all cases, we manually uplifted the potential to have zero cosmological constant at the minimum. The periodicity in the axionic direction and the constancy at large $a_2 \tau$ are manifest. If instead of $\tau$, we used the canonically-normalized field eq.(\ref{['eq:canon']}) which is amplified by $\sqrt{\mathcal{V}}$, the undulating nature in $\theta$ at large $\varphi$ becomes more evident. However, the canonically-normalized inflaton is trajectory dependent and not a global function. (b) $V(\tau)$ for $\theta$ at $\cos(a_2 \theta) = \pm 1$ shows how the flow for the positive value from large $\tau$ would be inward, but the flow would be outward for the negative value (and be unstable to $\theta$ perturbations). The dashed line is $V_\infty$, the $\tau\rightarrow\infty$ asymptote.
• Figure 5: Contour-plots of the potential including trajectories for several choices of initial values (denoted by filled circles) in field space $(\tau, \theta)$. The trajectories are evolved numerically until inflation ends at $\epsilon=1$. The number of e-folds is indicated next to the corresponding trajectory. During the last stages of inflation, the field always rolls along one of the valleys towards the minima which are located in the centre of the white circles. The maxima are located in the dark spots. Inflation in the axion direction can significantly enhance the amount of inflation over that obtained in pure $\tau$ inflation. Trajectories starting at large $\tau$ roll to the nearest valley, then to the minimum. But starting at intermediate $\tau$ with the axion sufficiently far from its minimum, we find the field can cross several $\theta$-ridges before settling into a valley. Another manifestation is the run-away character for $\tau$ if the axion is placed close to its maximum. (a) shows the simple pure-$\tau$ inflation if $\theta$ is set to its minimum, as in Conlon and Quevedo, for parameter set 1. (b) shows the complex evolution for sample general starting conditions, for parameter set 1, (c), (d) and (e) show the same for sets 2, 3 and 4, respectively.