Challenges for Large-Field Inflation and Moduli Stabilization

TL;DR

This work shows that embedding large-field chaotic inflation in string-inspired supergravity with Kähler moduli stabilization generically yields a universal, flattened inflaton potential due to modulus backreaction. When SUSY breaking is tied to the moduli, the heavy fields do not decouple, producing a bilinear soft mass term and subleading quartic corrections that shape inflation; all three concrete frameworks (KKLT, Kähler Uplifting, LVS) lead to a similar effective potential and observable predictions. A central result is a universal form $V(\varphi) \approx \tfrac{1}{2} m_\varphi^2 \varphi^2 - \tfrac{1}{4} \lambda \varphi^4$ with a local maximum at $\varphi_M = m_\varphi/\sqrt{\lambda}$, and inflation proceeds to the left of this maximum, yielding a lower bound $r \gtrsim 0.05$ for $60$ e-folds. Stability requires a large gravitino mass, $m_{3/2} \gg H$ (or $m_{3/2} > H \sqrt{\mathcal{V}}$ in LVS/Kähler Uplifting), which challenges the connection between SUSY breaking and inflation and underscores the sensitivity to initial conditions and modulus barriers. The results hint at a form of universality across distinct moduli-stabilization schemes and motivate further exploration of how string-compactification details influence observable inflationary signatures.

Abstract

We analyze the interplay between Kähler moduli stabilization and chaotic inflation in supergravity. While heavy moduli decouple from inflation in the supersymmetric limit, supersymmetry breaking generically introduces non-decoupling effects. These lead to inflation driven by a soft mass term, $m_\varphi^2 \sim m m_{3/2}$, where $m$ is a supersymmetric mass parameter. This scenario needs no stabilizer field, but the stability of moduli during inflation imposes a large supersymmetry breaking scale, $m_{3/2} \gg H$, and a careful choice of initial conditions. This is illustrated in three prominent examples of moduli stabilization: KKLT stabilization, Kähler Uplifting, and the Large Volume Scenario. Remarkably, all models have a universal effective inflaton potential which is flattened compared to quadratic inflation. Hence, they share universal predictions for the CMB observables, in particular a lower bound on the tensor-to-scalar ratio, $r \gtrsim 0.05$.

Figures (6)

• Figure 1: Effective inflaton potential in KKLT for $W_0 = 0.009$, $A=-0.75$, $a = \frac{2 \pi}{10}$, and $m=1.67 \times 10^{-5}$. With these parameters we find $T_0 = 10$ and $m_{3/2} = 10^{-4}$. The dotted line denotes a purely quadratic potential with $m_\varphi= 6 \times 10^{-6}$ imposed by COBE normalization. The dashed line is the effective potential Eq. \ref{['eq:EffPotKKLT2']} evaluated at all orders in $(a T_0)^{-1}$. This potential is valid only as long as the modulus remains stabilized. The solid line is obtained numerically by setting the modulus to its minimum value at each value of $\varphi$. Evidently, above the critical value $\varphi_\text{c} \approx 24$ the modulus is destabilized towards the run-away minimum at $T=\infty$ and the theory can not be described by Eq. \ref{['eq:EffPotKKLT2']} any longer.
• Figure 2: Scalar potential as defined by Eqs. \ref{['eq:WKKLT']} as a function of $T$ and $\varphi$, for the same parameter example as in Fig. \ref{['fig:PlotKKLT']}. Apparently, a minimum for the modulus exists for $\varphi \lesssim \varphi_\text{c} \approx 24$. Beyond this point the modulus runs away towards $T = \infty$ and can no longer be integrated out. For $\varphi < \varphi_\text{c}$ inflation may take place in the valley of the uplifted modulus minimum.
• Figure 3: Effective inflaton potential in Kähler Uplifting for $W_0 = 4.67$, $A=-3.4 \times 10^{-4}$, $a = \frac{2 \pi}{30}$, $m=8 \times 10^{-4}$, and $\xi = 0.0047$. With these parameters we find $T_0 = 11.9$, $m_{3/2} =0.04$, and $\eta_0 = 2 \times 10^{-5}$. The dotted line denotes a purely quadratic potential with $m_\varphi = 6 \times 10^{-6}$ imposed by COBE normalization. The dashed line is the effective potential Eq. \ref{['eq:KUEffPot']} evaluated at all orders in $\eta$. The solid line is obtained numerically by setting the modulus to its minimum value at each value of $\varphi$. In this case, modulus destabilization occurs at $\varphi_\text{c} \approx 19$. Again, Eq. \ref{['eq:KUEffPot']} and the dashed line are only meaningful for $\varphi < \varphi_\text{c}$.
• Figure 4: Effective inflaton potential in Kähler Uplifting for $W_0 = 0.23$, $A=-0.008$, $a = \frac{2 \pi}{30}$, $m=-1.37 \times 10^{-4}$, and $\xi = 2.29$. With these parameters we find $T_0 = 11.8$, $m_{3/2} =0.002$, and $\eta_0 = 0.01$. The dotted line denotes a purely quadratic potential with $m_\varphi= 6 \times 10^{-6}$ imposed by COBE normalization. The dashed line is the effective potential Eq. \ref{['eq:KUEffPot']} evaluated at all orders in $\eta$. The solid line is obtained numerically by setting the modulus to its minimum value at each value of $\varphi$. In this setup, modulus destabilization occurs at $\varphi_\text{c} \approx 20$. Again, Eq. \ref{['eq:KUEffPot']} and the dashed line are only meaningful for $\varphi < \varphi_\text{c}$.
• Figure 5: Effective inflaton potential in LVS for $W_0 = 1$, $A=0.13$, $a = 2 \pi$, $m=5.8 \times 10^{-4}$, and $\xi = 1.25$. With these parameters we find $T_0 = 0.75$, $\mathcal{V}_0 = 200$, and $m_{3/2} =0.005$. The dotted line denotes a purely quadratic potential with $m_\varphi= 6 \times 10^{-6}$ imposed by COBE normalization. The dashed line is the effective potential Eq. \ref{['eq:LVSEffPot']} evaluated at all orders in $a T_0$. The solid line is obtained numerically by setting the modulus to its minimum value at each value of $\varphi$. Since the barrier height and Hubble scale are the same as in the previous example, modulus destabilization occurs at $\varphi_\text{c} \approx 18$. Again, Eq. \ref{['eq:LVSEffPot']} and the dashed line are only meaningful for $\varphi < \varphi_\text{c}$. Notice that the difference between the dashed and the solid line is comparably large in this example. This is because the relatively small value of $\mathcal{V}_0$ limits the precision of the expansion in $\mathcal{V}^{-1}$.