Moduli backreaction on inflationary attractors

We investigate the interplay between moduli dynamics and inflation, focusing on the Kachru-Kallosh-Linde-Trivedi scenario and cosmological $\alpha$-attractors. General couplings between these sectors can induce a significant backreaction and potentially destroy the inflationary regime; however, we demonstrate that this generically does not happen for $\alpha$-attractors. Depending on the details of the superpotential, the volume modulus can either be stable during the entire inflationary trajectory or become tachyonic at some point and act as a waterfall field, resulting in a sudden end of inflation. In the latter case there is a universal supersymmetric minimum where the scalars end up, preventing the decompactification scenario. The gravitino mass is independent from the inflationary scale with no fine-tuning of the parameters. The observational predictions conform to the universal value of attractors, fully compatible with the Planck data, with possibly a capped number of $e$-folds due to the interplay with moduli.

Figure 1

Plot of the scalar potential defined by Eq. (12) as a function of $\mathrm{Re}(T)$ and $\mathrm{Re}(\mathrm{\Phi })$. The $\alpha$-KKLT branch, placed at $T_0$ with the potential value given by Eq. (13), is highlighted by the red dashed line. It develops an instability around $\mathrm{\Phi }=1$ where one can clearly appreciate the universal Minkowki minimum, placed at shifted values of $T$. (Parameters are $A=1$, $W_0=-0.0004$, $a=0.1$, $\alpha =1.5$.)

Figure 2

Effective potential $V(\mathrm{\Phi })$ (upper panel) and vacuum structure (lower panel) for the product separable case. The minimum of the potential in $T$ is denoted by the solid lines, the maximum with the dashed ones. The red line denotes the $\alpha$-KKLT branch at $T_0$ while the green line represents the non-SUSY branch defined by Eq. (15). (Parameters are $A=1$, $W_0=-0.0004$, $a=0.1$, $\alpha =1.1$.)

Figure 3

Effective potential $V(\mathrm{\Phi })$ (upper panel) and vacuum structure (lower panel) for the general coupling case. The minimum of the potential in $T$ is denoted by the solid line, the maximum by the dashed line. (Parameters are $A_{+}=1.1$, $A_{-}=1$, $W_0=-0.0004$, $a=0.1$ and $\alpha =1.1$.)

Figure 4

The effective scalar potential for product separable models with tuned profile (31) (upper panel) and generic profile (33) (lower panel). Note the presence of the universal Minkowski minimum in both cases. In the tuned setup, there is an additional Minkowski minimum before the waterfall point. (Parameters are $W_0=-0.0004$, $A=1$, $a=0.1$, $\alpha =1$.)

Figure 5

Simulation of scalar field dynamics of $\mathrm{\Phi }$ (upper panel) and $T$ (lower panel) versus time for the product coupling case with generic profile (33). Initial displacements in $T$ from the $\alpha$-KKLT trajectory are quickly dampened close to $\mathrm{\Phi }=0$. The waterfall effect is clearly visible in this picture: the scalars oscillate around the universal Minkowski minimum after producing inflation on the plateau. (Parameters are $W_0=-0.0004$, $A=1$, $a=0.1$, $\alpha =1$.)

Figure 6

Mass of $\mathrm{Im}(\mathrm{\Phi })$ for the product case examples with tuned profile (31) (upper panel) and generic profile (33) (lower panel) for $\alpha >1$. In both cases, there is a divergent positive mass at the boundary $\mathrm{\Phi }=0$, a large positive mass around the SUSY Minkowski vacuum and a mild destabilization at intermediate field values. (Parameters are $W_0=-0.0004$, $A=1$, $a=0.1$.)

Figure 7

Scalar potential along $T$-direction minimum (upper panel) and mass of $\mathrm{Im}(\mathrm{\Phi })$ (lower panel) for the general coupling case. There is a finite positive mass of order Hubble scale at the boundary $\mathrm{\Phi }=0$ and a large positive mass around the vacuum at the end of inflation. (Parameters are $W_0=-0.0004$, $A_{+}=6$, $A_{-}=1$, $a=0.1$, $\alpha =1$.)