Chaotic Inflation in Supergravity after Planck and BICEP2

TL;DR

This work assesses how chaotic inflation can be embedded in supergravity in light of Planck 2013 and BICEP2 observations. It leverages shift-symmetric Kähler potentials and a superpotential of the form $W=S f(Φ)$ to realize inflation with an inflaton potential $V(φ)=|f(φ/\sqrt{2})|^{2}$, enabling flexible shaping of $n_s$ and $r$ through the function $f$. By introducing small corrections to the quadratic potential and, separately, mild non-minimal couplings to gravity, the authors show that simple, well-controlled models can align with current data and even accommodate low-ℓ CMB power suppression. They also discuss critical issues in moduli stabilization for string/supergravity implementations, arguing that strong stabilization is essential to preserve predictive power while allowing large-field inflation compatible with observations. Overall, the paper demonstrates that chaotic inflation in supergravity provides a rich, adaptable framework capable of spanning a wide range of observationally viable inflationary scenarios.

Abstract

We discuss the general structure and observational consequences of some of the simplest versions of chaotic inflation in supergravity in relation to the data by Planck 2013 and BICEP2. We show that minimal modifications to the simplest quadratic potential are sufficient to provide a controllable tensor mode signal and a suppression of CMB power at large angular scales.

Figures (6)

• Figure 1: The green area describes observational consequences of inflation in the Higgs model (\ref{['minhiggs']}) with $v \gg 1$ ($a \ll 1$), for the inflationary regime when the field rolls down from the maximum of the potential. The continuation of this area upwards corresponds to the prediction of inflation which begins when the field $\phi$ initially is at the slope of the potential at $|v-\phi| \gg v$. In the limit $v\to \infty$, which corresponds to $a\to 0$, the predictions coincide with the predictions of the simplest chaotic inflation model with a quadratic potential ${m^{2}\over 2} \phi^{2}$.
• Figure 2: The potential $V(\phi) = {m^{2}\phi^{2}\over 2}\,\bigl(1-a\phi +a^{2}b\,\phi^{2})\bigr)^{2}$ for $a = 0.1$ and $b = 0.36$ (upper curve), 0.35 (middle) and 0.34 (lower curve). The potential is shown in units of $m$, with $\phi$ in Planckian units. For each of these potentials, there is a range of values of the parameter $a$ such that the observational predictions of the model are in the region of $n_{s}$ and $r$ preferred by Planck 2013. For $b = 0.34$, the value of the field $\phi$ at the moment corresponding to 60 e-foldings from the end of inflation is $\phi \approx 8.2$. Change of the parameter $a$ stretches the potentials horizontally without changing their shape.
• Figure 3: Predictions for $n_{s}(a)$ and $r(a)$ in at 55 e-folds the model with $V(\phi) = {m^{2}\phi^{2}\over 2}\,\bigl(1-a\phi +a^{2}b\,\phi^{2})\bigr)^{2}$ for various values of $b = 0.334 \ldots 5$. All curves have $a$ running from $0.001$ to $0.2$. The red ($b=0.34$) and green ($b=5$) balls correspond to $a = 0.01 \ldots 0.13$ in steps of $0.01$ from the joint start point $a=0.001$ outward. For $a = 0$ one recovers the predictions for the simplest chaotic inflation model with a quadratic potential for all $b$, while for $b=0.34$ and $a = 0.13$ the predictions almost exactly coincide with the predictions of the Starobinsky model and the Higgs inflation model (red balls). Conversely for $b\gtrsim 1$ and moderately small $a$ our model nicely traverses the BICEP2 constraints within the 1-$\sigma$ area (green balls).
• Figure 4: Grey: Exclusion contours for the microscopic parameters $(a,b)$ in the model with $V(\phi) = {m^{2}\phi^{2}\over 2}\,\bigl(1-a\phi +a^{2}b\,\phi^{2})\bigr)^{2}$ from the PLANCK+WP+BAO approximated exclusion limits on $n_s,r$. Dark grey and light grey denote the $68\%$ and $95\%$ confidence level exclusion contour plot for the microscopic parameters $(a,b)$, respectively, conditioned on $m$ chosen to keep COBE normalization of the curvature perturbation power. Blue-striped: Exclusion contours for the microscopic parameters $(a,b)$ in the same model from the BICEP2+PLANCK+WP+highL approximated exclusion limits on $n_s,r$. Dark blue and light blue denote the $68\%$ and $95\%$ confidence level exclusion contour plot for the microscopic parameters $(a,b)$, respectively, conditioned on $m$ chosen to keep COBE normalization of the curvature perturbation power.
• Figure 5: Comparison of a model (\ref{['polyn']}) with additional steepening from the $\xi$-dependent correction in the Käbler potential (\ref{['broken']}) with the Planck 2013 CMB temperature data. The grey line shows a reference prediction from a $\Lambda$CDM pure power-law power spectrum with $n_s=0.96$. The red line gives the prediction of our model with $a=0.105$ , $b=0.263$ and $\xi=-0.0055$. This model provides $n_s\simeq0.98$ and $r\simeq 0.1$ at $N_e=50$ e-folds before the end of inflation, and generates a power suppression at low-$\ell$ of about 10%. We see that the power suppression is confined to a relatively small range of low $\ell$'s, because the small $\xi$-correction generates an exponentially steep term in the scalar potential at large $\phi$.