Primordial non-Gaussianity from $G$ inflation

We present a comprehensive study of primordial fluctuations generated from $G$ inflation, in which the inflaton Lagrangian is of the form $K(\varphi ,X)-G(\varphi ,X)\square \varphi$ with $X=-(\partial \varphi {)}^2/2$. The Lagrangian still gives rise to second-order gravitational and scalar field equations, and thus offers a more generic class of single-field inflation than ever studied, with a richer phenomenology. We compute the power spectrum and the bispectrum, and clarify how the non-Gaussian amplitude depends upon parameters such as the sound speed. In so doing we try to keep as great generality as possible, allowing for non slow-roll and deviation from the exact scale invariance.

Figure 1

The non-Gaussian amplitude $\mathcal{A}(1,k_2/k_1,k_3/k_1)(k_2/k_1{)}^{-1}(k_3/k_1{)}^{-1}$ as a function of $k_2/k_1$ and $k_3/k_1$ for kinematicallly driven $G$ inflation. The amplitude is normalized to unity at an equilateral configuration, $k_2/k_1=k_3/k_1=1$. The parameters are given by $\sigma =0.36$, $c_s=0.03$, $\varrho =1$, and $\mathcal{I}/\mathcal{G}=1$, so that $r\simeq 0.17$. The size of non-Gaussianity is $f_{\mathrm{NL}}\simeq 210$.

Figure 2

The non-Gaussian amplitude $\mathcal{A}(1,k_2/k_1,k_3/k_1)(k_2/k_1{)}^{-1}(k_3/k_1{)}^{-1}$ as a function of $k_2/k_1$ and $k_3/k_1$ for kinematicallly driven $G$ inflation. The amplitude is normalized to unity at an equilateral configuration, $k_2/k_1=k_3/k_1=1$. The parameters are given by $\sigma =0.1$, $c_s=0.1$, $\varrho =60$, and $\mathcal{I}/\mathcal{G}=1$. The size of non-Gaussianity is $f_{\mathrm{NL}}\simeq 204$.

Figure 3

The non-Gaussian amplitude $\mathcal{A}(1,k_2/k_1,k_3/k_1)(k_2/k_1{)}^{-1}(k_3/k_1{)}^{-1}$ as a function of $k_2/k_1$ and $k_3/k_1$ for kinematicallly driven $G$ inflation. The amplitude is normalized to unity at an equilateral configuration, $k_2/k_1=k_3/k_1=1$. The parameters are given by $\sigma =0.1$, $c_s=0.1$, $\varrho =1$, and $\mathcal{I}/\mathcal{G}=300$. In this case the shape peaks in the folded configuration $k_1=2k_2=2k_3$.

Figure 4

The non-Gaussian amplitude $\mathcal{A}(1,k_2/k_1,k_3/k_1)(k_2/k_1{)}^{-1}(k_3/k_1{)}^{-1}$ as a function of $k_2/k_1$ and $k_3/k_1$ for potential driven $G$ inflation. The amplitude is normalized to unity at an equilateral configuration, $k_2/k_1=k_3/k_1=1$. The size of non-Gaussianity is $f_{\mathrm{NL}}=235/3888\simeq 0.06$.