Primordial perturbations from inflation with a hyperbolic field-space

We study primordial perturbations from hyperinflation, proposed recently and based on a hyperbolic field-space. In the previous work, it was shown that the field-space angular momentum supported by the negative curvature modifies the background dynamics and enhances fluctuations of the scalar fields qualitatively, assuming that the inflationary background is almost de Sitter. In this work, we confirm and extend the analysis based on the standard approach of cosmological perturbation in multifield inflation. At the background level, to quantify the deviation from de Sitter, we introduce the slow-varying parameters and show that steep potentials, which usually cannot drive inflation, can drive inflation. At the linear perturbation level, we obtain the power spectrum of primordial curvature perturbation and express the spectral tilt and running in terms of the slow-varying parameters. We show that hyperinflation with power-law type potentials has already been excluded by the recent Planck observations, while exponential-type potential with the exponent of order unity can be made consistent with observations as far as the power spectrum is concerned. We also argue that, in the context of a simple $D$-brane inflation, the hyperinflation requires exponentially large hyperbolic extra dimensions but that masses of Kaluza-Klein gravitons can be kept relatively heavy.

Figure 1

(Left) The time evolution of $\log (\varphi /L)$ in terms of $N\equiv \ln [a/a(t_{\mathrm{ini}})]$ in a model with $V(\varphi )=V_0\exp [\lambda \frac{\varphi }{M_{\mathrm{Pl}}}]$. The blue solid line denotes the numerical result, while the red dotted line denotes the analytic solution given by Eq. (51) with an appropriate choice of $A$. (Right) The time evolution of $ϵ$ in terms of $N$ in a model with $V(\varphi )=V_0\exp [\lambda \frac{\varphi }{M_{\mathrm{Pl}}}]$. The blue solid line denotes the numerical result, while the red dotted line denotes the analytic solution given by Eq. (54).

Figure 2

(Left) The time evolution of $\log (\varphi /L)$ in terms of $N\equiv \ln [a/a(t_{\mathrm{ini}})]$ in a model with $V(\varphi )=\frac{1}{2}{m}^2{\varphi }^2$. The blue solid line denotes the numerical result, while the red dotted line denotes the analytic solution given by Eq. (57) with an appropriate choice of $A$. (Right) The time evolution of $ϵ$ in terms of $N$ in a model with $V(\varphi )=\frac{1}{2}{m}^2{\varphi }^2$. The blue solid line denotes the numerical result, while the red dotted line denotes the analytic solution given by Eq. (59) with an appropriate choice of $A$.

Figure 3

(Left) The time evolution of the mode function $|\mathrm{Re}[u_{\chi }]|$ in terms of $\tilde{\tau }$ on subhorizon scales for $h=10$ and $\tilde{k}={10}^{-2}$ (blue). It is well approximated by (79) with $C_4=1/\sqrt{2\tilde{k}}$ and the other $C_i$s 0 (red), but slightly show the deviation in the amplitude as $\tilde{\tau }$ approaches to $-\sqrt{2}h/\tilde{k}$. (Right) The time evolution of the mode function $|u_{\chi }|$ on superhorizon scales in terms of $\tilde{\tau }$ for $h=10$ and $\tilde{k}={10}^{-2}$ multiplied by $(-\tilde{\tau })$.

Figure 4

(Left) The time evolution of $|u_{\chi }|$ normalized by $|u_{\chi }^0|$ for $\tilde{k}={10}^{-2}$ and various values of $h$. From the top to the bottom at small $|\tilde{\tau }|$, $h=10$, 8, 6, 4, 2, 0. (Right) The $h$-dependence of $|u_{\chi }|$ normalized by $|u_{\chi }^0|$ for $\tilde{k}={10}^{-2}$, evaluated at $\tilde{\tau }={\tilde{\tau }}_{\mathrm{late}}=-1$.

Figure 5

(Left) The power spectrum of ${\tilde{u}}_{\varphi }$ (blue) and ${\tilde{u}}_{\chi }$ (red) multiplied by ${\tilde{\tau }}^2$ for $h=4$, evaluated at $\tilde{\tau }={\tilde{\tau }}_{\mathrm{late}}=-1$. (Right) The power spectrum of ${\tilde{u}}_{\varphi }$ (blue) and ${\tilde{u}}_{\chi }$ (red) multiplied by ${\tilde{\tau }}^2$ for $h=10$, evaluated at $\tilde{\tau }={\tilde{\tau }}_{\mathrm{late}}=-1$.