Scalar suppression on large scales in open inflation

We consider two toy models of open inflation and investigate their ability to give a suppression of scalar power on large scales while also satisfying observational constraints on the spatial curvature of the Universe. Qualitatively we find that both models are indeed capable of fulfilling these two requirements, but we also see that effects of the quantum tunneling must be carefully taken into account if we wish to make quantitative predictions.

Figure 1

The general form of the potential associated with open inflation models. Inflation is preceded by tunneling from the false vacuum and the steepening of the potential in the vicinity of the barrier leads to a suppression of the scalar power spectrum.

Figure 2

Upper panel: $\varphi$ as a function of $\xi$ for the CDL instanton solution of Model 1. Lower panel: A plot of $|V_{\varphi \varphi }|/H^2$ as a function of $\xi$ for the CDL instanton solution of Model 1. We can see that for most of the trajectory the condition $|V_{\varphi \varphi }|>H^2$ is satisfied.

Figure 3

Evolution of the Hubble scale as a function of the number of $e$-foldings before the end of inflation (solid black curve). Also plotted are the evolution of the curvature scale (upper dashed line), the current Hubble scale as determined assuming that the observational constraint on ${\mathrm{\Omega }}_{\text{K}}$ is saturated (middle dot-dashed line) and the scale associated with $l=100$ (bottom dotted line). The left-most vertical dashed line corresponds to the time at which the curvature term becomes subdominant to the potential term in the Friedmann equation. The next two lines, from left to right, indicate the times at which the current Hubble scale and the scale associated with $l=100$ left the Hubble horizon.

Figure 4

Evolution of the kinetic (solid black curve), curvature (dashed red curve) and potential (dot-dashed blue curve) contributions to $H^2$ as a function of the number of $e$-foldings before the end of inflation. The three vertical dashed lines are as in Fig. 3.

Figure 5

Plot of the potential in Model 1 of Ref. [24], with the parameter choice shown in the main text. The three vertical lines are as in Fig. 3, except their order from left to right is reversed.

Figure 6

Upper panel: Evolution of the four slow-roll quantities $\epsilon$ (blue dot-dashed curve), ${\epsilon }_V$ (red dashed curve), ${\epsilon }_m$ (black dotted curve) and ${\dot{\varphi }}^2/2H^2$ (solid green curve) as a function of the number of $e$-foldings before the end of inflation. Lower panel: Evolution of the three slow-roll quantities $\tilde{\eta }\equiv -\eta /2+2\epsilon$ (blue dot-dashed curve), ${\eta }_V$ (red dashed curve) and ${\eta }_m$ (black dotted curve) as a function of the number of $e$-foldings before the end of inflation. In both panels the three vertical lines are as in Fig. 3.

Figure 7

The scalar and tensor power spectra as functions of the comoving scale $p$. In the case of the scalar spectrum we plot three variants of the analytic expression (14), namely (1) the full expression (solid blue curve), (2) Eq. (14) without the $p$-dependent suppression factor that distinguishes Eq. (14) from the standard flat-universe expression (red dot-dashed curve) and (3) Eq. (14) taking $t_{\mathcal{R},p}$ as the time when $a^2{H}^2=p^2+1$ instead of when Eq. (19) is satisfied (purple dotted curve). We also plot the numerical solution for ${\mathcal{R}}_c^p$ from Ref. [24] (upper solid black curve). The fitted curves use $c_1=4$. Finally, for comparison, we also plot the form of the spectrum associated with the cutoff model considered in Ref. [3] (solid orange curve). In the case of the tensor spectrum we simply plot the full expression (15) with $c_2=1$ (solid green curve) and the numerical solution from Ref. [24] (lower solid black curve). The vertical dashed line at $p=10$ corresponds to the current Hubble scale if we assume that observational constraints on ${\mathrm{\Omega }}_{\text{K}}$ are saturated.

Figure 8

The magnitude of the scalar power relative to the magnitude at the scale of transition between a blue- and red-tilted spectrum. In this plot the vertical dashed lines, from left to right, correspond, respectively, to (1) the current Horizon scale $p_{H_0}$ as determined assuming that ${\mathrm{\Omega }}_{\text{K}}=0.01$, (2) the scale associated with $l=100,p_{100}$, and (3) the scale at which the spectrum transitions from being blue- to red-tilted, $p_{\text{red}}$. On the upper horizontal axis we also give the number of $e$-foldings before the end of inflation that each given scale $p$ leaves the horizon.

Figure 9

Upper panel: $\varphi$ as a function of $\xi$ for the CDL instanton solution of Model 2. Lower panel: A plot of $|V_{\varphi \varphi }|/H^2$ as a function of $\xi$ for the CDL instanton solution of Model 2. We can see that for most of the trajectory the condition $|V_{\varphi \varphi }|>H^2$ is satisfied.

Figure 10

Plot of the potential in Model 2 of Ref. [24] with the parameter choice shown in the main text. The three vertical lines are as in Fig. 5.

Figure 11

The scalar and tensor power spectra as a function of the comoving scale $p$ for Model 2. The set of curves plotted is the same as in Fig. 7 for Model 1, except that solid black curves are now replaced with black dot-dashed curves.

Figure 12

The magnitude of the scalar power relative to the magnitude at the scale of transition between a blue- and a red-tilted spectrum for Model 2. In this plot the vertical dashed lines, from left to right, correspond, respectively, to (1) the current horizon scale $p_{H_0}$ as determined assuming that ${\mathrm{\Omega }}_{\text{K}}=0.01$, (2) the scale associated with $l=100,p_{100}$, and (3) the scale at which the spectrum transitions from being blue- to red-tilted, $p_{\text{red}}$. On the upper horizontal axis we also give the number of $e$-foldings before the end of inflation that each given scale $p$ leaves the horizon.

Figure 13

Schematic diagram showing the evolution of $\ln (1/H)$ (solid black curve) through the three stages of post-tunneling evolution referred to in the main text. Initially the curvature dominates, giving $1/H\propto a$. In the second stage the potential now dominates but the slope of the potential is too steep to allow for slow-roll inflation, so the Hubble rate continues to evolve, but less rapidly. Once the potential flattens out, a period of slow-roll inflation commences, where the Hubble rate is approximately constant. Also plotted are the curvature scale (blue dashed curve), the current Hubble scale (red dotted curve) and the scale associated with the onset of suppression (green long-dashed curve). The points at which these curves intersect the solid black curve corresponds to the horizon-crossing times. The shaded region corresponds to the fast-roll period.