Inflection points and the power spectrum

Inflection point inflation generically includes a deviation from slow-roll when the inflaton approaches the inflection point. Such deviations are shown to be generated by transitions between singular trajectories. The effects on the power spectrum are studied within the context of universality classes for small-field models. These effects are shown to scale with universality parameters, and can explain the anomalously low power at large scales observed in the cosmic microwave background. The reduction of power is related to the inflection point’s basin of attraction. Implications for the likelihood of inflation are discussed.

Figure 1

$\Xi$ as a function of $N$ for chaotic inflation on a quadratic potential (solid red line). $\varphi$ starts from rest and is shown (dashed, with arbitrary units) for qualitative comparison. Near $N=50$, slow-roll ends and $\varphi$ accelerates towards the minimum.

Figure 2

$\Xi$ as a function of $N$ for chaotic inflation on a quadratic potential (solid red line). $\varphi$ starts from fast-roll, and is shown (dashed, with arbitrary units) for qualitative comparison. $\Xi$ spikes as the field transitions from fast-roll to slow-roll. $\Xi$ spikes again as $\varphi$ accelerates towards the minimum of the potential.

Figure 3

$\Xi$ as a function of $N$ for an inflection point potential (solid red line). $\varphi$ starts from fast-roll, and is shown (dashed, with arbitrary units) for qualitative comparison.

Figure 4

$\Xi$ plotted for trajectories with various values of the cubic coupling $\alpha$. Each was started with the same, slow-roll boundary conditions. As $\alpha$ increases, the amplitude of $\Xi$ rapidly decreases. Here $\alpha$ ranges from 0.7 to 1.2 in increments of 0.1.

Figure 5

Initial conditions for the scalar field subject to the potential (14). Points in the shaded regions come to rest at the inflection point, others overshoot. The basin of attraction shrinks with smaller $\alpha$. The colored bands correspond to different values of $\alpha$: 2, 1, 0.8 and 0.7.

Figure 6

Trajectories of $\varphi$ started on slow-roll for the potential (14). The bottom curve uses $\alpha =0.6599$ and the top uses $\alpha =1.1599$. Note how the bottom curve abruptly approaches the inflection point, whereas the top curve is slowly asymptoting towards it.

Figure 7

The deviation from slow-roll, $\Xi$, is plotted versus $N$ for $\alpha =0.6599$ and $\alpha =1.1599$. The former curve, nearer to the critical value, exhibits the larger deviation from zero. Note the saturation $\Xi$ at $-3$.

Figure 8

$x_k$ plotted against the number of $e$-foldings (solid black curve). During slow-roll inflation, it asymptotes to unity. Deviation from slow-roll yields a spike in this quantity. $\varphi (N)$ is plotted (dashed, with arbitrary units) for reference.

Figure 9

Normalization of the power spectrum (in arbitrary units) during a spike in $\Xi$. Deviation from slow-roll yields a spike in this quantity. When separated, the red curve corresponds to the $H^{(2)}$ mode, the black is $H^{(1)}$. The magnitude of the Hankel function index, $|M|$ is plotted for reference.

Figure 10

The power spectrum $\alpha =0.7$, 0.8, 1.2 and 1.6. As $\alpha$ approaches the critical value ${\alpha }_c\approx 0.659$, the effect on the power spectrum is more pronounced.

Figure 11

The slow-roll derivation parameter $\Omega$ as a function of $\alpha$ (solid red curve), the cubic coupling during inflation. $\Omega$ diverges near ${\alpha }_C$ and decreases as $\alpha$ grows. The dashed line, plotted for comparison, represents a power law $1/N^{1.5}$.

Figure 12

The angular power spectrum $\alpha =0.7$, 0.8, 1.2 and 1.6. As $\alpha$ approaches the critical value ${\alpha }_c\approx 0.659$, the effect on the power spectrum is more pronounced.

Figure 13

Integrand of the quadrupole moment, $d{C}_2$ versus $N$, for various values of $\beta$, defined in Eq. (30). The solid red curve corresponds to the $\beta$ indicated in each panel. The grey dashed curves are the $d{C}_2$ from previous panels, replotted for reference. As $\beta$ increases, $d{C}_2$ shifts left. ${\Delta }^2$, which acts as an envelope, is also plotted for reference (thin, dashed curve). We chose $\alpha =0.7$ for this figure.

Figure 14

Plot of the relative quadrupole moment versus the $N-N_{\star }$ for the quartic potential with $\alpha =0.7$. Here $N_{\star }$ marks the end of the transition from chaotic to inflection point inflation. If the modes relevant for the CMB leave the horizon after $N_{\star }$, no appreciable reduction of power is noticed. Significant power reduction occurs if CMB modes leave at $N\approx N_{\star }-1$.

Figure 15

The basin of attraction is plotted with curves of constant initial ${\varphi }^{\prime }$. The red, black and orange curves, ordered from top to bottom in both plots, correspond to ${\varphi }^{\prime }=0$, $-1.9$, $-2.4$, respectively. Deeper into the basin, the more $e$-foldings of inflation occur above the inflection point. Note how little initial ${\varphi }^{\prime }$ suppresses $N_{\mathrm{ex}}$. We chose $\alpha =1$ in this figure. (a) Curves of constant initial velocity. (b) Curves of constant initial velocity within the basin of attraction.