Bump in the blue axion isocurvature spectrum

Blue axion isocurvature perturbations are both theoretically well motivated and interesting from a detectability perspective. These power spectra generically have a break from the blue region to a flat region. Previous investigations of the power spectra were analytic, which left a gap in the predicted spectrum in the break region due to the nonapplicability of the used analytic techniques. We therefore compute the isocurvature spectrum numerically for an explicit supersymmetric axion model. We find a bump that enhances the isocurvature signal for this class of scenarios. A fitting function of three parameters is constructed that fits the spectrum well for the particular axion model we study. This fitting function should be useful for blue isocurvature signal hunting in data and making experimental sensitivity forecasts.

Figure 1

Homogeneous background field values for a model with $c_{+}=2.235$, $c_{-}=0.9$, ${\theta }_{+}(t_{\mathrm{i}})=0.04$, $h=1$, $F_{\mathrm{a}}=7.9\times 10^{10}\mathrm{GeV}$, $H=9\times 10^9\mathrm{GeV}$, and ${\tilde{\mathrm{\Phi }}}_{+}({\eta }_i)=0.3M_{\mathrm{Pl}}$. Left: the fields relax to $\sim F_{\mathrm{a}}$ in $\sim 10$ e-folds. Right: after inflation ends at $H_{\mathrm{in}}t=55$, the fields undergo damped oscillation. On the horizontal axis of these plots, we have denoted $H_{\mathrm{in}}$ as the expansion rate at the initial time $t_i=0$ to emphasize the fact that the plots actually continue to the time period after the end of inflation when the expansion rate starts to decrease. In the text where we primarily discuss the approximately constant expansion rate during inflation, we denote $H_{\mathrm{in}}$ simply as $H$.

Figure 2

The transition from rolling to constant fields is not always monotonic. ${\tilde{\mathrm{\Phi }}}_{+}$ (thick lines) and ${\tilde{\mathrm{\Phi }}}_{-}$ (thin lines) overshoot their final values of Eq. (108) for sufficiently large $c_{+}$.

Figure 3

Power spectra ${\mathrm{\Delta }}_S^2(k)/{\omega }_a^2$ for $c_{+}=1.5$ and $H=6\times 10^9\mathrm{GeV}$ (left) as well as $c_{+}=2.235$ and $H=9\times 10^9\mathrm{GeV}$ (right), corresponding to spectral indices of $n_{\mathrm{I}}=2.27$ and $n_{\mathrm{I}}=3.76$, respectively (shown as dotted lines). The other parameters, $h=1$, ${\theta }_{+}(t_{\mathrm{i}})=0.04$, $c_{-}=0.9$, and $F_{\mathrm{a}}=7.9\times 10^{10}\mathrm{GeV}$ are the same for both models. (See Fig. 1 for an explanation of $H_{\mathrm{in}}$.) These power spectra are evaluated at $H_{\mathrm{in}}t=100$, long after the end of inflation.

Figure 4

Peaks of ${\mathrm{\Delta }}_S^2(k)/{\omega }_a^2$, calculated using the numerical solutions of Sec. 4a (thick lines) as well as a smoothed homogeneous solution interpolating Eqs. (106), (107), and (108) (thin lines). (See Fig. 1 for an explanation of $H_{\mathrm{in}}$.) These plots are evaluated at $H_{\mathrm{in}}t=100$, long after the end of inflation.

Figure 5

Fractional change in the power spectrum due to the variation of $h$ from 1. The remaining parameters are identical to those used in Fig. 3 (bottom). (See Fig. 1 for an explanation of $H_{\mathrm{in}}$.) These plots are evaluated at $H_{\mathrm{in}}t=100$, long after the end of inflation.

Figure 6

Field perturbations for a single mode after the end of inflation, $\eta =55$, for the $c_{+}=2.235$ model shown in Fig. 1. (See Fig. 1 for an explanation of $H_{\mathrm{in}}$.)

Figure 7

Tests of our fitting function for randomly chosen model parameters, as given in Table 2. For clarity, power spectra have been divided by “no-wiggle” spectra ${\mathrm{\Delta }}_{\mathrm{nw}}^2$ found by setting $\alpha =0$ in the fitting function. Top-left: small-$c_{+}$. Top-right: medium-$c_{+}$. Bottom: high-$c_{+}$.