Inflation that runs naturally: Gravitational waves and suppression of power at large and small scales

We point out three correlated predictions of the axion monodromy inflation model: the large amplitude of gravitational waves, the suppression of power on horizon scales and on scales relevant for the formation of dwarf galaxies. While these predictions are likely generic to models with oscillations in the inflaton potential, the axion monodromy model naturally accommodates the required running spectral index through Planck-scale corrections to the inflaton potential. Applying this model to a combined data set of Planck, ACT, SPT, and WMAP low-$\ell$ polarization cosmic microwave background (CMB) data, we find a best-fit tensor-to-scalar ratio $r_{0.05}={0.07}_{-0.04}^{+0.05}$ due to gravitational waves, which may have been observed by the BICEP2 experiment. Despite the contribution of gravitational waves, the total power on large scales (CMB power spectrum at low multipoles) is lower than the standard $\mathrm{\Lambda }\mathrm{CDM}$ cosmology with a power-law spectrum of initial perturbations and no gravitational waves, thus mitigating some of the tension on large scales. There is also a reduction in the matter power spectrum of 20–30% at scales corresponding to $k=10{\text{Mpc}}^{-1}$, which are relevant for dwarf galaxy formation. This will alleviate some of the unsolved small-scale structure problems in the standard $\mathrm{\Lambda }\mathrm{CDM}$ cosmology. The inferred matter power spectrum is also found to be consistent with recent Lyman-$\alpha$ forest data, which is in tension with the Planck-favored $\mathrm{\Lambda }\mathrm{CDM}$ model with a power-law primordial power spectrum.

Figure 1

Posteriors in the inflation model parameters ($p$, $b$, $\delta$, $f$) discussed in Sec. 3a and the tensor-to-scalar ratio $r$ at the pivot scale $k_{*}=0.05{\mathrm{Mpc}}^{-1}$. Contours enclose 68% and 95% of the total probability in each joint posterior. Note the small probability for $r=0.2$, which lies just outside the 99% confidence region. The correlation between the exponent $p$ and $r$ is a consequence of the $e$-folding constraint [Eq. (15)]. In contrast to the amplitude $b$ and phase shift $\delta$ which have peaked distributions, the axion decay constant $f$ is poorly constrained although it exhibits a mild correlation with $b$. Finally, note that the distributions in $r$ and $p$ exhibit very little dependence on $f$, and are thus fairly insensitive to the assumed prior in $f$.

Figure 2

Marginal posterior probability in four derived parameters. (a) The spectral index at the pivot scale $k_{*}=10{\mathrm{Mpc}}^{-1}$. The total spectral index $n_s$ (solid line) is plotted together with the unmodulated spectral index $n_{s,0}$ (dashed line) which excludes the oscillatory part. (b) Running of the spectral index ${\alpha }_{*}$ at the pivot scale. (c) The number of $e$-foldings $N$ from the time that mode $k_{*}$ left the horizon, to the end of inflation. (d) Period of oscillation in the power spectrum in terms of ${\log }_{10}k$.

Figure 3

Best-fit $TT$ angular power spectrum for the base Planck model with zero running (dark line), and axion monodromy model (red line). Error bars are shown for $2\le \ell <50$. These best-fit spectra are determined using a combination of $\text{Planck}+\text{lensing}+\mathrm{WP}+\text{high}\text{−}\ell$ data.

Figure 4

Posterior distribution in the tensor-to-scalar ratio $r$ for four different priors in $f$: three uniform priors covering different ranges in $f$ (solid, dashed, dot-dashed lines) and a log prior in $f$ (dotted line). The inferred probability in $r$ (and likewise $p$) is quite robust since it is insensitive to the assumed prior in $f$, while the other model parameters ($b,\delta$) are fairly well constrained.

Figure 5

Joint posteriors in the spectral index $n_s$ and the tensor-to-scalar ratio $r$, both evaluated at the pivot scale $k=0.05{\mathrm{Mpc}}^{-1}$. Constraints (68% and 95%) are shown for the constant running model (red) and axion monodromy model (blue). While the constant running model prefers $r=0$, in axion monodromy the highest probability occurs for $r\approx 0.07$ because a nonzero $r$ value is required to fit the running and spectral index well. Note also the constant running model allows for a higher $r$, primarily because it is not subject to the $e$-folding constraint.

Figure 6

Best-fit CMB $TT$ angular power spectrum at low multipoles for the base Planck model with zero running (red solid line), the axion monodromy model (blue dashed line), and the constant running model (magenta dotted line). Although axion monodromy and the constant running model exhibit a similar suppression of power at the lowest multipoles, the latter has a greater suppression of power in the range $10<\ell <50$, partly because there is no tensor contribution ($r=0$) and because it is not subject to the $e$-folding constraint.

Figure 7

Best-fit scalar power spectrum for axion monodromy (solid lines) vs constant running models (dashed lines), all normalized to 1 at $k=0.05{\mathrm{Mpc}}^{-1}$ for the sake of comparison. For $r=0.07$ (dark lines), axion monodromy achieves only slightly less suppression at low $k$ (corresponding to low $\ell$) compared to the constant running model. For $r=0.19$ (red lines) however, axion monodromy has milder negative running at low $k$ and thus cannot achieve enough suppression to make up for the tensor contribution at low $\ell$. This is a consequence of the $e$-folding constraint, which enforces a large exponent $p$ and limits the oscillation amplitude $b$ of the power spectrum.

Figure 8

Posterior in $r$ assuming two different $e$-folding priors. The solid line corresponds to a flat prior in $N$, which gives the number of $e$-foldings under the assumption that the oscillation in the potential continued with undiminished amplitude to the end of inflation. The dotted line corresponds to a flat prior in $N_0$, which approximates the number of $e$-foldings in the scenario that the oscillation in the potential died out rapidly after the modes observed in the CMB exited the horizon. The former scenario gives smaller probability for large $r$ values, since it excludes regions of parameter space for which oscillations dominate near the end of inflation, resulting in a large or infinite number of $e$-foldings. However, $r=0.2$ is located outside the 99% confidence region for the flat $N$ prior, and outside the 95% confidence region for the flat $N_0$ prior and thus is disfavored in either scenario.

Figure 9

The ratio of the matter power spectrum of axion monodromy compared to that of the base Planck model. The curves shown correspond to the best-fit model with the tensor-to-scalar ratio $r$ fixed to 0.07 (dark line), 0.13 (red dashed line), and 0.19 (blue dotted line). Note there is a reduction in power of $\sim 20–35\%$ at $k=10{\mathrm{Mpc}}^{-1}$, roughly the scale relevant to dwarf galaxy formation.

Figure 10

Constraints on the amplitude ${\mathrm{\Delta }}_L^2$ and slope $n_{\text{eff}}$ of the linear matter power spectrum at $z=3$ and $k=0.009(\mathrm{km}/\mathrm{s}{)}^{-1}$ where $k$ is given in velocity space. Grey contours are from Lyman-$\alpha$ forest data analyzed in Ref. [82]; green contours are from Lyman-$\alpha$ forest data from the BOSS survey [80]; red and blue contours are from the marginalized posterior probability for the axion monodromy model and the base Planck model (with zero running), respectively. Note that the more recent BOSS data are in tension with the base Planck model at greater than 2-sigma, whereas they are consistent with axion monodromy to within 1-sigma.

Figure 11

Posterior in the amplitude of the slow-roll parameter $\eta$. Since ${\eta }_{\max }\ll 1$ for all allowed regions of parameter space, the slow-roll expansion holds to good approximation, justifying the method used to derive the power spectra in Sec. 2b.

Figure 12

Joint three-dimensional posteriors in the amplitude $b$ and phase shift $\delta$, color coded by the tensor-to-scalar ratio $r$, running ${\alpha }_{*}$, and axion decay constant $f$ respectively. Note that outside the best-fit region, there are tails extending to large $b$, or small $b$ and extreme values of $\delta$. In these tails, $r$ (and correspondingly, the running ${\alpha }_{*}$) prefers to be small. Likewise, $f$ tends to take extreme values in the tails while preferring intermediate values in the best-fit region.