Axiverse Lampposts

Abstract

The string axiverse predicts a unique connection between the high scales approachable only through theory and the low energies within reach of experimental verification: a multitude of light, feebly interacting axions. In order to capture the collective effects of such an axion ensemble, we model the string axiverse by $N$ coupled axions with a simple assumption: hierarchical axion masses that arise from hierarchical instantons with statistically distributed axion couplings. In this limit, we find that axion field ranges, which determine late-time cosmological abundances, shrink as $1/\sqrt{N}$ as the number of axions grows. Moreover, the heaviest modes tend to align with the smallest kinetic eigenvalues, further reducing their field ranges. Interactions with the Standard Model (SM) are largely set by the kinetic structure and do not grow with $N$, thus suppressing detection prospects relative to the individual-axion expectation. The exceptions are the ensemble's lightest and heaviest states as well as the Quantum Chromodynamics (QCD) axion, which incur no such suppression. We further find that coupled axiverse dark matter has parametrically relaxed tuning on initial conditions when produced via long, low-scale inflation relative to independent axions and high-scale inflation. Taken together, these results sharpen the observational outlook: the most accessible signals typically come from the QCD axion and from heavy axions that make up small dark matter subcomponents. An anthropic plateau of comparable energy density states produces subdominant signals; meanwhile, if light axions have SM interactions independent of QCD, they can also be within reach of future direct-detection experiments.

Figures (11)

• Figure 1: Effective field ranges (GS decay constants $f_{\rm GS}$) for an ensemble of $N=50$ axions with an isotropic kinetic matrix, $\bm K=f^2\bm I$, and isotropic instanton charges drawn as $[\bm Q]_{ij}=[\bm r_i]_j\sim {\rm round}[{\cal N}(0,\sigma_r^2)]$, where ${\rm round}[\cdot]$ denotes rounding to the nearest integer. Points show $f_{\rm GS}$ from a single draw of this ensemble. Solid curves show the mean prediction from \ref{['eqn:meanfield-ni2']}, and shaded regions the central $95\%$ interval from \ref{['eqn:fGS_variance_estimate']}. All curves exhibit the trend expected from the isotropic scaling \ref{['eqn:fGS_isotropic']}: heavier axions have field ranges suppressed by $\sqrt{N-i+1}$, while the lightest axions have unsuppressed [and, for sufficiently small ${\rm var}([\bm r_i]_j)$, enhanced] field ranges relative to the nominal UV scale $f$. The spread also grows toward lighter axions, so typical realizations contain light modes with $f_{\rm GS}$ well above the mean. As $\sigma_r$ decreases (see legend), more entries round to zero and the charge matrix becomes sparse. In this sparse regime the mean prediction continues to capture the overall trend, while the spread estimate becomes less accurate as fluctuations become increasingly dominated by shot noise. In the extreme sparse limit (green curve), the isotropic scaling with ${\rm var}([\bm r_i]_j)\sim 1/N$ yields the enhancement $f_{{\rm GS},N}\propto \sqrt{N}\,f$ for the lightest mode described in Ref. Bachlechner:2017hsj (see text around \ref{['eqn:sparse']}). Because rounding distorts the variance when $\sigma_r$ is small, analytic curves use the post-rounding variance $\sigma_{\rm eff}^2 \equiv {\rm var}([\bm r_i]_j)$ (measured from the rounded charge matrix) in place of the input $\sigma_r^2$.
• Figure 2: Gram-Schmidt decay constants $f_{\rm GS}$ for ensembles of $N=50$ axions with instanton charges drawn in the fundamental basis as $[\bm r_i]_j \sim {\rm round}[{\cal N}(0,1)]$, where ${\rm round}[\cdot]$ denotes rounding to the nearest integer. The kinetic matrix eigenvalues are sampled from a power law, $f_i^2 = f_{\rm max}^2\, x^p$ with $x\sim{\cal U}(0,1)$, where $f_{\rm max}$ sets the UV scale in the fundamental basis. The case $p=0$ (blue) corresponds to an isotropic kinetic matrix, consistent with the behavior shown in \ref{['fig:GS_isotropic']}. Increasing $p$ introduces anisotropy: for $p=2$ (red) the distribution shifts downward and, as the heaviest axions are integrated out first, they preferentially "absorb" the smallest kinetic eigenvalues, reducing their field ranges, as expected from the strongly anisotropic limit [\ref{['eqn:fGS_anisotropic']}]. For $p=4$ (green) this effect is more pronounced, as a few especially small kinetic eigenvalues lead to a stronger suppression of the heaviest-mode field ranges. As a result, the kinetic term becomes more isotropic for lighter axions as the highly anisotropic directions are pruned, returning to the scaling expected from \ref{['eqn:fGS_isotropic']}. The solid lines indicate the mean estimate [\ref{['eqn:meanfield-ni2']}] while the shaded regions indicate the central $95\%$ interval estimated using \ref{['eqn:fGS_variance_estimate']}.
• Figure 3: Couplings $|f_{{\cal O},i}|$ of $N=300$ axions to a Standard Model operator written in the fundamental basis as $\bm q^{T}\bm\theta\,{\cal O}_{\rm SM}$, shown for two choices of the coupling vector $\bm q$. For the green curve, the components of $\bm q$ are drawn i.i.d. as $q_j\sim{\cal N}(0,1)$, which we take as a simple model for couplings (e.g. to photons) that are not directly tied to an instanton term in the axion potential. The solid pink curve shows the mean coupling after marginalizing over draws of $\bm q$, with the shaded pink band indicating the central $95\%$ interval. For the blue curve, $\bm q$ is identified with the QCD charge vector, so that the same combination $\bm q^{T}\bm\theta$ appears in the potential via $\bm q^{T}\bm\theta\,\tilde{G} G$. We take $\Lambda_{\rm QCD}$ to coincide with the 150th instanton scale in the hierarchy and set $\bm q=\bm r_{150}/\sigma_r$, where the division by $\sigma_r$ normalizes the component variance of $\bm q$ to unity for a direct comparison with the Gaussian case. The blue curve exhibits three regimes: for large masses, the QCD-induced term is negligible and the couplings are statistically indistinguishable from those of a random $\bm q$ (hence agreement with the green curve). At the QCD scale ($i=150$) , the axion aligned with the $150$th GS direction couples in proportion to the corresponding GS decay constant (up to trivial rescalings), leading to an enhancement relative to heavier modes by a factor $\sqrt{N-i+1}\simeq 12$. The dashed lime-green curve shows the analytic expectation for the inverse GS decay constants [\ref{['eqn:meanfield-ni2']}], illustrating the correspondence between the QCD axion's GS decay constant and coupling depending on where it falls in the instanton potential hierarchy. Finally, for axions lighter than the $150$th axion, their couplings are exactly zero in the limit of infinite hierarchy. We discuss finite hierarchy corrections in \ref{['subsubsec:QCD_axion_coupling']}. In this example, the kinetic matrix is chosen with eigenvalues $f_i/ f_{\rm max} \sim{\cal U}(0,1)$, and the instanton charges are drawn as $[\bm r_i]_j\sim {\rm round}[{\cal N}(0,\sigma_r^2)]$.
• Figure 4: The ratio of the GS decay constants to the corresponding coupling decay constants normalized by the ratio of the variances of the instanton charges and the coupling vector to enable direct comparison to the analytic expressions \ref{['eqn:signal-strength-isotropic', 'eqn:signal-strength-anisotropic']} assuming kinetic matrices which are isotropic versus strongly anisotropic, respectively. The green line in this figure corresponds exactly to the ratio of the dashed lime green line to the pink line in \ref{['fig:couplings_isotropic']}, i.e. the ratio of the mean coupling decay constants \ref{['eqn:coupling_decay_constant_average']} to the analytic expression for the GS decay constants \ref{['eqn:fGS-ESP']}. The orange line corresponds to the scaling expected in the isotropic limit \ref{['eqn:signal-strength-isotropic']}, while the blue line corresponds to the expected scaling in the anisotropic limit \ref{['eqn:signal-strength-anisotropic']}. Values of the green curve larger than $1$ (gray dashed) represent a relative suppression of detection prospects.
• Figure 5: The probability density $p(\theta_0)$ for an axion's initial misalignment angle $\theta_0$, assuming a cosine potential $V(\theta) =\Lambda^4(1 - \cos\theta)$ and accounting for a long period of inflation at the scale $H_I$. Provided inflation lasts long enough [\ref{['eqn:long-inflation']}], the axion distribution $p(\theta_0)$ relaxes to a von Mises distribution, whose width is determined by the scale of inflation, as given in \ref{['eqn:inflationtheta0-dist']}. The lower the scale of inflation, the more Gaussian and peaked at $\theta_0=0$ the the distribution will be. The higher the scale of inflation, the more closely the distribution for the initial conditions will resemble a flat distribution.