String axiverse

String theory suggests the simultaneous presence of many ultralight axions, possibly populating each decade of mass down to the Hubble scale ${10}^{-33}\mathrm{eV}$. Conversely the presence of such a plenitude of axions (an axiverse) would be evidence for string theory, since it arises due to the topological complexity of the extra-dimensional manifold and is ad hoc in a theory with just the four familiar dimensions. We investigate how several upcoming astrophysical experiments will be observationally exploring the possible existence of such axions over a vast mass range from ${10}^{-33}\mathrm{eV}$ to ${10}^{-10}\mathrm{eV}$. Axions with masses between ${10}^{-33}\mathrm{eV}$ to ${10}^{-28}\mathrm{eV}$ can cause a rotation of the cosmic microwave background polarization that is constant throughout the sky. The predicted rotation angle is independent of the scale of inflation and the axion decay constant, and is of order $\alpha \sim 1/137$ –within reach of the just launched Planck satellite. Axions in the mass range ${10}^{-28}\mathrm{eV}$ to ${10}^{-18}\mathrm{eV}$ give rise to multiple steps in the matter power spectrum, providing us with a snapshot of the axiverse that will be probed by galaxy surveys–such as BOSS, and 21 cm line tomography. Axions in the mass range ${10}^{-22}\mathrm{eV}$ to ${10}^{-10}\mathrm{eV}$ can affect the dynamics and gravitational wave emission of rapidly rotating astrophysical black holes through the Penrose superradiance process. When the axion Compton wavelength is of order of the black hole size, the axions develop superradiant atomic bound states around the black hole nucleus. Their occupation number grows exponentially by extracting rotational energy and angular momentum from the ergosphere, culminating in a rotating Bose-Einstein axion condensate emitting gravitational waves. For black holes lighter than $\sim {10}^7$ solar masses accretion cannot replenish the spin of the black hole, creating mass gaps in the spectrum of rapidly rotating black holes that diagnose the presence of destabilizing axions. In particular, the highly rotating black hole in the X-ray binary LMC X-1 implies an upper limit on the decay constant of the QCD axion $f_a\lesssim 2\times {10}^{17}\mathrm{GeV}$, much below the Planck mass. This reach can be improved down to the grand unification scale $f_a\lesssim 2\times {10}^{16}\mathrm{GeV}$, by observing smaller stellar mass black holes.

Figure 1

Map of the axiverse: The signatures of axions as a function of their mass, assuming $f_a\approx M_{\mathrm{GUT}}$ and $H_{\inf }\sim {10}^8\mathrm{eV}$. We also show the regions for which the axion initial angles are anthropically constrained not to over-close the Universe, and axions diluted away by inflation. For the same value of $f_a$ we give the QCD axion mass. The beginning of the anthropic mass region ($2\times {10}^{-20}\mathrm{eV}$) as well as that of the region probed by density perturbations ($4\times {10}^{-28}\mathrm{eV}$) are blurred as they depend on the details of the axion cosmological evolution (see Sec. 2c). $3\times {10}^{-18}\mathrm{eV}$ is the ultimate reach of density perturbation measurements with 21 cm line observations. The lower reach from black hole super-radiance is also blurred as it depends on the details of the axion instability evolution (see Sec. 2e). The region marked as “Decays,” outlines very roughly the mass range within which we expect bounds or signatures from axions decaying to photons, if they couple to $\overrightarrow{E}·\overrightarrow{B}$. We will discuss axion decays in detail in a companion paper.

Figure 2

Evolution of different physical momentum scales as a function of the scale factor. The horizontal long-dashed line corresponds to the axion mass. The short-dashed line shows Hubble. The dash-dotted line shows the evolution of the Jeans scale. Finally, three solid lines correspond to physical momenta of long and short modes and of the mode with $k=k_m$.

Figure 3

Suppression of the power spectrum observed today as a function of the comoving momentum. ${\delta }_k$ has been normalized to the value at large scales and we have assumed $\frac{{\Omega }_a}{{\Omega }_m}=0.01$.

Figure 4

A time evolution of the axion field for different initial conditions (left panel) and an enhancement factor $P(\theta )$ for the axion abundance as a function of a probability for different initial axion values (right panel).

Figure 5

The effective potential of Eq. (63). Depicted are the ergoregion, to the left, with the horizon at $r^{*}\to -\infty$, the centrifugal barrier (whose height depends on the angular momentum of a mode), the potential well to the right of it, and the asymptotic mass barrier which plays the role of the mirror that reflects the escaping Penrose fragment back. The relevant modes will be the states bound in the potential, and leaking through the barrier towards the horizon.

Figure 6

The Carnot cycle of the axionic instability: The black hole “feeds” the superradiant state forming an axion Bose-Einstein condensate. Axions from that state quantum-mechanically transition through graviton emission to a lower-energy nonsuperradiant state. The nonsuperradiant state decays into the black hole. Accreting matter around the black hole replenishes the rotational energy lost to gravitons and sustains this cycle.

Figure 7

The parameter space in the plane of axion and black hole masses where the superradiance leads to the black hole spindown assuming Eddington accretion (dark shaded region), and no accretion, i.e. the superradiance time scale is required to be faster than the age of the Universe (light shaded area). In both cases the upper bound has been calculated from (66) and the lower bound from (67).

Figure 8

The maximum allowed spin for a black hole as a function of its mass assuming there are two axions with mass $m_{a1}$ and $m_{a2}$ corresponding roughly to black hole masses of $2M_{\odot }$ and ${10}^6{M}_{\odot }$. This plot has been created using (66) (and the dependence of the superradiance rate on the black hole spin for heavy masses from 59) and (67) and is indicative.

Figure 9

A toy model potential encapsulating all the salient features of the superradiant instability: infinite potential barrier, representing the mass mirror; a potential well with $V=0$, a reasonable approximation for the modes near its top; a strongly repulsive Dirac $\delta$-function potential simulating the centrifugal barrier; and a potential $V=2m{\omega }_{+}\omega -m^2{\omega }_{+}^2$, modeling the dispersive properties of the waves far in the ergoregion as they approach the horizon, where they satisfy $k^2=(\omega -m{\omega }_{+}{)}^2$, such that the net potential there is much larger than $m^2{\omega }_{+}^2$ for large eigenvalues, (${\omega }^{\prime }$ on the figure), but is smaller than $m^2{\omega }_{+}^2$ for superradiant modes (denoted by $\omega$ on the figure). The key is that the superradiant modes have larger momentum outside the well than inside it.