A search for ultralight axions using precision cosmological data

Ultralight axions (ULAs) with masses in the range $10^{-33}\mathrm{eV}\le m_a\le 10^{-20}\mathrm{eV}$ are motivated by string theory and might contribute to either the dark-matter or dark-energy densities of the Universe. ULAs could suppress the growth of structure on small scales, lead to an altered integrated Sachs-Wolfe effect on cosmic microwave-background (CMB) anisotropies, and change the angular scale of the CMB acoustic peaks. In this work, cosmological observables over the full ULA mass range are computed and then used to search for evidence of ULAs using CMB data from the Wilkinson Microwave Anisotropy Probe (WMAP), Planck satellite, Atacama Cosmology Telescope, and South Pole Telescope, as well as galaxy clustering data from the WiggleZ galaxy-redshift survey. In the mass range $10^{-32}\mathrm{eV}\le m_a\le 10^{-25.5}\mathrm{eV}$, the axion relic-density ${\mathrm{\Omega }}_a$ (relative to the total dark-matter relic density ${\mathrm{\Omega }}_d$) must obey the constraints ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d\le 0.05$ and ${\mathrm{\Omega }}_a{h}^2\le 0.006$ at 95% confidence. For $m_a\gtrsim 10^{-24}\mathrm{eV}$, ULAs are indistinguishable from standard cold dark matter on the length scales probed, and are thus allowed by these data. For $m_a\lesssim 10^{-32}\mathrm{eV}$, ULAs are allowed to compose a significant fraction of the dark energy.

Figure 1

Marginalized 2 and $3\sigma$ contours show limits to the ultralight axion (ULA) mass fraction ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d$ as a function of ULA mass $m_a$, where ${\mathrm{\Omega }}_a$ is the axion relic-density parameter today and ${\mathrm{\Omega }}_d$ is the total dark-matter energy density parameter. The vertical lines denote our three sampling regions, discussed below. The mass fraction in the middle region is constrained to be ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d\lesssim 0.05$ at 95% confidence. Red regions show CMB-only constraints, while grey regions include large-scale structure data.

Figure 2

Adiabatic matter power spectra generated with the modified camb described in Sec. 3, with varying axion mass and energy-density fraction ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d$ at fixed total dark-matter density fraction ${\mathrm{\Omega }}_d$. Power is suppressed for modes that enter the horizon when the axion sound speed $c_s\sim 1$.

Figure 3

Evolution of the fractional DM density perturbation $\delta$ when ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d=1$ (solid), for a ULA mass of $m_a=10^{-26}\mathrm{eV}$ and a series of wave numbers $k$ (as shown in the figure), compared to standard CDM (dashed). The overall normalization of the mode amplitude is arbitrary here. The range of k-values encompasses different behaviors, with suppression of growth relative to CDM when $k\sim k_J(a)$, oscillation when $k>k_J(a)$ and growth as CDM when $k<k_J(a)$. This leads to an overall suppression of power for large-$k$ modes.

Figure 4

Evolution of the fractional dark-matter density perturbation with wave number $k=10^{-4}h{\mathrm{Mpc}}^{-1}$ for the 3 different ULA masses indicated compared to the standard CDM case (dashed). For these ULA masses, $k<k_m$ always, and so soon after $a>a_{\text{osc}}$, the mode behaves just as CDM.

Figure 5

Evolution of the integrated Sachs-Wolfe (ISW) source term [54] for a mode with $k=10^{-4}h{\mathrm{Mpc}}^{-1}$. The overall amplitude is arbitrary. Dark colored curves are generated using the modified camb described in the text. Lighter curves are generated using direct numerical integration of scalar-field perturbation EOMs. Green curves show the effect of choosing ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d=0.1$ (with all other parameters set to $\mathrm{\Lambda }\mathrm{CDM}$ values) with $m_a=10^{-32}\mathrm{eV}$. Blue curves are obtained assuming ${\mathrm{\Omega }}_{\mathrm{\Lambda }}=0$, ${\mathrm{\Omega }}_m=1$ and ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d=0.1$ with $m_a=5\times 10^{-32}\mathrm{eV}$.

Figure 6

CMB temperature power spectrum with varying ULA mass and energy density fraction ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d$. Here, as in Fig. 2, we introduce ULAs as a fraction of the dark matter, holding ${\mathrm{\Omega }}_d={\mathrm{\Omega }}_a+{\mathrm{\Omega }}_c$ fixed. Since ULAs have $w_a=-1$ for some time during the radiation era this changes the ratio of matter to radiation and alters the relative heights of the CMB acoustic peaks. For dark-matter like ULAs with the highest ULA masses, and lowest fractions, $C_{\ell }^{\mathrm{TT}}$ becomes indistinguishable from $\mathrm{\Lambda }\mathrm{CDM}$, with the $\mathrm{\Lambda }\mathrm{CDM}$ curve lying directly underneath the ULA curve.

Figure 7

CMB temperature power spectrum with varying ULA mass and energy density fraction ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d$. Here, we introduce the lightest ULAs as a fraction of the dark energy, holding ${\mathrm{\Omega }}_c{h}^2$ and $H_0$ fixed so that maintaining flatness while introducing ULAs reduces ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$. The lightest ULAs transition to matter-like behavior late in the lifetime of the Universe and can contribute to the dark energy. The visible effects come from the change in the age of the Universe, which changes the angular size of the sound horizon, and in changing the integrated effect of dark energy, which changes the amplitude of the ISW plateau. For dark-energy like ULAs with the lowest ULA masses, and lowest fractions, $C_{\ell }^{\mathrm{TT}}$ becomes indistinguishable from $\mathrm{\Lambda }\mathrm{CDM}$, with the $\mathrm{\Lambda }\mathrm{CDM}$ curve lying directly underneath the ULA curve.

Figure 8

CMB temperature power spectrum with varying ULA mass and energy density fraction ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d$. Here, we introduce the lightest ULAs as a fraction of the dark energy, holding ${\mathrm{\Omega }}_c{h}^2$ and ${\theta }_A$ fixed, which requires varying $H_0$. With the angular size of the sound horizon fixed, the ULAs only affect the CMB by altering the shape of the ISW plateau. Low values of $H_0\sim 60\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ were necessary in these examples to keep ${\theta }_A$ fixed, and certain cosmologies cannot be brought to fixed ${\theta }_A$ while maintaining an expanding universe with ${\mathrm{\Omega }}_{\mathrm{\Lambda }}\ge 0$.

Figure 9

Matter power-spectrum with varying ULA mass and energy density fraction ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d$. Here we introduce the lightest ULAs as a fraction of the DE, holding ${\mathrm{\Omega }}_c{h}^2$ fixed. No masses considered in this example cluster on scales where there are data, and we exclude them from the matter density used to define $P(k)$ [taking $a_{\text{bias}}=a_{\text{eq}}$ in Eq. (32)]. Left Panel: Fixed $H_0$, reducing ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ to maintain flatness. On scales shown the shape of $P(k)$ is unchanged, and the only effect comes from the reduction in the age of the Universe giving less growth time. Right Panel: Fixed ${\theta }_A$, which requires reducing $H_0$ as ULAs are introduced.

Figure 10

Theoretical galaxy clustering power spectra, varying the ULA mass fraction ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d$ at fixed ULA mass $m_a=10^{-27}\mathrm{eV}$. All non-ULA parameters are held at fiducial $\mathrm{\Lambda }\mathrm{CDM}$ values. An example selection of WiggleZ data are shown from the “9 hour region” [5, 157]. The theoretical curves have been computed at the same redshift ($z=0.6$) as the WiggleZ data, and multiplied by the window functions for the region in question. Left panel: Power spectra with galaxy bias fixed to its best-fit $\mathrm{\Lambda }\mathrm{CDM}$ value. Right panel: Power spectra, marginalizing over bias in the course of parameter-space exploration. At high values of ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d$, the preference is for higher values of the bias (to absorb the overall power suppression), explaining the upward trend in power at large scales in this panel.

Figure 11

Mass-dependent degeneracy of axions and CDM. Points are shown for a MultiNest chain and colored by $m_a$. If axions are light ($m_a<10^{-30}\mathrm{eV}$), they behave as dark energy. Therefore while the CDM density is unchanged as ${\mathrm{\Omega }}_a{h}^2$ increases, the dark-energy density ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ is reduced (see Fig. 15). If axions are heavy ($m_a>10^{-25}\mathrm{eV}$), they behave as dark matter, and so there is a perfect degeneracy between ${\mathrm{\Omega }}_c{h}^2$ and ${\mathrm{\Omega }}_a{h}^2$. For $m_a$ in the intermediate mass range, the axion energy density is constrained to be small.

Figure 12

Marginalized 2 and $3\sigma$ contours in the $m_a-{\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d$ plane for both the CMB-only (red) and $\mathrm{CMB}+\text{WiggleZ}$ (black) combinations of data sets. We obtain constraints of ${\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d\le 0.03–0.05$ at 95% confidence level over some seven orders of magnitude in $m_a$. The high-mass fluctuations/dips in the plane are due to sampling of the space rather than true features in the data set. For ultralight axions (ULAs) with masses $m_a\lesssim 10^{-20}\mathrm{eV}$, small-scale structure formation is suppressed [29, 30, 31, 32, 33, 34] on astronomically observable length scales.

Figure 13

Marginalized 2 and $3\sigma$ contours in the $m_a-{\mathrm{\Omega }}_a{h}^2$ plane for both the CMB-only and $\mathrm{CMB}+\text{WiggleZ}$ combinations of data sets. The left panel shows the contours with the axion density shown on logarithmic scale, while the right hand side shows the same contours on a linear scale. We obtain constraints of ${\mathrm{\Omega }}_a{h}^2\le 0.006$ at 95% confidence level over some seven orders of magnitude in axion mass $m_a$. Color code is as in Fig. 12.

Figure 14

Degeneracies between the axion parameters and other cosmological parameters. Color indicates value of indicated parameter, as shown by color bar to the right of each panel. Axion parameters are independent of the baryon density, as well as the normalization and tilt of the primordial power-spectrum, as can be seen in the top right, top left, and bottom left panels of the plot. In the bottom right panel, we see that axion parameters can be degenerate with the Hubble constant $H_0$ (in $\mathrm{km}{\mathrm{s}}^{-1}/\mathrm{Mpc}$) in the dark-energy like part of parameter space, at low values of $m_a$, where allowed values can drop to $H_0\lesssim 60\mathrm{km}\hspace{0.17em}{\mathrm{s}}^{-1}/\mathrm{Mpc}$.

Figure 15

Marginalized one-dimensional constraints on the axion parameters for various data sets. Left panel: the solid lines show the constraints when considering only CMB data, while the dashed lines (right panel) show the constraints when adding in WiggleZ data. In both panels the parameter constraints are shown for the axions sampled in separated mass bins. The black dot-dashed lines indicate the constraints obtained when combining the chains from the individual runs, weighted by the global run.

Figure 16

Comparison of marginalized 95% contours and locally defined one-dimensional limits on the axion fraction. The chains are binned in mass bins of $\mathrm{\Delta }{\log }_{10}(m_a/\mathrm{eV})=0.5$. The bars give the 95% upper percentile of the axion fraction. The solid line in each bar shows the location of the 50% percentile of the chain, and the two dotted lines show the 84% and 16% percentiles respectively.

Figure 17

The $m_a-{\mathrm{\Omega }}_a/{\mathrm{\Omega }}_d$ parameter space showing sample points for the CMB-only data, colored by the initial field displacement ${\varphi }_i/M_{\text{pl}}$. All points satisfy $\varphi /M_{\text{pl}}<\pi$ and so are consistent with sub-Planckian decay constants, $f_a<M_{\text{pl}}$, and the Weak Gravity Conjecture. Most points satisfy ${\varphi }_i/M_{\text{pl}}<1$ and so are consistent with $f_a<M_{\text{pl}}$. Regions with ${\varphi }_i/M_{\text{pl}}<0.01$ are consistent with a GUT-scale decay constant with no need for additional production mechanisms.