Axion cosmology revisited

The misalignment mechanism for axion production depends on the temperature-dependent axion mass. The latter has recently been determined within the interacting instanton liquid model, and provides for the first time a well-motivated axion mass for all temperatures. We reexamine the constraints placed on the axion parameter space in the light of this new mass function. Taking this mass at face value, we find an accurate and updated constraint $f_a\le 2.8(\pm 2)\times {10}^{11}\mathrm{GeV}$ or $m_a\ge 21(\pm 2)\mu \mathrm{eV}$ from the misalignment mechanism in the classic axion window (thermal scenario). However, this is superseded by axion string radiation which leads to $f_a\lesssim {3.2}_{-2}^{+4}\times {10}^{10}\mathrm{GeV}$ or $m_a\gtrsim {0.20}_{-0.1}^{+0.2}\mathrm{meV}$. In this analysis, we take care to precisely compute the effective degrees of freedom and, to fill a gap in the literature, we present accurate fitting formulas. We solve the evolution equations exactly, and find that analytic results used to date generally underestimate the full numerical solution by a factor 2–3. In the inflationary scenario, axions induce isocurvature fluctuations and constrain the allowed inflationary scale $H_I$. Taking anharmonic effects into account, we show that these bounds are actually weaker than previously computed. Considering the fine-tuning issue of the misalignment angle in the whole of the anthropic window, we derive new bounds which open up the inflationary window near ${\theta }_a\to \pi$. In particular, we find that inflationary dark matter axions can have masses as high as $0.01–1\mathrm{meV}$, covering the whole thermal axion range, with values of $H_I$ up to ${10}^9\mathrm{GeV}$. Quantum fluctuations during inflation exclude dominant dark matter axions with masses above $m_a\lesssim 1\mathrm{meV}$.

Figure 1

The mass for the QCD axion follows from the topological susceptibility, $m_a^2{f}_a^2=\chi$. The fit goes over to the dilute gas approximation for moderately high temperatures $T\approx 400\mathrm{MeV}$, in accordance with the IILM data. Note that the large errors are mostly due to the large uncertainties in the determination of $\Lambda$, used to set dimensions.

Figure 2

Shown are the mass for the QCD axion from IILM simulations (19), from a lattice-inspired fit that uses the IILM mass shifted towards higher temperatures to mimic the phase transition at $T_c^{\mathrm{lat}}\approx 160\mathrm{MeV}$, from the classic dilute gas approximation (DGA) by Turner [53] and its update by Bae et al. [52], and from the DGA derived in this paper (22). The simple power-law DGA axion masses are cutoff by hand once they exceed $m_a(T=0)$ and give a surprisingly good approximation to the full IILM result; we believe this is a coincidence. The differences that persist to high temperatures, between the update and our DGA model, arise from the slightly different quark masses. Our choice has the merit that the masses were determined self-consistently within the IILM at $T=0$ [51].

Figure 3

In an adiabatically evolving universe the scale factor and the temperature are related through the condition of constant entropy. Given the knowledge of the effective degrees of freedom $g_{*,S}$, it amounts to solving an implicit equation. The QCD phase transition occurs at around $T_{\mathrm{QCD}}\approx 180\mathrm{MeV}$, when the number of hadronic excitations rises very sharply, and $g_{*,S}$ is almost discontinuous; the would-be latent heat “reheats” the universe, which is clearly seen in the graph.

Figure 4

As long as the axion Compton wavelength is well outside the horizon, the axion zero mode is frozen; this corresponds to the late-time solution of (26) with $m_a$ neglected. The axion starts to feel the pull of its mass at $m_a\approx 3H$, and evolves to its minimum at ${\theta }_a=0$, i.e. the PQ mechanism to solve the strong $CP$ problem. After a few oscillations the axion number per comoving volume stays constant as long as the axion mass and the scale factor change slowly (adiabatic approximation). This is then used to extrapolate the result to today.

Figure 5

The effective degrees of freedom $g_{*,R}$ and $g_{*,S}$ are given for the temperature range up to $T\approx 100\mathrm{GeV}$. The decoupling of the neutrinos is included and manifests itself in the differences between $g_{*,R}$ and $g_{*,S}$ after $e^{\pm }$ annihilation, when $T_{\nu }\ne T_{\gamma }$. We followed closely [57], but included some minor changes to take into account a better understanding of the QCD phase transition from recent lattice studies (see main text). We have determined fits by using a sequence of smoothed step functions, see the Appendix. As seen from the graph, the fits are good, generally with an accuracy below 1%.

Figure 6

The anthropic axion is defined through its relation between $f_a$ and ${\theta }_a$ given a fixed ${\Omega }_a$. We display here the result for the case that axions form the dominant dark matter component of the universe, i.e. ${\Omega }_a=0.23$. The short-dashed lines correspond to the uncertainties in $f_a({\theta }_a)$ from the systematic errors in $\Lambda$, which is the dominant source of uncertainties. We also include the result of the analytic computation (33). Over many orders of magnitude the agreement is very good. More pronounced differences only show up at ${\theta }_a\to 0$ and ${\theta }_a\to \pi$. The latter is due to the fact that the adiabatic condition is not fulfilled because the potential becomes very flat and acts like a source of inflation, i.e. a rapidly changing scale factor. The differences between the analytic and numerical data at small ${\theta }_a$ follow from the different functional form of the axion masses: in the former case the axion mass is constant whereas in the latter it is slightly rising with temperature (see main text).

Figure 7

To analyze the differences between the different determinations of the axion abundance, we compute ${\Omega }_a$ on a fixed set of $\{f_a,{\theta }_a\}$ derived from the full numerical determination with the IILM axion mass, given that ${\Omega }_a={\Omega }_c$. We see that the analytic results are off by more than a factor of 2 in the regimes of small and large axion angles; for the case when inflation has not operated after PQ symmetry breaking, i.e. for $f_a<2.8\times {10}^{11}\mathrm{GeV}$, we can see that the numerical and analytic results are different by a factor of 3 (and more if the axion is not the dominant dark matter component). For the main part of parameter space the discrepancy is smaller but systematically an underestimate. The full numerical data using the lattice-inspired axion mass are very similar for most $f_a$ (by construction), and differ only for large $f_a$, as might have been expected. It is noteworthy that the numerical determination runs into the proper IILM axion mass already for rather low $f_a$’s before it extrapolates to today.

Figure 8

The allowed parameter space in the $H_I–f_a$ plane is plotted in white; the inflationary energy scale is defined by $E_I\equiv (H_I{m}_{\mathrm{Pl}}/\sqrt{8\pi /3}{)}^{1/2}$. The green curve in the upper left corner follows from the isocurvature constraint (51), when the axion is the dominant dark matter candidate; the dashed line corresponds to the (semi-)analytic computation (based on (31, 32, 33), taking fully into account the temperature dependence of $g_{*}$) together with (49), i.e. the constraint for a harmonic potential. If axions provide only a fraction of the dark matter content of the universe, the bound becomes weaker. For lower $f_a$, i.e. larger ${\theta }_a$, the anharmonic effects become important and the bound on $H_I$ weakens because anharmonic effects lead to smaller perturbations. For ${\theta }_a\to \pi$, the dependence of $H_I$ on ${\theta }_a$ can no longer be neglected and leads to the black curve. The lower green curve gives the lower bound for isocurvature production (very inefficient reheating is assumed [68]); beneath this curve, the axion angle is spatially varying (with root-mean-square fluctuation ${\theta }_a=\pi /\sqrt{3}$). The cyan wedge is excluded as it would lead to too much dark matter from axion string radiation. The bound from the nondetection of gravitational waves, i.e. $r<0.22$ [60], leads to the upper bound on the inflationary scale $H_I<1.26\times {10}^{14}\mathrm{GeV}$. Finally, $f_a$ is bounded from below by astrophysical considerations, i.e. axion emission from stars; we use $f_a>4\times {10}^8\mathrm{GeV}$, see 23, Chapter 3.

Figure 9

The allowed parameter space in the anthropic window. The dark matter axion lives on the face pointing towards the $H_I–f_a$ plane. Although not as clearly visible as for the right plot, the projection of said face into the $H_I–f_a$ plane corresponds to the anthropic window displayed in Fig. 8. The shading is calibrated to the axion density; this is clear for the right plot but is also true for the left plot. We see that the parameter space does not depend sensitively on ${\Omega }_a$ in the range ${\Omega }_a\in [{\Omega }_c,0.1{\Omega }_c]$; for smaller densities $H_I$ starts to grow rapidly, and eventually we reach the regime $⟨{\theta }_a⟩\ll {\sigma }_{{\theta }_a}\ll 1$ where the isocurvature bound can no longer be fulfilled. This gives another bound in the anthropic window, although a rather uninteresting one since the axion density has become totally negligible at that point.

Figure 10

Effect of the loop creation ratio $r\equiv \alpha /\kappa$ on the dark matter axion constraint. If $\alpha (t)$ the loop creation size at time $t$ is larger than $\kappa (t)$, the loop radiation backreaction scale, then the constraint is stronger and conversely for $\alpha <\kappa$. Note that in the second case with $\alpha \ll \kappa$, the dominant contribution arises from direct long string radiation, which again exceeds misalignment production.