Structure of axion miniclusters

The peak-patch algorithm is used to identify the densest minicluster seeds in the initial axion density field simulated from string decay. The fate of these dense seeds is found by tracking the subsequent gravitational collapse in cosmological $N$-body simulations. We find that miniclusters at late times are well described by Navarro-Frenk-White profiles, although for around 80% of simulated miniclusters a single power-law density profile of $r^{-2.9}$ is an equally good fit due to the unresolved scale radius. Under the assumption that all miniclusters with an unresolved scale radius are described by a power-law plus axion star density profile, we identify a significant number of miniclusters that might be dense enough to give rise to gravitational microlensing if the axion mass is $0.2\mathrm{meV}\lesssim m_a\lesssim 3\mathrm{meV}$. Higher resolution simulations resolving the inner structure and axion star formation are necessary to explore this possibility further.

Figure 1

Possible minicluster density profiles. NFW profiles (dashed) are never dense enough to lens due to the large scale radius for the measured concentration relation (see Fig. 4). For power-law (PL) profiles there is a critical lensing radius, $r_{\mathrm{len}}^{\mathrm{PL}}$, which gives the minimum impact parameter of a minicluster along the line of sight in order to create above-threshold lensing amplification. This critical radius should be larger than the estimated axion star radius (indicated by the gray region). For $m_a=50\mu \mathrm{eV}$ (upper panel) the estimated axion star radius is too large, while for $m_a=1\mathrm{meV}$, the estimated axion star radius can be significantly smaller. $N$-body simulations are rescaled to higher axion masses, and axion star radii are estimated from the measured velocity dispersion (see text for details).

Figure 2

Peak-patch comparison with $N$-body. Projection plot of the $N$-body particles (black) contained within the 400 largest halos located using the subfind halo finder at a redshift $z=3976.5$. Red circles indicate the most likely equivalent from the peak-patch data. The left panel shows Lagrangian coordinates, i.e. particles traced back to the initial density field, without displacement. The right panel shows the real-space (Eulerian) coordinates of the particles, which in peak-patch are found using first order Lagrangian displacements. In the right panel blue circles indicate the size and extent of the halos as found using subfind.

Figure 3

Minicluster mass function at $z=99$. Dashed lines indicate the portion of the mass function with resolved scale radius, i.e. known NFW halos. The vertical line indicates the minimum mass that a pure power-law halo must have in order to produce significant lensing amplification after accounting for wave optics effects and assuming a power-law profile [see Eq. (15)].

Figure 4

Minicluster concentrations. Press-Schechter predicted minicluster concentrations at different redshifts, assuming Gaussianity and NFW profiles. Points show $N$-Body results used for normalization. All masses were rescaled to the axion mass $m_a=1\mathrm{meV}$. Note that only those miniclusters in the highest mass bin have well resolved scale radius according to our criteria (see Ref. [36] for details).

Figure 5

Minicluster distribution. We use peak-patch to identify all miniclusters at $z=3976.5$ and measure their mass and overdensity at collapse redshift, ${\delta }_{\mathrm{col}}$ [see Eq. (b5)]. Overlaid points show the sample of miniclusters tracked in $N$-body simulations, which survived until $z=99$ (index matching fraction $f_{\mathrm{m}}>0.5$). All masses were rescaled to the axion mass $m_a=1\mathrm{meV}$.

Figure 6

Trends in minicluster survival. Two-dimensional histogram for index matching fraction $f_{\mathrm{m}}$ as a function of mass and ${\delta }_i(z_{\mathrm{col}})$ for all sample halos at $z=99$. The top panel shows $f_{\mathrm{m}}$ as a function of sample mass $M$ and $f_{\mathrm{m}}$ as a function of ${\delta }_i(z_{\mathrm{col}})$ is shown on the right. This figure uses the simulated value $m_a=50\mu \mathrm{eV}$.

Figure 7

Demonstration of survival metric. Random selection of 300 halos from the $(M,{\delta }_i)$ plane followed to $z=99$ in $N$-body simulations. We pot the radial density profile evaluated in spheres from the point of maximum density of the tagged particles. Those miniclusters with a high index matching fraction $f_{\mathrm{m}}$ have much more well-defined density profiles, indicating that $f_{\mathrm{m}}$ is a good metric for the minicluster surviving the merger process. The gray shaded region indicates the softening cutoff. This figure uses the simulated value $m_a=50\mu \mathrm{eV}$.

Figure 8

Minicluster density profiles with resolved scale radius. In total, 76 samples with a matching fraction $f_m>0.5$ are shown at $z=3976.5$ (top) and 48 samples with a matching fraction $f_m>0.75$ at $z=99$ (bottom). The line colors are given by ${\delta }_{\mathrm{col},\mathrm{i}}(z=3976.5)$. Left: NFW prediction calibrated from $c(M)$, where the shown density profiles are additionally multiplied with $r/r_s$ to highlight the turnover of the NFW profile at $r_s$. Right: power-law density profile with a slope parameter of $\alpha =3$ and its deviation from the sample. This figure uses the simulated value $m_a=50\mu \mathrm{eV}$.

Figure 9

Minicluster density with unresolved scale radius. In total, 25 samples with a matching fraction $f_m>0.5$ are shown at $z=3976.5$ (left) and 100 samples with a matching fraction $f_m>0.75$ at $z=99$. The density profiles are compared to single power law fits with slope parameters of $\alpha =9/4$ and $\alpha =2.9$, respectively. This figure uses the simulated value $m_a=50\mu \mathrm{eV}$.

Figure 10

Axion star mass and radius. Estimated by calculating $v_{\mathrm{vir}}$ for $N$-body halo samples with $f_{\mathrm{m}}>0.5$ and substituting into Eqs. (6) and (7), respectively. The $N$-body data were rescaled to the axion mass $m_a=1\mathrm{meV}$.

Figure 11

Conditions for microlensing. The potential lensing region, shown by the unshaded region is produced by imposing a minimum halo mass from wave effects (green), a minimum mass from axion stars (red), and a maximum mass from NFW profiles observed in $N$-body simulations (purple).

Figure 12

Fraction of samples meeting criteria for lensing. The fraction was calculated for lenses exactly halfway between the observer and the light source. Lensing and nonlensing halos were shown in Fig. 5 as yellow stars and orange circles, respectively.

Figure 13

Point mass magnification. Magnification due to lensing by a point mass as a function of impact parameter under the geometrical optics approximation.

Figure 14

Relative lensing tube size $\mathcal{R}$ for NFW profiles. White regions indicate where $\mathcal{R}=0$ identically, due to the shallow inner slope $r^{-1}$ preventing caustics. The effective mass within the tube, $M_{\mathrm{eff}}$, should exceed $M_{\min }$ to avoid wave optics effects suppressing magnification. NFW miniclusters with $c(M)$ calibrated to $N$-body simulations at $z=99$ and extrapolated to $z=0$ have $\mathcal{R}=0$ and $M_{\mathrm{eff}}\ll M_{\min }$ and thus cannot lens. This is the case for the “maximum” concentration, $c_{\max }$, discussed in the text.

Figure 15

Relative lensing tube size $\mathcal{R}$ for power-law profiles. The color map shows the lensing tube radius in units of the Einstein radius for a power law $\rho \propto r^{-2.9}$. The dashed line sets the enclosed mass within the lensing tube equal to be larger than the minimum mass accounting for wave optics effects. In this figure, the average overdensity is fixed from the initial overdensity, ${\delta }_i$, assuming spherical collapse.