Axion miniclusters made easy

We use a modified version of the peak patch excursion set formalism to compute the mass and size distribution of QCD axion miniclusters from a fully non-Gaussian initial density field obtained from numerical simulations of axion string decay. We find strong agreement with $N$-body simulations at significantly lower computational cost. We employ a spherical collapse model, and provide fitting functions for the modified barrier in the radiation era. The halo mass function at $z=629$ has a power-law distribution $M^{-0.6}$ for masses within the range ${10}^{-15}\lesssim M\lesssim {10}^{-10}{M}_{\odot }$, with all masses scaling as $(m_a/50\mu \mathrm{eV}{)}^{-0.5}$. We construct merger trees to estimate the collapse redshift and concentration mass relation, $C(M)$, which is well described using analytical results from the initial power spectrum and linear growth. Using the calibrated analytic results to extrapolate to $z=0$, our method predicts a mean concentration $C\sim \mathcal{O}(\mathrm{few})\times {10}^4$. The low computational cost of our method makes future investigation of the statistics of rare, dense miniclusters easy to achieve.

Figure 1

Projected square density of axion DM initial conditions. Field at $z={10}^6$, calculated using field theory simulations from Ref. [25]. The scale of the projected densities is set by the average square density along the line of sight.

Figure 2

Power spectrum of initial conditions. Data from Ref. [25]. The extrapolated adiabatic power spectrum (orange dot-dashed) is included for comparison. The adiabatic power will also be cut-off below $k_1$, since this is the axion Jeans scale. The inset graph shows the full extent of these two power spectra as well as the points at which their extrapolated curves meet. The adiabatic power spectrum is computed with CAMB and extrapolated with the given power law to large $k$ [61].

Figure 3

Modified critical overdensity. Calculated by solving the spherical collapse as given by Eq. (12) to determine the point of non-linear collapse and relating this to density from linear growth using Eq. (27). Performing a least-squares fit, we find that the collapse threshold saturates to the Einstein-de Sitter value 1.686 at late times, and is larger during the radiation era.

Figure 4

Concentration parameter estimates. Curves calculated by solving Eq. (38) using the initial power spectrum to estimate the collapse redshift and combining with the concentration estimation formalism of NFW (solid lines) via Eq. (40) and Eq. (42) using $f=0.01$ and the concentration estimation formalism of Bullock et al. (dashed lines) via Eq. (45) with $f={10}^{-5}$. The free parameters are determined using a least-squares fit to $N$-body results from Ref. [54] and are found to be ${\kappa }_{\mathrm{NFW}}=1.28\times {10}^5$ and ${\kappa }_{\mathrm{B}}=10.7$ respectively.

Figure 5

Evolution of DM halos calculated by peak patch. Calculated for a small section overlayed onto the initial density field. Black circles indicate the radii of regions in which the contained overdensity is equal to the critical overdensity as shown in Fig. 3. The section shown is a few percent of the total box size.

Figure 6

Minicluster mass function from peak patch. Solid lines show the HMF calculated using peak-patch with a redshift dependent collapse threshold calculated in 3b. Dotted lines show the HMFs at the same redshifts calculated using $N$-body techniques in Ref. [54].

Figure 7

Comparison of HMF with PS prediction. Solid line indicates the HMF calculated using peak-patch at $z=629$. Dotted and dot-dashed lines show the Press-Schechter predictions for the HMF at the same redshift using fitting functions PS and ST.

Figure 8

Number of minicluster halos. Top: total fraction of gravitationaly bound mass. Bottom: total number of miniclusters above mass scales in the range $\{{10}^{-15},{10}^{-10}\}{M}_{\odot }$. Solid lines indicate the values calculated using peak-patch while dotted lines show the values predicted using Press-Schechter. peak patch finds a large number of halos to form at high redshift from the strongly non-Gaussian density peaks, consistent with the results of $N$-body.

Figure 9

Example merger tree. Top: merger tree for a halo with a final mass of $2.35\times {10}^{-12}{M}_{\odot }$ showing all progenitors with a mass greater than ${10}^{-14}{M}_{\odot }$. The radius of the circles is proportional to the effective radius of the progenitor. Bottom: total mass of progenitors that have a mass greater that 0.01 times the final halo mass. The collapse redshift is then defined as the point at which this total equals half of the final mass, as introduced by Navarro et al. in Ref. [68].

Figure 10

Minicluster concentrations. Colour map indicates the concentration parameters calculated using peak patch merger trees. The solid green and dashed red lines show the concentrations from the power spectrum, assuming Gaussianity. The grey colour map shows halos calculated to have concentrations less than 50. Since these objects have only just formed it is thought that they have not had enough time to become as dense as the NFW prediction.