Astrophysical effects of scalar dark matter miniclusters

We model the formation, evolution and astrophysical effects of dark compact Scalar Miniclusters (“ScaMs”). These objects arise when a scalar field, with an axion-like or Higgs-like potential, undergoes a second-order phase transition below the QCD scale. Such a scalar field may couple too weakly to the standard model to be detectable directly through particle interactions, but may still be detectable by gravitational effects, such as lensing and baryon accretion by large, gravitationally bound miniclusters. The masses of these objects are shown to be constrained by the $\mathrm{Ly}\alpha$ power spectrum to be less than $\sim {10}^4{M}_{\odot }$, but they may be as light as classical axion miniclusters, of the order of ${10}^{-12}{M}_{\odot }$. We simulate the formation and nonlinear gravitational collapse of these objects around matter-radiation equality using an $N$-body code, estimate their gravitational lensing properties, and assess the feasibility of studying them using current and future lensing experiments. Future MACHO-type variability surveys of many background sources can reveal either high-amplification, strong-lensing events, or measure density profiles directly via weak-lensing variability, depending on ScaM parameters and survey depth. However, ScaMs, due to their low internal densities, are unlikely to be responsible for apparent MACHO events already detected in the Galactic halo. As a result, in the entire window between ${10}^{-7}{M}_{\odot }$ and ${10}^2{M}_{\odot }$ covered by the galactic scale lensing experiments, ScaMs may in fact compose all the dark matter. A simple estimate is made of parameters that would give rise to early structure formation; in principle, early stellar collapse could be triggered by ScaMs as early as recombination, and significantly affect cosmic reionization.

Figure 1

CDM power spectrum derived from BBKS [45] with white-noise power spectrum smoothed on scales $r_s\simeq 7\times {10}^{-2}\mathrm{Mpc}$, given by Eq. (20). The amplitude of the white-noise spectrum corresponds to the power spectrum for a ScaM of mass $M\simeq 4\times {10}^3{M}_{\odot }$.

Figure 2

Two-dimensional slice in the x-y plane of initial white-noise field energy density distribution. x-y coordinates are in $\eta$, where $\eta =1$ corresponds to a length $d_H(T_{\mathrm{trans}})$, one horizon size at the time of the phase transition. The z-axis is the initial white-noise over-density $\rho (\mathbf{x})/\overline{\rho }$, where $\overline{\rho }$ is the mean density in the box.

Figure 3

Density distribution after the phase transition, at $\eta =10$ for the axion-like potential. Axes are same as in Fig. 2. Note the highly nonlinear nature of the initial density perturbations. This distribution remains fixed until near matter-radiation equality when it evolves gravitationally into collapsed ScaMs; this density spectrum is the input for the $N$-body simulation.

Figure 4

Density distribution generated by the smoothed white-noise power spectrum. The size of the density fluctuations were chosen consistent with the analytic result for the Higgs-like potential of Sec. 2. Like Fig. 3, this distribution is input for the $N$-body simulation.

Figure 5

Snapshot of structure formation at $z=3000$ for axion-like ScaMs; the plot is colored according to the logarithm of the density. The scale here is $0.4h^{-1}\mathrm{kpc}$, in physical coordinates, although the result is expected to be invariant for any box size, given that we rescale the smoothing scale accordingly. Note that structure formation has already commenced before matter-radiation equality.

Figure 6

Same as Fig. 5, but for the Higgs-like potential at a redshift $z=1000$. This simulation was evolved to a lower redshift, as the initial over-density of the Higgs-like system is lower, and hence gravitational collapse occurs later. Although difficult to see from this rendering, the densities of the ScaMs are also lower.

Figure 7

Axion-like ScaM density profiles for five gravitationally collapsed ScaMs. Vertical axis is $\log (\rho (r)/{\rho }_{\mathrm{sph}})$, where ${\rho }_{\mathrm{sph}}(\delta =1)=280{\rho }_{\mathrm{eq}}$; that is, we normalized the density profile against the uniform density prediction of the spherical model, Eq. (14), with $\delta =1$. Horizontal axis is $\log (r/R_{\mathrm{sph}})$, where $R_{\mathrm{sph}}$ is the radius computed from ${\rho }_{\mathrm{sph}}$ and the total mass contained within the horizon at the phase transition, $d_H(T_{\mathrm{trans}})$.

Figure 8

Same as Fig. 7, but for the Higgs-like potential.

Figure 9

Axion-like ScaM Einstein radius for the enclosed mass, derived from the computed $N$-body density profile, versus ScaM radius, again normalized against the ScaM radius $R_{\mathrm{sph}}$ predicted by the spherical model. The diagonal line divides where $R_E>r$, when lensing is possible. The upper set of curves in each plot is the Einstein radius for cosmological scale lensing, the lower set of curves the Einstein radius for lensing at a distance $D=5\mathrm{Mpc}$, which may be relevant for MACHO-type experiments in nearby galaxy halos. This is shown for three different ScaM masses, marked at the lower right in each plot.

Figure 10

Same as Fig. 9, but for the Higgs-like potential.

Figure 11

Log mean surface density $\Sigma$ relative to the critical density ${\Sigma }_c$ as a function of log radius, averaged in annuli, normalized again to the prediction of the spherical model, $R_{\mathrm{sph}}$. $\Sigma /{\Sigma }_c\sim 1$ corresponds roughly to the onset of strong lensing, and $2\Sigma /{\Sigma }_c$ corresponds to the fractional weak-lens amplification. Higgs-like potential.

Figure 12

Same as Fig. 11, but for the axion-like potential.