Density profiles of ultracompact minihalos: Implications for constraining the primordial power spectrum

Enhanced density fluctuations on small scales would lead to the formation of numerous dark matter minihalos, so limits on the minihalo abundance can place upper bounds on the small-scale primordial power spectrum. In particular, the ultracompact minihalo (UCMH), a dark matter structure hypothesized to possess a $\rho \propto r^{-9/4}$ density profile due to its formation at $z\ge 1000$, has been used to establish an upper bound on the primordial power spectrum at scales smaller than 2 Mpc. The extreme slope of this density profile amplifies the observational signals of UCMHs. However, we recently showed via N-body simulations that the $\rho \propto r^{-9/4}$ density profile does not develop in realistic formation scenarios, throwing UCMH-derived power spectrum constraints into question. Instead, minihalos develop shallower inner profiles with power-law indices between $-3/2$ and $-1$. In this paper, we expand on that result and discuss its implications. Using a model that is calibrated to simulation results and predicts halo structures in spiked power spectra based on their formation times, we calculate new upper bounds on the primordial power spectrum based on limits on the dark matter annihilation rate within the Galaxy. We find that despite assuming shallower profiles, this minihalo model actually yields stronger constraints than the previous UCMH picture owing to its inclusion of all minihalos instead of only the earliest-forming ones.

Figure 1

The dimensionless primordial power spectrum of curvature fluctuations used in our UCMH simulations. The solid line shows the spike modification, while the dotted line shows the step. The vertical dashed line indicates the smallest $k$ (largest scale) accessible in the simulations.

Figure 2

A slice of the 7.4 kpc density field used as initial conditions for the primary simulation run. Lighter regions are denser. The circle indicates the spherical region for a high-particle-density simulation.

Figure 3

The density field for the primary run at different redshifts. Top: A $(0.24\mathrm{kpc}{)}^2\times 0.06\mathrm{kpc}$ slice showing the collapse of the UCMH near $z=1000$. The color scale is logarithmic in units of the background matter density. Bottom: The full $(7.4\mathrm{kpc}{)}^3$ projected density field at $z=715$. There is still only one halo, a testament to its rarity.

Figure 4

The projected density field of the primary simulation box at $z=100$. Top: The full 7.4 kpc field. Bottom: Expanded pictures of the UCMH. The left (right) panel shows the projected density field for the surrounding 1.5 kpc (0.3 kpc) cube. Note that the expanded pictures do not fully match the white boxes because they are projected over smaller depths.

Figure 5

The spherically averaged density profile of the UCMH in the primary density field at $z=400$, $z=200$, $z=100$, and $z=50$. The vertical axis is scaled by $r^{3/2}$ to reduce the vertical range and better exhibit asymptotic behaviors; this practice will be adopted without remark in later figures. The density profile approaches $\rho \propto r^{-3/2}$ at small $r$ and $\rho \propto r^{-3}$ at large $r$ and is fit well by Eq. (4) (solid curve). The solid vertical line shows the physical scale of the power spectrum spike at $z=1000$, while the vertical dashed lines show the halo virial radius at different redshifts. Inset: The same plot without y-axis scaling. A $\rho \propto r^{-9/4}$ curve is shown for comparison. We plot physical, not comoving, quantities. We show results from the vacuum-bounded sphere inside $r_{\mathrm{vir}}$ for $z\le 100$, and results from the full box otherwise. The smallest radius at each redshift is set by $r>2.8\epsilon$, where $\epsilon$ is the force-softening length parameter (see Appendix pp2), and contains $N>3000$ particles.

Figure 6

Radial density profiles at $z=400$, $z=200$, $z=100$, and $z=50$ of all nine UCMHs. These density profiles are cut off above the virial radius at each redshift. Evidently, all of our UCMHs possess similar density profiles to the one depicted in Fig. 5 (which is also plotted here).

Figure 7

The virial mass $M_{\mathrm{vir}}$ plotted against the scale factor $a$ for all nine UCMHs. For $a\gtrsim 2.5\times {10}^{-3}$ ($z\lesssim 400$), the growth appears logarithmic, presenting as a straight line on this plot. A $M_{\mathrm{vir}}\propto a$ reference curve is shown as a dashed line: these halos are growing more slowly than $M_{\mathrm{vir}}\propto a$.

Figure 8

A merger event and its result. Top: A $(1.5\mathrm{kpc}{)}^2\times 0.7\mathrm{kpc}$ (projected) region showing the first merger event experienced by the larger halo, which formed at $z\simeq 1000$. Bottom: The change in the density profile of this UCMH as a result of the merger. Mass is dispersed from the inner region.

Figure 9

The density profiles of a sample of 10 halos in the same simulation box at $z=100$. Top: The density profiles in physical coordinates. Bottom: The density profiles scaled to each halo’s formation time using Eqs. (5) and (6).

Figure 10

A halo at $z=100$ that collapsed at $z\simeq 1000$ from the step power spectrum shown in Fig. 1. Top: The projected full $(7.4\mathrm{kpc}{)}^3$ density field and an expanded picture of the 1.5 kpc cube surrounding the UCMH. Bottom: The density profile of this halo. It is fit well by the NFW profile.

Figure 11

The ratio $\mu (d)=3M(d)/(4\pi d^3{\overline{\rho }}_0)$ of the dark matter mass $M(d)$ within distance $d$ of Earth to the cosmological mean dark matter mass in an equal volume. An NFW profile is assumed for the Milky Way with parameters from Ref. [96]. Minihalos are assumed to follow this spatial distribution.

Figure 12

The upper bound on the integrated area ${\mathcal{A}}_0$ of a spike in the primordial curvature power spectrum centered at scale wave number $k_s$. Black curves use point sources, while red curves employ the diffuse flux. The shaded regions are ruled out in the new minihalo picture with shallower density profiles. The dashed lines show the corresponding constraints in the old UCMH model calculated using the abundance constraints in BSA. As another comparison, the dotted lines show the constraints using shallower density profiles while still restricting to UCMHs forming at $z\ge 1000$. While the new density profiles slightly weaken the upper bound, the inclusion of all minihalos ends up leading to stronger constraints.

Figure 13

The constraint on the integrated area $\mathcal{A}$ of the spiked matter power spectrum Eq. (22). Note that this is not the primordial power spectrum; see Fig. 12 for that constraint.

Figure 14

Possible upper bounds on the integrated area ${\mathcal{A}}_0$ of a spike in the primordial curvature power spectrum centered at scale wave number $k_s$ when mergers are taken into account. The black curve shows the bound with mergers neglected, which is the same constraint shown in Fig. 12. The blue curve shows the bound if minihalos develop NFW profiles with the same scale parameters $r_s$ and ${\rho }_s$, while the red curve additionally halves the number of halos.

Figure 15

The power spectrum growth from $z=8\times {10}^6$ to $z=996$: a comparison between Gadget-2 with an added radiation component and linear theory. In matter domination, the power spectrum would have instead grown by a factor of about $6\times {10}^7$.

Figure 16

Fractional overdensity fields evolved to $z=996$ from the same initial box at $z=8\times {10}^6$. The figure depicts a $(7.4\mathrm{kpc}{)}^2\times 1.5\mathrm{kpc}$ slice. The left and right panels show the results of linear theory and modified Gadget-2 respectively.

Figure 17

Radial density profiles averaged between $z=100$ and $z=99$; the shading indicates the root-mean-squared variance over this interval. Clockwise from top left: force-accuracy, integration-accuracy, softening-length, and particle-resolution convergence comparisons. The vertical solid (dotted) lines indicate $2.8\epsilon$ for the reference run (alternate runs), the range beyond which forces are exactly Newtonian. The dot-dashed line indicates $r_{\lim }$, the radius beyond which fluctuations are likely not averaged (see text for details). ($\overline{\rho }$ is the background matter density.)

Figure 18

The softening-length convergence test with $\rho$ and $r$ rescaled for each run to its Moore fitting parameters. The thick dashed line shows the fit, which is the same for all runs by construction. (Shading and vertical lines have the same meaning as in Fig. 17).

Figure 19

Root-mean-squared variance in the density profile across 16 snapshots between $z=100$ and $z=99$. The reference and high-particle-count runs are compared. (Vertical lines have the same meaning as in Fig. 17.)

Figure 20

The filament to the left of the UCMH (see Fig. 4) for different simulation parameters. This figure demonstrates the presence of artificial fragmentation: the filament fragments differently for different parameters.

Figure 21

A comparison between the UCMH simulated in a $(7.4\mathrm{kpc}{)}^3$ periodic box with a halo from the same initial overdense region simulated in a sphere of radius 0.92 kpc with vacuum boundary conditions. (Vertical lines have the same meaning as in Fig. 17).

Figure 22

Plots of the function $h$ defined in Eq. (d3) along with the integrands in Eq. (d5). Each function is scaled to its maximum value to emphasize the support of these functions. The black curves are relevant to point-source constraints, while the red curves determine diffuse constraints. This figure shows that constraints are set primarily by peaks between $2\sigma$ and $4\sigma$.