Ultracompact Minihalos: Probing Early Dark Matter

• Ultracompact minihalos are extremely dense dark matter substructures formed from significant primordial overdensities with steep profiles.
• Their formation and evolution, observable via pulsar timing, gamma-rays, and gravitational lensing, provide critical constraints on dark matter and early Universe conditions.
• Recent simulations indicate shallower density profiles than the classical r^(-9/4) model, prompting refined analyses of their survival and impact on reionization and structure formation.

Ultracompact minihalos (UCMHs) are a proposed class of extremely dense, early-forming dark matter subhalos that originate from significantly enhanced primordial density fluctuations. Their steep density profiles and isolation in the quasi-linear regime make them exceptional laboratories for probing small-scale features of the cosmological power spectrum, the nature of dark matter, and early-Universe physics. UCMHs are expected to yield pronounced gravitational and indirect-detection signatures, and they serve as competitive, sometimes unique, probes of primordial perturbations on scales inaccessible to other astrophysical observables.

1. Formation Conditions and Theoretical Motivation

UCMHs originate from overdense regions in the early Universe where the density contrast at horizon entry satisfies $\delta \equiv \delta\rho/\rho$ in the range

$0.0003 \lesssim \delta \lesssim 0.3$

If $\delta > 0.3$, the collapse produces a primordial black hole (PBH) instead of a minihalo (Yang et al., 2012, 0908.4082). Regions with initial overdensity above $\sim 10^{-3}$ decouple from the Hubble flow and collapse much earlier than regions producing standard large-scale structure ($\delta \sim 10^{-5}$). The abundance of UCMHs is exponentially sensitive to the tail of the initial perturbation distribution. In a Gaussian scenario, the formation probability is

$$\beta(M) = \int_{\delta_{\min}}^{\delta_{\max}} P(\delta) d\delta$$

where $\delta_{\min} \approx 10^{-3}$ (Aslanyan et al., 2015, Josan et al., 2010).

Early collapse occurs shortly after matter–radiation equality ($z_{eq} \approx 3400$), resulting in halos with virialization redshift $z_c \gtrsim 1000$ (0908.0735, Delos et al., 2018). UCMHs can also form around non-Gaussian PBHs or cosmic string loops (Nakama et al., 2019, Anthonisen et al., 2015).

The fraction of dark matter in UCMHs is directly related to the amplitude and non-Gaussianity of the small-scale primordial power spectrum, making constraints on UCMH abundances critical cosmological probes (Aslanyan et al., 2015, Yang et al., 2012).

2. Density Profiles and Structural Properties

Canonical UCMH models assume nearly pure radial infall, resulting in an analytic, steep density profile: $$\rho(r,z) = \frac{3 f_\chi M_{\rm UCMH}(z)}{16\pi R_{\rm UCMH}(z)^{3/4} r^{9/4}}, \quad r_c < r < R_{\rm UCMH}(z)$$ where $$f_\chi = \Omega_{\rm DM}/(\Omega_{\rm DM} + \Omega_b) \approx 0.83$$, $M_{\rm UCMH}(z)$ is the halo mass at redshift $z$, and $R_{\rm UCMH}(z)$ is the characteristic (virial) radius (Yang et al., 2012, 0908.4082, Josan et al., 2010). The truncation at small radii is set by dark matter annihilation (if present) or nonvanishing angular momentum.

Mass Growth

After matter–radiation equality, the mass of a UCMH evolves as

$M_{\rm UCMH}(z) = M_i \frac{1 + z_{eq}}{1 + z}$

where $M_i$ is the initial mass enclosed in the overdensity. Growth typically ceases around $z_f \sim 10$, when hierarchical clustering disrupts further radial infall (Yang et al., 2012, 0908.4082).

Profile Slope: Numerical Simulations

While the analytic $r^{-9/4}$ (Fillmore–Goldreich/Bertschinger) profile holds for isolated, perfectly spherically symmetric collapse, recent high-resolution N-body simulations show that, under realistic cosmological initial conditions, UCMHs develop shallower profiles:

• Isolated, self-similar seeds: $\rho \propto r^{-9/4}$
• “Spike” power spectrum, rare peaks: $\rho \propto r^{-3/2}$
• “Step” or hierarchical clustering: $\rho \propto r^{-1}$ (NFW) (Delos et al., 2017, Delos et al., 2018, Gosenca et al., 2017)

True “ultracompactness” is only achievable under strictly self-similar and isolated conditions, which cannot arise from a Gaussian primordial field. The per-halo annihilation luminosity for a $r^{-3/2}$ profile is suppressed by $\sim 10^2–10^4$ compared to the classical UCMH expectation at fixed mass (Delos et al., 2017, Delos et al., 2018).

3. Observational Signatures and Detection Channels

(A) Pulsar Timing Noise

The time-varying gravitational potential of a UCMH passing near the line of sight to a pulsar produces a fluctuating Shapiro delay in pulse arrival times. For a population of UCMHs, the cumulative effect is an additional Gaussian “noise” in measured period derivatives ($\dot{P}$). Statistical analysis of the ATNF pulsar catalogue with 1810 pulsars yields a 95% C.L. upper limit: $\log_{10} \sigma_{\dot{P}} \leq -17.05$ which constrains the present-day fraction $f_{\rm UCMH}$ to:

• $f \lesssim 10^{-5}$ at $M_{\rm UCMH} \sim 10^{-2} M_\odot$,
• $f \lesssim 10^{-8}$ at $M_{\rm UCMH} \sim 10^{3} M_\odot$ over the mass range $10^{-12}–10^3 M_\odot$ (Clark et al., 2015). This bound is independent of the dark matter particle physics.

(B) Indirect Detection: Gamma-rays, Neutrinos, and 21-cm

• Gamma-rays: Self-annihilating WIMPs in UCMHs can produce gamma-ray point sources or contribute to diffuse extragalactic backgrounds. Non-detections by Fermi-LAT and other instruments place upper limits on $f_{\rm UCMH} \lesssim 10^{-5}–10^{-6}$ for $m_\chi \lesssim 100\,\rm GeV$ WIMPs (Zhang et al., 2021, Yang et al., 2011, Josan et al., 2010).
• Neutrinos: WIMP or gravitino decay/annihilation in UCMHs can yield neutrino flux detectable by IceCube/DeepCore. Non-observation constrains $f_{\rm UCMH} \lesssim 10^{-3}–10^{-2}$ for heavy (TeV-scale) dark matter (Yang et al., 2013, Zheng et al., 2014).
• 21-cm: UCMH populations modify the heating and ionization history of the intergalactic medium (IGM) during the cosmic dark ages through enhanced annihilation energy injection. The resulting effects on the global 21-cm signal and fluctuation power spectrum can be probed by current and next-generation experiments (SKA). SKA is forecasted to reach ${\cal A}_\zeta \lesssim 10^{-6}$ for the small-scale curvature power (Furugori et al., 2020).

(C) Gravitational Lensing

• Microlensing: UCMHs act as extended, non-stellar MACHOs. High-precision photometric light curves can distinguish UCMH lensing events (boosted wings, symmetric shape) from point-mass lensing, offering constraints on ${\cal P}(k)$ at scales inaccessible to the CMB or Ly-alpha forest (0908.0735).
• Astrometric lensing: Gaia astrometry can detect UCMH-induced positional shifts, with potential limits on the curvature spectrum $${\cal P}_{\mathcal{R}}(k\sim 10^3\,\mathrm{Mpc}^{-1}) < 10^{-5}$$, competitive with gamma-ray and PBH bounds (Li et al., 2012).

4. UCMHs and the Primordial Power Spectrum

The abundance of UCMHs translates directly to ultra-small-scale ($k \sim 10^2 – 10^9\,\rm{Mpc}^{-1}$) constraints on the primordial curvature power spectrum ${\cal P}_{\mathcal{R}}(k)$. Gamma-ray non-observation, for instance, implies

$${\cal P}_{\mathcal{R}}(k) \lesssim 10^{-6.5} – 10^{-6} \quad (95\%\,\mathrm{C.L.},\,k\sim 10^1–10^6\,\mathrm{Mpc}^{-1})$$

(Josan et al., 2010, Yang et al., 2011, Zhang et al., 2021), substantially stronger than corresponding PBH non-formation constraints. Constraints derived using the classical $r^{-9/4}$ profile must be relaxed by orders of magnitude in light of recent simulation results revealing shallower density cusps. However, the inclusion of later-forming minihalos (with $\alpha \sim 3/2$) can partially compensate, sometimes producing even stronger net constraints (Delos et al., 2018).

5. Cosmological and Astrophysical Impact

UCMHs, due to their dense early formation, can substantially impact cosmic evolution:

• Ionization and Thermal History: Even tiny fractions ($f_{\rm UCMH} \gtrsim 10^{-7}$) can dominate over the homogeneous dark matter annihilation background, strongly heating and partially ionizing the IGM at $z \gtrsim 20$ (Zhang, 2010, Yang, 2016). This increases the Jeans mass, suppressing low-mass structure formation and affecting the molecular hydrogen fraction relevant for Population III star formation (Abe, 2022).
• Reionization: UCMHs may host early Pop III stars, providing an additional, possibly dominant, channel for high-redshift ionizing photons. Planck 2018 data limit the UCMH-induced amplitude of the primordial power spectrum to ${\cal A}_\zeta \lesssim 10^{-8}$ at $k \lesssim 50\,\mathrm{Mpc}^{-1}$ (Abe, 2022).
• Constraints on Exotic Physics: UCMH formation from early Universe mechanisms, such as phase transitions or cosmic string loops, enables direct probing of high-energy physics through the resultant present-day mass spectrum and abundance. For example, Fermi-LAT gamma-ray bounds restrict the string tension to $G\mu \lesssim 5 \times 10^{-8}$ under optimistic assumptions (Anthonisen et al., 2015).

6. Methodological Challenges and Theoretical Uncertainties

Recent numerical and semi-analytical advances have highlighted crucial caveats:

• Profile Realism: Realistic N-body cosmological simulations show that only under highly idealized, self-similar, and isolated conditions does the $r^{-9/4}$ profile emerge. In practice, most UCMHs assembled from Gaussian initial conditions develop shallower inner slopes ($r^{-3/2}$ or $r^{-1}$) due to lack of isolation, finite spike width, and background fluctuations (Delos et al., 2017, Gosenca et al., 2017, Delos et al., 2018).
• Survival and Disruption: Assumptions regarding the intact survival of UCMHs to $z=0$, neglect of tidal disruption in the Milky Way environment, and the efficiency of core annihilation processes affect the interpretation of observational limits (Josan et al., 2010, Yang et al., 2011). Conservative analyses truncate limits at masses where the expected number of UCMHs in the halo falls below $\sim 20$ (Clark et al., 2015).
• Dependence on DM Properties: Indirect detection limits are strongly dependent on the dark matter annihilation cross section and particle mass. Astrometric and gravitational lensing constraints are independent of microphysics, providing critical complementarity.

7. Synthesis and Future Directions

UCMHs serve as one of the most sensitive probes of small-scale primordial fluctuations, capable of constraining curvature spectrum amplitudes many orders of magnitude beyond those accessible to traditional CMB, Ly-alpha forest, or weak lensing measurements. Their significance as cosmological observables is highly robust under conservative assumptions if appropriate attention is given to realistic halo structure and survival, and if their signatures are interpreted in the context of full minihalo populations.

Future directions include:

• Systematic exploitation of pulsar timing array sensitivities (Clark et al., 2015, Nakama et al., 2019)
• 21-cm fluctuation observations targeting the unique heating/ionization pattern of UCMHs (Yang, 2016, Furugori et al., 2020)
• Precision gravitational lensing campaigns and Gaia-scale astrometric studies (Li et al., 2012, 0908.0735)
• Refinement of power-spectrum constraints using up-to-date simulation-informed UCMH and minihalo models (Delos et al., 2017, Delos et al., 2018)
• Joint exploitation of multi-messenger UCMH constraints to decisively test inflationary, non-Gaussian, and exotic early-Universe scenarios.

The evolving theoretical framework and cross-disciplinary observational program continue to advance UCMHs as a leading tool for mapping the microphysical and cosmological landscape of structure formation on the smallest scales.