Improved constraints on the primordial power spectrum at small scales from ultracompact minihalos

TL;DR

The paper develops a comprehensive framework to translate non-detections of ultracompact primordial minihalos (UCMHs) into tight constraints on the primordial power spectrum across scales far smaller than those probed by the CMB. It refines UCMH formation criteria, computes present-day abundances, and uses gamma-ray limits from Fermi-LAT, diffuse emission, and reionization data to constrain the amplitude and shape of primordial perturbations, including scale-free, stepped, and non-parametric spectra. The results show very strong small-scale bounds, e.g., n<1.17 for a scale-free spectrum and step-height limits p~10^-1–10^-12 depending on k_s and DM assumptions, effectively extending the constraint reach to over 20 orders of magnitude in wavenumber compared to large-scale probes. The work provides a practical toolkit to connect UCMH physics to $\mathcal{P}_{\mathcal{R}}(k)$ for a broad class of inflationary or early-Universe scenarios and highlights potential future DM-independent constraints via gravitational lensing.

Abstract

For a Gaussian spectrum of primordial density fluctuations, ultracompact minihalos (UCMHs) of dark matter are expected to be produced in much greater abundance than, e.g., primordial black holes. Forming shortly after matter-radiation equality, these objects would develop very dense and spiky dark matter profiles. In the standard scenario where dark matter consists of thermally-produced, weakly-interacting massive particles, UCMHs could thus appear as highly luminous gamma-ray sources, or leave an imprint in the cosmic microwave background by changing the reionisation history of the Universe. We derive corresponding limits on the cosmic abundance of UCMHs at different epochs, and translate them into constraints on the primordial power spectrum. We find the resulting constraints to be quite severe, especially at length scales much smaller than what can be directly probed by the cosmic microwave background or large-scale structure observations. We use our results to provide an updated compilation of the best available constraints on the power of density fluctuations on all scales, ranging from the present-day horizon to scales more than 20 orders of magnitude smaller.

Figures (7)

• Figure 1: The maximum allowed fraction of DM in the Milky Way contained in UCMHs, as a function of $k$ and the UCMH mass $M^0_\mathrm{UCMH}$. Here we show limits derived in this paper from Fermi-LAT searches for individual and diffuse DM sources. The UCMH mass is related to the mass contained inside the horizon when mode $k$ enters by Eq. (\ref{['mhor']}). All limits correspond to a 95% CL. Limits from searches for individual minihalos are based on non-observation of point or extended DM sources during one year of operation in all-sky survey mode.
• Figure 2: Upper limits on the mass variance $\sigma^2_{\chi,\mathrm{H}}$ at horizon entry ($aH=k$), implied by the present-day UCMH abundance limits presented in Fig. \ref{['fig:fUCMH_constraint']}, as well as the limits on $f$ at the time of matter-radiation equality derived from the CMB, see Eq. (\ref{['reion']}). This later limit refers to the impact of UCMHs upon reionisation Zhang:2010cj; larger values of $\sigma^2_{\chi,\mathrm{H}}$ correspond to a UCMH fraction that speeds up reionisation to the point where the integrated optical depth of the CMB ($\tau_\mathrm{e}$) is not consistent with the value measured by WMAP5 wmap5.
• Figure 3: Upper limit on the spectral index of the primordial power spectrum from gamma-ray searches for UCMHs, and the impact of UCMHs on reionisation. These limits assume $\delta^2_\mathrm{H}\propto k^{n-1}$, and take into account only the constraints on $\sigma_{\chi,\mathrm{H}}$ given in Fig. \ref{['fig:sigH_constraint']}, for wave numbers smaller than $k$. For comparison, we also show the resulting gamma-ray constraint if we were to assume $\delta_\chi^\mathrm{min}=10^{-3}$ (improved upon in Appendix \ref{['app:dmin']}) and use the over-simplified calculation of $\sigma_{\chi,\mathrm{H}}$GreenLiddle (corrected in Appendix \ref{['app:sh']}).
• Figure 4: Left: Constraint on the allowed height $p$ of a step in the primordial power spectrum from gamma-ray searches for UCMHs and impacts on reionisation, as a function of the location $k_s$ of such a step. Here $p$ refers to the dimensionless ratio of the power at the wavenumbers immediately above and below $k_s$. Our central curves assume the spectral index $n=0.968$ obtained from WMAP7 observations of large scales wmap, and shaded regions correspond to the 68% CL for this measurement ($\Delta n=0.012$). Right: Variation of the gamma-ray and reionisation constraints on $p$ with the kinetic decoupling scale of DM. These two limits in particular are sensitive to the cutoff in the DM halo mass function, as the strongest limits (as shown in the left panel) come from the smallest viable UCMHs, for all $k_s$.
• Figure 5: Left: 95% CL upper limits on the amplitude of primordial curvature perturbations $\mathcal{P}_\mathcal{R}$ (for a non-parametric, generalised spectrum) allowed by gamma-ray searches for UCMHs and impacts on reionisation. For comparison, we show the previous limits from Fermi non-observation of UCMHs derived in Ref. JG10, based on a simplified treatment of the statistics of non-detection, mass variance and minimum density contrast required to form a UCMH. Corresponding constraints on the generalised amplitude of primordial density perturbations can be obtained by multiplying these limits by a factor of 0.191, see Eq. (\ref{['pdelta_conversion']}). Right: The variation of the gamma-ray limit on $\mathcal{P}_\mathcal{R}$ with WIMP mass and the redshift of UCMH collapse, showing the impact of less conservative (but entirely plausible) choices for these parameters than our canonical $m_\chi=1$ TeV, $z_c=1000$.