Dark matter from inflationary quantum fluctuations

TL;DR

This work proposes a dark-matter mechanism in which a massive bosonic spectator field carries the DM content, seeded entirely by inflationary quantum fluctuations that yield small-scale isocurvature perturbations. The authors derive an exact transfer function for the field’s evolution through the radiation-dominated era, expressed via the confluent hypergeometric function, and provide limiting-case approximations that span relativistic/non-relativistic and sub-/superhorizon regimes. By assuming a sharply peaked (monochromatic) initial spectrum at $k_p$, they compute the relic density, map the viable region of parameter space in $(m,k_p,\beta_{\rm inf})$, and show that non-relativistic modes with $m>H_p$ naturally form copious subsolar-mass halos at high redshift. The analysis is complemented by BBN and CMB constraints, revealing a broad viable region and a smoking-gun prediction of abundant low-mass halos that could confirm a purely inflationary origin for dark matter.

Abstract

We explore a scenario in which dark matter is a massive bosonic field, arising solely from quantum fluctuations generated during inflation. In this framework, dark matter exhibits primordial isocurvature perturbations with an amplitude of ${\cal O}(1)$ at small scales that are beyond the reach of current observations such as those from the CMB and large-scale structure. We derive an exact transfer function for the dark matter field perturbations during the radiation dominated era. Based on this result, we also derive approximate expressions of the transfer function in some limiting cases where we confirm that the exact transfer function reproduces known behaviors. Assuming a monochromatic initial power spectrum, we use the transfer function to identify the viable parameter space defined by the dark matter mass and the length scale of perturbations. A key prediction of this scenario is copious formation of subsolar mass dark matter halos at high redshifts. Observational confirmation of a large population of such low-mass halos will support for the hypothesis that dark matter originated purely from inflationary quantum fluctuations.

Figures (3)

• Figure 1: Physical scale $\lambda\sim a/{k}$ versus the scale factor in logarithmic scales are plotted. The lower $m_{ min}^{-1}$ and upper $m_{ max}^{-1}$ bounds on the mass are defined by Eqs. \ref{['mass-lower-bound']} and \ref{['mass-upper-bound']} respectively. Avoiding large fluctuations at $\lambda\gtrsim\lambda_{\rm obs}$ to be consistent with CMB and LSS observations, we have to restrict our setup to the scales $\lambda\lesssim\lambda_{\rm obs}$. The blue solid line $m^{-1}$ represents a typical mass value which should lie in the range $m_{min}\lesssim{m}\lesssim{m}_{max}$. The solid lines $\lambda_{p,{\rm R}}\sim a/{k}_{p,{\rm R}}$ and $\lambda_{p,{\rm NR}}\sim a/{k}_{p,{\rm NR}}$ show two possible momenta $k_{p,{\rm R}}$ and $k_{p,{\rm NR}}$ at which the initial power spectrum can have a peak. The mode $k_{p,{\rm R}}$ is relativistic at the time of horizon re-entry since $\lambda_{p,{\rm R}}(\tau_{p,{\rm R}})<m^{-1}$ or $k_{p,{\rm R}}/a(\tau_{p,{\rm R}})>m$ while $k_{p,{\rm NR}}$ is non-relativistic since $\lambda_{p,{\rm NR}}(\tau_{p,{\rm R}})>m^{-1}$ or $k_{p,{\rm NR}}/a(\tau_{p,{\rm R}})<m$. The latter case, which corresponds to $m>H_{p,{\rm NR}}$, collapses to form subsolar mass dark matter halos at high redshifts.
• Figure 2: The function $|{\cal I}(k_p)|^2$ is monotonically decreasing as mass decreases and it is bounded as $0<|{\cal I}(k_p)|^2\leq \pi/[2\Gamma(3/4)^2]$. The tail for light masses $m\ll{H}_p$ is well described by $1/(2\sqrt{H_p/m})$. Moreover, up to masses $m\sim{H}_p$ can be approximated with the tail behavior $1/(2\sqrt{H_p/m})$ with error less than ${\cal O}\left(10^{-3}\right)$. For heavy masses $m\gg{H}_p$, the function is almost independent of the mass and approaches to $\pi/[2\Gamma(3/4)^2]\approx1.046$.
• Figure 3: Mass of dark matter versus the scale is plotted. The red regions are excluded by the observations while the gray regions are excluded by the theoretical constraints. For the upper bound on the mass $m\ll{10}^{22}\,\hbox{eV}$, we have considered $H_{\rm inf}=10^{13}\,\hbox{GeV}$ in \ref{['mass-upper-bound']} which corresponds to $r\sim0.03$. For lower values of tensor-to-scalar ratio, this bound becomes tighter. The lower bound on the mass $m \gtrsim 10^{-21}\,\hbox{eV}$ comes from the galaxy observations. The lower bound on the scale $k_p \gtrsim 10^3{k}_{\rm obs}\sim 10\,\hbox{kpc}^{-1}$ should be imposed to be consistent with the absence of large fluctuations in the CMB and LSS observations. The allowed region restricted to $m>H_p$, that corresponds to the modes which are non-relativistic at the time of horizon re-entry, will form subsolar mass dark matter halos at high redshifts.