Suppressing structure formation at dwarf galaxy scales and below: Late kinetic decoupling as a compelling alternative to warm dark matter

Warm dark matter cosmologies have been widely studied as an alternative to the cold dark matter paradigm, the characteristic feature being a suppression of structure formation on small cosmological scales. A very similar situation occurs if standard cold dark matter particles are kept in local thermal equilibrium with a, possibly dark, relativistic species until the Universe has cooled down to keV temperatures. We perform a systematic phenomenological study of this possibility, and classify all minimal models containing dark matter and an arbitrary radiation component that allows such a late kinetic decoupling. We recover explicit cases recently discussed in the literature and identify new classes of examples that are very interesting from a model-building point of view. In some of these models dark matter is inevitably self-interacting, which is remarkable in view of recent observational support for this possibility. Hence, dark matter models featuring late kinetic decoupling have the potential not only to alleviate the missing satellites problem but also to address other problems of the cosmological concordance model on small scales, in particular the cusp-core and too-big-too-fail problems, in some cases without invoking any additional input.

Figure 1

Schematic illustration of the elastic scattering of a DM particle $\chi$ with a (possibly dark) relativistic particle $\tilde{\gamma }$. Throughout this article, we use thick lines to denote heavy (nonrelativistic) particles and thin lines to denotes light (relativistic) particles of any spin.

Figure 2

Cutoff in the halo mass function resulting from DM scattering with a radiation component $\tilde{\gamma }$. This assumes $M_{\text{cut}}=M_n$ as introduced in Eqs. (3)–(4), i.e. an amplitude that scales with the energy $\omega$ of $\tilde{\gamma }$ as ${|\mathcal{M}|}^2\propto {\omega }^n$, a coupling strength roughly corresponding to the electroweak coupling and a DM mass of 100 GeV. For comparison, we also indicate the value that is roughly excluded by Ly-$\alpha$ data, with slightly smaller values allowing a potential solution to the missing satellites problem.

Figure 3

For a scattering amplitude ${|\mathcal{M}|}^2\simeq c_n(\omega /m_{\chi }{)}^n$ close to kinetic decoupling, this figure shows the value of $c_n$ that results in $M_{\text{cut}}=10^{10}{M}_{\odot }$, as a function of the DM mass $m_{\chi }$. From top to bottom, the lines correspond to $n=(4,2,0,-0.9)$, and we have throughout fixed $\xi =T_{\tilde{\gamma }}/T=0.5$ and $g_{\text{eff}}=3.36$. The color scale indicates the size of the matrix element for $\omega \to ⟨\omega {⟩}_{T_{\mathrm{kd}}}$.

Figure 4

DM self-interaction in dwarf-scale systems, induced by a direct coupling between light bosonic $\tilde{\gamma }$ and DM, as a function of ${\alpha }_{\chi }=g_{\chi }^2/4\pi$. From top to bottom, the curves show the case for a DM mass increasing from $m_{\chi }=1\mathrm{MeV}$ to $m_{\chi }=10\mathrm{TeV}$. We have throughout assumed $m_{\tilde{\gamma }}=100\mathrm{eV}$; lighter masses would give higher cross sections. The dashed parts of the curves indicate where the coupling strength $g_{\chi }$ is so large that $\chi \chi \to \tilde{\gamma }\tilde{\gamma }$ would deplete the DM abundance below the observed value. Values above the horizontal thick line are excluded by dwarf galaxy observations.

Figure 5

Minimal effective DM self-interaction resulting from the $\chi$–$\tilde{\gamma }$ interactions shown in Fig. 1.

Figure 6

Overview of results for the two-particle models we have considered, where the DM particle $\chi$ and the (dark) radiation particle $\tilde{\gamma }$ can be scalars, Dirac fermions and vectors, respectively. Here, TOP denotes the topology of the (dominant) DM-DR scattering amplitude. Late kinetic decoupling (LKD) indicates whether in these types of models a small-scale cutoff as large as $M_{\text{cut}}\sim 10^{10}{M}_{\odot }$ can be arranged. Thermal production (TP) indicates whether the observed DM density can be explained by thermal production via $\chi \chi \to \tilde{\gamma }\tilde{\gamma }$, and ${\sigma }_T$ indicates the type of the DM self-interaction rate (only for viable models). A white cell indicates that LKD is possible and that models of this type additionally satisfy the indicated property (e.g. TP). Dark gray indicates that the model is either ruled out, or that it is not possible to achieve LKD, for the reason stated. Here, $⟨{\sigma }_T{⟩}_{30}$ indicates the DM self-interaction strength at the scale of dwarf galaxy scales and $Z_2$ the assumed symmetry to stabilize DM. Operators with dimension (dim) larger than four map to the scalar/scalar four-point case if they lead to an approximately constant scattering amplitude; otherwise they are too small to lead to LKD. Note that $t$-channel scattering with vector DR is only possible in a non-Abelian gauge theory and hence is covered in the $SU(N)$ part.

Figure 7

Diagram illustrating a pointlike interaction of a DM particle $\chi$ with a (possibly dark) relativistic particle $\tilde{\gamma }$. Because we focus on unsuppressed interactions, only dimension-4 operators are considered, which restricts the analysis of this topology to bosonic particles.

Figure 8

As in Fig. 7, but in the presence of a $\chi \text{−}\chi \text{−}\tilde{\gamma }$ coupling, which leads to a resonance in the $s$- (left) and $u$-channel (right).

Figure 9

As in Fig. 8, but in the presence of an additional 3-$\tilde{\gamma }$ coupling, which leads to a resonance in the $t$-channel in addition to the $s/u$ resonances shown in Fig. 8.

Figure 10

Fermionic DM of mass $m_{\chi }$ scattering with scalar dark radiation of mass $m_{\tilde{\gamma }}$, with ${\alpha }_{\chi }\equiv g_{\chi }^2/(4\pi )$ and the cubic DR self-coupling fixed to $\mu =m_{\tilde{\gamma }}$. Solid lines show the parameter combinations that lead to $M_{\text{cut}}\simeq 10^{10}{M}_{\odot }$, for a DR to photon temperature ratio of $\xi =1$, 0.5, 0.1 (from top to bottom). Dashed lines give the constraints that result from DM self-scattering, for a given DM mass; everything left of the respective dashed curve is excluded.

Figure 11

Fermionic DM of mass $m_{\chi }$ scattering with non-Abelian DR of mass $m_{\tilde{\gamma }}$. For DR consisting of $SU(N)$ gauge bosons, we define ${\alpha }_N\equiv g_{\chi }^2\sqrt{(N^2-1)}/(4\pi )$. Solid lines show the parameter combinations that lead to $M_{\text{cut}}=10^{10}{M}_{\odot }$, for a DR to photon temperature ratio of $\xi =1$, 0.5, 0.1 (from top to bottom). Dashed lines show the constraints that result from DM self-scattering, for $N=2$ and a given DM mass; everything left of the respective curve is excluded. Larger values of $N$ result in weaker constraints.

Figure 12

Combinations of ${\alpha }_{\chi }$ and $m_{\chi }$ that lead to $⟨{\sigma }_T{⟩}_{30}/m_{\chi }=1{\mathrm{cm}}^2/\mathrm{g}$, for various mediator masses $m_{{\tilde{\gamma }}^{\prime }}$. The shaded areas show the range of parameters where the appearance of resonances in ${\sigma }_T$ allows multiple solutions to this condition. For comparison, we also show the value of ${\alpha }_{\chi }$ that results in the correct relic density when considering only the process $\chi \chi \to {\tilde{\gamma }}^{\prime }{\tilde{\gamma }}^{\prime }$, assuming that $\xi =0.5$ (see text for further details).