${\mathrm{N}}_{\mathrm{eff}}$ from an excited dark matter state

For a cold dark matter (DM) originating from the coannihilation processes, there will be excited state(s) in the dark sector that may decay or annihilate away shortly after their freeze-out. In this paper, we investigate the impact of the decay of these excited states on the effective number of neutrino species, $N_{\mathrm{eff}}$, which is an important cosmological parameter and will be tested by the future CMB-S4 project. We work in the framework of pseudo-Dirac DM mode via the Higgs portal. The relic density and the direct detection signal of the DM are calculated, which gives the available parameter space. Impacts of the excited state (the heavy component of the pseudo-Dirac fermion) on the decoupling of active neutrinos are investigated by solving Boltzmann equations of photon and neutrinos with a collision term induced by the decay of the excited state. The numerical result shows that the maximum value of $|\mathrm{\Delta }{N}_{\mathrm{eff}}|$ can reach the order $\mathcal{O}(0.1)$ with suitable mass and lifetime of the excited DM state, which may be probed by future CMB-S4 and SPT/SO projects.

Figure 1

$\mathrm{\Omega }{h}^2$ vs the mediator mass $m_{\widehat{s}}$ with the solid and dashed lines corresponding to $m_{\mathrm{DM}}=500$, 300 GeV, respectively. The red line represents the observed relic abundance $\mathrm{\Omega }{h}^2=0.12$ [33].

Figure 2

The exclusion limit for DM mass $m_{\mathrm{DM}}$ vs $m_{\widehat{s}}$. The light-green shaded region is excluded by the PandaX-4T and XENONnT direct detection experiments. The brown shaded regions indicate the relic density $\mathrm{\Omega }{h}^2>0.12$, which is divided into three cases: I) $2m_{\mathrm{DM}}<m_{\widehat{s}}$, II) $m_{\mathrm{DM}}<m_{\widehat{s}}<2m_{\mathrm{DM}}$, and III) $m_{\widehat{s}}<m_{\mathrm{DM}}$, where the last case (darker brown) has extra annihilation channels $({\chi }_i{\chi }_j\to \widehat{s}\widehat{s}(\widehat{s}\widehat{h}))$. The blue and black dashed lines denote $m_{\widehat{s}}=2m_{\mathrm{DM}}$ and $m_{\widehat{s}}=m_{\mathrm{DM}}$, respectively.

Figure 3

Evolution of comoving number density $Y=n/s$ with $x=m/T$. The solid line represents species ${\chi }_1$, where the corresponding $Y$ becomes fixed after freeze-out and it decreases when ${\chi }_1$ begins to decay. The dashed line denotes species ${\chi }_2$, which increases by the decay of ${\chi }_1$. The dotted line represents the evolution of particles at equilibrium. The green and orange indicate $m_{\mathrm{DM}}=4,300\mathrm{GeV}$, respectively.

Figure 4

$\mathrm{\Delta }{N}_{\mathrm{eff}}$ as a function of the lifetime ${\tau }_{{\chi }_1}$ that can decay into photons or neutrinos through ${\chi }_1\to {\chi }_2+2\gamma$ or ${\chi }_1\to {\chi }_2+4\nu$ processes, respectively. The green (blue) solid line describes that the decay product is photons (neutrinos). In panel a (b), the DM mass is $m_{\mathrm{DM}}=300$ (4) GeV.

Figure 5

Feynman diagram for the decay of ${\chi }_1\to {\chi }_2+2\gamma$.