Dark Matter II: Standard Model

The microscopic nature of dark matter is crucial yet still unanswered. The most successful description of the microscopic world is particle physics standard model (abbr. SM). However, standard model could not offer satisfactory answer for dark matter.

Fig. 1: the standard model stew

In order to be "invisible", a dark matter candidate has to be neutral, stable and massive. Most of standard model particles and hadrons are either unstable or charged, or both. Two possibilities remains in standard model, neutrinos and neutrons. But neutrons themselves are not stable. They has to form nuclei and in addition elements together with protons and/or electrons. More broadly, we consider a class of massive astronomical compact objects (abbr. MACHOs) composed of baryonic matter.

Neutrinos

In the simplest version of standard model, all neutrinos are left-handed massless particles. Neutrino  oscillation reveals that neutrinos have non-zero mass. The current experimental values $\left|\Delta m _{23}^2 \right| \simeq \left| \Delta m_{13}^2 \right|= 2.43^{+0.13}_{-0.13} \times 10^{-3} \; \mathrm{eV}^2$, $\Delta m_{12}^2 = 7.59^{+0.20}_{−0.21} \times 10^{−5} \; \mathrm{eV}^2$ gives a lower bound $\geq 0.05 $ eV for the heaviest neutrino. In the simplest case, neutrino masses are just $m_1 \simeq 0, m_2 \simeq 0.01 \;\mathrm{eV}, m_3 \simeq 0.05 \;\mathrm{eV} $. Of course it is possible that neutrino masses are nearly degenerate: $m_1 \sim m_2 \sim m_3 \gg \Delta m^2$. All kinematic measurement so far, however, has failed. Therefore a direct constraint of neutrino mass is particularly important.

Such upper bounds comes from cosmological probes. According to the big bang theory, neutrinos in cosmos were created in the early universe and decoupled from synthesis in the lepton epoch. The relic neutrino density is related to their masses [2]:\[\Omega_\nu = \frac{\sum_f m_{\nu_f}}{93.2 \;\mathrm{eV} H_0^2} \times (100 \;\mathrm{km s^{-1} Mpc^{-1}})^2. \] Presence of massive neutrinos affects structures in the Universe. Analysis of observational data from various cosmological probes based on $\Lambda$-CDM model, with moderate statistical error roughly gives 0.5 - 1 eV neutrino mass upper bound.

On the other hand, simple argument from Pauli exclusive principle (the so-called phase space argument) could offer a lower bound of fermionic dark matter mass. In order to be bounded by gravity, the Fermion velocity should be smaller than the escape velocity,\[ m_f \geq \left( \frac{9\pi\hbar^3}{4\sqrt{2} g M^\frac{1}{2} R^\frac{3}{2} G^{\frac{3}{2}}} \right)^{\frac{1}{4}} \] where $g \geq 1$ is the internal degrees of freedom. Carrying out the analysis to the actually dark matter dominating system, we can conclude the mass of fermionic dark matter $ \gtrsim 1 \;\mathrm{keV}$.

Moreover, if dark matter are neutrinos, they are hot matter. So far, hot dark matter models have various issues. First of all, hot dark matter tends to smooth out the matter fluctuation observed by Sloan Digital Sky Survey and other observations. It is also incompatible with the angular power spectrum of cosmic microwave background (See section 2.1). Secondly, in hot dark matter model (HDM), the structural formation is in the opposite order (i.e. large structures form early) of what we observed. Numerical simulation of galaxy formation in hot dark matter halo so far could not agree with observation.

Therefore, neutrinos are unlikely dark matter candidate.

Fig. 2: neutrino oscillation is caused by the difference between
flavor eigenstates and mass eigenstates.

MACHOs

MACHOs such as red dwarfs and brown dwarfs may also show high mass-to-light ratio. However, large amount of baryonic matter will breaks the baryonic density constraint put by various astronomical probes based on cosmological models. Furthermore, observation using Hubble Space Telescope shows halo red dwarfs and brown dwarfs density is about 0.25% ~ 0.67% of the halo density, hence insignificant. The EROS project searching Magellanic clouds for microlensing events caused by MACHOs also reported that MACHOs make up to less than 8% of the halo mass.

Fig. 3: Left: An artist vision of a Y-dwarf; Right: An artist vision of a red dwarf.

In summary, standard model particles are almost ruled out as dark matter candidates.