# Cosmology Lecture 10

### The Dark Side of the Universe

There is evidence that dark matter exists:

• Zwicky (1933): measurements of ${\displaystyle \sigma _{\nu }}$ (the total mass) of galaxies in the Coma Cluster showed:
${\displaystyle {M \over L}\approx 300h{M_{\odot } \over L_{\odot }}\,\!}$

This means that there is lots of mass that doesn’t appear in luminosity (the mass is 300 times what you’d expect if all the stars were sun-like).

• Dynamics (70’s, Rubin and others): the rotation curves of galaxies is flat. That is, ${\displaystyle v^{2}\propto {M(r) \over r}}$, and this is constant, so ${\displaystyle M(r)\propto r}$.
• Structure formation: A baryonic universe can’t grow structure until decoupling (CMB era) has occurred, since rapid collisions with photons prevents gravitational collapse. But if that’s when gravitational collapse started, there wouldn’t have been time to form the structure we see today. Since dark matter does not couple, it could start collapsing earlier and suck the baryonic matter down with it.
• CMB fluctuations give us a measure of ${\displaystyle \Omega _{m}}$.
• Gravitational lensing probes luminous and dark matter.

### Nature of Dark Matter

If we calculate the mass of luminous matter we get ${\displaystyle \Omega _{stars}+\Omega _{gas}\approx .003}$ (Fukugita et al, 98). For baryons (p,n), ${\displaystyle \Omega _{b}h^{2}\approx .024}$, as calculated from Big Bang Nucleosynthesis. Overall, matter is estimated to constitute ${\displaystyle \Omega _{m}\sim .2\to .4}$ (baryons and non-baryons). This brings us to the two levels of the “dark matter problem”.

• ${\displaystyle \Omega _{b}\gg \Omega _{luminous}\Rightarrow }$ some baryons must be “dark”. This is the “Baryon Dark Matter problem”. What are these “dark baryons”? Candidates are: (a) MACHOs (compact stellar-mass objects), searched for by micro-lensing. These could be faded white dwarfs (${\displaystyle \sim {\frac {1}{2}}M_{sun}}$) or brown dwarfs (“failed stars” with ${\displaystyle M<.08M_{sun}}$). So far it seems that these cannot account for the mass density difference between baryonic and luminous matter. (b) Black holes, neutron stars: too few. (c) Planets: ${\displaystyle M_{Jupiter}\sim {1 \over 300}M_{sun}}$, too little mass. (d) Dust: radiates, bounded by metallicity constraints. ${\displaystyle \to }$(e) Diffuse warm gas: ${\displaystyle T\leq 10^{6}K}$. Can’t detect this very well. It will emit in the soft x-ray range. This one is still a viable option.
• The “Non-Baryonic Dark Matter Problem”: ${\displaystyle \Omega _{m}\gg \Omega _{baryon}}$ tells us that there is mass coming from non-baryons. Some options here are hot dark matter (massive neutrinos, for example), and cold dark matter (SUSY=SUper SYmmetry particles). See PS#5 for why neutrinos are “hot”. In order for neutrinos to work out as hot dark matter, they need to have mass.

### Massive ${\displaystyle \nu }$ (as Hot Dark Matter)

• 1. Current lab limits (based on weak-decay kinematics): ${\displaystyle m_{\nu _{\tau }}<18.2MeV}$ based on ${\displaystyle \tau }$ decays: ${\displaystyle \tau ^{-}\to \nu _{\tau }+pions}$. ${\displaystyle m_{\nu _{\mu }}<190keV}$ based on ${\displaystyle \Pi ^{+}}$ decays: ${\displaystyle \Pi ^{+}\to \mu ^{+}+\nu _{\mu }}$. ${\displaystyle m_{\nu _{e}}<3eV}$ based on tritium decays.
• 2. Neutrino oscillations (in vacuum) measures ${\displaystyle \Delta m^{2}=m_{1}^{2}+m_{2}^{2}}$. When solving the interaction Lagrangian, there are mass eigenstates and weak-interaction eigenstates, which do not have to be identical. Considering ${\displaystyle \nu _{1},\nu _{2}}$ as the mass eigenstates of a 2-generation interaction, and ${\displaystyle \nu _{e},\nu _{\mu }}$ as the weak eigenstates, then:
${\displaystyle {\begin{pmatrix}\nu _{e}\\\nu _{\mu }\end{pmatrix}}={\begin{pmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{pmatrix}}{\begin{pmatrix}\nu _{1}\\\nu _{2}\end{pmatrix}}\,\!}$

The Weak interaction Lagrangian is ${\displaystyle S=\int {d^{4}x{\mathfrak {L}}}}$, where ${\displaystyle {\mathfrak {L}}}$ is the Lagrangian density. Then:

${\displaystyle {\mathfrak {L}}_{weak}\propto {\bar {\nu }}_{e}\gamma ^{\alpha }(1-\gamma _{5})eW_{\alpha }+{\bar {\nu }}_{\mu }\gamma ^{\alpha }(1-\gamma _{5})\mu W_{\alpha }+\overbrace {m_{1}{\bar {\nu }}_{1}\nu _{1}+m_{2}{\bar {\nu }}_{2}\nu _{2}} ^{mass\ terms}\,\!}$

Where the “mass terms” are what’s relevant for determining neutrino oscillations.