# Cosmology Lecture 12

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### Wrapping Up Massive ${\displaystyle \nu }$

So far we’ve discussed (1), (2), and (3) below:

• Direct Lab upper limits.
• ${\displaystyle \nu }$ oscillations
• Supernova ${\displaystyle \nu }$
• Cosmology (PS #5): There is a relic ${\displaystyle \nu }$ background from 1 second after the Big Bang,
${\displaystyle T_{\nu ,0}=1.945K\sim \left({4 \over 11}\right)^{1 \over 3}T_{\gamma ,0}\,\!}$

We calculated that the neutrino mass required to close the universe is:

${\displaystyle \Omega _{\nu }h^{2}={\sum _{i=e,\mu ,\tau }{m_{i}} \over 93.5eV}\,\!}$

(for ${\displaystyle m_{i}\ll 1MeV}$). This puts a pretty tight limit on the mass of neutrinos, given that we observe an open universe. The assumption we made about mass allowed us to make the assumption that ${\displaystyle \nu }$’s were relativistic when they decoupled from the rest of the universe.

If ${\displaystyle m_{\nu }>1MeV}$, then the # density of neutrinos will have a ${\displaystyle e^{-{m_{\nu }c^{2} \over kT}}}$ suppression, since ${\displaystyle \nu {\bar {\nu }}}$ will annihilate to ${\displaystyle Z^{0}}$. Lee-Weinberg (1977) showed that ${\displaystyle \Omega _{\nu }\propto m_{\nu }^{-2}}$ for ${\displaystyle m_{\nu }\gg 1MeV}$. Since this is a square function, there are two values for which ${\displaystyle \Omega _{\nu }=1}$. If we want ${\displaystyle \Omega _{\nu }<1}$, we have that either ${\displaystyle m_{\nu }>2GeV}$ or ${\displaystyle m_{\nu }<100eV}$.

### Cold Dark Matter (CDM)

The two best known candidates for CDM are:

• WIMPs (Weakly Interacting Massive Particles): The lightest super-symmetric particle is in the range of 10-100 GeV.
• Axions: These were “introduced” to resolve the strong CP problem in QCD. The problem was that non-perturbative effects in QCD led to a CP,T,P violation. Which would predict an excessively large electric dipole moment for the neutron. Axions were invented to suppress this effect. In terms of the Lagrangian density:
${\displaystyle L_{QCD}=L_{pert}+\underbrace {{\bar {\Theta }}{g^{2} \over 32\pi ^{2}}\overbrace {G^{a \atop \mu \nu }} ^{gluon \atop fields}{\tilde {G}}_{\mu \nu }^{a}} _{violates\ CP}\,\!}$

The CP violating term predicted an electric dipole moment of the neutron of ${\displaystyle d_{n}\sim 10^{-15}{\bar {\Theta }}\ e\cdot cm}$. Experimental results show that ${\displaystyle d_{n}<.63\cdot 10^{-25}e\cdot cm\Rightarrow {\bar {\Theta }}<10^{-10}}$. Thus, the axion mass required to suppress this is ${\displaystyle 10^{-5}eV}$. Why are we calling these cold? Clearly, these are still relativistic particles even today. However, since axions couple with photons only very weakly, they were never really in thermal equilibrium.

### The Lumpy Universe: Structure Formation

The universe has a dichotomy between being initially nearly smooth (as per the CMB), and quite lumpy (as in galaxy clusters). The general belief is that the lumpiness of the universe is caused by small perturbations amplified by gravitational instability. We will concern ourselves with the time-evolution of small perturbations to uniform (Robertson-Walker-Friedmann mode) parameters such as ${\displaystyle \rho }$ (density), ${\displaystyle P}$ (pressure), ${\displaystyle {\vec {v}}}$ (fluid velocity), and ${\displaystyle \phi }$ (gravitational potential).