Cosmology Lecture 12

Wrapping Up Massive Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu }

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

• Direct Lab upper limits.
• $\nu$ oscillations
• Supernova Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu }
• Cosmology (PS #5): There is a relic Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } background from 1 second after the Big Bang,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega _\nu h^2={\sum _{i=e,\mu ,\tau }{m_ i}\over 93.5eV} \,\!}

(for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu } ’s were relativistic when they decoupled from the rest of the universe.

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_\nu >1MeV} , then the # density of neutrinos will have a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{-{m_\nu c^2\over kT}}} suppression, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu \bar\nu } will annihilate to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z^0} . Lee-Weinberg (1977) showed that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega _\nu \propto m_\nu ^{-2}} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_\nu \gg 1MeV} . Since this is a square function, there are two values for which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega _\nu =1} . If we want Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega _\nu <1} , we have that either $m_{\nu }>2GeV$ or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_ n\sim 10^{-15}\bar\Theta \ e\cdot cm} . Experimental results show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_ n<.63\cdot 10^{-25} e\cdot cm\Rightarrow \bar\Theta <10^{-10}} . Thus, the axion mass required to suppress this is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho } (density), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P} (pressure), Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec v} (fluid velocity), and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi } (gravitational potential).