1. Neutrino Oscillations and Fugacity in the Early Universe
Copyright 2015 Symmetry Magazine, Artwork by Sandbox Studio, Chicago with Ana Kova
Neutrino Oscillations and Fugacity in the Early Universe
Speaker : Chun-Yen Chen (National Taiwan University)
Advisor : Pisin Chen (National Taiwan University, LeCosPA Center)
Co-Author : Cristopher Gauthier

2. Introduction : Neutrino Oscillation & Fugacity
We all now know that neutrinos has non-zero mass and so will oscillate, mainly having two types
1. between active and sterile ν
=> Planck’s data for suggested no sterile neutrinos
2. between different flavors
Birrell et al. have suggested that we should consider an additional cosmic parameter – the primordial neutrino fugacity, which will change the primordial neutrino distribution and also affect BBN and CMB.
Dolgov, A.D. et al. Nucl.Phys. B632 (2002) hep-ph/ MPI-PHT
2015 Nobel prize gave to the research for Neutrino Oscillation
Dolgov et al. have studied about the constraint on degeneracy of neutrino with oscillation.
Therefore we try to find out the relation between the fugacity and the final number density of neutrinos and anti-neutrinos for each flavor at the end of hadron epoch, and to see how and why the fugacity will change them.
Birrell, Jeremiah et al. Mod.Phys.Lett. A28 (2013) arXiv: [astro-ph.CO]

3. Introduction : Fugacity of Neutrinos
For Gibbs free energy of neutrinos:
In general, we can have independent chemical potential for neutrino and anti-neutrino
where and
Therefore the Fermi-Dirac distribution can be rewrite as
And the and can be rewritten as
degeneracy
fugacity

4. Introduction : Big Bang Nucleosynthesis
About 3min after the Big Bang, many light elements were created by the Big Bang Nucleosynthesis (BBN).
BBN is affected by two key parameters:
Baryon-photon ratio η
Neutron-proton ratio
The larger n/p+ ratio is, the more 4He would be created

5. Quantum kinetic equation of neutrinos
Using Boltzmann equation to get the quantum kinetic equation of neutrino, which is so called as “spin-precession formula”
where the and are defined in the SU(2) space with two-flavor model
and similar for anti-neutrinos.
Dolgov, A.D. et al. Nucl.Phys. B632 (2002) hep-ph/ MPI-PHT
Vacuum oscillation
MSW effect
forward scattering
Collisional decoherence

6. Approximation To analysis the QKE of neutrino we solve it from
where ,and and are new angular variables defined by
By solving the approximate eigenvalues and eigenvectors to 1st order in D ,

7. Approximation
Including only the relevant non-oscillation term, and taking the thermal average of the difference between the neutrino and anti-neutrino, we get a linear approximation solution
where and and
with

8. Approx. Result : D = 0
The solid lines assume a non-zero fugacity and the dashed lines assume zero fugacity for both flavor.

9. Approx. Result : With D
The solid lines assume a non-zero fugacity and the dashed lines assume zero fugacity for both flavor.

10. Numerical Result

11. Conclusion and Future Work
Including the freedom of fugacity of neutrinos and anti-neutrinos will affect the magnitude of the lepton asymmetry for each flavor in the early universe.
And this will also affect the BBN and so the abundance of the light elements.
On the other hand, we can give a constraint for the primordial neutrino fugacity by the observation data of the light elements abundance and know the evolution of neutrinos and anti-neutrinos before the decoupling.

12. Thank You For Listening
Speaker : Chun-Yen Chen

13. Introduction : Big Bang Nucleosynthesis
About 3min after the Big Bang, many light elements were created by the Big Bang Nucleosynthesis (BBN).
BBN is affected by two key parameters:
Baryon-photon ratio η
Neutron-proton ratio
The larger n/p+ ratio is, the more 4He would be created