Determinacy & Chaos in Neutrino Oscillations in the Early Universe
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1 Determinacy & Chaos in Neutrino Oscillations in the Early Universe Kev Abazajian University of Maryland K. A. & P. Agrawal, arxiv:0806.nnnn
2 Light Sterile Neutrinos: Why? LSND + MiniBooNE -> no sterile or N steriles A light sterile outside of the sensitivity still could arise as a low mass state of a see-saw like model One primary motivation for this possibility is the successful r-process in supernova wind nucleosynthesis by elimination of the alpha effect and achievement of high neutron densities with 0.01 ev 2 < δm 2 < 10 ev 2 (McLaughlin et al, 2003) Or, simply intellectual curiosity of an unresolved problem in oscillation-based lepton number generation
3 Neutrinos in the Early Universe The Two Neutrino Case
4
5 Vacuum Oscillation
6 Vacuum Oscillation e+/- / µ+/- background
7 Vacuum Oscillation Thermal Potential (finite temperature effects) e+/- / µ+/- background
8 Vacuum Oscillation Thermal Potential (finite temperature effects) e+/- / µ+/- background Neutrino Self-Potential
9
10 f W +/-,Z 0 Z 0 f
11 Collisions and Decoherence
12 z The Neutrino Self-Potential I J J P p x V eff y
13 z The Neutrino Self-Potential I J J P p x V eff y
14 History Thermalization of sterile neutrino at temperatures 1 MeV < T < 100 MeV occurs for mass splittings 10-6 < δm 2 and increasing mixing angles (Langacker 1989) Resonant lepton number generation could occur and even avoid scenarios of thermalization: Foot, Thomson & Volkas, 1996 Confirmed by X. Shi (1996) and found resonant production to be chaotic All work thus far used a single-momentum (averaged) quantum rate equation solution of the evolution
15 Lyapunov exponent can be positive at least intermittently (X. Shi, 1996) Information loss found it to be a chaotic system with increased sin 2 2θ was studied by Braad & Hannestad (2000)
16 Oscillations, Chaos & The Sign of the lepton number Enqvist, Kainulainen & Sorri (1999) (single momentum solution)
17 Full momentum space solution and stablization? Full momentum space solution can stabilize the oscillation as the resonance occurs for smaller momenta, and lepton number is smoothly created as <pres> moves up the neutrino distribution (Di Bari & Foot, 2000) Region of rapid oscillations still exists Kainulainen & Sorri (2002) find this region continues to exhibit chaotic behavior (sensitivity to initial conditions) Di Bari & Foot (2000)
18 Full momentum space solution and stablization? Full momentum space solution can stabilize the oscillation as the resonance Enqvist et 1999 occurs for smaller momenta, and lepton number is smoothly created as <pres> moves up the neutrino distribution (Di Bari & Foot, 2000) Region of rapid oscillations still exists Kainulainen & Sorri (2002) find this region continues to exhibit chaotic behavior (sensitivity to initial conditions) Di Bari & Foot (2000)
19 Chaos in Binary Black Hole Inspiral r dφ/dt dr/dt θ2 PN expansion of orbits yields coupled, precessing systems, sensitivity to ICs, chaos, and fractals in initial parameter space (J. Levin 2000) θ1
20 Oscillations, Chaos & The Sign of the lepton number Enqvist, Kainulainen & Sorri (1999) (single momentum solution)
21 Sign of the asymmetry
22 Subspace Zoom (1)
23 Subspace Zoom (2)
24 Subspace Zoom (3)
25 Sign of the asymmetry
26
27 Serpinsky Triangle: The Fractal Dimension
28 Correlation Dimension An estimate of the fractal dimension (Grassberger & Procaccia 1983) The probability that two states are close Correlation function in multiple dimensions -> correlation dimension is the value of the C(ε) power law For example, the power law of the 2PCF of galaxies was (and still is) used to argue for the fractal nature of the universe (despite that there are features in the 2PCF now)
29 Correlation Dimension For widest range of sterile neutrino parameter space: Correlation Dimension: 1.05 => up to 1.35 for higher sin 2 2θ => Interruption of trajectory of chaotic walk in phase space with expansion of universe larger mixing angles allow for more oscillations
30 Determinacy of L & Conclusions Fractals are generated in the parameter space of lepton number generation in the early universe Experimental measurement of δm 2, sin 2 2θ may not be able to determine the sign of the lepton asymmetry Future work with full momentum calculations will likely require a reformulation of the system of equations to make them reliably numerically solvable
31 Determinacy of L & Conclusions Fractals are generated in the parameter space of lepton number generation in the early universe Experimental measurement of δm 2, sin 2 2θ may not be able to determine the sign of the lepton asymmetry Future work with full momentum calculations will likely require a reformulation of the system of equations to make them reliably numerically solvable
32 Determinacy of L & Conclusions Fractals are generated in the parameter space of lepton number generation in the early universe Experimental measurement of δm 2, sin 2 2θ may not be able to determine the sign of the lepton asymmetry Future work with full momentum calculations will likely require a reformulation of the system of equations to make them reliably numerically solvable
33 Determinacy of L & Conclusions Fractals are generated in the parameter space of lepton number generation in the early universe Experimental measurement of δm 2, sin 2 2θ may not be able to determine the sign of the lepton asymmetry Future work with full momentum calculations will likely require a reformulation of the system of equations to make them reliably numerically solvable
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