Notes on Tests of Matter/Vacuum Oscillation Program: MATTOSC
RWE Starting 11 June 97
The program currently propagates a neutrino state, initially
a nu-e, through the sun, and to the earth. Currently, the neutrino state
originates at the center of the sun. The program uses a Bullirch-Stoer
stepping routine, found in Numerical Recipes, and double precision variables.
It is written in the language most suitable for scientfic computing.
The following tests have been performed. The results of each test
is in a postscript file, on umdgrb.umd.edu/~ellswort
NOTE: IN THE GRAPHS SPECIFIED BELOW, THE INDIVIDUAL POINTS ARE NOT
THE INTEGRATION STEPS. THE INTERVAL BETWEEN POINTS ON THE GRAPHS
CORRESPOND TO MANY HUNDREDS OF INTEGRATION STEPS.
1) Does the program give the correct vacuum oscillation amplitude and
period? A modified version of the program, with density = 0 everywhere,
was run. Results are compared with expectations in the file
PREDSOCVAC.PS. The upper dashed line is the predicted peak-to-peak
oscillation; the vertical dashed line is the time for 10 predicted
periods.
2)Comparison with predictions for constant density (taken to be that
of sun center) is in PREDOSCCEN.PS.
3)Stability is studied in DIFFSTEPS.PS. The nu-mu probability vs.
distance is plotted for the use of 50 steps/oscillation period, and
of 100 steps per oscillation period.
4) The oscillations close to the sun center are seen in CLOSETOSUN.PS.
In this graph, we can see both the expected increase in period as the
density decreases, and the increase in oscillation amplitude, AND
a "baseline" shift, i.e. the oscillation is about an increasing value.
While this set of parameters is referred to as "non-adiabatic", meaning
that the major part of the transition occurs in a small number
of oscillation periods, the graphs shows that some effect is occurring
far from the main "step" in probability.
5) The "step" for several neutrino energies is shown in 235MEV.PS.
As expected, for lower energies, P(nu-mu) rises to a larger value.
The step also occurs at a different distance for different energies.
Note 1 light second = 300 megameters.
6) Same as (5), except for 5, 7 and 10 MeV, are in 5710MEV.PS.
7) The step for one of the sets of parameters of the matter oscillation graph
in the book by Bohm and Vogel, is shown in VOGELPAR.PS
8) A routine was written to analytically propagate a state, once it
was outside the sun, to any distance. The routine was tested by
comparing its output from 4 to 4.03 light seconds, starting at 3 ls,
with the output of the program integrating directly. The comparison is
in VACPROP.PS. Note that both results have the expected vacuum oscillation
period for these parameters, 6.50 E-03 sec, but the amplitude and
baseline are NOT that of vacuum oscillations. This is because the
matter oscillation has altered the "inital" state for vacuum oscillations.
9) AMPLITUDES.PS plots the real vs imag parts of the nu-e and nu-mu
amplitudes. It illustrates the transitions, but does not show much
quantitative.
10) RHOCRIT.PS shows what happens when the density is set everywhere
to be rho-critical. The nu-mu and nu-e probabilities undergo
oscillations about 0.5. THERE IS NO BASELINE SHIFT. This shows
that a time-changing density, passing throught the rho-critical, is
necessary for a baseline shift.
11) (No graph.) When the initial state is a nu-mu, the transition
is reversed. The final state is mainly nu-e, with complementary
probabilities. The process drives low prob to higher prob.
12) If the initial phase is changed from that for (1,0) to
(.7071, .7071) for nu-e, there is NO CHANGE in the final (earth)
probability.
13) SPECTRA.PS shows the Nu-e survival probabilities at earth,
for the Non-adiabatic solution, and Large Angle I.
14) To study the transitions, a set of data with random initial
states was generated. The graph MOTRANS1.PS shows the final state (at
earth) nu-e probability (yaxis) vs the initial state nu-e probability.
(In this set, the initial state has random phase as well as magnitude
of amplitude.) The "Non-adiabatic" parameters, with Enu = 5 MeV,
were used.
15) MOTRANS2.PS shows the same thing, except for fixed phase (=0.)
Here we get a single curve for Pfinal vs. Pinitial. MOTRANS3.PS
shows the same for Large Angle I solution,(5 MeV); MOTRAN7MEV.PS
shows Non-Ad at 7 MeV, phase = 0.