MC~ELLAR
-165-
The rates of the relevant reactions, such as
NEUTRINO OSCILLATIONS IN THE EARLY UNIVERSE
v
Bruce H. J. McKellar and Henry Cranek
School of Physics, University of Melbourne,
Parkville, Victoria, Australia 3052
+ n ...
e
are then determined by replacing
(4)
Dve
n
We consider the influence of neutrino mixing and oscillations on
the primordial helium abundance. We find appreciable effects only
for neutrino chemical potentials of order kT.
lue il2
Pi
(5)
1:
A (n ... p) [n 12
i-I i
ei
(6)
Pee - 1:
i-I
We can then write the rate for n ...
The primordial helium abundance in the Universe is determined
principally by the temperature at which the reactions involving
electron neutrinos inter-converting protons and neutrons go out of
equilibrium. In the presence of neutrino mixing and oscillations
there is the possibility that the conversion between neutrino flavours will influence this equilibrium and change the helium abundance.
We investigate this possibility in this paper.
We begin by noting that the relevant temperatures are about
1 MeY, at which time the radius of the universe is of order lOam. As
long as ~2 ~ 10- 8 ey 2 individual oscillations will DOt be important,
and we can concern ourselves with average values of the density of
the different neutrino flavours. We will assume this to be the case.
Next we define our notation for the neutrino mass eigenstates
v1,v2, •• ,vi, •• ,vn and the neutrino weak interaction eigenstates
v v~, •• ,vC, ••• v~, with the usual unitary transformation between the
e
two
(1)
i
The neutrino density matrix P is assumed to be diagonal in the
mass eigenstate basis
Pi j - 6 i j Pi
+ e ,
in the usual formulae with P
ee
_ the leading diagonal element of the density matrix in the C baa is
ABSTllACT
"C - 1: UCivi
p
(2)
P:
n
A(O ... p)
where
Ai(n'" p)
A
J {I - m~
dq
}~
.
(Q
+ q)2 q2
(Q+m )2
e
[1 +
exp
{-(.!l - Ci ) } ]
Tv
[1 +
exp
(-~)l
Ty
Clearly neutrino mixing has no effect if all of the neutrino
chemical potentials are equal, since then Ai(n ... p) and Ai(P'" n)
are independent of i (assuming that all of the neutrino masses are
negligible compared to T - 1 MeV), and
\ueil2 - 1. Physically
f
this is to be expected since with equal neutrino densities any
electron neutrino flux lost to oscillation into another neutrino is
exactly compensated by that neutrino oscillating into electron
neutrinos.
In particular to get significant effects we will require
C - 1 for at least some of the neutrinos. This runs counter to
i
the conventional wisdom that the net lepton number of the Universe
should be of order of the net baryon number. As Dv _ ~l
the value
_i
with
(3)
Pi
ny
~i- 1 reouires a net lepton number for the ith lepton of order the
e
£i - Cil
exp Tv
+1
[
in a system of units with k • 1, and where ~i • ~i ,
T"
chemical potential for the neutrinos of masS mi'
~i
being the
-)
number of photons, whereas "vi - n B - 10- ny re~uires ~i - 10 •
However, appreciable valu~s of ~i5have been investigated by Waloner
et a1 3 , Yahil and Beaudet , Lind~ and Dimopolous and Feinberg.
Dimopolous and Feinberg 2 also investigate some scenarios which could
lead to net lepton numbers much greater than the net baryon number.
The motivation of Linde and Dimopolous and Feinberg for considering ~vi - 1 was that it provides a way of evading the constraint
MCKELLAR
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imposed on the number of neutrino types by the helium abundance. If
the low values of the primordial helium abundance recently proposed'
are established then such an evasion may be necessary to accommodate
as few as ) neutrinos. In equilibrium the neutron to proton ratio
C,2
is
.
a
e
II equll.
since
~
\Ie
•
~
n
~n
- mnlT
.
3"
e-Q/T -
N~--3
Cve
- - - - Nv--6
eE:p - mplT
•
~
e
+
~
P
, and
(7)
~ ... O.
e
< e - ~\I e
p equil.
and can be made as small as we please if ~\le is a free parameter,
Thus
!!.
independently of the number of neutrino species and the temperature
at which the neutrino reactions go out of equilibrium. To obtain
the neutron fraction, and hence the helium fraction, it is necessary to follow the evolution of the ratio through the non equilibrium
phase using the rate equation
dX
n
dT •
~
({A(n
+
p) + A(p
+
n)lXn -
A(p
+
n)l
(8)
At T _ 0.1 MeV the dentrium bottleneck breaks open and the neutrons
are essentially all converted to Helium, so we can set the Helium
fraction y ... 2X
MeV)
n(0.1
1 + Xn(O.l MeV)
We discuss the no mixing case in detail elsewhere.
When neutrino mixing is possible we have many more parameters
at our disposal. For illustrative purposes we have taken ~\I, aDd
one other neutrino chemical potential, say ~\I ' as non zero.
Only
2
\I, and \12 are assumed to mix with \Ie and we s~t a · lu e,1 • We
assume a total of N neutrinos which are of Kajoraroa type, or of
Dirac type with Ga « GL•
For a • 0.15 we show the results in figure 1 for N\I • ) aDd
Nv • 6.Plotted are iso-Helium-fraction lines for Y • 0.2 and Y - 0.3
on a ~, ~2 plot. For large values of ~, and ~2 the results are
independent of N\I at fixed (~',~2) as we expect. However this
increase can be compensated by increasing either ~, or ~2 or both.
The introduction of neutrino mixing simply increases the number
of parameters available to fit the data. The helium abundance is no
longer a probe of the number of light neutrinos unless the mixing
angles and the chemical potentials can be fixed.
FIG 1
HELIUM
ABUNDANCE
-2
FOR a=O.75
-3
-167-
It is possible that the deuterim abundance may further fix the
parameters, as it did in ref. 3. We plan to investigate this
question further. We also intend to investigate the possibility
that neutrino oscillations play a cosmological role at a time when
the diameter of the universe is of the same order as the oscillation
length.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
See eg G. Steigman, Ann Rev Nucl Part Sci 29. 313 (1979).
This is a straight forward generalisation of the equations
quoted by S. Dimopolous and G. Feinberg, Phys Rev D20, 1283
(1979).
a.v. Wagoner, W.A. Fowler and F. Hoyle, Science 155. 1369
(1967).
---A. Yahil and G. Beaudet, Astrophys J. 206, 26 (1976).
A.D. Linde, Phys Lett B83. 311 (1979).--F.W. Stecker, Phys. Re;-Lett 44. 1237 (1980).
H. Granek and B.H.J. McKellar:-University of Melbourne preprint
UK-p-so/67.
LOVELESS
BUBBLE CHAMBER NEUTRINO OSCILLATION PROPOSALS
R.J. Loveless
University of Wisconsin, Madison, WI 53706
I.
Wisconsin-Athens-Fermilab-Padova Proposal
A.
Introduction
We have proposed a search for the oscillation of ve + v T or
v + n where n is a new species of neutrino (possibly a singlet)
wftich does not interact. The neutrino detector will be the Fermilab
IS' BC with a heavy neon-Hydrogen mix and beam will be enriched with
v
The oscillation probability can be written:
e•
2
2
L
P(v + VB) • 6
- sin 29 sin (1.27 Ed)
e
eB
.
222
w~ere ~ (eV ) • mve-mvs, L(m) • length, E(MeV) • v e energy, and 9 •
m1xing angle.
estimate this experiment would be sensitive to ~ 2-3%
oscillation probability if v + v and ~ 10% if v e + n. Previous
e
T
results on v oscillations are:
e
BEBC expl
P(V + vel • 1.00±0.14
e
Los Alamos exp2
P(v + vel • 1.0910.40
e
Figure 1 shows a graphical comparison of these results and the
results on v~ + v experiments.
e
'I
Ve-V.
~t.
101L:""""!SEBC
}l
Q.
".
~
......
:.tJQ.
V~ -VJI
L.os
Alamos
v.
W·i;;Exfi'i:l
102
I
·1
~---.
Emu!•
....
CB
"T
-tGGM
! v~,
16'1",,,,
10- 1
•
, .
"""
10·'
I '
1.,.,,1
10-1
".
f
LoS
Alamos
_T_
"",1.
v,
t
•••'
10·'
LIE (m/MeV)
Figure 1) Oscillation probabilities for previous v and v experiments
as a function of LIE.
e
~