Asp ects of Neutrino
Physics in the Early
Universe
Ph D Thesis
Steen Hannestad
Institute of Physics and Astronomy
University of Aarhus
Octob er
Contents
Preface v
Intro duction
Primordial Nucleosynthesis
Thesis Outline
Particle Dynamics in the Expanding Universe
Global Evolution Equations
The Boltzmann Equation
Basic Big Bang Nucleosynthesis
Theory
Observations
Dirac and Ma jorana Neutrino Interactions
De nition and Prop erties of Neutrino Fields
Neutrino Weak Interactions
Mixed Neutrinos
Decaying Neutrinos
Neutrino decoupling in the Standard Mo del
Fundamental Equations
Numerical Results
Discussion and Conclusions
Massive Stable Neutrinos in BBN
Numerical Calculations
Mass Limits
Decaying Neutrinos in BBN
Necessary formalism
Numerical Results
Nucleosynthesis E ects
Conclusion iii
Neutrino Lasing
The Basic Scenario
The Level Laser
Numerical Results
Discussion
Conclusions
Outlo ok
Acknowledgements
A Mathematical Formalism
A Dirac Matrices and Their Algebra
A Dirac Spinors
A Creation and Annihilation Op erators
B Phase space Integrals
C English Resum e
Bibliography
Preface
The present thesis is a summation of the work carried out during my Ph D
study The bulk of the work has b een p erformed in collab oration with my su
p ervisor Jes Madsen at the Institute of Physics and Astronomy at the University
of Aarhus However some of the work has b een done during a six month stay
at the Max Planck Institut fur Physik in Munich during the spring and summer
of where I also had the pleasure of collab orating with Georg Ra elt in the
eld of sup ernova neutrino physics
Chapters contain an intro duction to various key issues connected with
neutrino physics and Big Bang nucleosynthesis Chapters are essentially
reprints of pap ers as describ ed b elow
Finally there is a thesis conclusion in Chapter and an outlo ok to p ossible
future work in Chapter There are three app endices at the end app endices
A and B contain various mathematical details and App endix C contains a brief
summary of the thesis work
Note that throughout I use natural units where c ~ k
B
List of Publications
I S Hannestad and J Madsen Neutrino Decoupling in the Early Universe
Published in Phys Rev D
I I S Hannestad and J Madsen Nucleosynthesis and the Mass of the Tau
Neutrino Published in Phys Rev Lett Erratum published
in Phys Rev Lett
I I I S Hannestad and J Madsen Nucleosynthesis and the Mass of the Tau
Neutrino Revisited Published in Phys Rev D
IV S Hannestad and J Madsen A Cosmological Three Level Neutrino Laser
Published in Phys Rev D v
vi Preface
V S Hannestad Tau Neutrino Decays and Big Bang Nucleosynthesis accepted
for publication in Phys Rev D
VI S Hannestad Can Neutrinos be Majorana Particles hep ph not
intended for publication
VI I S Hannestad and G Ra elt Supernova Neutrino Opacity from Inverse
Nucleon Nucleon Bremsstrahlung in preparation
The contents of pap er I have b een describ ed in detail in Chapter pap ers II
and III have b een summarised in Chapter pap er IV is contained in Chapter
and nally Chapter describ es pap er V Some of the contents of pap er VI
have b een included in Chapter Note that pap er VI I ab out sup ernova neutrino
physics has not b een included in the thesis The eld it covers is somewhat
disjoint from that of the thesis and for this reason I have left it out
Chapter I
Intro duction
One of the most succesful areas of mo dern physics is the interface b etween parti
cle physics and astrophysics This eld includes such phenomena as sup ernovae
active galactic nuclei and p erhaps most notably the early Universe It is also
a relatively new eld that has only existed for less than years so that there
are still big white sp ots left on the map making it a very interesting eld of
research
The present thesis deals with di erent asp ects of the physics of the parti
cles called neutrinos in the early Universe esp ecially in connection with the
formation of light elements
Neutrinos
Neutrinos are chargeless spin fermions b elonging to the family of leptons
There are three known sp ecies of neutrinos corresp onding to the three known
charged leptons They are group ed together in three generations
e
e
with corresp onding antiparticles These three generations corresp ond to the
three quark generations
u c t
d s b
It is not known whether there are additional generations of fermions but since
the fermion masses increase with generation any new generation would likely
b e extremely heavy In any case they would have to b e so in order not to have
b een detected in accelerator exp eriments
Neutrinos are unique among the known fermions in that they do not p ossess
electric charge meaning that they have no electromagnetic interactions at tree
level The only interactions that neutrinos have are weak interactions and since
Chapter Intro duction
the weak coupling constant G is very small neutrinos interact only very weakly
F
with the other fermions
Neutrinos can b e pro duced on earth for example in nuclear reactors How
ever the most constant neutrino source we have is the Sun where electron
neutrinos are pro duced by the nuclear burning of hydrogen via the reaction
p He e
e
yielding a neutrino ux of more than cm s on earth The most p owerful
neutrino sources known are the core collapse typ e I I sup ernovae They emit
a burst of neutrinos from the neutronisation reaction
p e n
e
lasting a few seconds During this p erio d roughly ergs are emitted in
neutrino radiation giving the sup ernova a neutrino luminosity comparable to
the optical luminosity of the whole visible Universe
Even though neutrinos are so abundant in the Universe their very weak
interactions mean that they are extremely di cult to detect in the lab oratory
Typically one needs many tons of detection material to achieve a detection rate
of a few p er day even with the huge neutrino ux coming from the Sun
For the same reason very little is known ab out the prop erties of neutrinos In
the standard mo del of particle physics neutrinos are strictly massless However
there is no a priori reason why this should b e so and in many extensions of the
standard mo del neutrinos do indeed have mass A great numb er of exp eriments
have b een made in order to constrain fundamental parameters like mass and
magnetic moment of the di erent neutrino sp ecies However most of these
exp eriments are ine cient b ecause of the very low counting statistics and this
is where indirect exp eriments enter the picture Neutrinos are thought to b e
of fundamental imp ortance in triggering the sup ernova explosions for example
Therefore observations of sup ernovae yield imp ortant information on neutrino
prop erties The other lab oratory where fundamental neutrino physics can b e
prob ed is the early Universe Here neutrinos are roughly as abundant as photons
and should therefore contribute signi cantly to for example the cosmic energy
density This in turn leads to stringent limits on parameters like neutrino mass
Neutrinos are also of fundamental imp ortance in the formation of light elements
in the early Universe Constraints on neutrino prop erties coming from the study
of light element formation form the bulk part of the present work and will b e
describ ed in much more detail later
Primordial Nucleosynthesis
For many years cosmology was a research eld strongly lacking in observational
input However during the last decade numerous improvements have b een
made on this For example the cosmic microwave anisotropies were discovered
in by the COBE satellite and small scale anisotropies are exp ected
Primordial Nucleosynthesis
to b e observed by the next generation of satellites exp ected to y within
years time This has op ened up a completely new window on the physics of
the early Universe since the physical comp osition of the Universe at the time of
creation of the background radiation is imprinted in these anisotropies Thus
a new precision determination of such essential parameters as the cosmological
density parameter and the Hubble parameter H can b e exp ected
Another eld under rapid development is that of large scale galaxy redshift
surveys This is primarily due to new telescop e technology and observational
techniques that automate the pro cesses that earlier had to b e done manually
In this way redshifts of millions of galaxies can b e obtained and the physical
structure of the Universe mapp ed out This again tells something ab out the
matter content of the Universe b ecause that a ects the way structures form
If the content of the Universe is mainly in the form of very light m eV
particles all imhomogeneities on small scales will have b een smeared out by the
free streaming of these particles This means that the rst structures to form are
very large roughly on the size of sup erclusters of galaxies Only subsequently
will these huge structures fragment to form galaxies This typ e of scenario is
called hot dark matter HDM The other p ossibility is that the matter content
is mainly in the form of very heavy particles These are almost at rest so that
no free streaming takes place and therefore the rst structures to form are of
subgalactic size From these subgalactic size initial structures larger and larger
structures are build This typ e of scenario is called hierarchical clustering or
cold dark matter CDM In Chapter we will go into a closer discussion of
these di erent mo dels
Both the ab ove techniques use the inhomogeneity and structure in our Uni
verse to say something ab out its comp osition There is a completely di erent
typ e of prob e that can b e used for this purp ose namely that of Big Bang nu
cleosynthesis These last few years have also seen strong improvements on the
determination of di erent nuclear abundances in the Universe and these pre
cision measurements have had a strong impact on Big Bang nucleosynthesis
Historically Big Bang nucleosynthesis was rst discussed by Gamow in
His idea was that as the Universe co ols down and expands nuclear reac
tions would pro ceed to build up abundances of the di erent nuclei Gamow
prop osed that essentially all the nuclei observed to day originated in the Big
Bang but it later b ecame clear that this is not p ossible The reason is that
there exist no stable elements with mass numb ers and and that heavier el
ements therefore must originate from three b o dy reactions like the triple alpha
pro cess that o ccurs in stars It turns out however that the numb er density of
particles at the time of nucleosynthesis is much to o low for such reactions to b e
e ective and that therefore essentially no nuclei heavier than A are formed
All the heavy elements must therefore have b een formed subsequently in stars
and sup ernovae However we also know that almost all of the helium in the
Universe to day must have b een formed in the primordial nucleosynthesis phase
as it turns out that only a very small fraction can have b een pro duced in stars
later on
Chapter Intro duction
QCD Phase n−p Inflation Transition Freeze−Out CMBR EW Phase Neutrino Transition Decoupling BBN Now
14 2 −1 −1 −4
10 GeV 10 GeV 100 MeV 2 MeV 0.8 MeV 10 MeV 10 eV 10 eV
Figure Timeline showing di erent imp ortant events during the evolution
of our Universe The scale shown is the temp erature of the photon plasma the
drawing is not to scale
The predicted abundances of light elements dep end strongly on the physical
comp osition of the Universe at the ep o ch of nucleosynthesis meaning that if
the primordial abundances can b e somehow determined observationally we can
prob e the Universe at very early times
In Fig we show a timeline marking some of the imp ortant events during
the evolution of the Universe and in the following we try go give a brief sketch
of the event taking place during the ep o ch of primordial nucleosynthesis At
a temp erature of roughly MeV the plasma of quarks and gluons undergo es
a phase transition to a gas consisting of hadrons Since there is some degree of
baryon asymmetry in the Universe some protons and neutrons remain of this
gas even at temp eratures much lower than the transition temp erature These
protons and neutrons are kept in thermo dynamic equilibrium via the reactions
until a temp erature of roughly MeV and since the time temp erature relation
is roughly
t s T MeV
this corresp onds to a time of roughly s During this p erio d the neutron to
proton ratio is determined by their mass di erence alone through the Boltzmann
factor
n
m m T
n p
e
p
When the temp erature drops b elow MeV however the weak reactions are
to o slow to maintain equilibrium and the proton neutron fraction remains xed
except for the free decay of neutrons At roughly the same time the weak
reactions keeping neutrinos in equilibrium with the cosmic plasma of photons
electrons and p ositrons freeze out and the neutrinos b ecome sterile
Primordial Nucleosynthesis
As the temp erature drops to MeV the free nucleons start forming light
nuclei in abundance Deuterium is formed rst but is quickly pro cessed into
He and the end result is that roughly of the baryons are in He and a much
smaller fraction on the level is in the form of nuclei like D He
and Li When the temp erature reaches MeV nuclear reactions freeze out
b ecause the numb er densities b ecome to o small and primordial nucleosynthesis
nishes
A theoretical calculation of the exp ected yields of di erent nuclei from Big
Bang nucleosynthesis is rather easily done once the input physics is known
The basic input that is needed to calculate the nucleosynthesis scenario is a
sp eci cation of the content of the Universe as a function of time or temp erature
By content we mean all particle sp ecies present and their relative abundance
as well as such more exotic comp onents as vacuum energy etc
Then if it is p ossible to determine somehow the primordial abundances from
observations this will put strong limits on the p ossible con guration of the
Universe during nucleosynthesis Unfortunately the main problem with using
nucleosynthesis as a prob e lies exactly in the observations What is observable
to day is primarily the element abundances in stellar atmospheres interstellar
clouds etc This means that the observations are p erformed in the lo cal Universe
which has an age of roughly years If one wants to relate the values observed
lo cally to the primordial values one has to take into account p ossible chemical
evolution since primordial nucleosynthesis to ok place The element comp osition
changes with time b ecause of nuclear burning in stars sup ernovae cosmic ray
pro cesses etc and therefore it is usually extremely di cult to pin down the
primordial values by observations
With the intro duction of a new generation of telescop es like the American
Keck telescop e and the Hubble Space Telescop e is has b ecome p ossible to study
light element abundances in much more distant systems Until now these ob
servations have concentrated on observing deuterium in what is called QSO
absorption systems These are huge clouds of gas lying in the line of sight to a
distant quasar When light from the quasar passes through such a system ab
sorption lines are pro duced by the material in the clouds and these can b e used
to determine element abundances in the absorbing cloud The advantage of this
is that the clouds lie at redshifts larger than z so that they are observed
at an earlier ep o ch than the present Universe Furthermore these absorption
systems are thought to b e systems that have not yet collapsed to form galaxies
and that there has therefore b een very little chemical pro cessing in them This
gives rise to the hop e that the abundances observed in such systems re ect the
primordial much more closely than those observed lo cally In Fig we show
an example of a sp ectrum obtained from this typ e of observation repro duced
from Ref Unfortunately the results that have hitherto b een obtained are
quite controversial and do not agree with each other The main conclusion must
b e that this typ e of observations is an extremely promising technique but that
it is at present to o early to use the results at face value Hop efully the coming
few years will resolve the issue
Chapter Intro duction
Figure Sp ectrum obtained by Burles and Tytler at the Lick observatory of
an absorb er at redshift z The strong dip to the left is the Lyman limit
This gure was reprinted from Ref
Now the standard mo del of particle physics together with the theory of gen
eral relativity and the so called cosmological principle stating that the Universe
should b e homogeneous and isotropic on large scales constitute what is known
as the standard mo del of cosmology Although there are still things that cannot
b e explained by this mo del like the initial density p erturbations needed to seed
galaxy formation and the asymmetry b etween particles and anti particles it
do es account for almost all the observed features of the physical Universe At
the time of nucleosynthesis the standard mo del is that of a plasma of photons
electrons and p ositrons in complete thermo dynamic equilibrium together with
three massless neutrino sp ecies that have completely decoupled from the elec
tromagnetically interacting plasma In addition to this there is a small amount
of baryons present but since the exact amount is unknown the baryon content
is usually expressed in terms of an adjustable parameter
n n
B
B
n
where the value of is roughly
Thesis Outline
One of the amazing things ab out Big Bang nucleosynthesis is that the pre
dicted yields are actually very close to the observed abundances of light nuclei
This can b e taken as evidence that the standard mo del is at least fundamen
tally correct It also puts very strong b ounds of any physics b eyond the standard
mo del during nucleosynthesis and gives us an opp ortunity to constrain such new
physics by means that are complementary to lab oratory exp eriments One ex
ample is the limit to the numb er of massless neutrino sp ecies present during
nucleosynthesis Here observations are able to give the constraint
N
whereas the numb er of light neutrinos measured at LEP from the decay of Z
is
N
There are numerous other such constraints coming from nucleosynthesis and
some of them will b e discussed in more detail in Chapters A classic exam
ple is the p ossibility for a heavy neutrino where the mass constraints from
nucleosynthesis can b e used to complement the accelerator limits This is very
imp ortant to exp erimentalists trying to measure directly the mass and lifetime
of the neutrino b ecause the nucleosynthesis constraints can b e used to exclude
large mass lifetime intervals thus making direct searches easier
Recently there has b een added a twist to this standard picture of primordial
nucleosynthesis As the observations have gradually improved over the years
there has b een growing evidence that something may not b e quite right after
all The trouble is that it app ears that the amount of helium predicted by BBN
calculations is somewhat larger than what is observed This has led some to
the conclusion that standard BBN is wrong and that some new element must
b e added However there still seems to b e quite large systematic uncertainties
in the helium observations so that it may after all turn out that the discrepancy
is due to some observational feature rather than new physics Still it may also
turn out that there is indeed such a nucleosynthesis crisis Therefore it is of
considerable interest to explore ways of changing the primordial abundances
relative to the standard value One such p ossibility is the decay of a heavy
tau neutrino during nucleosynthesis a p ossibility discussed in more detail in
Chapter
Thesis Outline
The remainder of the thesis is structured in the following way Chapter con
tains a review of the dynamics of an expanding Universe as well as a discussion
of the Boltzmann equation that describ es particle interactions b etween di erent
sp ecies Chapter deals with the concepts of Big Bang nucleosynthesis theory
Furthermore it contains a discussion of the observational asp ects In Chapter
we review the de nition of Dirac and Ma jorana neutrinos and discuss the dif
ferent p ossible interactions of neutrinos from the standard weak interactions to
Chapter Intro duction
neutrino decays Armed with the basic physical concepts intro duced in Chap
ter we pro ceed in Chapters to apply the formalism to actual problems
Chapter is ab out neutrino freeze out in the standard mo del Chapter deals
with massive but stable neutrinos in connection with primordial nucleosynthe
sis By use of nucleosynthesis arguments the tau neutrino mass is constrained
to b e b elow roughly MeV if it is stable during nucleosynthesis In Chapter
we go one step further to deal with p ossible neutrino decays during Big Bang
nucleosynthesis It is shown that such decays can lead to signi cant changes in
the primordial abundances Chapter deals with a somewhat di erent asp ect
of neutrino physics in the early Universe namely that of neutrino lasing This
e ect comes from the out of equilibrium decay of neutrinos to a nal state that
contains b osons Due to the stimulated emission factors in the phase space in
tegrals for b osons such decays can pro duce large amounts of very cold b osons
resembling a condensate Chapters and contain a thesis conclusion and an
outlo ok to p ossible future research resp ectively Finally there are three app en
dices the rst two containing mathematical detail and the third containing an
english resum e of the thesis work
Chapter II
Particle Dynamics in the Expanding Universe
Global Evolution Equations
Our present understanding of the evolution of our Universe is based on what is
called the Standard Hot Big Bang mo del There are two essential assumptions
that go into this mo del namely that
Einsteins theory of general relativity is correct
The Universe is homogeneous and isotropic on large scales R Mp c
This is usually referred to as the cosmological principle
The most general structure of the line element ds then has the well known
Rob ertson Walker shap e
dr
dt r d r sin d ds R t
k r
where R is a quantity with dimension l R is usually referred to as the scale
factor and it is a measure of the expansion of the Universe The quantity k
measures the sign of gaussian curvature of the Universe so that it can take the
values k The global evolution of a homogeneous and isotropic Universe
is therefore describ ed in terms of R and k alone
Using the two ab ove assumptions the Einstein equation
G GT
reduces to two scalar equations in R and k the comp onent giving the
Friedmann equation
k G R
R R
Chapter Particle Dynamics in the Expanding Universe
and the i i comp onent giving the equation
k R R
GP
R R R
The two equations Eqs and assume a p erfect uid form of the
stress energy tensor
T U U P g U U
where and P are the cosmic energy density and the pressure resp ectively
The second of the two ab ove equations Eq involves the second deriva
tive of the scale factor R It would b e of interest to nd instead an equation
involving only the rst derivative There are basically two ways to establish the
relation b etween time temp erature and scale factor in the early Universe during
the ep o ch of nucleosynthesis using only rst order equations The simplest way
to do it is via entropy conservation
As long as there is thermo dynamic equilibrium or all distribution functions
are describ ed by equilibrium distributions as in the case of totally decoupled
massless neutrinos entropy is conserved
This means that
P R
dS d
T
where and P are the total energy density and pressure contributions from all
particle sp ecies present This conservation equation can b e used to calculate
the relation b etween the scale factor and the photon temp erature
In the following we shall assume that the neutrinos are completely decou
pled from the electromagnetic plasma long b efore the electrons and p ositrons
annihilate This only intro duces a small error in the nal results An error that
can b e quanti ed by use of the more correct equation of energy conservation
that is describ ed b elow
We can then rewrite the entropy conservation equation in the following way
Without loss of generality we can put R t where t is the starting p oint
of the calculation By using T R we then get
P
e e
C onstant T T t R R
T
where the constant can b e determined at the initial time Thus we now have
the scale factor as a function of photon temp erature but in order to calculate
it as a function of time we parametrise it as R t f t t where f t is the
deviation from the standard expansion of a radiation dominated Universe We
1
Eq can b e used to calculate the nal temp erature of completely decoupled neutrinos
1 3
The result is T T If Boltzmann statistics is used for all particle sp ecies Eq
2 2
+P
4 4
3 3 3 3
e e
reads R T T t R C onstant and the result is T T
0
T 3 15 3 15
1 3
Global Evolution Equations
make the following go o d approximation f df dt t and the Friedmann
equation Eq then takes the form
G dR
t
dt R
This means that we can calculate t T and thus R t We have now established
a relationship b etween time temp erature and scale factor
However although the approximations made are very go o d for massless neu
trinos this is not necessarily the case for massive neutrinos In this case there
can b e signi cant deviations from the ab ove equations
A pro cedure that do es not involve any approximations on this level is to use
energy conservation From the conservation of stress energy
T
one nds the following equation if one assumes a p erfect uid form of the stress
energy tensor
d d
R p R
dt dt
where again and P are the total energy density and pressure contributions of
all particles This equation is an expanding Universe equivalent of the st law
of thermo dynamics
However we are not necessarily dealing with a p erfect uid Only if all
particle distributions are describ ed by equilibrium distribution functions or all
particle sp ecies are either massless or completely non relativistic not a mix of
relativistic and non relativistic particles will the uid b e p erfect Since we are
considering the p ossibility of non equilibrium pro cesses with semi relativistic
particles there is no reason to assume that the p erfect uid approximation
should b e valid
Phenomena such as heat conduction shear viscosity and bulk viscosity may
o ccur in such a scenario However Weinb erg has shown that in a homoge
neous and isotropic Universe only the bulk viscosity can b e non zero
Now in all that follows we shall assume that all particle sp ecies are describ
able by single particle distribution functions This approach of course assumes
that the particles do not interact apart from instantaneous collisions so that a
single particle distribution function makes sense at all However we shall always
use this approach in the present work For massless neutrinos it is de nitely
justi ed but for massive neutrinos that can oscillate this is not necessarily the
case Furthermore we shall assume that all phase space distributions are
homogeneous in co ordinate space and isotropic in phase space
f x p t f p t
To see the e ect of bulk viscosity we pro ceed in the following way We write
the distribution functions as
f f
i i i
Chapter Particle Dynamics in the Expanding Universe
f is an equilibrium distribution of the normal form
i
f
i
E
i
exp
T
i
and parametrises the deviation from equilibrium There is however an
i
ambiguity in the de nition of f as there are two unknown parameters and
i i
T These are usually xed by letting
i
Z Z
f d p f E d p
i i i i
so that and n n The pressure is not determined by these
i i i i
conditions We still have
Z
d p p
f P
i i i
E
Now if the s are not to o large as would b e exp ected unless some very extreme
i
pro cesses were o ccuring the deviation from a p erfect uid can b e describ ed by
rst order term in the stress energy tensor T
T g U U U
where is called the co e cient of bulk viscosity The total stress energy tensor
must then assume the form
R
T U U P g U U
R
R
g U U U U P
R
On the other hand the spatial comp onents of the stress energy tensor are
Z Z
X X
d p d p p p
aa
P T f f
i i i i
E E
i i
i
Since we are in a comoving co ordinate system U and U This gives
Z
X
d p R p
aa
T P P f
i i
R E
i
or
Z
X
p R d p
f
i i
E
R
i
2
When dealing with general relativistic tensors Greek letter denominate all four spacetime
comp onents whereas Roman letters only denominate the three spatial comp onents
Global Evolution Equations
The energy conservation equation with bulk viscosity is then
R d
R P R R
dt R
d
R P
TOT
dt
Altogether then the energy conservation equation is unchanged if we just use
the total pressure de ned as
Z
X
d p p
P f
TOT i
E
i
As noted by Fields et al bulk viscosity can b e a p otentially serious prob
lem if one uses the standard way of solving the Boltzmann equation namely
by assuming complete kinetic equilibrium at all times that is working only
with distributions of the f form Using our metho d bulk viscosity is not a
i
problem b ecause we use the true distribution functions f directly without any
i
assumptions Thus when we calculate the pressure P the bulk viscosity is au
tomatically taken into account We can then still use the conservation equation
in our non equilibrium case
We now intro duce the quantity r ln R in which case the conservation
equation can b e rewritten as
d
dr
dT
dT P
In addition to this we again use the Friedmann equation
r
G dR
R
dt
or in terms of r
p
dr
G
dt
Combining the two equations we get
dT dr dt
dt dr dT
We thus have the two fundamental equations and describing
the time evolution of temp erature and scale factor without any approximations
One should p erhaps also note here that the problem of bulk viscosity do es
not app ear in the Friedmann equation since it only involves the energy density
and not the pressure
3
Note that we have actually made an approximation by assuming that a rst order bulk
viscosity term is su cient If the distributions deviate largely from kinetic shap es a rst
order term might not b e adequate Of course there is also still the approximation of a single
particle distribution function which may not b e adequate either
Chapter Particle Dynamics in the Expanding Universe
The Boltzmann Equation
The formalism describ ed ab ove describ es fully the global expansion of the Uni
verse assuming knowledge of and P as functions of time For a completely
non interacting plasma these quantities are easy to calculate but since the early
Universe consists of a plasma of di erent interacting sp ecies the situation is not
so simple We need a transp ort equation that describ es the coupling of di er
ent particle sp ecies so as to calculate their relative abundances etc Such an
equation together with the evolution equations ab ove is enough to at least in
principle describ e the whole evolution of the Universe The equation needed
for this purp ose is the Boltzmann transp ort equation
L f C f C f
coll dec
for the evolution of the single particle distribution function f of a particle
sp ecies In a homogeneous and isotropic Universe the Boltzmann equation as
sumes a fairly simple form the left hand side of the ab ove equation is the
Liouville op erator in an expanding Universe
dR f f
p L f
t dt R p
The right hand side of Eq involves the sp eci c physics of the decay
pro cess as well as p ossible other collision pro cesses such as elastic scattering and
annihilation This set of equations together with the two equations Eqs
and in principle allows us to follow the evolution of all particle sp ecies
as well as the global expansion of the Universe However these equations are
usually tremendously di cult to solve except for a few sp ecial cases For that
reason one usually resorts to approximations for the Boltzmann equation If we
lo ok at su ciently early times where the electromagnetic interaction timescale
is very short compared to the expansion timescale of the Universe T eV we
can always consider electromagnetically interacting particles such as photons
and e as b eing in p erfect thermo dynamic equilibrium
For particles that do not interact strongly but are not completely decoupled
either such as neutrinos the situation is much more complicated If we restrict
ourselves to lo ok only on particle reactions that is reactions like pair anni
a b b or elastic scattering a b a b the rst term on the hilation a
right hand side C can b e written as a dimensional phase space integral
coll
Z
C f d p d p d p f f f f
coll
E
p p p p S j M j
where f f f f f f f f f f f f is the
phase space factor including Pauli blo cking of the nal states and d p
d p E S is a symmetrisation factor of for each pair of iden
tical particles in initial or nal states and j M j is the weak interaction
The Boltzmann Equation
matrix element squared and appropriately spin summed and averaged p is
i
the four momentum of particle i The standard way of simplifying the Boltz
mann equation for this case is to assume that elastic scattering reactions are
always much more imp ortant than numb er changing reactions If this is the
case one may assume that the given sp ecies is in kinetic equilibrium and that
its distribution functions is of equilibrium shap e with an added pseudo chemi
cal p otential This allows us to lo ok only on the numb er changing reactions
since the elastic scattering integrals all b ecome zero for distributions in kinetic
equilibrium kinetic equilibrium is sometimes also referred to as scattering equi
librium If one furthermore assumes Boltzmann statistics for all the particle
sp ecies present
E T z
f p e
where z is the pseudo chemical p otential it is p ossible to arrive at a relatively
simple set of evolution equations one for each sp ecies present by integrating
the Boltzmann equation
X
h v i n n n H n
ij i i
i eq i
j
where n is the thermo dynamic equilibrium numb er density and the sum is
eq
over all annihilation channels h jv ji is the thermally averaged cross section for
the numb er changing reactions
p
Z
p
s
CM
s s m K s ds h jv ji
ij
ij
m TK m T T
2
m
where it has b een assumed that all annihilation pro ducts are massless The K s
i
are mo di ed Bessel functions of the second kind and s is de ned in the standard
way as s p p with p b eing the four momentum of particle i
i
A step up from this approach is to relax the assumption of Boltzmann statis
tics which was rst done by Kainulainen and Dolgov and subsequently de
velop ed further by Fields Kainulainen and Olive In this case the particle
distributions are of the form
f p
E T z
e
This step is done at the exp ense of not b eing able to write the right hand side
of the Boltzmann equation in a simple form However since errors induced by
using Boltzmann statistics instead of Fermi Dirac or Bose Einstein are typically
of order it may b e necessary to use the full statistics The Boltzmann
equation then takes the form
X
n H n C
i i ij
In this case the left hand side is
T
i
n H n H J x z x J x z
i i i i i i
i
Chapter Particle Dynamics in the Expanding Universe
dz T
i i
J x z T J x z
i i i i
T dT
i
where x m T and
i i
p
Z
2 2
x y z
e
n
p
J x z dy y x y
n
2 2
x y z
e
The right hand side is expressed as an integral of the same typ e as Eq but
integrated over momentum space for the incoming particle This integral
must then b e evaluated numerically Even for this more complicated equation
the evolution of particle sp ecies is relatively easy to follow numerically b ecause
one only needs to follow the functions z T which are functions of temp erature
i
alone
However most of the present work deals with p ossible deviations from these
simple equations Eq and Eq If the distribution functions deviate
signi cantly from kinetic equilibrium one is forced to use the full Boltzmann
equation which is numerically rather complicated b ecause one needs to use a
grid in momentum space and solve for the distribution function on this grid
meaning that e ectively one has to solve a huge numb er of coupled di erential
equations App endix B is devoted to describing the analytical integration of the
collision terms for the case of neutrinos interacting through the weak interaction
We now return to the original Boltzmann equation and lo ok at the second
term on the right hand side C In general this term will have the same form
dec
as the collision term Eq namely a large phase space integral of the
involved matrix element However we shall only need to lo ok at the case where
the decay is to a two particle nal state a b c We also restrict ourselves to
lo ok only at decays isotropic in the rest frame of the parent particle In this case
the collision term can b e simpli ed tremendously b ecause the decay strength is
fully describ ed by the rest frame decay lifetime The decay sp ectrum is then
determined by kinematics alone To b e sp eci c we lo ok at the decay
H F
where H and F are b oth fermions and is a scalar particle The decay terms in
the Boltzmann equation have b een calculated by Starkman Kaiser and Malaney
to b e
+
Z
E
m
H
C f dE f f f
dec H H F
m E p
H H
E
+
Z
E
H
m g
H
H
dE f f f C f
H H F dec F
g m E p
F F F
E
H
+
Z
E
H
g m
H
H
C f dE f f f
dec H H F
g m E p
E H
The Boltzmann Equation
where f f f f f f f f f and m m
H F H F F H
H
m m m m m is the lifetime of the heavy neutrino and g
F F H
is the statistical weight of a given particle The integration limits are
m m
H
E m m E E
i i i
H
m
i
E m
i i
and
m
E m m p E E
H i H H
i
m
H
where the index i F
The formalism presented in this chapter gives us all the equations needed to
follow the evolution of neutrinos in the early Universe that will b e discussed in
Chapters
Chapter Particle Dynamics in the Expanding Universe
Chapter III
Basic Big Bang Nucleosynthesis
Theory
At high temp eratures the baryon content of the Universe is in the form of a
free quark gluon plasma However at a temp erature of MeV this plasma
undergo es a phase transition to a gas of b ound quark systems At rst this
plasma consists almost solely of pions but as the temp erature drops the pions
disapp ear and the gas then only consists of neutrons and protons that are left
over b ecause of the baryon asymmetry of our Universe These protons and
neutrons are kept in thermo dynamic equilibrium by the reactions
n p e
e
n p e
e
n e p
e
n e p
e
During this era the ratio of protons to neutrons is determined by the Boltzmann
factor
n
exp m m T
n p
p
alone Now since the rate for these interactions is roughly
G T
F
and the expansion timescale of the Universe is
r
H
H
G
which is prop ortional to T in a radiation dominated Universe one arrives at
the relation
T MeV
H
Chapter Basic Big Bang Nucleosynthesis
so that at a temp erature of roughly MeV the average time for n p conversion
will b e longer than the expansion time of the Universe At this p oint the con
versions can no longer maintain equilibrium and the neutron to proton fraction
freezes out From this p oint on the ratio of neutrons to protons remains con
stant except for the change due to free neutron decays However shortly after
this freeze out o ccurs all free neutrons are b ound in nuclei thereby preventing
further neutron decay
Because of the high entropy of the Universe free nucleons are preferred down
to very low temp eratures only at T MeV do es deuterium pro duction
commence This deuterium is quickly pro cessed into He but at higher A very
little happ ens The reason for this is easy to understand Because formation
of nuclei is suppressed down to very low temp eratures the numb er density is
very low so that triple reactions are unlikely Furthermore the Coulomb barrier
against nuclear reactions is typically very high compared with typical thermal
energies and there exist no stable A and A nuclei Small amounts of Li
Be and B are pro duced but b eyond this essentially nothing In Fig we show
the primary reaction network for primordial nucleosynthesis and in Fig we
show the build up of light nuclei as the temp erature drops
The nal abundances stemming from BBN dep end sensitively on the baryon
to photon ratio
n n
B
B
n
Therefore BBN in principle allows us to determine if the primordial abun
dances can somehow b e determined Furthermore the outcome esp ecially the
abundance of He dep ends on the numb er of particle sp ecies present during
nucleosynthesis The reason can b e seen from the Friedmann equation
G
H
If the numb er of particle sp ecies is increased during nucleosynthesis this in
creases and thereby H for given temp erature This again means that H
is decreased so that the weak interactions decouple earlier than usual Thus
increasing during nucleosynthesis has the e ect of leaving more neutrons to
survive than normally meaning that in the end more helium is formed There
fore one can use Big Bang nucleosynthesis to put limits on novel physics by
limiting the numb er of particle sp ecies that can b e present during nucleosyn
thesis In Fig we show the primordial abundances after nucleosynthesis
b oth as functions of the baryon to photon ratio and of the equivalent numb er of
neutrinos N This quantity is just a measure of the energy density present
eq
in neutrinos and all other non electromagnetic interacting particles during nu
cleosynthesis and is de ned as
P
X
N
eq
0
Theory
7 Be
7 Li
3 4 He He
3 p D H
n
Figure The most imp ortant reactions for primordial nucleosynthesis
where indicates a standard massless neutrino sp ecies X indicates novel
particles from physics b eyond the standard mo del that may b e present during
nucleosynthesis such as ma jorons axions etc
Finally it should also b e mentioned that nucleosynthesis is very sensitive
to the electron neutrino distribution b ecause this enters directly into the
conversion rates through the pro cess rate integrals of typ e
Z
f E jM j E f
e e n pe
e e e
n pe
e
d p d p p p d p p p
e n p e n
e e
where d p d p E Thus one can also put strong limits on physical
i i i
pro cesses that change the electron neutrino distribution such as massive or
unstable neutrinos This is a p oint we shall return to in more detail later
Chapter Basic Big Bang Nucleosynthesis
Figure The mass fraction of He as well as the numb er densities of di erent
light nuclei relative to H as a function of cosmic temp erature
Observations
A very imp ortant but also very di cult part of this game is the determination
of primordial abundances from observations in the present day Universe The
observations are made di cult by the fact that often the abundances of the
relevant elements such as lithium are extremely small However even more
di cult is the interpretation of the results The trouble is that the present day
Universe has evolved since the Big Bang Most of the material in the present day
Universe has b een pro cessed through stars and sup ernovae changing its chem
ical comp osition We will try to give a very brief resum e of the observational
status of the most imp ortant Big Bang nucleosynthesis elements
The observation of He is relatively easy in the sense that the abundance of
He is large in all astrophysical systems However since the primordial helium
abundance is only weakly logarithmically dep endent on it is also necessary to
pin down the primordial fraction to very high precision if one wants to measure
the baryon to photon ratio Since He is extremely stable it is very di cult to
destroy and one can assume that it is only pro duced not destroyed We also
know that no elements with A are pro duced in the Big Bang Therefore
Observations
Figure Relic abundances of di erent nuclei as a function of the baryon to
photon ratio n n for di erent values of N The full lines
B eq
indicate the standard value N the dotted ones indicate N and the
eq
dashed ones N
eq
Chapter Basic Big Bang Nucleosynthesis
Figure Observed abundance of He in a numb er of extragalactic HI I regions
as a function of metallicity The gure has b een repro duced from Ref
if one nds a system with very low abundance of these heavy elements relative
to the solar values it should indicate that very little pro cessing has taken place
and one may hop e that the measured helium abundance in these systems re ects
the primordial abundance closely These observations are typically done in large
extragalactic HI I regions In Fig we show a typical example of such a set
of observations The di culties in pinning down the helium mass fraction
Y m He m H
P
from observations should b e obvious lo oking at the gure At present there are
two pap ers that re ect the most up to date observational constraints Unfortu
nately these two pap ers do not agree with each other on the value found First
there are the results of Olive and Steigman who analyse existing data from
extragalactic HI I regions Doing this they nd a primordial helium value of
Y
P
where the rst uncertainty is of purely statistical nature and the second is an
estimated systematical uncertainty The results of Izotov et al predict a
much higher value for the primordial helium abundance
Y
P
In Fig we show the results of Izotov et al corresp onding to those shown in
Fig
Observations
Figure The results of Izotov et al for the abundance of helium The
gure has b een repro duced from Ref
There are two problems in b oth these results First of all the usual way of
getting the primordial value for helium is to do a linear t to the observational
data and interpret the y axis crossing p oint as the primordial abundance This
assumes that a linear t makes sense at all which is by no means obvious The
other problem is the lack of very metal p o or HI I regions Ideally one should
nd just one extremely metal p o or ob ject and use the value obtained for helium
in this system as the primordial one This would eliminate the need for a linear
t using more metal rich ob jects but has unfortunately not b een p ossible so
far Thus it is not clear what the primordial abundance of helium actually is
Hop efully the coming years will show an increase in the numb er of metal p o or
ob jects known so that a more precise value can b e found There are several
ongoing pro jects with this goal
For deuterium the observations used to b e done in the solar neighb ourho o d
However since D is very fragile and no known pro cess can pro duce signi cant
amounts of deuterium these observations give a lower limit to the primordial
abundance The values that have b een obtained from such observations are
typically
D H
In the last few years it has b ecome p ossible to observe deuterium in quasar
absorption systems at signi cant redshift Since these systems have evolved very
little since the Big Bang one can hop e to pin down the primordial deuterium
abundance precisely Unfortunately the results so far are very controversial
Some of the observations indicate a high deuterium fraction in these clouds
Chapter Basic Big Bang Nucleosynthesis
around
D H
However there are other observations mainly those by Tytler et al that
indicate a relatively low deuterium value
D H
which is more consistent with the lo cal value This low value seems more reli
able since the analysis by Tytler et al has b een extremely careful Nevertheless
only a few systems have so far b een observed and more observations are greatly
needed It also app ears that there are problems in accomo dating a high pri
mordial deuterium value in mo dels of chemical evolution At present it thus
seems very di cult to draw any conclusions ab out the primordial value of D
He has b een used together with D in lo cal observations to put an upp er limit
on the combined abundance of D and He The argument here is that it seems
di cult to destroy D without at the same time pro ducing He However these
results dep end strongly on chemical evolution mo dels and therefore seem not
to o reliable either
For Li the situation is also quite muddy Until the early s it was thought
imp ossible to determine the primordial Li abundance from observations b ecause
Li is b oth pro duced and destroyed in astrophysical environments However in
results app eared by Spite and Spite indicating that in a certain group
of very old metal p o or halo stars the Li abundance is constant and do es not
dep end on metallicity This abundance is then interpreted as the primordial
abundance sub ject to rather small evolution e ects compared with normal main
sequence stars The most recent published determination of the Li abundance
gives a value of
Li H stat syst
but the systematical uncertainties are of course di cult to estimate since they
involve chemical evolution The trouble is that there may still have b een sig
ni cant depletion even in the plateau stars so that the value observed there is
smaller than the primordial value by some unknown factor It therefore seems
premature to use Li measurements to say anything ab out the value of In
Fig we show a typical measurement of the Li abundance in a group of old
halo stars At high masses almost all Li remains whereas at lower masses the
convection zone extends deep enough to disso ciate Li
In conclusion there is only one element namely He whose primordial abun
dance is known to any real precision However the trouble is that even here
there is controversy as to the primordial abundance so that a precision deter
mination of from the observational value of Y is not p ossible However the
abundance is still known su ciently well that one may put rather stringent
limits on new physics At present one may fairly certainly conclude that
N
eq
Observations
Figure The abundance of Li in old halo stars shown as a function of
surface temp erature The Spite plateau is clearly seen to the left at high surface
temp eratures The gure has b een repro duced from Ref
so that only the equivalent of one extra neutrino sp ecies may b e present during
BBN This b ound allows for a fairly large uncertainty in the observational pa
rameters However the value of N favoured is much lower than this even
eq
somewhat b elow The reason for the low value is that the primordial he
lium fraction observed is to o low compared to the theoretical predictions This
may p oint to a coming crisis for standard Big Bang nucleosynthesis so that
new physics like decaying neutrinos must b e invoked to bring consistency At
this p oint however it is probably to o early to say that there is any real crisis
for the standard mo del of primordial nucleosynthesis Finally if the primordial
value of deuterium is ever determined to within a few tens of p ercent this will
pin down the value of very accurately b ecause the predicted D abundance
is a very steeply varying function of Until then it is very likely that not
much progress will b e made in determining the baryon to photon ratio using
Big Bang nucleosynthesis Nevertheless as previously mentioned there now
seems to b e some real hop e that it will actually b e p ossible to measure the
primordial deuterium abundance within the foreseeable future
Note that in Chapters nucleosynthesis limits will b e used several times
to constrain mo dels of neutrinos However for historical reasons the limits
used in the di erent chapters do not necessarily coincide but rather re ect the
Chapter Basic Big Bang Nucleosynthesis
observational status at the time where the sp eci c pap er was written None
of the conclusions change in any signi cant way by using the most up to date
observational values
Chapter IV
Dirac and Majorana Neutrino Interactions
Neutrinos are an integral part of the standard mo del of particle physics They
are spin leptons without electromagnetic charge As far as we know they
are also massless although there is no fundamental reason why this should b e so
In fact there are a numb er of indications that neutrinos have non zero masses
These all come from the fact that if neutrinos are massive the di erent avour
states can mix b ecause these avour states do not necessarily corresp ond to
mass eigenstates
The rst piece of evidence is from the solar neutrinos Neutrinos emitted
from the Sun through the pp reaction can b e observed on earth using di erent
techniques A common feature of all these exp eriments is that fewer neutrinos
than exp ected from standard solar mo del calculations are seen It also app ears
that there is no astrophysical solution to this problem in the sense that no
solar mo dels have b een able to solve this discrepancy in a satisfactory way The
trouble is that the neutrino emission is very sensitive to the central temp erature
of the Sun but if that is changed in order to explain the neutrino ux other
visible e ects will o ccur The explanation that seems most likely is that the
electron neutrinos emitted from the Sun mix with muon neutrinos which are
not observed on earth so that this app ears as an apparent de cit in the numb er
of neutrinos emitted
The second place where evidence for neutrino mass is thought to b e seen
is the atmospheric neutrinos These are pro duced by cosmic rays that
interact with nuclei in the atmosphere to pro duce and the corresp onding
e
antineutrinos Water Cerenkov detectors like Sup er Kamiokande see a strong
de cit in the numb er of the observed ratio compared with the exp ected
e
ux and this can again b e taken as evidence for neutrino oscillations The last
indication is from the accelerator exp eriment at the LSND facility In this
case evidence for neutrino oscillations is claimed from a b eam
There has b een a large numb er of terrestrial exp eriments lo oking for neutrino
masses For the endp oint of the tritium decay sp ectrum is used to lo ok for
e
a non zero mass b oth b ecause the decay is very simple and the Q value is very
low Q keV A small neutrino mass should show up as a de cit
Chapter Dirac and Ma jorana Neutrino Interactions
in the electron sp ectrum near the endp oint energy However it turns out that
there is apparently a small surplus of electrons in this region corresp onding to
a negative neutrino mass squared This mystery has b een tried resolved for
example by invoking interactions with cosmic background neutrinos but this
e ect should b e to o small by many orders of magnitude to explain the observed
e ect So far no go o d explanation has b een pro duced
For the pion decay has b een used to constrain the mass
This is done by measuring the decay energy sp ectrum of the decay pro duct
muons
To measure the mass decay of leptons is used The sp eci c decay is
and the measured quantities are the invariant mass as well as the
energy of the hadronic decay pro ducts Altogether the present upp er mass
limits are
eV m
e
keV m
MeV m
De nition and Prop erties of Neutrino Fields
Only two physical neutrino states are known the left handed neutrino and
the right handed anti neutrino The other fermions are known to b e Dirac
particles having b oth left and right handed particles and antiparticles but
p ossibly neutrinos are describable in a theory containing only two comp onents
We distinguish b etween these two fundamental ways of building the theory of
neutrinos the four comp onent Dirac neutrino and the two comp onent Ma jorana
neutrino Let us start out from the de nition of the neutrino elds The Dirac
eld is de ned in the same way as for the other fermions
Z
h i
X
d p
y
ip x ip x
f u e f v e
s s s D
s
E
s
where u and v are the plane wave solutions to the Dirac equation for p ositive
and negative energy resp ectively
p m u p
s
p m v p
s
y
f is the particle annihilation op erator and f is the creation op erator for an
s
s
tiparticles
f p jp si j i
s
y
f p j i jp si
s
1
This section b orrows heavily from Ref
De nition and Prop erties of Neutrino Fields
We use the standard normalisation of the Dirac spinors
y
0 0
p E u p u
ss s
s
y
0 0
p E v p v
ss s
s
y
0
p v p u
s
s
The Ma jorana eld on the other hand is de ned using only two comp onents
Z
X
d p
ip x y ip x
f u e f v e
D s s s
s
E
s
where is a phase called the creation phase factor It can b e chosen at will but
it is not always advantageous to cho ose it as For this reason it is usually
kept as a free parameter to de ne The obvious di erence b etween Dirac and
Ma jorana elds is that the Ma jorana eld is de ned by only two comp onents
Essentially this must also mean that a Ma jorana particle is its own antiparticle
To show this we need a few de nitions We start with a spinor eld x
Under a Lorentz tranformation
x x x
the spinor eld transforms as
i
x x exp
where
i
To de ne a conjugate eld that transforms in this way it is not enough to use
The reason is that do es not transform in the same way as under
Lorentz transformations instead it ob eys the transformation law
0
i
x exp
which is not the same as Eq We de ne the conjugate eld as
C
where by we mean the column vector
y
B C
y
B C
B C
y
A
y
Chapter Dirac and Ma jorana Neutrino Interactions
C is some op erator which will in a moment b e identi ed with the charge
conjugation op erator For consistency C must ob ey the condition
C C
So far we have not discussed things in a sp eci c representation basis but we
now shift to the Dirac representation in which
I
i
i
I
i
where are the Pauli matrices In this basis we can cho ose C i so that
i
it ob eys the condition Eq Under the op eration of charge conjugation
any fermion eld transforms as
C C C
C
where is a phase factor Let us now check the e ect of using our op erator
C
C on the Dirac eld to see that we can indeed make the identi cation C C
We do this by assuming C C and from there show that this identi cation
makes sense The right hand side of Eq gives
C i
C C
and if we use the plane wave expansion
Z
X
d p
ip x y ip x
f C u e f v e
s s
C C s s
E
s
since
u v
v u
On the other hand the left hand side gives
Z
h i
X
d p
y
ip x D ip x
C f C u e C C C f C v e
s s s
s
E
s
Putting the two equations together we obtain the following relations for the
creation and annihilation op erators
C f C f
s
C s
y
y
f C C f
C
s s
The e ect of C on a state jp si is then
C jp si jp si C
Neutrino Weak Interactions
Altogether then this means that the e ect of C is to change a physical state
into a corresp onding state for the antiparticle justifying our identi cation of
the op erator C with the charge conjugation op erator C
Finally we are now ready to apply the same pro cedure to the Ma jorana
eld Here one nds that
M M
C C
C
where we have used our plane wave de nition of the Ma jorana eld Eq
The Ma jorana eld is thus apart from a phase factor invariant under charge
conjugation In exactly the same way as for the Dirac eld one may show that
the following relations hold for the creation and annihilation op erators
C f C f
s C s
y y
C f C f
C
s s
If we now apply this to a free Ma jorana state we get
C jp si jp si
C
so we see that a free Ma jorana state is an eigenstate of charge conjugation
Similar relations apply for CP and CPT sp eci cally for CPT one nds the
relation
s
CPT jp si jp si
CPT
where is again a phase This indeed means that the left and right handed
CPT
states are particle and anti particle in the usual sense since application of the
CPT op erator on a left handed Ma jorana state turns it into a right handed
state with the same momentum
Neutrino Weak Interactions
Let us now lo ok at the interactions of neutrinos to see the di erence b etween
Dirac and Ma jorana neutrinos At su ciently low energies neutrinos interact
via a current current interaction b ecause of the large masses of the gauge b osons
W Z so that the weak hamiltonian density to lowest order in the coupling
constant has the form
G
F
p
j x j x H
weak
This interaction is describ ed by the standard V A theory In the present work
we are only interested in purely leptonic interactions meaning that the terms
we are interested in are the leptonic charged current
X
CC
j x x
i
i e
Chapter Dirac and Ma jorana Neutrino Interactions
the charged fermion neutral current
X
NC
j x g g x
V A i
i
e
and the neutrino neutral current
X
NC
x x j
i
i
e
g and g are the charged fermion vector and axial vector neutral current cou
V A
plings resp ectively They are given in terms of the Weinb erg angle sin
W
as
C sin
V W
C
A
The S matrix is then given by
Z
S i d xH
weak
and from this all matrix elements are calculable
There is no a priori reason why Dirac and Ma jorana neutrinos should b ehave
the same under these interactions However b ecause of the left handedness of
the weak interactions it turns out that interactions that are di erent for Dirac
and Ma jorana neutrinos always scale as some p ower of neutrino mass This
means that for massless neutrinos Dirac and Ma jorana neutrinos are indistin
guishable However if neutrinos are massive Dirac and Ma jorana neutrinos
b ehave di erently For example there is no conserved lepton numb er if neutri
nos are Ma jorana particles b ecause neutrinos can ip their spin direction and
thereby violate lepton numb er by two units If we lo ok at Ma jorana neutri
nos it is straightforward to show that they p ossess no neutral current vector
interactions This is true b ecause of the relation
c
c
which holds for any fermion eld However since the ma jorana eld is invariant
under C up to a phase this implies that
M M
M M
M
Furthermore there is the di erence that the Ma jorana eld can b oth create
M
and destroy neutrino states as opp osed to the Dirac eld that only contains
the annihilation op erator for neutrinos
One should p erhaps think that this di erence leads to observable e ects even
for massless neutrinos However for massless neutrinos there is no detectable
Neutrino Weak Interactions
di erence b etween Ma jorana and Dirac neutrinos Neutrinos are either created
in pairs via neutral current reactions or as a single neutrino or anti neutrino via
charged current reactions In the case of neutral current reactions the neutrino
pair is not b orn with de nite helicity This has nothing to do with lack of vector
current interactions but is rather an e ect of wave packet sup erp osition and
is indep endent of whether neutrinos are of Dirac or Ma jorana typ e We now
lo ok at the detection of neutrinos created via neutral current reactions Since
the charged current is still left handed even for Ma jorana neutrinos if these
neutrinos are detected via charged current reactions only the left handed com
p onent will interact and one therefore detects either particles or antiparticles
but not b oth The left handedness of the charged current thus reintro duces
the vector current for Ma jorana neutrinos making them b ehave as Dirac parti
cles This means that there is absolutely no detectable di erence b etween Dirac
and Ma jorana neutrinos in this case If they are detected via neutral current
reactions there is no way to measure their helicity anyway and again there is
no detectable di erence Next we lo ok at neutrinos created via charged current
reactions These are b orn with de nite helicity and even if detected by neu
tral current reactions still b ehave as if they do p ossess vector neutral current
interactions b ecause of the de nite helicity of their initial state
Also of course the two sup er uous states for Dirac neutrinos the right
handed neutrino and the left handed anti neutrino are completely sterile if
neutrinos are massless b ecause of the factor in the weak current Thus
they are not relevant for the massless standard mo del neutrinos
This discussion ab out the p ossible di erences b etween Dirac and Ma jorana
neutrinos have given rise to some discussion recently It was claimed that
the measurement of the e vector neutral current coupling constant by the
CHARM II collab oration giving
C
V
excludes the neutrino as b eing a Ma jorana particle In light of the discussion
ab ove this cannot b e the case of course since the muon neutrinos observed by
CHARM II are pro duced by the charged current decays of charged kaons and
pions and therefore they are b orn with de nite helicity Therefore they
are observed as having vector current interactions even if they are Ma jorana
particles
For massive neutrinos the situation is much more complex Here all the
four states for Dirac neutrinos are active so that Ma jorana and Dirac neutrinos
b ehave di erently Also their weak interaction cross sections are di erent In
the early Universe it is therefore imp ortant to distinguish b etween Dirac and
Ma jorana neutrinos if their mass su ciently large For Dirac neutrinos if the
mass is b elow roughly MeV the two sterile states decouple prior to the
QCD phase transition so that their numb er density is diluted by a large factor
b ecause of the entropy release from the phase transition In this case they do not
contribute to the cosmic energy density during nucleosynthesis Also the mass
is then su ciently low that the Dirac and Ma jorana annihilation cross sections
Chapter Dirac and Ma jorana Neutrino Interactions
are practically identical until long after they have decoupled Therefore at
these low masses the two typ es are again indistinguishable For higher masses
one needs to do a full calculation for all four states of Dirac neutrinos and
one can exp ect that Dirac and Ma jorana neutrinos have a completely di erent
impact on nucleosynthesis In Fig we show the relic numb er density of
massive neutrinos during nucleosynthesis times neutrino mass This is a go o d
measure of the energy density contributed during BBN by a massive neutrino
either Ma jorana or Dirac It is seen that the Dirac neutrino in most of the
mass region contributes much more than the corresp onding Ma jorana neutrino
b ecause it has four interacting states For large masses however the Ma jorana
annihilation cross section is smaller than the Dirac one so that the Ma jorana
neutrinos actually contribute more to the cosmic energy density The cross
section for annihilation into massless nal states is for non relativistic neutrinos
given by
X
G m
F
C C h jv ji
Dirac
V i A i
i
X
G m
F
h jv ji C C
Ma jorana
V i A i
i
where is the relative velo city of the incoming particles This shows the strong
di erence b etween Dirac and Ma jorana neutrinos in the non relativistic limit
The cross section is velo city dep endent for Ma jorana neutrinos whereas it is not
for Dirac neutrinos As the neutrino kinetic energy b ecomes smaller and smaller
relative to the mass the Dirac cross section is increased relative to the Ma jorana
one and this is the reason why very heavy Ma jorana neutrinos contribute more
than Dirac neutrinos at high mass
In the remainder of this work we shall only b e concerned with Ma jorana
typ e neutrinos The go o d thing ab out these is that one need only consider the
two usual states Furthermore since there is very little lepton asymmetry in the
early Universe Ma jorana neutrinos e ectively b ehave as one single unp olarised
particle sp ecies Interaction cross sections and the like are therefore relatively
easy to calculate in this case
Mixed Neutrinos
If neutrinos are massive they may in principle mix The reason is that the
states pro duced by the weak interaction are not necessarily mass eigenstates
In general there exists a unitary transformation b etween the weak interaction
avour eigenstates and the mass eigenstates
j i U j i
f m
where U is a unitary matrix Therefore if neutrinos have mass the freely
propagating neutrinos may oscillate b etween di erent avours This is very
Decaying Neutrinos
Figure The numb er relic numb er density times mass of a massive neutrino
de ned in units of the numb er density of a massless neutrino sp ecies r m
n n m m The full line is for Dirac neutrinos and the dotted for Ma jorana
neutrinos The discontinuity at m MeV for Dirac neutrinos indicate that
ab ove this mass all four states are in equilibrium b elow the QCD phase transition
temp erature The gure has b een reprinted from Ref
imp ortant for our understanding for example of the solar neutrino problem
They are of course p otentially very interesting for BBN as well However in
the present work we shall not deal with neutrino oscillations at all We shall
b e lo oking at mixed neutrinos in the context of neutrino decays but neglect the
p ossible oscillations b etween avours This particular area is however de nitely
worthy of further study
Decaying Neutrinos
In the standard mo del neutrinos do not have mass and therefore they are ab
solutely stable However once a mass term is intro duced this is no longer so
Also if the mass is larger than roughly eV for any neutrino it must b e
unstable with a lifetime shorter than the Hubble expansion time in order not to
overclose the Universe but there are many p ossible decay mo des for such
Chapter Dirac and Ma jorana Neutrino Interactions
neutrinos For example within the standard mo del the decay e e is
i e
m and that the mixing amplitude p ossible at tree level provided that m
e
i
b etween the two neutrino avours is di erent from zero Another p ossibility is
the neutrino radiative decay which is mediated by interactions at lo op
i j
level If there exists a avour violating neutral current there is also the decay
to a three neutrino nal state Further b eyond the standard mo del
i j k k
there are decays like where is some new particle like the ma joron
i j
There is a large numb er of mo dels for physics b eyond the standard mo del
that predict di erent neutrino lifetimes to the di erent decay mo des Our ap
proach will b e to discuss the exp erimental limits to the di erent decay mo des
not so much to go into sp eci c physical mo dels of neutrino decays
We start out with the simplest p ossible decay namely e e This
i e
decay mo de is relatively easy to constrain b ecause the decay pro ducts interact
electromagnetically If we restrict ourselves to the three known neutrino gener
ations only the neutrino mass is not required to b e b elow the threshold for
this decay mo de Within the standard mo del the decay lifetime is given by
m s jU j m m
e
e MeV
where jU j is the mixing b etween electron and tau neutrino and is a phase
e
space factor dep endent on the tau neutrino mass
q q q q q
q
q q ln
q
Strong limits on this decay come from SN A b ecause of the very intense
tau neutrino pro duction from this source Furthermore a tau neutrino with mass
in the relevant region should decay su ciently early that the decay pro ducts do
not overclose the Universe These limits have b een summarised by Ra elt
and e ectively this decay mo de has b een excluded by existing data In Fig
we show the excluded regions for this decay from b oth SN A cosmological
and other constraints
Next we turn to the radiative decay of heavy neutrinos Unless
i j
the neutrino p ossesses a small electromagnetic charge the radiative decay of
neutrinos pro ceed via lo op graphs Therefore the decay lifetime is exceedingly
long in all viable mo dels that have so far b een constructed From an argument
using the co oling of red giant stars it is p ossible to constrain severely the charge
of neutrinos The reason is that if neutrinos do p ossess charge they couple to
the photon at tree level and therefore plasma excitations can pro duce neutrino
pairs at a much higher rate than in the standard mo del This leads to a very
fast co oling of stars in con ict with observations From this argument an
upp er b ound of q e on the neutrino charge can b e derived In
conclusion it do es not seem p ossible to construct mo dels that predict neutrino
radiative decay lifetimes shorter than the lifetime of the Universe
Decaying Neutrinos
Figure Exclusion plot for the decay e e The excluded region
e
indicated by the arrow to the right is from limits to the galactic p ositron anni
hilation line ux This gure has b een repro duced from Ref
The radiative decay is relatively easy to constrain exp erimentally b ecause
of the emitted photon that interacts electromagnetically In the most general
case the decay rate can b e expressed in terms of an e ective electromagnetic
transition moment as
m
Bounds can b e placed on this transition moment from observations of SN A
These come from the gamma ray observations made by the solar maximum
mission satellite and have b een discussed in great detail by for example Ra elt
However the most interesting limits come from cosmological considerations
esp ecially from observations of the cosmic microwave background and the dif
fuse photon background These mainly concern neutrinos with long lifetime
but since it is probably imp ossible to construct viable physical mo dels that pre
dict short neutrino lifetimes to radiative decay this is exactly the region we are
interested in constraining The relevant limits have b een compiled by for exam
ple Kolb and Turner Fig shows the excluded regions repro duced from
Chapter Dirac and Ma jorana Neutrino Interactions
Figure Exclusion plot for radiative decay of neutrinos from cosmological
considerations The excluded regions are within the b oxes This gure has b een
repro duced from Ref
If we discard the cosmologically allowed region for low masses on the the
oretical grounds discussed ab ove and use the MeV upp er limit on allowed
neutrino masses the only allowed region is in the upp er left corner of Fig
Thus the lifetime to radiative decay is constrained to b e larger than s
meaning that this particular decay mo de is completely uninteresting for BBN
However such a neutrino may still b e interesting for other reasons for example
as an explanation of the Gunn Peterson e ect Observations indicate that
at high redshift a surprisingly small fraction of the hydrogen is in the form of
neutral hydrogen and this is what is usually referred to as the Gunn Peterson
e ect This lack of neutral hydrogen indicates some ionising source which may
b e an early generation of stars Another explanation could b e a massive neu
trino with mass just large enough to ionise hydrogen decaying on a timescale
of roughly s
As for the decay to a three neutrino nal state this is very di cult to
constrain exp erimentally b ecause the decay pro ducts are invisible However it
of course requires the existence of a avour violating neutral current Such a
Decaying Neutrinos
current would also give rise to pro cesses like e e e which are strongly
constrained exp erimentally The b est constraints on the decay itself come
from BBN considerations the most recent calculation b eing that of Do delson
Gyuk and Turner In general neutrinos with mass in the MeV region are
excluded if their lifetime is to o long For very light neutrinos m MeV BBN
do es not in general exclude this typ e of decay b ecause it only b ecomes imp ortant
after BBN has already taken place
Finally in the present work we shall only deal with neutrino decays of the
typ e Exactly as for the previous decay mo de to three neutrinos
i j
this decay mo de is of course not easy to constrain since the decay pro ducts are
invisible For the class of mo dels called ma joron mo dels the ab ove coupling to
also leads to pro cesses like neutrinoless double b eta decay and from this some
constraints can b e put on the p ossible decay strength However these are very
weak and at the moment the b est constraints come from the BBN considerations
discussed in more detail in Chapters and
Chapter Dirac and Ma jorana Neutrino Interactions
Chapter V
Neutrino decoupling in the Standard Mo del
Essentially all of this chapter has app eared as the pap er Neutrino Decoupling
in the Early Universe Phys Rev D
Nonequilibrium thermo dynamics is fundamental to the understanding of the
early Universe There are many examples of this the quark hadron phase tran
sition the electroweak phase transition in ation etc Another example is the
decoupling of neutrinos from the electromagnetic plasma at a temp erature of
a few MeV The normal assumption is that neutrinos decouple completely
b efore the temp erature approaches the mass of the electron When the temp er
ature gets b elow the electron rest mass the electrons and p ositrons annihilate
and pump energy and entropy into other particles But since the neutrinos are
decoupled they do not share in this transfer Their nal distribution is there
fore still describ ed by an equilibrium distribution but with a lower temp erature
than the photon temp erature T T
However this is not the complete story The fact that the decoupling tem
p erature is of the same order as the electron mass means that neutrinos must
to some degree share the entropy transfer This phenomenon is known as neu
trino heating Because it has consequences for Big Bang Nucleosynthesis it has
b een investigated many times in the literature All of these investiga
tions have found that the change in neutrino energy density is of the order
relative to the standard case where total decoupling is assumed This is to o
small a change to have any noticeable e ect on nucleosynthesis
One p ossible problem is however that they all assume that Boltzmann
statistics is a go o d approximation for all particle distribution functions This
means that e ects that dep end on quantum statistics such as Pauli blo cking
of nal states are neglected Furthermore they also neglect the electron mass
in the weak interaction matrix elements This approximation is valid as long
as the temp erature is high but p ossibly not when it is comparable to or lower
than the electron mass
Since the neutrino distributions are imp ortant b oth for nucleosynthesis and
Chapter Neutrino decoupling in the Standard Mo del
structure formation mo dels it is of considerable interest to see if neutrino heat
ing has any consequences for either
Fundamental Equations
As explained in Chapter the equation necessary to describ e all the di er
ent particle interactions is the Boltzmann equation Since we are lo oking at
standard weak interactions at low energy all pro cesses can b e calculated using
simple V A theory as explained in Chapter To rst order in the weak coupling
constant G all interactions are two particle scatterings and annihilations and
F
the matrix elements therefore are all of the same typ e As an example we show
the matrix element for the neutral current pro cess
i j i j
G
F
p
u u u u M
i j i j
All the p ossible reactions and their corresp onding matrix elements have b een
summarised in Tables and Higher order corrections to these matrix
elements as well as electromagnetic corrections from plasma pro cesses are ne
glected The collision integral on the right hand side of the Boltzmann equation
then has the form given in Eq Assuming standard weak interactions as
describ ed in Chapter it is p ossible to analytically integrate this dimensional
integral down to two dimensions App endix B This makes the integral much
easier to evaluate numerically
The problem now is to evaluate R t This can basically b e done in two ways
using either entropy or energy conservation see Chapter As explained in
Ref the end result is almost the same for massless neutrinos
Numerical Results
We have solved the coupled Boltzmann equations together with the entropy
conservation equation to obtain numerical results The equations were solved
numerically by using a grid of p oints in momentum space and a Runge Kutta
integrator to evolve the relevant quantities forward in time To compare with
previous results we have solved for several di erent cases m and
e
FD statistics m and FD statistics m and MB statistics
e e
m and MB statistics In all cases we assume zero chemical p otentials for
e
the particles involved at the initial temp erature Case is the approximation
used in previous studies Case is the correct approach
For each of these scenarios we have calculated the energy density deviation
as a function of the photon temp erature T de ned as
0
is the energy density of a neutrino sp ecies that has decoupled long b efore
0
e annihilation
For electron neutrinos the result is shown in Fig whereas Fig shows
the result for muon and tau neutrinos In the limit of MB statistics and zero
Numerical Results
Table Possible electron neutrino pro cesses and the corresp onding matrix
elements for massless Dirac neutrinos is de ned equal to the particle for
which we calculate C is the other incoming particle is de ned as either
coll
particle going out scattering or the outgoing lepton annihilation is
de ned as either particle going out scattering or the outgoing antilepton
sin and annihilation We have intro duced the quantities C
W V
C where sin p is the four momentum of particle i
A W i
Pro cess S j M j
G p p p p
e e e e
F
p p p p G
e e e e
F
p p p p G
e i e i
F
G p p p p
e i e i
F
C C p p p p e e G
A V e e
F
C C p p p p
A V
C C m p p
e
V A
e e G C C p p p p
e e A V
F
C C p p p p
A V
p p C C m
e
V A
e e G C C p p p p
e e A V
F
C C p p p p
A V
C C m p p
e
V A
G p p p p
e e i i
F
electron mass case we repro duce the results of Ref to within
This is quite reassuring as we use a completely di erent pro cedure for
solving the Boltzmann equations Do delson and Turner used a rst order
p erturbation expansion
p t p t f p f
i i
and
T T t
p t and t They then solved the p erturbation equations for
i
Using the correct quantum statistics and non zero electron mass case
the nal result for the deviation in energy density is roughly
for
e
for
Generally it is seen that the deviation is smaller if Fermi statistics is used This
is b ecause the Pauli blo cking factors in the collision integral Eq damp en
the interactions Furthermore using the correct electron mass also lessens the
deviation b ecause the structure of the matrix elements changes
Chapter Neutrino decoupling in the Standard Mo del
Table Possible neutrino pro cesses and matrix elements The de nition
of particle numb ers is the same as in table
Pro cess S j M j
p p p p G
i i i i
F
p p p p G
i j i j
F
G p p p p
i i i i
F
p p p p G
i j i j
F
p p p p G
i e i e
F
G p p p p
i e i e
F
e e G C C p p p p
i i V A
F
C C p p p p
A V
C C m p p
V A
e
e e G C C p p p p
i i V A
F
C C p p p p
A V
C C m p p
V A
e
C C p p p p e e G
V A i i
F
C C p p p p
A V
p p C C m
V A
e
G p p p p
i i e e
F
G p p p p
i i j j
F
Apart from the changes in energy density the shap e of the distribution
functions are changed from their equilibrium form We parametrise this change
by using an e ective temp erature for the neutrino distributions de ned as
p
FD
T p
e
ln f p
for Fermi Dirac statistics and
p
MB
T p
e
ln f p
for Maxwell Boltzmann statistics
Fig shows the e ective temp erature using Fermi statistics b oth with
and without electron mass incorp orated Note that the e ective temp erature
of a completely decoupled neutrino is T T Fig shows the same
but using Boltzmann statistics Notice the general o set in e ective temp era
ture compared to Fig This is b ecause the nal e ective temp erature of a
decoupled neutrino is T T if Boltzmann statistics is used see Chapter
We see that regardless of the statistics used the e ective temp erature rises
with momentum for medium and high momentum states As noted in Refs
this is not surprising b ecause the weak cross sections are much larger
for large momenta The shap e of the e ective temp erature curve is the same in
Discussion and Conclusions
Figure The evolution of for electron neutrinos The solid curve
corresp onds to case the dashed to case the dotted to case and the
dot dashed to case
all four cases but the actual numerical values are slightly di erent b eing higher
if the electron mass is neglected in C f This e ect re ects that the neutrino
coll
energy density is slightly higher for massless electrons as seen in Fig
Discussion and Conclusions
We have calculated the decoupling of massless neutrinos in the early Universe
by solving the Boltzmann equations Our metho d di ers from previous studies
in that it makes no approximations for the neutrino distribution functions and
the weak interaction matrix elements Although our results do not deviate
dramatically from these previous calculations we show that the inclusion of FD
statistics and non zero electron mass reduce the e ect of neutrino heating on
the relativistic neutrino energy density by almost for compared to the
e
results of Do delson Turner
We nd that the neutrino distribution after decoupling is non thermal at
the level This is also in relatively go o d agreement with previous results
The e ective temp erature grows with momentum for medium and large
Chapter Neutrino decoupling in the Standard Mo del
Figure The evolution of for muon and tau neutrinos The curve
lab els are as in Fig
momenta b ecause of the momentum dep endence of the weak interactions
The slight heating of neutrinos relative to the standard scenario has con
sequences for Big Bang nucleosynthesis We have changed the nucleosynthesis
co de of Kawano to include the e ect of neutrino heating As discussed in
Ref there are several e ects that combine to change the nucleosynthesis
scenario First of all the energy density in neutrinos is changed However this
has the consequence of lowering the energy density in photons and e b ecause
of energy conservation The result is if we use FD statistics and non zero elec
tron mass that the photon temp erature go es up b ecause of e annihilations by
a factor of instead of the usual a change of Further
more the weak rates are changed b ecause of the di erent abundances of e and
With the relevant changes to the co de we obtain a change in the primordial
e
He abundance Y of Y This is in go o d agreement with Ref
P
They calculate a change of but with a larger neutrino
heating Other authors have found similar values A change in He
abundance of is much b elow current observational accuracies
The systematic uncertainty in the primordial He abundance is estimated to b e
as large as Y or more than a factor of larger sys
Discussion and Conclusions
Figure The e ective neutrino temp erature after complete e annihilation
using FD statistics The solid line is for and case the dashed is for
e
also case The dotted is for and case the dot dashed for and case
e
than the change induced by neutrino heating
Another consequence of neutrino heating is that it increases the numb er
density of neutrinos If one of the neutrino sp ecies has a mass it will therefore
contribute slightly more than usually exp ected to the cosmological density
parameter In general it is p ossible to put a mass limit on a light neutrino that
is completely decoupled long b efore e annihilation The mass density of
such a sp ecies to day is
n m
Expressing this in terms of the photon density n T one gets
n
T m
n
where x is the Riemann zeta function If the neutrinos decouple long b efore
e annihilation this can b e translated into the normal textb o ok relation
g m
h
eV
Chapter Neutrino decoupling in the Standard Mo del
Figure The e ective neutrino temp erature after complete e annihilation
using MB statistics The solid line is for and case the dashed is for
e
also case The dotted is for and case the dot dashed for and case
e
using a present photon temp erature of K h is the dimensionless Hubble
constant and g for one avour of neutrino and antineutrino Since obser
vations demand that h we have a mass limit on any given light
neutrino Because of neutrino heating this limit is changed by a small amount
Using FD statistics and non zero m the nal numb er density of neutrinos
e
after decoupling deviates from the standard case by n n for and
e
n n for For electron neutrinos this changes Eq to
m g
e e
h
e
eV
whereas for muon or tau neutrinos we nd
g m
h
eV
As this is a very small change to the standard value neutrino heating do es not
alter the usual conclusion that electron neutrinos can not contribute more than
eV ab out using the current exp erimental upp er limit to its mass m
e
to whereas muon and tau neutrinos can
Discussion and Conclusions
The main conclusion is that it is safe to ignore neutrino heating when doing
nucleosynthesis and structure formation calculations However one should note
that neutrino heating is a non negligible e ect if one wants to construct a high
precision BBN co de It should also b e noted that since this work was p erformed
new results by Dolgov et al have app eared They use the same basic
approach as ours but with much higher numerical precision and their results
con rm ours very nicely Their results for the neutrino energy density change
compared to our result of and are
e e e e
They nd a resulting increase in the primordial and
He pro duction of Y which is somewhat larger than our result
Y However their implementation into the nucleosynthesis
co de di ers from ours to some extent which may explain this small di erence
Chapter Neutrino decoupling in the Standard Mo del
Chapter VI
Massive Stable Neutrinos in BBN
Essentially all of this chapter has app eared as the pap er Nucleosynthesis and
the Mass of the Tau Neutrino Revisited Phys Rev D
The p ossibility of heavy neutrinos in the MeV region has b een investigated
many times in the literature primarily b ecause such a neutrino could have very
interesting consequences for b oth cosmology and sup ernovae In Chapter
we discussed the mass limits on the di erent neutrino sp ecies The b est current
exp erimental limit to the neutrino mass comes from the ALEPH collab ora
MeV with CL Primordial nucleosynthesis puts tion and is m
stringent limits on the mass and lifetime of the neutrino The
b ounds are most stringent for neutrinos in the MeV region b ecause their mass
is comparable to the weak decoupling temp erature This can b e seen from the
following argument As long as massive neutrinos are kept in weak equilib
rium with the relativistic sp ecies their numb er density is that of a sp ecies in
thermo dynamic equilibrium That is if the mass is much smaller than the tem
p erature the numb er density is roughly equivalent to that of a massless sp ecies
whereas if the mass is much larger than the temp erature the numb er density is
exp onentially suppressed
n
exp m T
n
massless
As so on as the massive neutrinos decouple their numb er density stays constant
relative to that of a massless sp ecies However their energy density starts grow
ing as the temp erature drops b ecause the energy density b ecomes dominated by
the rest mass term In this case the energy density scales as
mT
NR
whereas for relativistic sp ecies the energy density scales as
T
R
Chapter Massive Stable Neutrinos in BBN
b ecause of the redshift of the particle wavelength At late times the rest mass
term therefore dominates the cosmic energy density completely
Now if the rest mass of the neutrino is much smaller than the decoupling
temp erature its numb er density after decoupling is roughly that of a massless
sp ecies However since nucleosynthesis takes place not long after weak freeze
out the rest mass term will not have b ecome imp ortant at that time and the
neutrino b ehaves e ectively as a massless sp ecies during nucleosynthesis On
the other hand if the rest mass is much larger than the freeze out temp erature
the numb er density is exp onentially suppressed by a large factor Therefore
even though the rest mass is large the numb er density is su ciently small that
such a neutrino do es not play a role during nucleosynthesis If such a neutrino
sp ecies is one one the three known sp ecies we therefore e ectively have only
neutrino sp ecies present during nucleosynthesis This is the reason why the
e ect is exp ected to b e strongest for neutrinos with mass m few MeV
Some of the most recent investigations of the e ects of a massive Ma jorana
neutrino in the MeV region on nucleosynthesis are those of Kolb et al
Dolgov and Rothstein and Kawasaki et al For a standard Ma jorana
neutrino that is stable on the timescale of nucleosynthesis Kolb et al nd an
excluded mass interval of MeV whereas Kawasaki et al nd an exclusion
interval of MeV Combined with the exp erimental data this means that
a massive neutrino with mass greater than MeV and lifetime greater
than s is excluded by nucleosynthesis However these calculations except
use the integrated Boltzmann equation Furthermore they assume that
the distribution functions are kept in kinetic equilibrium at all times by scat
tering reactions Kawasaki et al have used the full equations but
assuming only annihilation interactions no scattering Furthermore they use
Boltzmann statistics It is therefore imp ortant to see how these limits change
when we use the full set of Boltzmann equations with all p ossible interactions
Below we calculate the relic abundance of a massive Ma jorana neutrino with
lifetime longer than the time of nucleosynthesis s We then use nucle
osynthesis to put limits on the allowed mass range and nd that the resulting
limits are more stringent than those normally obtained from the integrated
Boltzmann equation but less stringent than found from the full Boltzmann
equations without inclusion of scattering
Numerical Calculations
As discussed in Chapter the fundamental way to go ab out calculating the
relic abundance of massive tau neutrinos is to solve the Boltzmann equation
Eq In addition to this we use the two indep endent evolution equations
for the cosmic expansion the Friedmann equation Eq and the equation
of energy conservation Eq Now with resp ect to the right hand side
collision integral Eq the matrix element for all the weak pro cesses can
b e calculated using the theory outlined in Chapter Since we are lo oking at
Ma jorana neutrinos and the lepton asymmetry of the early Universe is very
Numerical Calculations
small L we can as previously explained assume these neutrinos to
b e e ectively one single unp olarised sp ecies In this case the interaction matrix
elements are rather simple to calculate and are given in Table
Table Possible neutrino pro cesses and the corresp onding matrix elements
is de ned equal to the particle for which we calculate C is the other
coll
incoming particle is de ned as either particle going out scattering or
the outgoing lepton annihilation is de ned as either particle going out
scattering or the outgoing antilepton annihilation We have intro duced the
sin and C where quantities C and C If then C
W A V A e V
sin and C For sin If then C
W A W V
otherwise m p pro cesses involving neutrinos the parameter m m
i
is the four momentum of particle i
Pro cess S j M j
p p p p G
F
p p p p
m p p
G p p p p
F
p p p p
m p p
G p p p p
i i
F
p p p p
m p p
G p p p p
i i
F
p p p p
m p p
e e G f p p p p
F
p p p p
m p p g C C
V A
f p p m g C C m
e
V A
e e G f p p p p
F
p p p p
m p p g C C
V A
C C m f p p m g
e
V A
e e G f p p p p
F
p p p p
m p p g C C
V A
C m f p p m g C
e
V A
G p p p p
i i
F
p p p p
m p p
We parametrise the momenta of all particles in terms of the parameter z
Chapter Massive Stable Neutrinos in BBN
Figure The relic numb er density of massive neutrinos normalised to a
massless sp ecies times mass r m The solid curve is calculated using the full
Boltzmann equation and all interactions Dotted and dashed curves are adopted
from and
p T and use a grid of p oints to cover the range in z The system of
equations is then evolved in time by a Runge Kutta integrator giving us the
distribution function of each particle sp ecies as well as the scale factor R as a
function of photon temp erature By doing it this way we have the advantage
that we know the sp eci c distribution functions for all neutrino sp ecies whereas
a treatment using the integrated Boltzmann equation has to make an assumption
of kinetic equilibrium giving only a rough approximation to the true distribution
functions
Fig shows the nal relic density of b efore decay b ecomes imp ortant
r m is de ned as r m m n n m We have compared our results to
those of Kolb et al and Dolgov and Rothstein who use the integrated
Boltzmann equation For small masses it is not imp ortant whether we use
the integrated or the full Boltzmann equation The reason is that the neutrino
decouples b efore annihilation really commences therefore the treatment of the
annihilation terms is not very imp ortant But for higher masses our results
exceed those obtained in earlier investigations This is to b e exp ected since our
Numerical Calculations
Figure The distribution of neutrinos of mass MeV at a photon tem
p erature of MeV The solid curve includes all p ossible interactions whereas
the dashed dotted curve only includes annihilations The dotted curve is a ki
netic equilibrium distribution with the same numb er density as that of the solid
curve and T T whereas the dashed curve has the same numb er and energy
densities as the solid with T and adjusted accordingly
inclusion of the full neutrino sp ectra should reduce the annihilation of neutrinos
as discussed b elow thereby p ermitting more neutrinos to survive Thus the
inclusion of sp ectral distortions leads in fact to a strengthening of previous limits
on MeV neutrinos from the integrated Boltzmann equation However for very
high masses there is still a di erence b etween our result and those of Kolb et
al and Dolgov and Rothstein At rst one should exp ect the results to
converge b ecause at high masses the deviations from equilibrium are exp ected
to b e small b ecause of the e ectiveness of elastic scatterings However the
calculations of Refs and use Boltzmann statistics for all particles This
means that Pauli Blo cking of nal states is neglected increasing the annihilation
rate and giving a smaller nal abundance Our results app ear to agree well with
those of Fields et al who also use Fermi statistics although a direct comparison
is not p ossible
In Fig we show the distribution of a MeV The di erence b etween
Chapter Massive Stable Neutrinos in BBN
the distributions with and without scattering interactions is striking If only
annihilations are taken into account the high momentum states are almost
empty b ecause they cannot b e re lled On the other hand the low momentum
states are highly p opulated b ecause they annihilate more slowly and cannot b e
scattered away That this should b e so can b e seen directly from the thermally
averaged annihilation cross section for heavy Ma jorana neutrinos In case of
non relativistic neutrinos the cross section for pair annihilation into massless
sp ecies is
X
G m
F
C C h jv ji
M
A V
i i
i
where is the relative velo city of the incoming particles and the summation
index i runs over all annihilation channels The annihilation cross section is
therefore strongly increasing with neutrino kinetic energy and thus the high
momentum states annihilate more quickly Therefore calculations based on the
full Boltzmann equations without scattering underestimate the annihilation rate
Our neutrino distribution functions now di er less from kinetic equilibrium
than found in Deviations by a factor of app ear frequently when compared
to a kinetic equilibrium distribution f exp E T with the
same numb er density and T T as often assumed whereas the distribution is
close to that of a kinetic distribution with T and determined by the numb er
and energy densities
The resulting and distributions are signi cantly heated relative to the
e
standard case with a massless For eV mass or this changes the present
e
day contribution to the cosmic density to h m eV with for
a massless and for
e
MeV h is the Hubble parameter in units of km s for m
Mp c Later decay of can further increase the value of
Furthermore the and distribution functions b ecome non thermal b e
e
cause the pairs created by annihilation of neutrinos cannot thermalise suf
ciently b efore the electron and muon neutrinos decouple completely This
non thermality can have a signi cant e ect on the pro duction of light elements
esp ecially He b ecause the n p converting reactions dep end not only on the
energy density of neutrinos but also on the sp eci c form of the distribution
function for which enters in the n p conversion rate integrals Overall the
e
increase in the cosmic expansion rate due to the extra energy density in and
massless neutrinos which increases He pro duction is to some extent com
p ensated by a decrease in neutron fraction due to the change in the n p rate
integrals Notice also that for tau neutrino masses b elow MeV there is very
little e ect from heating since almost no tau neutrinos annihilate b efore weak
e
freeze out For very large masses the e ect is again small The reason here is
that although there is a signi cant non thermal contribution to the distri
e
bution from annihilations this is washed out b ecause the electron neutrinos
stay in thermo dynamic equilibrium with the photon plasma until long after the
annihilations of have ceased The biggest e ect is in the intermediate regime
Mass Limits
where annihilations cease roughly at the same time as the electron neutrinos
drop out of equilibrium Also one should note that at low tau neutrino masses
N found from our approach is roughly the same as that found from bulk
eq
heating The di erence b etween bulk heating and our approach is that bulk
heating neglects residual annihilations of tau neutrinos after weak decoupling
For low tau neutrino masses the pro duced electron neutrinos have energy not
much ab ove Q the proton neutron mass di erence Therefore they do not dis
the turb the neutron proton ratio very much On the other hand for large m
electron neutrinos have energy much larger than Q Now since the absorption
cross section at high energy is the same on neutrons and protons and there are
much fewer neutrons present the end result is that more neutrons are pro duced
This again pro duces more helium and therefore these residual annihilations go
in the direction of p ositive N For very large masses the e ect disapp ears
eq
b ecause there are so few of these residual annihilations that they do not e ect
BBN at all
Mass Limits
In order to derive mass limits on we need to calculate the predicted abun
dances of the di erent light elements and compare them with observations To
do this we have changed the nucleosynthesis co de of Kawano to incorp orate
a massive Ma jorana neutrino As previously mentioned we assume that decay
has no e ect during nucleosynthesis s Note that we not only have
to change the energy density of the neutrino but also the energy density in
of electron massless neutrinos as well as the sp eci c distribution function f
e
neutrinos in the nucleosynthesis co de
Fig illustrates the consequences for Big Bang nucleosynthesis in terms
of the equivalent numb er of massless neutrinos N needed to give a similar
eq
pro duction of He at a baryon to photon ratio The revised
results with and without inclusion of the change in the numb er density are in
e
ne agreement with recent results based on the integrated Boltzmann equation
by Fields Kainulainen and Olive Including also the actual shap e of the
distribution to some extent comp ensates for the e ect on the energy density
e
a result also found by Dolgov Pastor and Valle though we disagree by
several equivalent neutrino sp ecies with the total N found in and to some
eq
extent also with the di erential changes in N quoted in We note that a
eq
small overall increase in N due to these e ects was estimated in a fo otnote in
eq
This agrees with our results for large m
In conclusion our results show that no MeV neutrino with mass in the
region b etween MeV and the the exp erimental upp er limit of MeV and
lifetime longer than the timescale of nucleosynthesis is p ermitted unless more
than equivalent massless neutrino avours can b e accomo dated by nucleosyn
thesis observations Currently there is a lot of controversy as to the value of
N allowed by nucleosynthesis As discussed in Chapter there app ears to
eq
b e a consensus that the maximum allowed value is de nitely not larger than
Chapter Massive Stable Neutrinos in BBN
Figure Equivalent numb er of massless neutrinos shown as N N
eq eq
The dashed curve is without heating of the dotted with only the electron
e
neutrino numb er density changed and the solid curve with the full distribution
of
e
This means that we can fairly safely say that a tau neutrino with mass larger
than a few tenths of an MeV is not allowed by Big Bang nucleosynthesis if its
lifetime is longer than s
Chapter VI I
Decaying Neutrinos in BBN
The contents of this chapter have b een submitted to Phys Rev D
In the last chapter we discussed the impact of a massive but stable tau neutrino
on nucleosynthesis From cosmology we have the well known limit on stable
low mass m GeV neutrinos
m g
h
eV
using a present photon temp erature of K h is the dimensionless Hubble
constant and is the cosmological density parameter g for one avour
of neutrino and antineutrino Since observations demand that h
we have a mass limit on any given stable neutrino Thus any neutrino with
mass in the range eV MeV is necessarily unstable In Chapter we
discussed in some detail the p ossible generic typ es of neutrino decay First of
all there is the standard mo del decay e e if the mass is larger than
i j
m and the mixing angle b etween the two neutrinos is di erent from zero A
e
avour changing neutral current can also lead to the decay There
i j j j
can also b e other more exotic mo des of decay for example decay via emission
of scalars or pseudo scalars This decay mo de is generic for example in the
ma joron mo dels of neutrino mass
The e ect of such unstable tau neutrinos on nucleosynthesis have b een in
vestigated many times in the literature the most recent inves
tigations b eing those of Do delson Gyuk and Turner and Kawasaki et al
Do delson Gyuk and Turner have p erformed a detailed study of several
p ossible decay mo des in the context of non relativistic decays whereas Kawasaki
et al have p erformed a calculation using the full Boltzmann equation for the
decay mo de In all cases it is found that it is p ossible to change
signi cantly the primordial abundances via decay of the tau neutrino
1 +
Note that this relation changes slightly if the heating of neutrinos from e e annihilation
is included as discussed in Chapter
Chapter Decaying Neutrinos in BBN
In the present pap er we fo cus on the decay
e
where is assumed to b e a Ma jorana particle and is a light scalar or pseu
doscalar particle This di ers from the decay in that it includes an
electron neutrino in the nal state Since enters directly into the weak inter
e
actions that interconvert neutrons and protons this decay can p otentially alter
the outcome of nucleosynthesis drastically Indeed the non relativistic results
of Do delson Gyuk and Turner indicate that the primordial helium abundance
Y can b e changed radically either increasing or decreasing Y dep ending on
P P
the mass and lifetime of the tau neutrino
Now in the last few years evidence has b een gathering that the standard
picture of the way light nuclei are formed in the early Universe may b e facing
a crisis The main p oint is that helium is overpro duced relative to the other
light nuclei so that the standard theoretical predictions are only marginally con
sistent with the observational results Other measurements of the primordial
helium abundance do yield somewhat higher values and the unknown sys
tematical errors b oth in observations and in chemical evolution calculations may
however b e larger than presently assumed so that it is p erhaps premature to
talk of a real crisis for Big Bang nucleosynthesis
Our approach will not b e so much to discuss the sp eci c limits from nu
cleosynthesis since these are still quite uncertain as it will b e a discussion of
the di erences b etween our way of solving the Boltzmann equations and those
previously used Nevertheless in light of the p ossibility that some new element
is missing from the standard nucleosynthesis calculations we think that it is im
p ortant to try and nd metho ds of changing the nucleosynthesis predictions by
including plausible new physics in the calculations One p ossible way of doing
this is to include a massive and unstable tau neutrino
In order to obtain go o d ts to the observational data it is as just mentioned
necessary to lower the helium abundance somewhat compared to the other light
nuclei This can b e achieved by having relatively low mass tau neutrinos decay
while they are still relativistic or semi relativistic However this is exactly the
region where the non relativistic formalism breaks down b ecause it assumes a
delta function momentum distribution of the decay pro ducts and neglects in
verse decays It is therefore of signi cant interest to investigate this decay using
the full Boltzmann equation in order to calculate abundances in this parameter
region
In the present chapter we calculate the exp ected primordial abundances
for a tau neutrino mass in the range MeV In Sec we describ e
the necessary formalism needed for this calculation In Sec we discuss
our numerical results Sec contains a description of our nucleosynthesis
calculations compared to the observational data and nally Sec contains a
summary and discussion
Necessary formalism
Necessary formalism
As usual the basic formalism is the one describ ed in chapter namely that of
the Boltzmann equation Again since we only consider particle interactions
like C can b e written as in Eq
weak
Since we are only lo oking at Ma jorana neutrinos the decay terms are quite
simple Since there is almost no net lepton numb er in the early Universe the
Ma jorana neutrino is e ectively an unp olarised sp ecies However this means
that there can b e no preferred direction in the rest frame of the parent particle
Therefore the decay is necessarily isotropic in this reference frame The decay
terms in this case were given in Chapter as
+
Z
E
m
f f dE f C f
dec
e
p m E
E
+
Z
E
m
g
C f f f f dE
dec
e e
p g m E
E
e e e
+
Z
E
m
g
C f f f f dE
dec
e
g m E p
E
m m f f f f f f f f where f
e e e
is the lifetime of the heavy neutrino and g m m m m m
e e
and g g is the statistical weight of a given particle We use g
e
corresp onding to This assumption is not signi cant to the present
investigation Furthermore we shall assume that the masses of and are
e
e ectively zero during nucleosynthesis
The integration limits are given by
m m
E m m E E
i i i
m
i
E m
i i
and
m
m m p E E E
i
i
m
H
where the index i
e
These Boltzmann equations are then solved together with the Friedmann
equation and the equation of energy conservation as discussed in Chapter
Chapter Decaying Neutrinos in BBN
Figure The electron neutrino distribution at asymptotically low temp era
ture after complete decay in units of the distribution of a standard massless
neutrino The tau neutrino mass is MeV The full line is for s the
dotted for s the dashed for s and the dot dashed for
s
Numerical Results
We have solved the Boltzmann equation for the evolution of distribution func
tions together with the energy conservation equation Eq and the Fried
mann equation Eq Sp eci cally we have solved for masses of MeV
and lifetimes larger than s
In Fig we show the evolution of energy density in neutrinos and the
pseudoscalar particle for a tau neutrino mass of MeV The energy density
evolves quite di erently in the di erent cases Since the energy density in a
non relativistic sp ecies only decreases as R compared to R for relativistic
particles the rest mass energy of the tau neutrino will dominate completely at
late times if it is stable If it decays the rest mass energy is transferred into
relativistic energy so that the total energy density no longer increases relative
to that of a single standard massless neutrino sp ecies This di erence is clearly
seen b etween di erent tau neutrino lifetimes
Numerical Results
Figure The electron neutrino distribution at asymptotically low temp era
ture after complete decay in units of the distribution of a standard massless
neutrino The tau neutrino mass is MeV The full line is for s the
dotted for s the dashed for s and the dot dashed for
s
In Fig we show the sp ectral distribution of the electron neutrino for a
tau neutrino mass of MeV and di erent lifetimes To understand this plot
b etter we can de ne a relativity parameter for the decay
m
MeV s
A particle shifts from relativistic to non relativistic at a temp erature of roughly
T m When the Universe is radiation dominated
t T
s MeV
Therefore if the decay is relativistic
m
s t T m
MeV
Chapter Decaying Neutrinos in BBN
Thus if the decay is relativistic whereas if it is non relativistic
i i
For lifetimes of and s the relativity parameters are resp ectively
and For non relativistic decays the decay neutrino dis
tribution assumes a rather narrow shap e coming from the delta function energy
distribution For very short lifetimes the decay installs an equilibrium b etween
and b ecause of rapid inverse decays This can lead to a signi cant
e
depletion of high momentum electron neutrinos as also noted by Madsen
who treated this case of very short lifetimes using equilibrium thermo dynamics
Nucleosynthesis E ects
In order to estimate the e ect on nucleosynthesis we have employed the nu
cleosynthesis co de of Kawano mo di ed in order to incorp orate a decaying
neutrino This includes taking into account the changing energy density as well
as the change in electron neutrino distribution
A decaying tau neutrino can a ect nucleosynthesis in several di erent ways
Firstly the cosmic energy density is changed Since the cosmic expansion rate
is given directly in terms of this energy density via the Friedmann equation Eq
it is also changed It is a well known fact that increasing the energy density
leads to an earlier freeze out of the n p conversion and therefore pro duces more
helium whereas decreasing the energy density decreases the helium fraction
This is the e ect discussed by Kawasaki et al namely that an MeV neutrino
decaying into sterile daughter pro ducts while still relativistic or semirelativistic
can actually decrease the cosmic energy density thereby decreasing the helium
abundance
However there is also another another e ect stemming from the change in
electron neutrino temp erature Since the electron neutrino enters directly into
the n p pro cesses this can b e called a rst order e ect and is p otentially much
more imp ortant than the second order e ect of changing the energy density
This e ect of change in the electron neutrino temp erature has already b een
discussed by several authors in the context of non relativistic decays
If the decay is non relativistic the energy of a pro duced electron neutrino
is m If this energy is signi cantly ab ove the energy threshold for the two
pro cesses
n e p
e
and
p e n
e
the decaying tau neutrinos will act to pro duce more He The reason
is that the absorption cross section at high energies is the same on neutrons
and protons Since there are many more protons present than neutrons more
neutrons will b e pro duced In the end this leads to a higher helium fraction
However if the mass of the decaying neutrino is b elow this threshold the pro
duced electron neutrinos will stimulate the conversion of neutrons into protons
Nucleosynthesis E ects
Figure Helium abundance contours as a function of tau neutrino mass and
lifetime for a baryon to photon ratio of The full line is Y
P
the dotted is Y and the dashed is Y The value in the
P P
standard mo del for this is Y
P
thereby actually decreasing the He abundance This e ect then comp etes with
the rest mass e ect which increases Y
P
If the decay is relativistic the electron neutrinos are pro duced at roughly
thermal energies E ectively this amounts to increasing the electron neutrino
temp erature This in turn leads to a decrease in helium pro duction If the
decay takes place at high temp eratures it is b ecause b eta equilibrium is kept
for a longer time whereas if the decay takes place at temp eratures b elow the
threshold for proton to neutron conversion it still leads to lower helium abun
dance b ecause an increase in the electron neutrino temp erature stimulates the
conversion of neutrons to protons over the inverse reaction
In Fig we show contour lines for the helium abundance as function of
neutrino mass and lifetime It is seen that helium can b e signi cantly suppressed
relative to the standard case if the lifetime is short enough and increased if the
mass and lifetime are b oth high If the mass and lifetime are b oth su ciently
high the helium abundance is instead increased Notice also that even for small
masses of the order MeV the helium abundance can b e changed signi cantly
compared to the standard value for rather a large range of lifetimes
Our Fig should b e compared with for example the results of Terasawa
and Sato or Do delson Gyuk and Turner obtained using non relativistic
theory The most straightforward comparison is with Figs b and b in Ref
Chapter Decaying Neutrinos in BBN
For long lifetimes the di erence is quite small as would b e exp ected since
this is the non relativistic limit However for short lifetimes the di erence is
signi cant For very short lifetimes our calculated He abundance is larger than
that of Terasawa and Sato The reason is that if one uses the full Boltzmann
equation in this case decays and inverse decays will bring the particle distribu
tions into equilibrium as discussed in Sec Thus if one keeps on going to
shorter and shorter lifetimes nothing new happ ens since it is already decay in
equilibrium Therefore our curve for He attens out instead of decreasing for
very short lifetimes For somewhat longer lifetimes our He abundance is on the
other hand smaller than that found by Terasawa and Sato The reason here
is that the decay pro duces a p eak of very low momentum electron neutrinos
and that these states are not upscattered b ecause the weak interactions have
already frozen out In the end this pro duces a somewhat colder electron neu
trino distribution than would have b een obtained using non relativistic theory
and therefore predicts less helium This low momentum p eak can b e seen in
Fig for the example of a MeV neutrino For non relativistic decays
this p eak disapp ears b ecause low momentum states are not energetically acces
sible In essence our predicted curve for the He abundance is therefore much
atter at short or intermediate lifetimes than what one would obtain using the
non relativistic formalism For very long lifetimes our calculation ts fairly well
with that obtained by Terasawa and Sato as could b e exp ected
In Fig we show the abundance of D He and Li for a sp eci c example
of m MeV We see that the abundances of these elements only change by
relatively small amounts even for great variations in neutrino lifetime Thus
the main e ect of the decay is to lower the helium abundance while leaving the
other abundances more or less unchanged
The calculated abundances for di erent masses and lifetimes are compared
with observational limits Unfortunately there is a great deal of controversy con
nected with these However since our main emphasis is on the di erences b e
tween our approach to solving the Boltzmann equations and the non relativistic
approximations previously used and not so much on the sp eci c nucleosynthe
sis limits to tau neutrino mass and lifetime we will not go into to o much detail
regarding this p oint For He we use the value calculated by Hata et al of
Y
P
For deuterium the situation is somewhat complicated From measurements in
the lo cal interstellar medium one can obtain a deuterium abundance of
D H
which can b e viewed as a lower limit to the primordial abundance However
some recent results from QSO absorption systems seem to indicate a primordial
value much higher than this
D H
Nucleosynthesis E ects
Figure The abundance of D He and Li as a function of tau neutrino
lifetime The curves have b een calculated for m MeV and
The full line shows D H the dashed shows D He H and the
dot dashed shows Li H
Other similar observations yield much lower values closer to the lo cal one
In light of the controversy of using deuterium results from these measurements
we use the lo cally obtainable lower limit in the present pap er From evolu
tion arguments one can also obtain an upp er limit to the primordial D He
abundance of
D He H
Finally for the abundance of Li we use a b ound of
Li H
obtained by Copi Schramm and Turner
Altogether these are the observational values which the theoretical predic
tions should b e able to repro duce In the standard mo del the theoretical predic
tions are only marginally consistent with observations b ecause helium is over
pro duced compared to the other light nuclei In our scenario this problem is
resolved by having the tau neutrino decay during nucleosynthesis into an elec
tron neutrino nal state
In Fig we show the allowed region of lifetime versus mass for the tau
neutrino using the ab ove constraints In all the mass interval from MeV
it is p ossible to obtain a t to the observed abundances Note however that
Chapter Decaying Neutrinos in BBN
Figure Allowed region of tau neutrino mass and lifetime The allowed
region is b etween the two full lines
for masses in the high end of this region a t can only b e obtained in a very
narrow lifetime interval This is b ecause of the very steep dep endence of Y
P
on the lifetime in this region For lower masses a go o d t can b e obtained in a
broad region of lifetimes
Another imp ortant fact is that since the helium abundance is lowered with
out disturbing greatly the other abundances the upp er and lower b ound on the
baryon to photon ratio is now given essentially only by the limits coming
from D He and Li This also means that a relatively high value for can b e
accommo dated ab out coming from requiring that Li should not b e
overpro duced
In Fig we show the allowed region of as a function of tau neutrino
lifetime for three di erent masses We have also plotted the upp er and lower
limits to from the standard calculation
Conclusion
We have studied the decay of a relatively low mass tau neutrino into an electron
neutrino and a scalar or pseudoscalar particle using the full Boltzmann equation
It was found that the primordial helium abundance Y can change drastically
P
compared to the standard value This is in concordance with the ndings of
previous authors who used a non relativistic treatment Our actual
Conclusion
Figure Allowed regions of for di erent tau neutrino masses
and lifetimes The regions inside the full lines are allowed regions The vertical
dotted lines show the consistency interval for in the standard mo del for our
chosen observational constraints
numerical values di er signi cantly from those previously obtained by use of
non relativistic formalism but the general trend is the same namely that low
mass neutrinos decaying while relativistic or semi relativistic lower the helium
abundance
The decay we have studied di ers completely from the decay
studied by Kawasaki et al b ecause the electron neutrinos directly a ect
the weak reaction rates that interconvert neutrons and protons Only if much
more reliable estimates of the primordial abundances are develop ed will it b e
p ossible to discern b etween the two di erent decay mo des
Our aim has mainly b een to discuss the di erences b etween using the full
Boltzmann formalism and using the non relativistic approximation in doing
these calculations We have not done very detailed statistical analysis in or
der to obtain strict nucleosynthesis limits
However it was shown that a go o d t to the observed primordial abundances
can b e achieved for a large range of di erent masses and lifetimes Given the
p ossibly large unknown systematical errors in the observations and chemical
evolution mo dels it is p erhaps to o early to talk of a real crisis for Big Bang
nucleosynthesis However once the observational b ounds b ecome more strict
there might very well turn out to b e such a crisis
In light of this p ossible discrepancy b etween observed and predicted abun
Chapter Decaying Neutrinos in BBN
dances we still feel that it is imp ortant to explore p ossible ways to change the
light element abundances via plausible intro duction of new physics The tau
neutrino decay into an electron neutrino nal state is just such a p ossibility
Perhaps one should also nally note that even if the helium abundance turns
out to have b een signi cantly underestimated a tau neutrino decay of the typ e
we have discussed can still make nucleosynthesis predictions t the observations
but for completely di erent values of mass and lifetime
Chapter VI I I
Neutrino Lasing
Essentially all of this chapter has app eared as the pap er A Cosmological Three
Level Neutrino Laser Phys Rev D
In this chapter we wish to pro ceed further with a detailed discussion of neutrino
decays in the early Universe As discussed in detail in Chapter there are
several p ossible decay mo des for example the simple radiative decay
Another or the avour changing neutral current weak decay
p ossibility is the decay
H F
where H is a heavy neutrino F is a light neutrino and is a light scalar particle
for example the ma joron It is not essential that the b oson should have spin
but it simpli es calculations signi cantly All of these decay mo des have b een
investigated in an early Universe context We cho ose to fo cus on the last
decay typ e in order to investigate some very interesting non equilibrium e ects
that p ertain to this decay It turns out that under the right circumstances the
heavy neutrino will decay by stimulated decay into F and instead of just
decaying thermally This runaway pro cess leads to the formation of a very low
momentum comp onent of the scalar particle while the rest are in an almost
thermal distribution The scenario has b een describ ed by Madsen and
subsequently in more detail by Kaiser Malaney and Starkman the idea
b eing that the particle could make up the dark matter in our Universe One
of the currently favoured mo dels of structure formation is the so called Mixed
Dark Matter MDM mo del This is essentially a Cold Dark Matter CDM
mo del but with a small matter content of Hot Dark Matter HDM that lessen
the uctuations on small scales The particles could t b oth roles the cold
comp onent acts as cold dark matter and the thermal comp onent as hot dark
matter
There are however several problems with this scenario First of all the
favoured mixed dark matter mo dels consist of roughly baryons HDM
Chapter Neutrino Lasing
and CDM If there is to o much HDM structure formation will b e a ected
so as to pro duce to o little clustering on small scales if normalised to COBE
data The simple scenario of Refs predicts only ab out CDM
and HDM which is incompatible with the observed structure
We develop further this mo del and in Sec we list the relevant equations
for calculating the particle distribution functions resulting from the decay In
Sec we describ e a new way of pro ducing neutrino lasing that is very similar
to an atomic level laser Here it is p ossible to achieve lasing without invoking
new neutrinos and to increase the CDM fraction which however still remains
well b elow Sec contains a description of the numerical results from
our investigation of this level mo del and nally Sec contains a discussion
of the p ossible consequences of this typ e of decays as well as a discussion of
various astrophysical limits on them From this discussion it will b ecome clear
that lasing decays are most likely to take place after the ep o ch of nucleosynthesis
T MeV Of course they are also required to take place b efore matter
radiation equality T eV if the b osons are to act as ordinary MDM
The Basic Scenario
As previously we use the Boltzmann equation Eq to follow the evolution
of the di erent sp ecies as the Universe expands and co ols We shall assume that
the term C can b e neglected but p ossible limitations to this assumption will
coll
b e discussed later For simplicity we assume that the decay is isotropic in the
rest frame see however Refs for a discussion of the case of anisotropic
decays
The decay terms are then given by Eqs and
+
Z
E
m
a
dE f f f C f
a F dec a
m E p
a a
E
+
Z
E
a
g m
a
a
dE f f f C f
a a F dec F
g m E p
F F F
E
a
+
Z
E
a
m g
a
a
dE f f f C f
a a F dec
g m E p
E
a
where f f f f f f f f f m m m m
a F a F F a
a
F
is the lifetime of the heavy neutrino and g is the statistical m m m
a
F
weight of a given particle We use g g and g corresp onding to
a F
The integration limits are
m m
a
E E E m m
i i i
a
m
i
m E
i i
1
All calculations may b e redone using g The changes are not very signi cant and
tend to lessen the lasing e ect describ ed b elow
The Basic Scenario
and
m
E m m p E E
a i a a
i
m
a
where the index i F
For non relativistic decays E m T This means that the resulting
H
fraction of low momentum b osons is vanishing b ecause they are kinematically
inaccessible However for relativistic decays this is not the case Here the min
imum b oson energy is roughly E p m m m p E which
H H
H H H
can b e seen from Eq This corresp onds to b osons b eing emitted back
wards relative to the direction of motion of the parent particle In relativistic
decays it is thus p ossible to access very low energy b oson states This is essential
for the phenomenon describ ed b elow
Something very interesting can happ en if there is initially an overabundance
of H over F To see this let us lo ok at the structure of the decay term for the b o
son Eq Following Ref we make the following rough approximation
f f The decay term then reduces to
+
Z
E
H
m
H
f dE f f C f
H H F dec
m E p
E
H
If f f this indicates an exp onential growth of f This e ect has b een called
H F
neutrino lasing b ecause it is very similar to normal lasing the overabundance
of H over F corresp onding to a p opulation inversion Exactly as in a laser the
pro cess saturates when f f The remaining neutrinos will then decay to
H F
an almost thermal distribution as they go non relativistic
Note that in all our actual calculations we use a zero initial abundance of
b osons b efore the decay commences This is not essential for the lasing phe
nomenon but the nal fraction of cold b osons is decreased if there is an initial
thermal p opulation In Ref it was assumed that the b osons decoupled prior
to the QCD phase transition so that their numb er density was depleted by a
factor of relative to the other sp ecies
The e ect will b e larger for low momentum states b ecause of the E p
term This means that if the low momentum states are accessible the b osons
will cluster in these states and pro duce something similar to a Bose condensate
For this to happ en it is a necessary condition that the decay is relativistic
b ecause otherwise no low momentum states can b e accessed
This phenomenon was rst investigated in Refs However Ref
used a somewhat di erent approach namely equilibrium thermo dynamics
where the decays can lead to the formation of a true Bose condensate In the
ab ove scenario following Refs no assumption of equilibrium has to b e
made
2
Note that as mentioned we have only considered isotropic decays but as mentioned in
Ref even if the decay is fully p olarised and the b osons are emitted backwards there will
only b e a rather small increase in the numb er of low momentum b osons If the b osons are
preferentially emitted in the forward direction there will b e no lasing Refs have go o d
discussions of these p olarisation e ects
Chapter Neutrino Lasing
The Level Laser
It was assumed in Refs that F is a hitherto unknown neutrino that
decouples prior to the QCD phase transition for example a right handed com
p onent of one of the known neutrinos This is the only way of achieving lasing
in the ab ove level decay However it is actually p ossible to obtain lasing
with only the three known neutrinos The e ect is somewhat similar to an
atomic level laser where particles are pump ed from the ground state to the
top level From there they decay rapidly to the lasing state and from the lasing
state they decay by stimulated decay to the ground state Exactly the same
situation can o ccur for neutrinos if we have a three generation mass hierarchy
the stable particles and are of course required to have m m m
e
e
masses b elow the cosmological limit for stable particles m eV and can
therefore b e considered essentially massless in the calculations Let us assume
that we have the same generic decay mo de as b efore The p ossible ma joron
emitting decays are then
e
e
We shall assume that the second decay mo de Eq is suppressed so that
it is negligible relative to the rst Eq This assumption is not ruled out
by the physical mo dels describ ed later
If the decays are non relativistic nothing exciting happ ens and the decays
are thermal However if all the decays are relativistic and the free decay of
pro ceeds much faster than that of then we can achieve lasing The
neutrino now functions as the top level in a normal laser it feeds particles to
the lasing state Particles in this state then decay to the ground state by
e
f stimulated decay The condition for the lasing pro cess to work is that f
e
but b ecause of the fast decay of the neutrino this is automatically ful lled for
and m a large temp erature range b etween m
Let us for the moment assume that the decays completely b efore b egins
to decay It is then easy to see why the amount of lasing increases if has
decayed relativistically We now start with a non thermal distribution of muon
neutrinos If decays while still relativistic the distribution of will b e
signi cantly colder than thermal If we again lo ok at the structure of Eq
we see that for n given the integral increases if the distribution of H gets
H
factor in the numb er density integral This means that colder b ecause of the p
the relativistic decay to muon neutrinos enhances the lasing from to The
e
e ect is opp osite if the decay is non relativistic and the lasing diminishes or
disapp ears completely Fig shows the momentum distribution of the light
neutrino after complete decay of the heavy neutrino We see exactly that the
3
We cho ose to fo cus alone on the ma joron decays In principle there might b e other relevant
decay mo des as mentioned previously
Numerical Results
Figure The distribution function of the light neutrino F after complete
decay of the heavy neutrino H for di erent lifetimes of the heavy neutrino
using sp eci cally m MeV m MeV and m MeV The full
H F
curve is for s the dotted for s and the dashed for s
H H H
light neutrino distribution gets colder if the decay is relativistic and hotter if
it is non relativistic Of course it is always a necessary condition for lasing to
o ccur that decays relativistically b ecause otherwise it cannot p opulate low
momentum b oson states at all
Numerical Results
We have investigated numerically the b ehaviour of this level decay for dif
ferent parameters by solving explicitly the coupled Boltzmann equations Our
numerical routine uses a grid in comoving momentum space and evolves forward
in time using a Runge Kutta integrator The time evolution of the scale factor
R and the photon temp erature T are found by use of the energy conservation
equation d R dt pd R dt and the Friedmann equation H G
In Fig we show an example of the numb er density evolution of the
di erent involved particles It is seen that in a large temp erature interval the
numb er density is slightly higher than that of This leads to a very fast
e
Chapter Neutrino Lasing
Figure The evolution of numb er density for the di erent involved particles
The full curve is the dotted the dashed and the dot dashed The
e
MeV and MeV m m MeV m parameters used are m
e
s s
growth of the b oson numb er density and is identi ed with the region where
lasing takes place The nal numb er density of and are the same b ecause
e
of neutrino numb er conservation neglecting a p ossible small initial abundance
of b ecause even though the abundance starts out higher three s are
e
pro duced for every two
e
Since b oth and are almost massless there are essentially four free pa
e
rameters in our equations The rst is the mass scale de ned for example
by the neutrino mass This mass scale is unimp ortant for the shap e of the
sp ectra this is only true if the decays take place b elow the weak decoupling
temp erature T MeV If the free decay temp erature is larger than T
dec dec
the distribution will quickly thermalise Only well b elow T can the non
dec
equilibrium lasing pro ceed but now with an initial thermal distribution In
our numerical work we cho ose that the free decay temp erature to b e lower than
the decoupling temp erature In Fig we show an example of changing the
by a factor of two This leads to an almost identical nal normalisation m
Numerical Results
Figure Final b oson distribution for two di erent decay mo dels The full
MeV m m MeV m curve is for the parameters m
e
MeV s The dashed curve is for m s MeV and
s Note that s MeV and MeV m m m
e
and which is orders of magnitude although we have chosen values for m
e
larger than the exp erimental cosmological limit this do es not in uence the re
sults as long as the values are small compared to all other relevant masses and
temp eratures
b oson distribution The other three parameters are
m
MeV s
m
MeV s
m
m
The rst two parameters describ e how relativistic the free decays of the par
ticles are A thermal particle shifts from relativistic to non relativistic at a
Chapter Neutrino Lasing
Figure The momentum sp ectrum of after complete decay for di erent
and The decay parameters The full curve is
and The dashed is for dotted is for
and and The dot dashed is for
and The The dot dot dot dashed is for
long dashed is a thermal Bose distribution with the same numb er density
temp erature of roughly T m When the Universe is radiation dominated
t T
s MeV
Therefore if the decay is relativistic
m
s t T m
MeV
Thus if the decay is relativistic whereas if it is non relativistic
i i
The third parameter is just the dimensionless mass
Fig shows the nal b oson distribution after complete decay for several
di erent choices of decay parameters for comparison we have also shown a
thermal Bose distribution with the same numb er density It is seen that in
and order to get a large fraction of cold particles we need to have
Numerical Results
Figure Final momentum distribution of The full curve is for
and and The dotted curve is for
The reason for this was describ ed in the previous section Fig
shows the e ect of changing the relative mass of the two heavy neutrinos We
m see here that to get a large cold fraction we also need to have m
This is also understandable if we lo ok at the allowed momentum range of the
pro duced b osons from Eq
s
p
p m
p
T T T T
If is close to one we can get very cold b osons This means that the b osons will
tend to cluster in low momentum states but b ecause of energy conservation we
get a very hot distribution On the other hand if then the lowest
b oson states are unaccessible and we get instead a colder distribution This
in turn leads to more lasing in the lasing state and even though the fraction
of cold b osons coming from the initial decay gets smaller this is more than
comp ensated for by the larger fraction coming from the decay This is indeed
why the cold fraction increases when decreases
Chapter Neutrino Lasing
Figure The temp erature evolution of the distribution The decay pa
and The dotted curve is at rameters are
T MeV the dashed at T MeV and the dot dashed at T MeV
The full curve is the nal distribution after complete decay
Thus if we want to study the region of parameter space with p ossible in
terest for structure formation scenarios we have essentially limited ourselves
to the region of ultrarelativistic decays with a large mass di erence b etween
the two heavy neutrinos It is also interesting to follow the evolution of the
b oson distributions with temp erature to see how the di erent states p opulate
Fig shows the temp erature evolution of the b oson distribution for the decay
and parameters
The minimum p ossible b oson energy at a given temp erature is given by Eq
where instead of we use and for we use An approximate
T corresp onding roughly lower limit can b e calculated by setting p
to the mean value from a normal thermal distribution This lower limit is
an increasing function with decreasing temp erature and indicates roughly the
momentum where the distribution decouples decoupling here means that the
states are kinematically inaccessible at a given temp erature For the mo del
shown in Fig we get p T for photon temp eratures
of and MeV We see that the lower limit at a given temp erature
Numerical Results
Figure The temp erature evolution of the distribution The decay pa
rameters are as in Fig The full curve is the initial thermal distribution
The dotted curve is at T MeV the dashed at T MeV and the
dot dashed curve is at T MeV
corresp onds roughly to the momentum where the distribution decouples Of
course the lower limit will always b e somewhat higher than the actual decoupling
p oint but it helps to understand the evolution of the sp ectra Fig shows
the corresp onding evolution of We see the b efore mentioned e ect of
decay pro ducing a cold distribution and thereby enhancing lasing
However even in the case where the decays are ultrarelativistic and the mass
ratio small the fraction of cold b osons is only mo derate For example in the
the fraction of b osons m and m case where
with p T is only If the b oson was to act as the dark matter
H
in our Universe a much larger fraction would b e needed Note however that
even if the obtainable cold fraction is to o small it is comparable with or larger
than that obtained in the mo del of Refs Furthermore it is achieved
without invoking an unknown neutrino state
Chapter Neutrino Lasing
Discussion
We have develop ed further the scenario originally prop osed in Refs
by intro ducing a physically di erent way of obtaining neutrino lasing However
this mo del still su ers from one of the same shortcomings that the original
mo del did namely that it is imp ossible to achieve su ciently large fractions of
cold b osons for structure formation
An interesting consequence of the level decay is that the numb er density
of electron neutrinos is increased by a factor of almost three relative to the stan
dard case If the decay takes place b efore or during Big Bang Nucleosynthesis
BBN then it can signi cantly change the outcome b ecause the electron neu
trinos are very imp ortant for the neutron to proton conversion reactions The
higher than usual numb er of electron neutrinos will tend to keep equilibrium
longer than in the standard case meaning that less neutrons survive to form
helium The helium abundance will then b e lower than in the standard case
Furthermore the ratio of low energy electron neutrinos is signi cantly higher
than normal This favours the neutron to proton reaction b ecause of the mass
di erence b etween the neutron and the proton and leads to the opp osite e ect of
the one describ ed by Gyuk and Turner where a massive neutrino decays
to electron neutrinos thereby enhancing the neutron fraction and allowing a
very high baryon density without violating the BBN constraints However in
their scenario the decaying neutrino is non relativistic and the resulting electron
neutrinos therefore have very high energy the opp osite of our case
Decays pro ducing lasing e ects are however not likely to take place during
BBN The reason is that neutrinos are kept in equilibrium by the weak reaction
until roughly T MeV All neutrino distributions are then of equilibrium
form and no lasing will take place If the mass of any of the heavy neutrinos
is of the same order or larger than this temp erature we will essentially have a
non relativistic decay after decoupling but this will not pro duce lasing Thus
the heaviest neutrino in our scenario must b e substantially lighter than MeV
thus making it improbable that it should decay during BBN In general neutrino
decays a ecting BBN may well take place but then they will not pro duce lasing
e ects Therefore we have not treated this case in the present pap er
If we turn to more sp eci c physical mo dels the most natural candidate for
the light scalar particle is the ma joron There exist several ma joron mo dels
all of them designed to explain the very small neutrino masses compared to the
other fermions In the simplest mo dels the decay is forbidden at
tree level making the lifetime to o long to b e of interest However there are
more complicated mo dels where the lifetime can b e short In the simplest case
the ma joron couples to the neutrino via a Yukawa interaction
g
c
h c J L i
From this generic interaction several limits to the coupling strength g and
therefore to the lifetime can b e derived For example neutrinoless double
Discussion
decays give an upp er limit on g of roughly Other limits can b e
derived from the neutrino signal of SN A but as mentioned in Ref
it is very di cult to get a clear picture of the current limits A coupling
constant of gives a restframe decay lifetime of
m
s
g m eV
This limit is rather weak and leaves op en a large region of parameter space for
relativistic decays in the early Universe Note that for our mo del to work the
decay must b e suppressed This corresp onds to the coupling g b eing
e e
very small compared to the other coupling constants
Finally we have assumed that the scattering and annihilation reactions b e
tween neutrinos and ma jorons are weak compared with the decay reaction If
as we have assumed the decay is not forbidden at tree level this is the case
b ecause the scattering and annihilations all are of order g whereas the decays
are of order g Were the annihilation and scattering reactions to b e imp or
tant we would not obtain any cold b osons b ecause the low momentum states
would not b e decoupled Therefore it is absolutely necessary that the decays
are stronger than the scattering and annihilation reactions Note that even if
the decay is of order g and the scattering and annihilation g the decay rate
will b e damp ed by a factor m E which can b e very large if the decay
is extremely relativistic Thus in the limit of we have to account for
scatterings and annihilations but would have to b e extremely large b ecause
of the very small value of g
As far as the neutrino masses are concerned they are naturally explained
by see saw mo dels Then only the left handed comp onents participate in the
interactions b ecause of the very large mass of the right handed comp onents
The p ossible Ma joron decays are then exactly
L L
i j
R L
j i
as required in our lasing mo del
In conclusion what we have shown is that it is p ossible to generate large non
equilibrium e ects even starting from identical neutrino distributions Very few
ne tunings are necessary in order to achieve lasing Thus these e ects may
b e present many places in early Universe physics b oth b ecause the decay typ es
are generic and b ecause the early Universe is an inherently non equilibrium
environment This was also noted in Ref but our sp eci c mo del shows
that the relevant range of mo dels is much larger than originally thought It is
therefore of considerable interest to search for other physical scenarios where the
lasing pro cess and other similar non equilibrium pro cesses may b e e ective
Chapter Neutrino Lasing
Chapter IX
Conclusions
In the present work we have discussed in detail the e ects of neutrinos on pri
mordial nucleosynthesis Neutrinos are imp ortant in several ways for the way in
which nucleosynthesis pro ceeds First of all they contribute to the cosmic energy
density which in turn controls the expansion rate of the Universe Furthermore
the electron neutrinos directly enter the weak conversion rates b etween neu
trons and protons Since neutrinos in uence nucleosynthesis in such a strong
way primordial nucleosynthesis can b e used as an e ective prob e of neutrino
physics
Starting out with physics entirely within the standard mo del we have p er
formed a detailed calculation of the e ect of neutrinos on nucleosynthesis This
calculation was describ ed in Chapter The p oint is that contrary to the usual
assumption neutrinos have not decoupled completely from the cosmic plasma
when the electrons and p ositrons annihilate Therefore they share to some de
gree the entropy transfer that takes place from the e plasma to the other
sp ecies This heats up the neutrino distribution relative to a totally decoupled
sp ecies and in uences nucleosynthesis The change is minute but nevertheless
imp ortant if one wants to build a very accurate nucleosynthesis co de This typ e
of calculation has b een p erformed b efore but our approach di ers by the fact
that we use the full set of Boltzmann equations without any approximations We
have found a somewhat smaller neutrino heating e ect than the previous state
of the art calculation done by Do delson and Turner Since our calculation
was p erformed new results have app eared by Dolgov Hansen and Semikoz
that con rm our results They have used the same approach as us but with
higher numerical precision
Going b eyond the standard mo del we have lo oked at neutrinos that are
massive but with lifetime much longer than the timescale for nucleosynthesis
This was done in Chapter Nucleosynthesis limits on massive neutrinos turn
out to b e complementary to limits obtainable in lab oratories and used together
they exclude a neutrino mass ab ove MeV if the neutrino lifetime is longer
than s Again our approach di ers from earlier calculations in that we
use the full Boltzmann equations It do es turns out however that our results
Chapter Conclusions
do not di er widely from those obtained by Fields Kainulainen and Olive
who used the integrated Boltzmann equation but included the correct Fermi
statistics for all neutrino sp ecies
Since any neutrino with mass ab ove the cosmologically safe limit of roughly
eV must b e unstable with lifetime shorter than the Hubble time it is natural
to discuss the p ossibility of neutrinos decaying during nucleosynthesis Neutrino
decays can b e split into a few generic typ es Esp ecially there is a di erence
b etween the case where decay pro ducts interact electromagnetically and the case
where they are invisible There exist very strong limits on the rst typ e of
decay making it extremely unlikely that these decays should play any role during
BBN However this is not the case for the second typ e of decay Such decays
have b een discussed in detail most recently by Do delson Gyuk and Turner
However their approach is to use the integrated Boltzmann equation in
a non relativistic approximation For neutrinos decaying su ciently late when
they have already b ecome non relativistic this is a very go o d approximation
For relativistic or semi relativistic decays this not the case Here one needs to
study the full Boltzmann equation since there can b e signi cant deviations from
kinetic and chemical equilibrium This has b een done previously for the very
simple decay by Kawasaki et al In Chapter we presented
the calculation for the case of of which is much more complicated
e
b ecause the decay pro duct electron neutrinos enter into the conversion pro cess
b etween neutrons and protons as previously mentioned In our calculation we
have found signi cant quantitative di erences b etween our calculation and those
p erformed with the non relativistic approximation We have also discussed the
p ossibility that a low mass tau neutrino decaying with short lifetime to an
electron neutrino nal state can bring nucleosynthesis predictions into b etter
agreement with observations
Lastly we have lo oked at a neutrino pro cess which is somewhat more exotic
namely that of neutrino lasing The fundamental idea was originally intro duced
by Madsen who studied a p ossible stimulated decay of some neutrino sp ecies
into a neutrino plus scalar particle nal state If this stimulated decay pro ceeds
in equilibrium it gives rise to the p ossibility of forming a Bose condensate of the
decay pro duct b osons This could p otentially b e very interesting for mo dels of
structure formation in the early Universe The reason is that b oth cold and hot
dark matter is apparently needed to t observations of the present day large
scale structure By having the ab ove mentioned stimulated decay some of the
b osons are in a Bose condensate whereas the rest are in a thermal distribution
If this b oson has a small mass it can act as b oth cold and hot dark matter
thereby explaining these mixed dark matter mo dels with only one new particle
The problem is however that only a relatively small fraction of the b osons end
up in the condensate roughly whereas a fraction closer to is needed to
t observational data The scenario was calculated through in more detail using
Boltzmann equation formalism by Starkman Kaiser and Malaney They
found that a true Bose condensate do es not form but that instead there will b e
a p eak of very low momentum b osons in a pseudo condensate The fraction
of particles in this low momentum p eak is roughly the same as that found by
Madsen for the equilibrium decay that is much to o small to b e interesting for
structure formation
We have develop ed a new scenario for the lasing decay using only equilibrium
initial distributions The trouble for the original scenario was that an initial
p opulation inversion is needed b etween the heavy and light neutrino Otherwise
the stimulated decay is never initiated and the nal distribution is thermal
Originally this p opulation inversion was invoked by having the light neutrino
decouple very early prior to the QCD phase transition so that its abundance
would b e diluted by the entropy released in the subsequent phase transition
However if there is a three neutrino mass hierarchy we have shown that it
is p ossible to achieve lasing with just the three known neutrino sp ecies In
this scenario the heaviest neutrino acts as the pumping level in a normal three
level laser whereas the next sp ecies in the hierarchy is the actual lasing level
When the heaviest sp ecies decays while still relativistic it pro duces a p opulation
inversion b etween the rst and second generation so that a lasing mechanism
o ccurs naturally We have presented thorough calculations of this scenario in
Chapter It was found that the amount of cold b osons pro duced is not much
higher than in the original scenario so that the mo del is still not viable as an
explanation of mixed dark matter However it do es show that the range of
mo dels where the lasing phenomenon can o ccur is much larger than originally
thought
In conclusion the present work has presented detailed calculations of some
of the physical mechanisms that can take place during or close to the era of
Big Bang Nucleosynthesis It was found that neutrinos play a very imp ortant
role for the evolution of our Universe and that cosmological observations can
therefore b e used to prob e neutrino physics in regions that terrestrial exp eri
ments cannot reach However there are still signi cant problems for example
the observational determinations of the primordial abundances are still quite
uncertain This again means that primordial nucleosynthesis is still not at a
su cient level of precision that one may start discussing constraints on ne
p oints in the input physics This issue will hop efully b e resolved in the coming
few years as the observations of for example QSO absorption systems are im
proved By determining the deuterium abundance accurately one should b e able
to pin down the baryon to photon ratio very precisely so that rm predictions
can b e made for the abundance of the other light elements We should then b e
in a p osition to really start testing the physical comp osition of the Universe at
an age of seconds using Big Bang nucleosynthesis
Finally a few words should p erhaps b e said ab out a p ossible new approach
to determining the fundamental cosmological parameters namely from the mea
suring of small scale anisotropies in the cosmic microwave background This will
b e done within the next few years by two satellites the American MAP and the
Europ ean PLANCK formerly known as COBRAS SAMBA missions
According to exp ectations these should b e able to p erform a precision mea
surement of the uctuation amplitude b eyond the rst Dopplerp eak thereby
Chapter Conclusions
enabling an accurate determination of cosmological parameters like H etc
This would yield a value for indep endent of nucleosynthesis arguments so
that the eld of Big Bang nucleosynthesis would enter a new precision era by
means of this external input
Chapter X
Outlo ok
There are a numb er of issues which have not b een addressed at all in the present
work that may b e very interesting for future work The rst is entirely within
the standard mo del and deals with the precision to which one can really predict
the helium abundance theoretically During the past year or two more and
more emphasis has b een put on the need to really understand the theoretical
uncertainties in the nucleosynthesis calculations This p oint is esp ecially con
cerning when one lo oks at He b ecause this is probably the only light element
where we can ever hop e to measure the primordial abundance to su cient ac
curacy There are a numb er of uncertainties in the standard nucleosynthesis
calculations that need to b e addressed This was rst discussed by Dicus et
al some years ago and since then there has b een an increasing numb er
of calculations of these e ects Primarily the error in the standard calculation
comes from neglect of nite nucleon masses The neutrino heating e ect dis
cussed in Chapter is another example of a missing element However there
are many more such ne p oints like medium e ects etc The normal calculation
assumes that all pro cesses pro ceed in vacuum which is of course not true Many
things are changed when physical pro cesses o ccur in a dense medium for ex
ample normally massless sp ecies like neutrinos and photons acquire an e ective
mass Even though many of these e ects are minute they are still interesting
from a theoretical p oint of view and may improve our understanding of physics
in the early Universe
Another imp ortant issue is the p ossibility of neutrino oscillations during
primordial nucleosynthesis This is of course p otentially imp ortant for massive
neutrinos such as a heavy tau neutrino Esp ecially if electron neutrinos are
involved in the oscillations one can obtain quite large e ects on nucleosynthesis
Last if we lo ok at neutrino physics more generally it would also b e very
interesting to do studies of neutrinos in the sup ernova environment This eld
is somewhat disjoint from neutrino physics in the early Universe however many
of the techniques applied are similar Also what we can learn from studying
neutrinos in sup ernovae is in many ways complementary to what one can learn
from neutrino physics in the early Universe For this reason this typ e of study is
Chapter Outlo ok
a very interesting future prosp ect In many ways the sup ernova environment is
not nearly as well understo o d as the early Universe Sup ernova physics couples
a lot of complicated phenomena together in an almost intractable way and
pro cesses such as the transp ort of neutrinos in a very dense medium are not
well understo o d from a theoretical p ersp ective Thus a lot of work remains to
b e done in this eld For instance presently the precise mechanism that leads
to the visible sup ernova blast instead of complete collapse to a black hole is not
understo o d but is b elieved to b e connected with the way neutrinos propagate
through the medium surrounding the young neutron star
Acknowledgements
There are a great numb er of p eople to whom thanks are due First of all I wish
to thank my thesis sup ervisor Jes Madsen for his guidance during the years and
for making me cho ose this eld of work by b eing such an inspiring lecturer
The help and guidance of Georg Ra elt has b een invaluable and he also
made p ossible my stay at the Max Planck Institut fur Physik in Munich for six
months during
There are a numb er of p eople at the Institute of Physics and Astronomy
whom I also wish to thank for interesting discussions and go o d company in
general my o ce mates Michael Christiansen and Georg B Larsen as well as
my other friends at the institute
Del B of my Ph D study has b een nanced by the Theoretical Astrophysics
Center under the Danish National Research Foundation and they have also
agreed to hire me after I have nished my Ph D
In the non physics community I wish to thank my parents for all their sup
p ort and esp ecially my friend Jesp er who has b een very go o d at keeping me
from doing physics all the time Thanks also to all my other friends
App endix A
Mathematical Formalism
A Dirac Matrices and Their Algebra
The Dirac matrices are de ned from the Dirac equation for a free particle
p m A
and they ob ey the commutation relations Cli ord algebra
f g g A
From these matrices one can de ne the a matrix as
i A
where ob eys the relations
A
and
f g A
so that is its own inverse and furthermore it anticommutes with all the other
matrices
Several di erent representations are used for the Dirac matrices dep end
ing on convenience in a given calculation The standard choice is the Dirac
representation where
I
i
A
i
I
i
and
I
A
I
The matrices are the usual Pauli spin matrices de ned as
i
i
A
i
App endix A Mathematical Formalism
Other choices than the Dirac representation are p ossible for example the
chiral representation in which
I
i
A
i
I
i
In this representation is
I
A
I
There are many di erent such representations which may b e chosen for conve
nience in a given situation
Another quantity which is often imp ortant is the commutator b etween Dirac
matrices
i
A
which has the prop erties
g i A
and
A
A Dirac Spinors
For free particles the solutions to the Dirac equation have the form of plane
waves of p ositive and negative energy
ip x
x ue A
ip x
x v e A
where u and v are Dirac spinors that are themselves solutions to the Dirac
equation
p m u A
p m v A
Explicitly these spinors are given by
p
s
A E m u p s
p
s
E m
p
p
s
E m
E m A v p s
s
where is a two comp onent spinor eld s is the helicity which can b e
s
Therefore the spinor can assume the values
s
A
A Creation and Annihilation Op erators
This gives the normalisation
y
0 0
A p E u p u
ss s
s
y
0 0
A p E v p v
ss s
s
y
0
p A v p u
s
s
of the Dirac spinors
A Creation and Annihilation Op erators
y
The fermion creation and annihilation op erators f and f as well as the cor
y
f and f are de ned so as to ob ey the resp onding op erators for antiparticles
anti commutation relations
y
0
ff p s f p s g p p A
ss
y
0
f p s f p s g p p A f
ss
with all other anti commutators b eing zero The action of the creation and
annihilation op erators on physical states has the following e ect
f p s jp si j i A
y p s
f j i j p s i A
f p s jp si j i A
y p s
f j i j p s i A
where by j i we mean the vacuum state
App endix A Mathematical Formalism
App endix B
Phase space Integrals
This app endix shows how to reduce the integral in Eq from to dimen
sions Except for the inclusion of mass terms most of what can b e found in this
app endix was originally develop ed by Yueh Buchler for use in sup ernova
calculations
The collision integrals all have the form of Eq The innermost integral
can b e rewritten using the equality
d p
d p p m p B
E
The integral over d p is now done using the function Hereafter p is xed as
p p p p p p p p p p B
Now the following angles are intro duced
p p
cos B
p p
p p
cos B
p p
p p
B cos
p p
cos cos sin sin cos
Thus we obtain
dp d cos d B d p p
d p p dp d cos d B
The integration over d is carried out using the function We use that
f B m p
App endix B Phase space Integrals
From the standard theory of functions we use
Z Z
X
B d f d
i
df
i
d
i
where are the ro ots of f The derivative is evaluated to
i
df
p p sin sin sin B
d
sin is found as cos where
i i
E E p p cos cos Q
B cos
i
p p sin sin
E E p p cos
p p sin sin
E E p p cos
p p sin sin
and we have intro duced Q m m m m The equation for sin will
i
have two solutions one in the interval and one in the interval
Since the r h s of Eq B is symmetric in we can multiply by two and
integrate over The limits on the integral d cos come from demanding
that cos But this means that
p p sin sin sin B
Notice that this is the same as demanding that
df
B
d
Finally we end up with
Z
df
B d f
df
d
i
d
i
The derivative can b e written as
p
df
a cos b cos c B
d
i
where
B a p
b p p p Q B
c Q Q Q B
cos p p
and we have intro duced the following parameters in order to limit the amount
of space required to write out the formulae
E E E E E E B
p p cos B
p p B
Notice that Eqs B B and B reduce to those of Yueh Buchler
in the limit of zero masses
Now any one of the p ossible matrix elements only include pro ducts of
momenta All the p ossible pro ducts of these momenta are calculated b elow
p p E E p p cos B
p p E E p p cos B
p p m E E p p cos B
E E p p cos
p p E E p p cos B
E E p p cos Q
p p E E p p cos m Q B
p p E E p p cos m Q B
Because all the ab ove pro ducts are analytically integrable over d cos the in
tegrals over this parameter can now b e carried out by use of the fundamental
relation
Z
dx
p p
ax bx c b ac B
a
bx c ax
The step function comes from demanding that there b e a real integration inter
val This actually also insures that the ro ots of ax bx c are not outside the
fundamental integration interval of b ecause the step function singles out
the physical situations where d cos can not b e outside this interval Integrat
ing over d is no problem as there is no dep endency on this parameter The
nal integration that can b e done is the one over d cos Any one of the p ossi
ble pro ducts of momenta is analytically integrable over d cos The solution of
b ac gives the integration interval The solutions are
Q B cos p
1
2
p p Q p p p p
If there is to b e a real integration interval b oth of these solutions must b e
real The fundamental interval is of course but the real limits are
sup cos and inf cos There can only b e a real integration
min max
interval if b oth and are real numb ers and
App endix B Phase space Integrals
Note that there are seemingly two places where divergences o ccur in the
collision integral The rst one is for p p cos but in this case there
is no problem b ecause although the integrand b ecomes in nite it is integrably
in nite The second place is for p To see what happ ens here we have
to go back to the fundamental integral Eq B If p then a p p
but then there can b e no divergence as p no longer app ears in any denominator
Clearly there is still the p ossibility of divergence if m also b ecause of the
E term Fortunately it turns out that b ac if m p so that
the rate b ecomes equal to in this case as it ought to b e Finally the collision
integral can b e written as
Z
p dp p dp
S B C f
coll
E E E
f f f f F p p p A
where A is the parameter space allowed that is the space de ned by requiring
real and F is the matrix element integrated over d cos and d cos
Z
F p p p j M j d cos d cos B
App endix C
English Resum e
The goal of the present work has b een to investigate asp ects of the interface
b etween particle physics and cosmology This eld is one of the most promising
in the whole area of mo dern physics since in many cases the whole Universe can
b e used as a giant particle physics lab oratory We have fo cused on the particles
called neutrinos and their imp ortance in the early Universe Neutrinos are
weakly interacting spin leptons essentially equivalent to electrons except
for the fact that they apparently do not p ossess either mass or electric charge
Three generations of neutrinos are known to exist and corresp onding
e
to the charged leptons e and Since neutrinos interact so weakly very little
is actually known ab out them They are extremely di cult to detect in the
lab oratory even though they are present in great abundance everywhere b eing
pro duced continuously by for example the Sun For this reason one must often
turn to indirect means of studying neutrinos One such p ossibility is to use the
early Universe as a neutrino lab oratory
There is no a priori reason why neutrinos should b e completely massless
and a large part of this thesis has b een devoted to the study of massive neu
trinos in cosmology It turns out that neutrinos are intimately connected with
the formation of light nuclei when the Universe was ab out second old the
phenomenon usually referred to as Big Bang Nucleosynthesis BBN Studies of
the outcome of BBN can therefore tell us a lot ab out the prop erties of neutrinos
Features that are completely unreachable in terrestrial lab oratories
The rst four chapters of the thesis have b een devoted to an intro duction to
the fundamental concepts needed Chapter is a general intro duction Chap
ter contains a discussion of the evolution equations for the expansion of the
Universe as well as a review of the Boltzmann equation the fundamental trans
p ort equation relating di erent particle sp ecies in the early Universe Chapter
reviews Big Bang Nucleosynthesis b oth the theoretical and the observational
asp ects In the theoretical section the in uence of neutrinos and other particles
on nucleosynthesis is discussed In the observational part we lo ok at how the
di erent light element abundances are observed as well as discuss the problems
connected with these observations In Chapter the physics of neutrinos is
App endix C English Resum e
discussed We start out with the de nitions of quantum neutrino elds and
discuss the weak interactions of neutrinos emphasising the similarities and dif
ferences of Dirac and Ma jorana neutrinos In the nal section neutrino decays
are discussed primarily the exp erimental constraints that exist on such decays
In the next four chapters we pro ceed with a more sp eci c discussion of neu
trino physics in the early Universe Chapter contains a thorough calculation of
the in uence of massless standard mo del neutrinos on primordial nucleosynthe
sis We nd that the neutrinos distributions are heated slightly relative to the
standard calculations b ecause energy is transferred from electrons and p ositrons
as these b ecome non relativistic and annihilate Our calculation has b een more
careful than previous ones taking into account the full quantum statistics of
neutrinos After our results app eared new and numerically more precise results
by Dolgov Hansen and Semikoz have app eared that con rm our results
Chapter go es b eyond the minimal standard mo del in that neutrinos are
allowed to have mass However we still assume that they only p ossess standard
mo del weak interactions and furthermore we assume their lifetime against de
cay to b e su ciently long not to have any e ect on nucleosynthesis s
Massive neutrinos a ect primordial nucleosynthesis quite dramatically mainly
b ecause their non zero rest mass energy can come to dominate the cosmic en
ergy density at temp eratures b elow the rest mass This again changes the cosmic
expansion rate and generally leads to a strong increase in helium pro duction
We nd that tau neutrinos with mass larger than roughly MeV are ruled
out by combining our nucleosynthesis arguments with the lab oratory mass limit
of MeV for the tau neutrino This is consistent with previous results but
our treatment has included use of the full Boltzmann equation following each
momentum mo de in time instead of the usual approach where one assumes scat
tering equilibrium for the involved neutrinos Our calculations were p erformed
for Ma jorana neutrinos only but the limits can b e exp ected to b e roughly the
same for Dirac neutrinos also ruling out a massive Dirac neutrino if its rest
mass is larger than roughly MeV still assuming a su ciently long lifetime
against decay
In Chapter we pro ceed to lo ok at the p ossibility of neutrino decays during
nucleosynthesis This phenomenon has b een discussed many times in the liter
ature but we lo ok at one very interesting range in parameter space that has so
far not b een treated prop erly namely that of a low mass tau neutrino decaying
to an electron neutrino nal state For the last few years there have b een some
indication that the standard picture of BBN may b e lacking something The
problem is that apparently to o much helium is pro duced to b e consistent with
observations A decay of the typ e we discuss can help solve this problem since
it lowers the helium abundance from nucleosynthesis if the mass and lifetime of
the decaying neutrino are b oth su ciently low Again we have used the full set
of Boltzmann equations as opp osed to previous treatments For the sp eci c de
cay we lo ok at this is the only way to obtain sensible results b ecause the decay
do es not necessarily install equilibrium in the particle distributions We have
found a large range of allowed neutrino decay lifetimes for masses in the range
MeV As in the previous chapter we only lo oked at Ma jorana neutrinos
b ecause there are several additional complications for Dirac neutrinos
Lastly in Chapter we have lo oked at a di erent asp ect of neutrino physics
in the early Universe Here we study the p ossibility of making dark matter
from neutrino decays Dark matter is the generic term for the mass known
to b e present in the Universe but not identi able as normal baryonic matter
This matter is for example needed to pro duce the present day structure of
the Universe such as galaxies etc It turns out that to b e consistent with
observations most of the dark matter should b e in very heavy particles called
cold dark matter However a small fraction of light particles is needed termed
hot dark matter Some years ago it was suggested by Madsen that this typ e of
scenario could p ossibly b e pro duced by stimulated decay of massive neutrinos
to a nal state containing some scalar particle It turned out however that the
relative fraction of cold and hot dark matter was wrong most of the dark matter
ended up in the form of hot dark matter We have develop ed this scenario to
a new mo del with more than one neutrino decay but it still app ears that to o
much hot dark matter is pro duced
App endix C English Resum e
Bibliography
G F Smo ot et al Astrophys J Lett L
See for example R Giovanelli and M P Haynes in Annual Review of
Astronomy and Astrophysics
G Gamow Phys Rev
Much more extensive information can b e found in recent reviews such as
H Reeves Rev Mo d Phys R A Malaney and G J
Mathews Phys Rep M S Smith L H Kawano and R
A Malaney Astrophys J Suppl
S Burles and D Tytler Rep ort no astro ph
K A Olive and D Thomas Astropart Phys
R M Barnett et al Particle Data Group Phys Rev D
N Hata et al Phys Rev Lett
J Bernstein Kinetic Theory in the Expanding Universe Cambridge Uni
versity Press
S Weinb erg Gravitation and Cosmology John Wiley and Sons
G G Ra elt Stars as Laboratories for Fundamental Physics University
of Chicago Press
B D Fields K Kainulainen and K A Olive Rep ort no hep ph
Bibliography
R V Wagoner in Physical Cosmology Les Houches edited by
R Balian J Audouze and D N Schramm Les Houches Summer Scho ol
Pro ceedings Vol p
P Gondolo and G Gelmini Nucl Phys B
K Kainulainen and A D Dolgov Nucl Phys B
G D Starkman N Kaiser and R A Malaney Astrophys J
K A Olive and G Steigman Astrophys J Suppl
Y I Izotov T X Thuan and V A Lip ovetsky Astrophys J Suppl
A Songaila et al Nature London
M Rugers and C J Hogan Astron J
J K Webb et al Nature
D Tytler S Burles and D Kirkman Rep ort no astro ph
F Spite and M Spite Astron Astrophys
P Bonifacio and P Molaro Mon Not R Astron So c
see for example J Bahcall talk given at th Texas Symp osium on Rela
tivistic Astrophysics Chicago IL Rep ort no astro ph
See for example T K Gaisser Talk given at the th International Con
ference on Neutrino Physics and Astrophysics Neutrino Helsinki Fin
land Rep ort no hep ph
C Athanassop oulos et al Phys Rev Lett
K Grotz and H V Klap dor The Weak Interaction in Nuclear Particle
and Astrophysics Adam Hilger
Bibliography
K Assamagan et al Phys Rev D
D Buskulic et al Phys Lett B
B Kayser F Gibrat Debu and F Perrier The Physics of Massive Neutri
nos World Scienti c Lecture Notes in Physics Vol
D Bailin and A Love Introduction to Gauge Field Theory Adam Hilger
R Plaga Rep ort no hep ph S Hannestad Rep ort no
hep ph B Kayser Reoprt no hep ph A
Ho efer Rep ort no hep ph S H Hansen Rep ort no hep
ph
see the pap er by R Plaga in Ref
P Vilain et al CHARM II collab oration Phys Lett B
E W Kolb and M S Turner The Early Universe Addison Wesley
P J E Peebles Principles of Physical Cosmology Princeton University
Press
see for example A L Mellott and D W Sciama Phys Rev Lett
see for example R N Mohapatra and P B Pal Massive Neutrinos in
Physics and Astrophysics World Scienti c Lecture Notes in Physics
Vol
S Do delson G Gyuk and M S Turner Phys Rev D
S Do delson and M S Turner Phys Rev D
A D Dolgov and M Fukugita Phys Rev D
D A Dicus et al Phys Rev D
Bibliography
M A Herrera and S Hacyan Astrophys J
N C Rana and B Mitra Phys Rev D
S Hannestad and J Madsen Phys Rev D
L Kawano Rep ort no FERMILAB Pub A unpublished
B D Fields S Do delson and M S Turner Phys Rev D
C J Copi D N Schramm and M S Turner Phys Rev Lett
R Cowsik and J McClelland Phys Rev Lett
A D Dolgov S H Hansen and D V Semikoz Preprint no TAC
Rep ort no hep ph
For recent reviews see G Gyuk and M S Turner Nucl Phys B Pro c
Suppl G Gelmini and E Roulet Rep Prog Phys
D A Dicus et al Phys Rev D
E W Kolb and R J Scherrer Phys Rev D
R J Scherrer and M J Turner Ap J
A D Dolgov K Kainulainen and I Z Rothstein Phys Rev D
E W Kolb et al Phys Rev Lett
A D Dolgov and I Z Rothstein Phys Rev Lett
M Kawasaki et al Nucl Phys B
A D Dolgov S Pastor and J W F Valle Rep ort no hep ph
Bibliography
S Hannestad and J Madsen Phys Rev Lett
J Bernstein L S Brown and G Feinb erg Phys Rev D
In the published version of Ref the same e ect will b e accounted for
with very go o d agreement with our results Kimmo Kainulainen private
communication
N Terasawa and K Sato Phys Lett B
E W Kolb and R J Scherrer Phys Rev D
R J Scherrer and M S Turner Astrophys J R J Scherrer
and M S Turner ibid
J Madsen Phys Rev Lett
M Kawasaki K Kohri and K Sato Rep ort no astro ph
S Hannestad and J Madsen Phys Rev D
C J Copi D N Schramm and M S Turner FERMILAB Pub A
Recent treatments are M Kawasaki G Steigman and H S Kang Nucl
Phys B G Gyuk and M S Turner in Neutrino Pro
ceedings of the th International Conference on Neutrino Physics and As
trophysics Eilat Israel edited by A Dar et al Nucl Phys Pro c Suppl
B
N Kaiser R A Malaney and G D Starkman Phys Rev Lett
See for example A Klypin J Holtzman J Primack and E Reg os Astro
phys J
G B Larsen and J Madsen Phys Rev D
Bibliography
G Gyuk and M S Turner Phys Rev D
Z G Berezhiani et al Phys Lett B
See for example C W Kim and A Pevsner Neutrinos in Physics and
Astrophysics Harwo o d Academic Publishers
W R Yueh and J R Buchler Astrophysics Space Science