Abstract
Cosmological implications of neutrinos are reviewed. The following subjects are discussed at a different level of scrutiny: cosmological limits on neutrino mass, neutrinos and primordial nucleosynthesis, cosmological constraints on unstable neutrinos, lepton asymmetry of the universe, impact of neutrinos on cosmic microwave radiation, neutrinos and the large scale structure of the universe, neutrino oscillations in the early universe, baryo/leptogenesis and neutrinos, neutrinos and high energy cosmic rays, and briefly some more exotic subjects: neutrino balls, mirror neutrinos, and neutrinos from large extra dimensions.
April, 2002 INFN2002
Neutrinos in cosmology
A.D. Dolgov
[.05in] INFN, sezzione di Ferrara
via Paradiso, 12, 44100  Ferrara, Italy ^{1}^{1}1Also: ITEP, Bol. Cheremushkinskaya 25, Moscow 113259, Russia.
Content

Introduction

Neutrino properties.

Basics of cosmology.
3.1. Basic equations and cosmological parameters.
3.2. Thermodynamics of the early universe.
3.3. Kinetic equations.
3.4. Primordial nucleosynthesis. 
Massless or light neutrinos
4.1. GersteinZeldovich limit.
4.2. Spectral distortion of massless neutrinos. 
Heavy neutrinos.
5.1. Stable neutrinos, GeV.
5.2. Stable neutrinos, GeV. 
Neutrinos and primordial nucleosynthesis.
6.1. Bound on the number of relativistic species.
6.2. Massive stable neutrinos. Bounds on .
6.3. Massive unstable neutrinos.
6.4. Righthanded neutrinos.
6.5. Magnetic moment of neutrinos.
6.6. Neutrinos, light scalars, and BBN.
6.7. Heavy sterile neutrinos: cosmological bounds and direct experiment. 
Variation of primordial abundances and lepton asymmetry of the universe.

Decaying neutrinos.
8.1. Introduction.
8.2. Cosmic density constraints.
8.3. Constraints on radiative decays from the spectrum of cosmic microwave background radiation.
8.4. Cosmic electromagnetic radiation, other than CMB. 
Angular anisotropy of CMB and neutrinos.

Cosmological lepton asymmetry.
10.1. Introduction.
10.2. Cosmological evolution of strongly degenerate neutrinos.
10.3. Degenerate neutrinos and primordial nucleosynthesis.
10.4. Degenerate neutrinos and large scale structure.
11.4. Neutrino degeneracy and CMBR. 
Neutrinos, dark matter and large scale structure of the universe.
11.1. Normal neutrinos.
11.2. Lepton asymmetry and large scale structure.
11.3. Sterile neutrinos.
11.4. Anomalous neutrino interactions and dark matter; unstable neutrinos. 
Neutrino oscillations in the early universe.
12.1. Neutrino oscillations in vacuum. Basic concepts.
12.2. Matter effects. General description.
12.3. Neutrino oscillations in cosmological plasma.
12.3.1. A brief (and noncomplete) review.
12.3.2. Refraction index.
12.3.3. Loss of coherence and density matrix.
12.3.4. Kinetic equation for density matrix.
12.4. Nonresonant oscillations.
12.5. Resonant oscillations and generation of lepton asymmetry.
12.5.1. Notations and equations.
12.5.2. Solution without backreaction.
12.5.3. Backreaction.
12.5.4. Chaoticity.
12.6. Activeactive neutrino oscillations.
12.7. Spatial fluctuations of lepton asymmetry.
12.8. Neutrino oscillations and big bang nucleosynthesis.
12.9. Summary. 
Neutrino balls.

Mirror neutrinos.

Neutrinos and large extra dimensions.

Neutrinos and lepto/baryogenesis.

Cosmological neutrino background and ultrahigh energy cosmic rays.

Conclusion.

References.
1 Introduction
The existence of neutrino was first proposed by Pauli in 1930 [1] as an attempt to explain the continuous energy spectrum observed in betadecay [2] under the assumption of energy conservation. Pauli himself did not consider his solution to be a very probable one, though today such observation would be considered unambiguous proof of the existence of a new particle. That particle was named “neutrino” in 1933, by Fermi. A good, though brief description of historical events leading to discovery can be found in ref. [3].
The method of neutrino detection was suggested by Pontecorvo [4]. To this end he proposed the chlorineargon reaction and discussed the possibility of registering solar neutrinos. This very difficult experiment was performed by Davies et al [5] in 1968, and marked the discovery neutrinos from the sky (solar neutrinos). The experimental discovery of neutrino was carried out by Reines and Cowan [6] in 1956, a quarter of a century after the existence of that particle was predicted.
In 1943 Sakata and Inouë [7] suggested that there might be more than one species of neutrino. Pontecorvo [8] in 1959 made a similar conjecture that neutrinos emitted in betadecay and in muon decay might be different. This hypothesis was confirmed in 1962 by Danby et al [9], who found that neutrinos produced in muon decays could create in secondary interactions only muons but not electrons. It is established now that there are at least three different types (or flavors) of neutrinos: electronic (), muonic (), and tauonic () and their antiparticles. The combined LEP result [10] based on the measurement of the decay width of boson gives the following number of different neutrino species: , including all neutral fermions with the normal weak coupling to and mass below GeV.
It was proposed by Pontecorvo [11, 12] in 1957 that, in direct analogy with oscillations, neutrinos may also oscillate due to transformation. After it was confirmed that and are different particles [9], Maki, Nakagawa, and Sakata [13] suggested the possibility of neutrino flavor oscillations, . A further extension of the oscillation space what would permit the violation of the total leptonic charge as well as violation of separate lepton flavor charges, and , or flavor oscillations of Majorana neutrinos was proposed by Pontecorvo and Gribov [14, 15]. Nowadays the phenomenon of neutrino oscillations attracts great attention in experimental particle physics as well as in astrophysics and cosmology. A historical review on neutrino oscillations can be found in refs. [16, 17].
Cosmological implications of neutrino physics were first considered in a paper by Alpher et al [18] who mentioned that neutrinos would be in thermal equilibrium in the early universe. The possibility that the cosmological energy density of neutrinos may be larger than the energy density of baryonic matter and the cosmological implications of this hypothesis were discussed by Pontecorvo and Smorodinskii [19]. A little later Zeldovich and Smorodinskii [20] derived the upper limit on the density of neutrinos from their gravitational action. In a seminal paper in 1966, Gerstein and Zeldovich [21] derived the cosmological upper limit on neutrino mass, see below sec. 4.1. This was done already in the frameworks of modern cosmology. Since then the interplay between neutrino physics and cosmology has been discussed in hundreds of papers, where limits on neutrino properties and the use of neutrinos in solving some cosmological problems were considered. Neutrinos could have been important in the formation of the largescale structure (LSS) of the universe, in big bang nucleosynthesis (BBN), in anisotropies of cosmic microwave background radiation (CMBR), and some others cosmological phenomena. This is the subject of the present review. The field is so vast and the number of published papers is so large that I had to confine the material strictly to cosmological issues. Practically no astrophysical material is presented, though in many cases it is difficult to draw a strict border between the two. For the astrophysical implications of neutrino physics one can address the book [22] and a more recent review [23]. The number of publications rises so quickly (it seems, with increasing speed) that I had to rewrite already written sections several times to include recent developments. Many important papers could be and possibly are omitted involuntary but their absence in the literature list does not indicate any author’s preference. They are just “large number errors”. I tried to find old pioneering papers where essential physical mechanisms were discovered and the most recent ones, where the most accurate treatment was performed; the latter was much easier because of available astroph and hepph archives.
2 Neutrino properties.
It is well established now that neutrinos have standard weak interactions mediated by  and bosons in which only lefthanded neutrinos participate. No other interactions of neutrinos have been registered yet. The masses of neutrinos are either small or zero. In contrast to photons and gravitons, whose vanishing masses are ensured by the principles of gauge invariance and general covariance respectively, no similar theoretical principle is known for neutrinos. They may have nonzero masses and their smallness presents a serious theoretical challenge. For reviews on physics of (possibly massive) neutrinos see e.g. the papers [24][30]. Direct observational bounds on neutrino masses, found kinematically, are:
(1) 
(2) 
(3) 
while cosmological upper limit on masses of light stable neutrinos is about 10 eV (see below, Sec. 4.1).
Even if neutrinos are massive, it is unknown if they have Dirac or Majorana mass. In the latter case processes with leptonic charge nonconservation are possible and from their absence on experiment, in particular, from the lower limits on the nucleus lifetime with respect to neutrinoless double beta decay one can deduce an upper limit on the Majorana mass. The most stringent bound was obtained in HeidelbergMoscow experiment [35]: eV; for the results of other groups see [25].
There are some experimentally observed anomalies (reviewed e.g. in refs. [24, 25]) in neutrino physics, which possibly indicate new phenomena and most naturally can be explained by neutrino oscillations. The existence of oscillations implies a nonzero mass difference between oscillating neutrino species, which in turn means that at least some of the neutrinos should be massive. Among these anomalies is the well known deficit of solar neutrinos, which has been registered by several installations: the pioneering Homestake, GALLEX, SAGE, GNO, Kamiokande and its mighty successor, SuperKamiokande. One should also mention the first data recently announced by SNO [36] where evidence for the presence of or in the flux of solar neutrinos was given. This observation strongly supports the idea that is mixed with another active neutrino, though some mixing with sterile ones is not excluded. An analysis of the solar neutrino data can be found e.g. in refs. [37][42]. In the last two of these papers the data from SNO was also used.
The other two anomalies in neutrino physics are, first, the appearance seen in LSND experiment [43] in the flux of from decay at rest and appearance in the flux of from the decay in flight. In a recent publication [44] LSNDgroup reconfirmed their original results. The second anomaly is registered in energetic cosmic ray air showers. The ratio of fluxes is suppressed by factor two in comparison with theoretical predictions (discussion and the list of the references can be found in [24, 25]). This effect of anomalous behavior of atmospheric neutrinos recently received very strong support from the SuperKamiokande observations [45] which not only confirmed deficit but also discovered that the latter depends upon the zenith angle. This latest result is a very strong argument in favor of neutrino oscillations. The best fit to the oscillation parameters found in this paper for oscillations are
(4) 
The earlier data did not permit distinguishing between the oscillations and the oscillations of into a noninteracting sterile neutrino, , but more detailed investigation gives a strong evidence against explanation of atmospheric neutrino anomaly by mixing between and [46].
After the SNO data [36] the explanation of the solar neutrino anomaly also disfavors dominant mixing of with a sterile neutrino and the mixing with or is the most probable case. The best fit to the solar neutrino anomaly [42] is provided by MSWresonance solutions (MSW means MikheevSmirnov [47] and Wolfenstein [48], see sec. 12)  either LMA (large mixing angle solution):
(5) 
or LOW (low mass solution):
(6) 
Vacuum solution is almost equally good:
(7) 
Similar results are obtained in a slightly earlier paper [41].
The hypothesis that there may exist an (almost) new noninteracting sterile neutrino looks quite substantial but if all the reported neutrino anomalies indeed exist, it is impossible to describe them all, together with the limits on oscillation parameters found in plethora of other experiments, without invoking a sterile neutrino. The proposal to invoke a sterile neutrino for explanation of the total set of the observed neutrino anomalies was raised in the papers [49, 50]. An analysis of the more recent data and a list of references can be found e.g. in the paper [24]. Still with the exclusion of some pieces of the data, which may be unreliable, an interpretation in terms of three known neutrinos remains possible [51, 52]. For an earlier attempt to “satisfy everything” based on threegeneration neutrino mixing scheme see e.g. ref. [53]. If, however, one admits that a sterile neutrino exists, it is quite natural to expect that there exist even three sterile ones corresponding to the known active species: , , and . A simple phenomenological model for that can be realized with the neutrino mass matrix containing both Dirac and Majorana mass terms [54]. Moreover, the analysis performed in the paper [55] shows that the combined solar neutrino data are unable to determine the sterile neutrino admixture.
If neutrinos are massive, they may be unstable. Direct bounds on their lifetimes are very loose [10]: sec/eV, sec/eV, and no bound is known for . Possible decay channels of a heavier neutrino, permitted by quantum numbers are: , , and . If there exists a yetundiscovered light (or massless) (pseudo)scalar boson , for instance majoron [56] or familon [57], another decay channel is possible: . Quite restrictive limits on different decay channels of massive neutrinos can be derived from cosmological data as discussed below.
In the standard theory neutrinos possess neither electric charge nor magnetic moment, but have an electric formfactor and their charge radius is nonzero, though negligibly small. The magnetic moment may be nonzero if righthanded neutrinos exist, for instance if they have a Dirac mass. In this case the magnetic moment should be proportional to neutrino mass and quite small [58, 59]:
(8) 
where is the Fermi coupling constant, is the magnitude of electric charge of electron, and is the Bohr magneton. In terms of the magnetic field units G=Gauss the Born magneton is equal to . The experimental upper limits on magnetic moments of different neutrino flavors are [10]:
(9) 
These limits are very far from simple theoretical expectations. However in more complicated theoretical models much larger values for neutrino magnetic moment are predicted, see sec. 6.5.
Righthanded neutrinos may appear not only because of the leftright transformation induced by a Dirac mass term but also if there exist direct righthanded currents. These are possible in some extensions of the standard electroweak model. The lower limits on the mass of possible righthanded intermediate bosons are summarized in ref. [10] (page 251). They are typically around a few hundred GeV. As we will see below, cosmology gives similar or even stronger bounds.
Neutrino properties are well described by the standard electroweak theory that was finally formulated in the late 60th in the works of S. Glashow, A. Salam, and S. Weinberg. Together with quantum chromodynamics (QCD), this theory forms the so called Minimal Standard Model (MSM) of particle physics. All the existing experimental data are in good agreement with MSM, except for observed anomalies in neutrino processes. Today neutrino is the only open window to new physics in the sense that only in neutrino physics some anomalies are observed that disagree with MSM. Cosmological constraints on neutrino properties, as we see in below, are often more restrictive than direct laboratory measurements. Correspondingly, cosmology may be more sensitive to new physics than particle physics experiments.
3 Basics of cosmology.
3.1 Basic equations and cosmological parameters.
We will present here some essential cosmological facts and equations so that the paper would be selfcontained. One can find details e.g. in the textbooks [60][65]. Throughout this review we will use the natural system of units, with , , and each equaling 1. For conversion factors for these units see table 1 which is borrowed from ref. [66].
In the approximation of a homogeneous and isotropic universe, its expansion is described by the FriedmanRobertsonWalker metric:
(10) 
For the homogeneous and isotropic distribution of matter the energymomentum tensor has the form
(11) 
where and are respectively energy and pressure densities. In this case the Einstein equations are reduced to the following two equations:
(12)  
(13) 
where is the gravitational coupling constant, , with the Planck mass equal to GeV. From equations (12) and (13) follows the covariant law of energy conservation, or better to say, variation:
(14) 
where is the Hubble parameter. The critical or closure energy density is expressed through the latter as:
(15) 
corresponds to eq. (13) in the flat case, i.e. for . The presentday value of the critical density is
(16) 
where is the dimensionless value of the present day Hubble parameter measured in 100 km/sec/Mpc. The value of the Hubble parameter is rather poorly known, but it would be possibly safe to say that with the preferred value [67].
The magnitude of mass or energy density in the universe, , is usually presented in terms of the dimensionless ratio
(17) 
Inflationary theory predicts with the accuracy or somewhat better. Observations are most likely in agreement with this prediction, or at least do not contradict it. There are several different contributions to coming from different forms of matter. The cosmic baryon budget was analyzed in refs. [68, 69]. The amount of visible baryons was estimated as [68], while for the total baryonic mass fraction the following range was presented [69]:
(18) 
with the best guess (for ). The recent data on the angular distribution of cosmic microwave background radiation (relative heights of the first and second acoustic peaks) add up to the result presented, e.g., in ref. [70]:
(19) 
Similar results are quoted in the works [71].
There is a significant contribution to from an unknown dark or invisible matter. Most probably there are several different forms of this mysterious matter in the universe, as follows from the observations of large scale structure. The matter concentrated on galaxy cluster scales, according to classical astronomical estimates, gives:
(20) 
A recent review on the different ways of determining can be found in [74]; though most of measurements converge at , there are some indications for larger or smaller values.
It was observed in 1998 [75] through observations of high redsift supernovae that vacuum energy density, or cosmological constant, is nonzero and contributes:
(21) 
This result was confirmed by measurements of the position of the first acoustic peak in angular fluctuations of CMBR [76] which is sensitive to the total cosmological energy density, . A combined analysis of available astronomical data can be found in recent works [77, 78, 79], where considerably more accurate values of basic cosmological parameters are presented.
The discovery of nonzero lambdaterm deepened the mystery of vacuum energy, which is one of the most striking puzzles in contemporary physics  the fact that any estimated contribution to is 50100 orders of magnitude larger than the upper bound permitted by cosmology (for reviews see [80, 81, 82]). The possibility that vacuum energy is not precisely zero speaks in favor of adjustment mechanism[83]. Such mechanism would, indeed, predict that vacuum energy is compensated only with the accuracy of the order of the critical energy density, at any epoch of the universe evolution. Moreover, the noncompensated remnant may be subject to a quite unusual equation of state or even may not be described by any equation of state at all. There are many phenomenological models with a variable cosmological ”constant” described in the literature, a list of references can be found in the review [84]. A special class of matter with the equation of state with has been named ”quintessence” [85]. An analysis of observational data [86] indicates that which is compatible with simple vacuum energy, . Despite all the uncertainties, it seems quite probable that about half the matter in the universe is not in the form of normal elementary particles, possibly yet unknown, but in some other unusual state of matter.
To determine the expansion regime at different periods cosmological evolution one has to know the equation of state . Such a relation normally holds in some simple and physically interesting cases, but generally equation of state does not exist. For a gas of nonrelativistic particles the equation of state is (to be more precise, the pressure density is not exactly zero but ). For the universe dominated by nonrelativistic matter the expansion law is quite simple if : . It was once believed that nonrelativistic matter dominates in the universe at sufficiently late stages, but possibly this is not true today because of a nonzero cosmological constant. Still at an earlier epoch () the universe was presumably dominated by nonrelativistic matter.
In standard cosmology the bulk of matter was relativistic at much earlier stages. The equation of state was and the scale factor evolved as . Since at that time was extremely close to unity, the energy density was equal to
(22) 
For vacuum dominated energymomentum tensor, , , and the universe expands exponentially, .
Integrating equation (13) one can express the age of the universe through the current values of the cosmological parameters and , where sub refers to different forms of matter with different equations of state:
(23) 
where , , and correspond respectively to the energy density of nonrelativistic matter, relativistic matter, and to the vacuum energy density (or, what is the same, to the cosmological constant); , and yr. This expression can be evidently modified if there is an additional contribution of matter with the equation of state . Normally because and . On the other hand and it is quite a weird coincidence that just today. If and both vanishes, then there is a convenient expression for valid with accuracy better than 4% for :
(24) 
Most probably, however, , as predicted by inflationary cosmology and . In that case the universe age is
(25) 
It is clear that if , then the universe may be considerably older with the same value of . These expressions for will be helpful in what follows for the derivation of cosmological bounds on neutrino mass.
The age of old globular clusters and nuclear chronology both give close values for the age of the universe [72]:
(26) 
3.2 Thermodynamics of the early universe.
At early stages of cosmological evolution, particle number densities, , were so large that the rates of reactions, , were much higher than the rate of expansion, (here is crosssection of the relevant reactions). In that period thermodynamic equilibrium was established with a very high degree of accuracy. For a sufficiently weak and shortrange interactions between particles, their distribution is represented by the well known Fermi or BoseEinstein formulae for the ideal homogeneous gas (see e.g. the book [87]):
(27) 
Here signs ’+’ and ’’ refer to fermions and bosons respectively, is the particle energy, and is their chemical potential. As is well known, particles and antiparticles in equilibrium have equal in magnitude but opposite in sign chemical potentials:
(28) 
This follows from the equilibrium condition for chemical potentials which for an arbitrary reaction has the form
(29) 
and from the fact that particles and antiparticles can annihilate into different numbers of photons or into other neutral channels, . In particular, the chemical potential of photons vanishes in equilibrium.
If certain particles possess a conserved charge, their chemical potential in equilibrium may be nonvanishing. It corresponds to nonzero density of this charge in plasma. Thus, plasma in equilibrium is completely defined by temperature and by a set of chemical potentials corresponding to all conserved charges. Astronomical observations indicate that the cosmological densities  of all charges  that can be measured, are very small or even zero. So in what follows we will usually assume that in equilibrium , except for Sections 10, 11.2, 12.5, and 12.7, where lepton asymmetry is discussed. In outofequilibrium conditions some effective chemical potentials  not necessarily just those that satisfy condition (28)  may be generated if the corresponding charge is not conserved.
The number density of bosons corresponding to distribution (27) with is
(30) 
Here summation is made over all spin states of the boson, is the number of this states, . In particular the number density of equilibrium photons is
(31) 
where 2.728 K is the present day temperature of the cosmic microwave background radiation (CMB).
For fermions the equilibrium number density is
(32) 
The equilibrium energy density is given by:
(33) 
Here the summation is done over all particle species in plasma and their spin states. In the relativistic case
(34) 
where is the effective number of relativistic species, , the summation is done over all species and their spin states. In particular, for photons we obtain
(35) 
The contribution of heavy particles, i.e. with , into is exponentially small if the particles are in thermodynamic equilibrium:
(36) 
Sometimes the total energy density is described by expression (34) with the effective including contributions of all relativistic as well as nonrelativistic species.
As we will see below, the equilibrium for stable particles sooner or later breaks down because their number density becomes too small to maintain the proper annihilation rate. Hence their number density drops as and not exponentially. This ultimately leads to a dominance of massive particles in the universe. Their number and energy densities could be even higher if they possess a conserved charge and if the corresponding chemical potential is nonvanishing.
Since was very close to unity at early cosmological stages, the energy density at that time was almost equal to the critical density (22). Taking this into account, it is easy to determine the dependence of temperature on time during RDstage when and is given simultaneously by eqs. (34) and (22):
(37) 
For example, in equilibrium plasma consisting of photons, , and three types of neutrinos with temperatures above the electron mass but below the muon mass, MeV, the effective number of relativistic species is
(38) 
In the course of expansion and cooling down, decreases as the particle species with disappear from the plasma. For example, at when the only relativistic particles are photons and three types of neutrinos with the temperature the effective number of species is
(39) 
If all chemical potentials vanish and thermal equilibrium is maintained, the entropy of the primeval plasma is conserved:
(40) 
In fact this equation is valid under somewhat weaker conditions, namely if particle occupation numbers are arbitrary functions of the ratio and the quantity (which coincides with temperature only in equilibrium) is a function of time subject to the condition (14).
3.3 Kinetic equations.
The universe is not stationary, it expands and cools down, and as a result thermal equilibrium is violated or even destroyed. The evolution of the particle occupation numbers is usually described by the kinetic equation in the ideal gas approximation. The latter is valid because the primeval plasma is not too dense, particle mean free path is much larger than the interaction radius so that individual distribution functions , describing particle energy spectrum, are physically meaningful. We assume that depends neither on space point nor on the direction of the particle momentum. It is fulfilled because of cosmological homogeneity and isotropy. The universe expansion is taken into account as a redshifting of particle momenta, . It gives:
(41) 
As a result the kinetic equation takes the form
(42) 
where is the collision integral for the process :
(43)  
Here and are arbitrary, generally multiparticle states, is the product of phase space densities of particles forming the state , and
(44) 
The signs ’+’ or ’’ in are chosen for bosons and fermions respectively.
It can be easily verified that in the stationary case (), the distributions (27) are indeed solutions of the kinetic equation (42), if one takes into account the conservation of energy , and the condition (29). This follows from the validity of the relation
(45) 
and from the detailed balance condition, (with a trivial transformation of kinematical variables). The last condition is only true if the theory is invariant with respect to time reversion. We know, however, that CPinvariance is broken and, because of the CPTtheorem, Tinvariance is also broken. Thus Tinvariance is only approximate. Still even if the detailed balance condition is violated, the form of equilibrium distribution functions remain the same. This is ensured by the weaker condition [88]:
(46) 
where summation is made over all possible states . This condition can be termed the cyclic balance condition, because it demonstrates that thermal equilibrium is achieved not by a simple equality of probabilities of direct and inverse reactions but through a more complicated cycle of reactions. Equation (46) follows from the unitarity of matrix, . In fact, a weaker condition is sufficient for saving the standard form of the equilibrium distribution functions, namely the diagonal part of the unitarity relation, , and the inverse relation , where is the probability of transition from the state to the state . The premise that the sum of probabilities of all possible events is unity is of course evident. Slightly less evident is the inverse relation, which can be obtained from the first one by the CPTtheorem.
For the solution of kinetic equations, which will be considered below, it is convenient to introduce the following dimensionless variables:
(47) 
where is the scale factor and is some fixed parameter with dimension of mass (or energy). Below we will take MeV. The scale factor is normalized so that in the early thermal equilibrium relativistic stage . In terms of these variables the l.h.s. of kinetic equation (42) takes a very simple form:
(48) 
When the universe was dominated by relativistic matter and when the temperature dropped as , the Hubble parameter could be taken as
(49) 
In many interesting cases the evolution of temperature differs from the simple law specified above but still the expression (49) is sufficiently accurate.
3.4 Primordial nucleosynthesis
Primordial or big bang nucleosynthesis (BBN) is one of the cornerstones of standard big bang cosmology. Its theoretical predictions agree beautifully with observations of the abundances of the light elements, , , and , which span 9 orders of magnitude. Neutrinos play a significant role in BBN, and the preservation of successful predictions of BBN allows one to work our restrictive limits on neutrino properties.
Below we will present a simple pedagogical introduction to the theory of BBN and briefly discuss observational data. The content of this subsection will be used in sec. 6 for the analysis of neutrino physics at the nucleosynthesis epoch. A good reference where these issues are discussed in detail is the book [89]; see also the review papers [90, 91] and the paper [92] where BBN with degenerate neutrinos is included.
The relevant temperature interval for BBN is approximately from 1 MeV to 50 keV. In accordance with eq. (37) the corresponding time interval is from 1 sec to 300 sec. When the universe cooled down below MeV the weak reactions
(50)  
(51) 
became slow in comparison with the universe expansion rate, so the neutrontoproton ratio, , froze at a constant value , where MeV is the neutronproton mass difference and MeV is the freezing temperature. At higher temperatures the neutrontoproton ratio was equal to its equilibrium value, . Below the reactions (50) and (51) stopped and the evolution of is determined only by the neutron decay:
(52) 
with the lifetime sec.
In fact the freezing is not an instant process and this ratio can be determined from numerical solution of kinetic equation. The latter looks simpler for the neutron to baryon ratio, :
(53) 
where is the axial coupling constant and the coefficient functions are given by the expressions
(54) 
(55) 
These rather long expressions are presented here because they explicitly demonstrate the impact of neutrino energy spectrum and of a possible charge asymmetry on the ratio. It can be easily verified that for the equilibrium distributions of electrons and neutrinos the following relation holds, . In the high temperature limit, when one may neglect , the function can be easily calculated:
(56) 
Comparing the reaction rate, with the Hubble parameter taken from eq. (37), , we find that neutronsproton ratio remains close to its equilibrium value for temperatures above
(57) 
Note that the freezing temperature, , depends upon , i.e. upon the effective number of particle species contributing to the cosmic energy density.
The ordinary differential equation (53) can be derived from the master equation (42) either in nonrelativistic limit or, for more precise calculations, under the assumption that neutrons and protons are in kinetic equilibrium with photons and electronpositron pairs with a common temperature , so that . As we will see in what follows, this is not true for neutrinos below MeV. Due to annihilation the temperature of neutrinos became different from the common temperature of photons, electrons, positrons, and baryons. Moreover, the energy distributions of neutrinos noticeably (at per cent level) deviate from equilibrium, but the impact of that on light element abundances is very weak (see sec. 4.2).
The matrix elements of transitions as well as phase space integrals used for the derivation of expressions (54) and (55) were taken in nonrelativistic limit. One may be better off taking the exact matrix elements with finite temperature and radiative corrections to calculate the ratio with very good precision (see refs. [93, 94] for details). Since reactions (50) and (51) as well as neutron decay are linear with respect to baryons, their rates do not depend upon the cosmic baryonic number density, , which is rather poorly known. The latter is usually expressed in terms of dimensionless baryontophoton ratio:
(58) 
Until recently, the most precise way of determining the magnitude of was through the abundances of light elements, especially deuterium and , which are very sensitive to it. Recent accurate determination of the position and height of the second acoustic peak in the angular spectrum of CMBR [70, 71] allows us to find baryonic mass fraction independently. The conclusions of both ways seem to converge around .
The light element production goes through the chain of reactions: , , , , , etc. One might expect naively that the light nuclei became abundant at because a typical nuclear binding energy is several MeV or even tens MeV. However, since is very small, the amount of produced nuclei is tiny even at temperatures much lower than their binding energy. For example, the number density of deuterium is determined in equilibrium by the equality of chemical potentials, . From that and the expression (30) we obtain:
(59) 
where MeV is the deuterium binding energy and the coefficient 3/4 comes from spin counting factors. One can see that becomes comparable to only at the temperature:
(60) 
At higher temperatures deuterium number density in cosmic plasma is negligible. Correspondingly, the formation of other nuclei, which stems from collisions with deuterium is suppressed. Only deuterium could reach thermal equilibrium with protons and neutrons. This is the so called ”deuterium bottleneck”. But as soon as is reached, nucleosynthesis proceeds almost instantly. In fact, deuterium never approaches equilibrium abundance because of quick formation of heavier elements. The latter are created through twobody nuclear collisions and hence the probability of production of heavier elements increases with an increase of the baryonic number density. Correspondingly, less deuterium survives with larger . Practically all neutrons that had existed in the cosmic plasma at were quickly captured into . The latter has the largest binding energy, MeV, and in equilibrium its abundance should be the largest. Its mass fraction, , is determined predominantly by the ratio at the moment when and is approximately equal to . There is also some production of at the level (a few). Heavier elements in the standard model are not produced because the baryon number density is very small and threebody collisions are practically absent.
Theoretical calculations of light elements abundances are quite accurate, given the values of the relevant parameters: neutron lifetime, which is pretty well known now, the number of massless neutrino species, which equals 3 in the standard model and the ratio of baryon and photon number densities during nucleosynthesis, (58). The last parameter brings the largest uncertainty into theoretical results. There are also some uncertainties in the values of the nuclear reaction rates which were never measured at such low energies in plasma environment. According to the analysis of ref. [95] these uncertainties could change the mass fraction of at the level of a fraction of per cent, but for deuterium the “nuclear uncertainty” is about 10% and for it is could be as much as 25%. An extensive discussion of possible theoretical uncertainties and a list of relevant references can be found in recent works [93, 94]. Typical curves for primordial abundances of light elements as functions of , calculated with the nucleosynthesis code of ref. [96], are presented in fig. 1.
Another, and a very serious source of uncertainties, concerns the comparison of theory with observations. Theory quite precisely predicts primordial abundances of light elements, while observations deals with the present day abundances. The situation is rather safe for because this element is very strongly bound and is not destroyed in the course of evolution. It can only be created in stars. Thus any observation of the presentday mass fraction of gives an upper limit to its primordial value. To infer its primordial value , the abundance of is measured together with other heavier elements, like oxygen, carbon, nitrogen, etc (all they are called ”metals”) and the data is extrapolated to zero metallicity (see the book [89] for details). The primordial abundance of deuterium is very sensitive to the baryon density and could be in principle a very accurate indicator of baryons [97]. However deuterium is fragile and can be easily destroyed. Thus it is very difficult to infer its primordial abundance based on observations at relatively close quarters in the media where a large part of matter had been processed by the stars. Recently, however, it became possible to observe deuterium in metalpoor gas clouds at high redshifts. In these clouds practically no matter was contaminated by stellar processes so these measurements are believed to yield the primordial value of . Surprisingly, the results of these measurements are grouped around two very different values, normal deuterium, [98][100], which is reasonably close to what is observed in the Galaxy, and high deuterium, [101][105]. The observed variation may not be real; for example, uncertainties in the velocity field allow the D/H ratio in the system at [105] to be as low as in the two highz systems [106][108]. An interpretation of the observations in the system at under the assumption of a simple single component [107] gives . With the possibility of a complicated velocity distribution or of a second component in this system a rather weak limit was obtained [107], . However, it was argued in the recent work [109] that the observed absorption features most probably are not induced by deuterium and thus the conclusion of anomalously high deuterium in this system might be incorrect. On the other hand, there are systems where anomalously low fraction of deuterium is observed [110], . An analysis of the data on D and and recent references can be found in [111]. It seems premature to extract very accurate statements about baryon density from these observations. The accuracy of the determination of light element abundances is often characterized in terms of permitted additional neutrino species, . The safe upper limit, roughly speaking, is that one extra neutrino is permitted in addition to the known three (see sec. 6.1). On the other hand, if all observed anomalous deuterium (high or low) is not real and could be explained by some systematic errors or misinterpretation of the data and only “normal” data are correct, then BBN would provide quite restrictive upper bound on the number of additional neutrino species, [112]. For more detail and recent references see sec. 6.1.
4 Massless or light neutrinos
4.1 GersteinZeldovich limit
Here we will consider neutrinos that are either massless or so light that they had decoupled from the primordial plasma at . A crude estimate of the decoupling temperature can be obtained as follows. The rate of neutrino interactions with the plasma is given by:
(61) 
where is the cross section of neutrinoelectron scattering or annihilation and means thermal averaging. Decoupling occurs when the interaction rate falls below the expansion rate, . One should substitute for the the crosssection the sum of the crosssections of neutrino elastic scattering on electrons and positrons and of the inverse annihilation in the relativistic limit. Using expressions presented in Table 2 we find:
(62) 
where , are the 4momenta of the initial particles, and are the coupling to the lefthanded and righthanded currents respectively, and , plus or minus in stand respectively for or . The weak mixing angle is experimentally determined as .
Process  
















We would not proceed along these lines because one can do better by using kinetic equation (48). We will keep only direct reaction term in the collision integral and use the matrix elements taken from the Table 2. We estimate the collision integral in the Boltzmann approximation. According to calculations of ref. [113] this approximation is good with an accuracy of about 10%. We also assume that particles with which neutrinos interact, are in thermal equilibrium with temperature . After straightforward calculations we obtain:
(63) 
Using the expression (49) and integrating over we find for the decoupling temperature of electronic neutrinos MeV and MeV. This can be compared with the results of refs. [114, 115]. On the average one can take and