Cosmological Neutrinos

Within the context of hot big-bang cosmology, a cosmic background of presently low energy neutrinos is predicted to exist in concert with the photons of the cosmic background radiation. The number density of the cosmological neutrinos is of the same order as that of the photons of the cosmic background radiation. That makes neutrinos the second most abundant particle species in the universe. In the early universe, when these neutrinos were highly relativistic, their effects in determining the ultimate structure and evolution of the universe were significant.

In the beginning...there was light.
-Genesis, 1:1-1:3 Within the context of hot big-bang cosmology, a cosmic background of presently low energy neutrinos is predicted to exist in concert with the photons of the cosmic background radiation.The number density of the cosmological neutrinos is of the same order as that of the photons of the cosmic background radiation.That makes neutrinos the second most abundant particle species in the universe.In the early universe, when these neutrinos were highly relativistic, their effects in determining the ultimate structure and evolution of the universe were significant.

Introduction
In Chapter 1 we outlined the history of neutrino physics and astrophysics and discussed the present state of neutrino mass measurements.This chapter will present a primer on the cosmological neutrino background that is a relic of the hot big bang.We briefly summarize the implications of the cosmological neutrino background on neutrino physics and vice versa.The predicted light element production in the early big-bang, assuming standard weak interaction neutrino physics, is in excellent agreement observations of the cosmic abundances of helium, deuterium and lithium in the early universe.Therefore, we will not consider the effects of more exotic physics in this chapter.
Fig. 1.The blackbody spectrum of the cosmic background radiation as first measured by the Cosmic Background Explorer (COBE). 1 The red points around the peak were measured by the diffuse infrared background explorer (DIRBE) on the COBE satellite.The cosmic microwave background radiation was first detected in 1966 by Penzias and Wilson 4 and elucidated by R.H. Dicke and P.J.E.Peebles and P.G.Roll, and D.T. Wilkinson. 5At a meeting of the American Physical Society in 1990, to a spontaneous standing ovation, John Mather, the leader of the COBE group, announced their observational results showing a full-blown perfect 2.73 K black body spectrum of the cosmic background radiation (CBR) as shown in Figure 1.This was solid proof of the hot big-bang picture of the evolution of the universe. 1In this picture, the early universe began in a hot, dense state that expanded and cooled adiabatically with time, explaining the cosmological origin and abundances of helium, deuterium, and lithium.An implication of this scenario was the predicted existence of a presently ultracold 1.95 K cosmic neutrino background coexisting with the background of cosmic blackbody radiation.

The cosmic background radiation (CBR) and the cosmic neutrino background (CνB)
In the hot big-bang scenario, at a time when T > T dec , where T dec , is the decoupling temperature (see below), the "radiation" in the universe, consisted of photons, electrons, positrons, and neutrinos, that were all in thermal equilibrium.At that time, the ratio of neutrino density to photon density in thermal equilibrium was given by where ω γ = 2 is the number of photon degrees of freedom, ω ν is the number of neutrino degrees of freedom, f , taken to be 2 per flavor that were in thermal equilibrium when T ∼ 1 MeV (only ν L and νR meet this criterion).Thus, the relation (1) is independent of the Dirac or Majorana nature of the neutrino.Here, F ( ν ) and B( γ ) designate the Fermi-Dirac distribution and Planck blackbody distribution respectively.The factor of 3/4 comes from the ratio of the integrals of the Fermi-Dirac distribution and the Planck distribution.† The decoupling temperature, T dec , when the neutrinos fell out of thermal equilibrium, occurred when the νe weak interaction rate was equal to the expansion rate of the universe.‡ Thus, an estimate of the decoupling temperature can be found by equating the thermally averaged value of the ν e e weak interaction rate, Γ νee , and the expansion rate, H.
It follows from the weak interaction theory of Fermi that for T <∼ 100 GeV, † In this discussion, we assume that the neutrino chemical potential, µν , that could generally appear in F ( ν ), is equal to zero.This is consistent with the very strong limits on µν e .We note that the strong bounds on µν e apply to all flavors, since neutrino oscillations lead to approximate flavor equilibrium before big-bang nucleosynthesis (BBN) occurs.Thus, µν f = 0 is consistent with the agreement between the BBN predictions and observations.We also take the effective number of neutrinos to be three, also consistent with observations.‡ At T ∼ 3 MeV neutrinos have already gone out of equilibrium with electrons because the cross section for νen → e − p is smaller than that for e + e − → γγ.
where n ν is the neutrino number density and G 2 F = g 4 /(32M 4 W ) is the square of the Fermi constant, with g being the weak coupling constant and M W = 80 GeV being the W -boson mass in electroweak theory.
The expansion rate (Hubble parameter), H, is given by where ρ is the total energy density and M P = c/G N is the Planck mass with G N being Newton's constant.
In the radiation dominated era ρ is given by given in terms of the number of neutrino flavors, f § .Equations ( 3) and ( 4) show that the expansion rate of the early universe is partially determined by the number of neutrino flavors, f .In natural units ( = c = 1), This gives H T 2 /M P .By roughly equating equations ( 2) and (3) we find that the temperature of the universe at decoupling was, T dec 1 MeV.
Shortly after the neutrinos decoupled from the radiation field, when the photon temperature dropped below the electron mass, i.e., T m e 1/2 MeV this enabled irreversible e + e − annihilation via e + e − → 2γ to occur, with the energy release going into the CBR, raising its temperature.One can assume that this entropy transfer did not affect the temperature of the neutrinos because they were already completely decoupled.Therefore, when T < m e , additional photons were created by e + e − annihilation.Thus, a new factor multiplying the photon number density is determined by the additional entropy per unit volume added to the photon component, viz., 11/3.Thus, for T < m e , From that point on, the ratio between the temperatures of relic photons and neutrinos became T γ /T ν = (11/4) 1/3 1.4, as follows from equations (1) and (5).Taking the present value of the CBR T 0 = 2.73 K, this relation gives T ν,0 = 1.95 K.
For almost all of its history the cosmic photon background radiation field expanded isotropically with the mean photon with redshift, z.This leads to the § We assume here the non-existence of sterile neutrinos It follows from equation ( 5) that the neutrino number density per flavor is determined by the temperature, T ν This further leads to the relation In the non-relativistic limit, ρ ν (m ν T ν ) = m ν n ν , so that the contribution of massive neutrinos to the energy density in the non-relativistic limit is a function of the mass (or the sum of masses of all neutrino states, given Σm i T ν ).At the present time the CMB temperature is measured to be T 0 =2.73 K corresponding to a photon density n γ 4 × 10 8 m −3 .Thus, from equation ( 5), n ν 3.4 × 10 8 m −3 , i.e., more than one neutrino for each man, woman and child in the United States in a single cubic meter!

Neutrino mass and cosmological mass density
Let us begin by giving the usual definitions relating the Hubble parameter to the overall mass density in the universe.The Hubble parameter relates the average expansion velocity of a point in space such as a galaxy to its distance.Locally, it is designated by the constant, H 0 .While there is some controversy about its exact value, depending on the measurement technique used, 6 we take it here to to be H 0 ∼ 70 km/s/Mpc.We also define h = H 0 /(100 km/s/Mpc) and take h 0.7.
The critical mass density needed to gravitationally close the universe with zero curvature is given by H 0 being the Hubble parameter and G N being the gravitational constant.Because neutrinos have mass, cosmic background neutrinos are a form of dark matter.Thus, they contribute to the overall mass density of the universe.Defining the ratios Ω m ≡ ρ m /ρ c , where ρ m is the total mass density of the universe, we can also define the contribution of cosmological neutrinos to the mass density of the universe.If we define the parameters Ω ν = ρ ν /ρ c and h = H 0 in units of 100 km/s/Mpc, the present contribution to the matter density of f ν neutrino species with standard weak interactions is given by where < m ν > is the average mass over the number of neutrino flavors, f ν .Neutrino oscillation observations shed some light on neutrino masses.They indeed prove that neutrinos have masses.However, the neutrino oscillation periods are determined by parameters involving the differences between the squares of the neutrino masses, e.g., (m 2 2 -m 2 1 ), so that the individual masses themselves are not determined by the oscillations (see Chapter 2).The oscillation phenomenon proves that neutrinos have mass eigenstates that are combinations of their flavor eigenstates and that neutrinos have non-zero masses.
Although we do not know the masses of the neutrino mass eigenstates, we know that they are all much lighter than those of the other known particles.As of this writing, the Karlsruhe Tritium Neutrino experiment (KATRIN), by studying the endpoint energy of the electron energy spectrum from tritium β-decay, has placed an upper limit on the neutrino-mass scale of 1.1 eV at a 90% confidence level.? On thother hand, neutrino oscillation experiments have placed a lower limit of 0.06 eV on the sum of the neutrino masses.8,9 This mass range shows that Ω ν h 2 1, as follows from equation (13).If we take < m ν >∼ 1 eV and f ν = 3 in equation ( 13), we obtain a value for Ω ν of ∼ 0.03.However, although neutrinos cannot account for a large portion of the dark matter in the universe, they are presently the only identified component of the dark matter.

Big Bang Nucleosynthesis
As we have seen, at present neutrinos have little effect on the mass density of the universe.However, in the early radiation dominated universe, when they were relativistic they played a significant role in the dynamics of the universe.Their effects in determining the ultimate structure and evolution of the universe were significant.
It follows from equations ( 3) and ( 4) that the expansion rate in the early universe is determined by the number of neutrino flavors f .The abundances of primordial helium, deuterium, and lithium made in the first few minutes after the big bang are determined by this expansion rate. 10In particular, the abundance of helium is sensitive to the available abundance of neutrons which is in turn determined by the n/p ratio at the "freezeout" temperature T f when Γ νe > H.In order to account for the observed abundances, one obtains a value of f 3, viz., the known number of neutrino flavors. 11,12A more precise determination of the number of neutrino flavors comes from a determination of the mass width of the Z boson using the LEP accelerator at CERN. 13 The following discussion therefore implicitly assumes f = 3 in equation ( 4): Before T ∼ 1 MeV protons and neutrons are kept in thermal equilibrium by the weak interactions: n + e + ↔ p + νe .
The neutrons and protons are non-relativistic, i.e., T m n , m p ) so they have a Maxwell-Boltzmann distribution and their relative number densities are given by As long as T (m n − m p ) = 1.3 MeV, N n ∼ N p .However, at a later time, when T < (m n − m p ), then N n < N p .A detailed calculation shows that once T fo ∼ 0.8 MeV, the abundance ratio N n /N p ∼ 0.2.
The production of the nuclei of the light elements then occurs through the reaction sequence chain of reactions: D + D → 4 He , ...
The destruction of nuclei by Wien tail of the blackbody photon distribution effectively stops once the temperature is less than ∼ 0.1 MeV.At this point the neutron to proton ratio drops to N n /N p 0.18.Then, most of the remaining neutrons form 4 He.For a further discussion see Ref. 14.

Blackbody Temperature Perturbations
During the era of radiation dominance the gravitational effects induced by inhomogeneities in the photon and neutrino backgrounds were comparable.The faster cosmological expansion owing to the neutrino background produces effects on the acoustic and damping angular scales of the cosmic microwave background.(See equation ( 4)).Following decoupling (T < T dec ), the neutrinos free stream at a velocity v ν c.The gravitational effect of neutrino background perturbations suppresses the acoustic peaks in the microwave background for the multipoles with l 200 and enhances the amplitude of matter fluctuations on these scales.In addition, the perturbations of relativistic neutrinos generate a unique phase shift in the acoustic oscillations of the CBR that for adiabatic initial conditions cannot be caused by any other standard physics.The origin of the shift is an effect of the neutrino free-streaming velocity exceeding the sound speed of the photonbaryon plasma. 15This phase shift in the acoustic oscillations of the CBR has been detected and it provides more evidence for only three neutrino flavors. 16

Galaxy and Structure Formation
Cosmological neutrinos do not cluster like cold dark matter or baryons.8][19] Mass fluctuations on scales smaller than these scales will thus grow at a smaller clustering rate than that expected. 20Their corresponding Jeans mass is larger than the mass of the largest galaxies and can be on the scale of galaxy clusters.Observational indications of such large-scale clustering and implications for neutrino mass include gravitational lensing studies 21 and the number density of clusters of galaxies. 22,23stronomical observations bearing on cosmological parameters Ω and h play an important role in neutrino physics and astrophysics.The power spectrum of the CMB is a function of the sum of the neutrino masses, m ν,tot and other cosmological parameters.By comparing data from the Planck satellite with simulations of the development of structure in the universe the Planck Collaboration obtained an upper limit on the sum of the neutrino mass states of m ν,tot < 0.12 eV. 24,25 more detailed treatment of the dynamical effects of the cosmological neutrino background is beyond the scope of this short treatment.For more extensive treatments reviewing the role of the CνB in cosmology, the reader is referred to papers by Dolgov 26 and Lesgourgues and Pastor. 20

Detection via the Z-boson resonance
A potential scheme for detecting cosmological neutrinos was suggested via the use of the Z-boson resonance. 27The IceCube detector has observed the W − boson resonance, 28 also known as the Glashow resonance, 29 from neutrinos interacting with electrons in ice.At the 6.3 PeV resonance energy E νe = M 2 W /2m e = 6.3 PeV, electrons in the IceCube volume provide enhanced target cross sections for νe 's through the W − resonance channel via νe + e − → W − → shower.
In addition to the Glashow resonance, the standard model predicts a Z-boson resonance via the neutral current interaction ν + ν → Z → shower.The corresponding resonance energy is E ν = M 2 Z /2m ν ≥∼ 2 × 10 13 GeV, taking m ν ≤ 0.2 eV.In order to produce a neutrino of that energy, it would require the interaction of a proton with an energy an order of magnitude greater. 30As there is no known mechanism for accelerating a proton to such an energy in an astronomical object, direct detection of cosmological neutrinos via the Z-boson resonance is doubtful.

Detection via neutrino capture
As of this writing, there has been no direct detection of the very large number of very low energy cosmological background.However, there is a proposed concept for such direct detection.The proposed experiment, called PTOLOMY (Princeton Tritium Observatory for Light, Early-universe Massive-neutrino Yield) ?,31 is based on detection via the neutrino capture processes on β-unstable tritium nucleus The experimental signature of neutrino capture is a peak in the electron spectrum that is displaced by 2m ν above the beta-decay endpoint.The signal would exceed the background from β-decay if the energy resolution is less than 0.7m ν .The capture rate depends on the nature of the neutrino mass.In principle, this provides a test that can distinguish between Dirac neutrinos and Majorana neutrinos.Assuming the neutrinos are non-relativistic, for a 100 g 3 He target the capture rate for unclustered Dirac neutrinos is ∼ 4 yr −1 and twice that for Majorana neutrinos, taking helicity into account. 32At present there is not enough 3 H available for the sensitivity needed, however, a prototype pathfinder experiment is in the works.

Fig. 2 .
Fig. 2. The small-scale PTOLEMY prototype installed at the Princeton Plasma Physics Laboratory (February 2013).Two horizontal bore NMR magnets are positioned on either side of a MAC-E filter vacuum tank.The tritium target plate is placed in the left magnet in a 3.35T field, and the RF tracking system is placed in a high uniformity 1.9T field in the bore of the right magnet with a windowless APD detector and in-vacuum readout electronics.(Courtesy of the PTOLEMY collaboration.)