Neutrinos in Cosmology

Suggested reading:

C. Giunti and C.W. Kim, Fundamentals of Neutrino Physics and Astrophysics, Oxford University Press (2007; 728 pages)

S. Hannestad, Primordial neutrinos, astro-ph/0602058A. D. Dolgov, Cosmology and neutrino properties, astro-ph/0803.2887, Cosmological implications of neutrinos, astro-ph/0208222

Elisa BernardiniDeutsches Elektronen-Synchrotron DESY (Zeuthen)

Introduction★ Of all known particles neutrinos have the weakest interactions and the

smallest possibly non-vanishing mass

★ Due to these properties, the neutrino is the second most abundant particle in the universe after photons

★ Their total number density is:

★ for all neutrino flavors.

★ In comparison, the average cosmological number density of normal baryonic matter is:

★ The average energy of cosmic neutrinos is very low, about 6.1 oK~5x10-4 eV

★ These neutrinos constitute the cosmic neutrino background (CνB).

★ It is not directly observable, due to the very weak interactions:

‣ if ν’s are massless:

‣ is they have non-zero mass:

★ In any case, the interaction products have very weak effects in matter (are they detectable??)

! ! G2F (T 0

!)2 " 10!64 cm2

! ! G2F m!

2 " 10!56(m!/ 1 eV)2 cm2

Neutrino fluxes

★ Neutrinos are normally assumed to undergo only usual weak interactions with (V-A)-coupling to the W and Z bosons

★ Therefore, if mν=0, only left-handed neutrinos νL do interact, while right-handed neutrinos νR are sterile

★ If on the other hand mν≠0, right-handed neutrinos would also be coupled to intermediate bosons, with strength suppressed as (E/mν)2

★ If neutrinos are mixed and massive, additional sterile neutrinos could be abundantly produced in the early universe

★ From experiments, it is known that there are at least (at most) three neutrinos families and from LEP data the number of light (νe, νμ, ντ) neutrinos flavors is 2.993±0.011

A few ingredients

★ The neutrino oscillation probability in the 3x3 oscillation case is quite complicated, by always depends on δm2

★ From the combination of all currently available data on neutrino oscillations we have two important squared mass differences:

★ If neutrino masses are hierachical:

★ However, neutrino masses could also be degenerate:

★ Since oscillation experiments depend only on δm2, if ν are degenerate they are not useful for determining the absolute scale mass

normal inverted

Bounds on ν masses from Cosmology

★ A little about cosmology

★ Relation between cosmological neutrinos and the CMBR

★ Role of neutrinos in big-bang nucleosynthesis (BBN) and limits on the number of neutrino species and possible neutrino degeneracy

★ Cosmological impact of neutrino oscillations

A (very) little about cosmology★ The universe is known to expand

according to the Hubble law, where V is the velocity of a distant object and r is the distance to it:

★ Data on far-away type Ia supernovae: their speed V determined from the measured red-shift was less than expected from Hubble’s law: far away galaxies are moving slower than closer ones → the Universe has in some recent times accelerated

★ The present day value of H is:

V = Hr with H = a/a

H0 = 100h km/s/Mpc with h = 0.72± 0.08

Hubble diagram from astro-ph/0303428. Data from High-z SN Team and The supernova Cosmology Project

redshift z

The fate of the Universe

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★ The Friedmann equation describes the competition between the attractive gravitational force and the expansion of the Universe

★ The gravitational attraction at the center of an arbitrary sphere cut out of the universe is proportional to the average density of matter

★ The measured value of the Hubble constant (H) yields the kinetic energy of the expansion of the sphere. If the present density is below the critical value at which the expansion and gravitational attraction balance, gravity cannot halt the expansion, and the universe must be open. The critical (or closure) energy density is proportional to H2 and is:

★ Contributions of different forms of matter into cosmological energy density are typically presented in terms of dimensionless parameters Ωj=ρj/ρc

★ The total energy density is close to the critical value, i.e. Ωtot ≈ 1

★ The dominant part of the energy density is made of by vacuum energy (or an unknown form of matter) such that is exerts a negative pressure (induce accelerated expansion) and Ωvac ≈ 0.7

ordinary matter ~ 4%

neutrinos 0.1 - 2%

Acts gravitationally like ordinary matter, probably new form of

weakly interacting particle Dominates dynamics of galaxies, clusters, larger

structures and decelerates cosmic expansion

Exerts negative pressure and accelerated cosmic expansion, while plays no role on small scale (homogeneous)

Cosmic Microwave Background Radiation

★ Discovered in 1965 by Penzias & Wilson, it is one of the main sources of information on the history of the Universe

★ It was generated after the temperature of the expansing Universe had become low enough to allow electrons and nuclei to form atoms (“recombination”)

★ Before recombination, matter and photons were in thermal equilibrium

★ At the recombination, the number of free electrons dropped and the Universe became transparent to photons (photon decoupling)

Spectrum of the microwave background radiation as measured by COBE

T rec! ! 0.3 eV

trec! ! 3.8" 105 yrzrec! ! 1.1" 103

CMBR anisotropies★ The anisotropies are fluctuations

of the temperature of the CMBR in different directions

★ It is convenient to expand these fluctuations in spherical harmonics:

★ and calculate the two point correlation function:

★ The set of Cl forms the “angular power spectrum

The angular power spectrum of the CMBR temperature anisotropy in terms of the angular scale (or multipole moment). Data from WMAP (2006), Acbar (2004) Boomerang (2005), CBI (2004) and VSA (2004).

the first peak is close to 200, which is evidence for a flat universe:

the other peaks have amplitudes that give information on the amount of total and baryonic matter:

Neutrino thermal equilibrium★ In a gaseous ensemble of atoms or molecules, thermal equilibrium is

obtained when the state of the system (# particles at a given energy) does not change with time

★ In an expanding universe the temperature is constantly changing and to be in equilibrium particles must adjust the energy faster than the time it takes to change the temperature, i.e. the interaction rate must be faster than the expansion rate of the universe:

★ Neutrinos interact only weakly

★ Interactions with nucleons were negligible when 1 MeV ≤ Tγ≤ 100 MeV since the number density of nucleons was << density of relativistic e+e-

★ Interaction rate is Γint~GF2T5:

‣ at small energies compared to the W mass, the cross section is ~ GFE2

‣ the energy is of the order of the temperature T (relativistic particles)

‣ the number density of relativistic particles is of the order of T3

!int > H(t)

Neutrino thermal equilibrium ★ The expansion rate (radiation dominated universe ρ~T4) is H~Tγ2/MPl

★ The condition for thermal equilibrium:

★ requires that T≥ 1 MeV (corresponding to t≤1 sec) for MPl=1.2x1019 GeV

★ This coincides with the temperature where light element synthesis occur: observation of light elements abundances provide important information on v’s as well

★ It also means that neutrino decouple at a temperature slightly higher than the electron mass me, so that when e+e- annihilation occurs at T~me/3 the Tν is not affected, while Tϒ is heated by a factor (11/4)1/3, since e+e-

became non-relativistic and their entropy is transferred to photons, so that:

★ This prediction is not yet been checked experimentally

T v!dec! ! (MPlG

2F )!1/3 ! 1 MeV

Temperature evolution★ Since ν’s were relativistic at the point of decoupling, the neutrinos

temperature after decoupling evolves in time as Tν~R-1

★ Same applies to the photon temperature after e+e- annihilation so that the previous ratio is maintained today

★ Therefore, from the measured temperature of the CMBR, we can infer the neutrino temperature today:

★ Similarly, it is possible to relate the number density of neutrinos to photons’ based on entropy conservation arguments, finding that for each neutrino flavor, we have:

★ Inserting today’s measured number density of photons in the CMBR, we find:

T 0! =

! 411

"1/3T 0

" = (1.945± 0.001) oK = (1.676± 0.001)! 10!4 eV

n! + n! = 3/11! n"

n0! + n0

! ! 112 cm!3

~1/6

Big-Bang Nucleosynthesis (BBN)★ Nucleosynthesis is the first step in the formation of heavy elements

out of neutrons and protons in the universe and responsible for the formation of 2He, 3He, 4He, 7Li

★ Shortly after neutrino decoupling the weak interactions which kept neutrons and protons in statistical equilibrium freeze out

★ Again, by requiring:

★ we find the freeze-out temperature, that is:

★ At a temperature of about 0.2 MeV deuterium starts to form and all free neutrons are rapidly processed into 4He, whose final abundance is given by:

★ nn/np is determined at the freeze out temperature roughly by:

★ YP is ultimately a function of g* (number of internal degrees of freedom, ~10.8 for T>0.2 eV) and Ωbh2

YP primordial mass fraction of 4He

Big-Bang nucleosynthesis

Primordial abundances of light elements as expected in Big Bang nucleosynthesis, relative to hydrogen and (upper) mass abundances of 4He, as a function of the ratio of baryons to photons,

compared to today’s measurements

★ The value of YP inferred from astronomical data (observing 4He in clouds of ionized hydrogen, the most metal poor and extrapolating to zero metallicity) is (PDG):

★ The theoretical prediction is:

exp

YP = 0.2484+0.0004!0.0005

theo

★ We know (LEP data) that there are only three light neutrino flavors, however it is possible that the effective number of neutrinos Nν is larger if there are sterile neutrinos

★ In such a case, the energy density of the Universe is increased by a change of g* given by:

★ g* (from photons, electrons, positrons and neutrinos) is then (10.75 is the SM value):

★ This would ultimately give a higher n/p freeze-out temperature and therefore a higher YP

★ By comparing the theoretical prediction and the experimental value, a bound is obtained:

★ Good agreement with the SM and additional degrees of freedom are constrained (but not very strongly)

Limit on number of ν’s from BBN

!g! = 7/4!N!

g! = 10.75 + 7/4!N!

|!YP | ! 0.009! |!N! | ! 0.82

Limit on ν asymmetry from BBN★ In the Standard Cosmological Model it is assumed that neutrinos are

far from degenerate and have asymmetries:

★ which are zero or very small

★ Large neutrino asymmetries have two effects on BBN:

‣ electron-neutrino asymmetry modifies the freeze-out n/p ratio to:

‣ large neutrino chemical potential increase the energy density in the Universe and lead to faster expansion

★ From the previous bounds it is found:

− ξe/Tfreeze

Effect of oscillations

★ Since neutrinos are almost maximally mixed, a chemical potential in one flavor is shared with the other flavors, so that during BBN all three flavors have almost equal chemical potential

★ A bound on the chemical potential of νe applies therefore to all species

★ The strongest bound on the neutrino asymmetry comes from BBN:

★

0.04

Relic neutrinos influence several cosmological epochs

Cosmological limit on neutrino masses (GZ bound)

★ Non relativistic massive neutrinos may give a significant contribution to the energy density of the Universe

★ Knowing the number density of neutrinos in the present day Universe, we can calculate their energy density and require that it should not exceed the measure energy density

‣ number density:

‣ we obtain:

★ where h = H/100km/s/Mpc is the dimensionless Hubble parameter and Ωm is the fraction of the energy of matter relative the critical energy density, ρc = 3H2m2Pl/8π

★ From WMAP we know that the Universe is not over-closed by matter

★ using observational data:

★ we obtain:

★ assuming almost equal masses of neutrinos (as we obtain from neutrino oscillation results):

★ The “Gershtein & Zeldovich (GZ) limit” Gershtein, Zeldovich JETP Lett. 4 (1966) 120.

★ These limits are less stringent than those obtained from trizium β-decay:

★ but stronger constrains can be derived from the effect of cosmic neutrinos on the formation of structures in the universe, for which we can achieve sensitivities

Cosmological data

★ Even though BBN provides interesting bounds on neutrino physics, observations of the late-time universe has provided very strong constraints on cosmological parameters and neutrino physics too

★ In particular neutrinos affect the growth of cosmic clustering and this clustering can be accurately measured combining data from:

‣ Large Scale Structures (LSS)

‣ Cosmic Microwave Background (CMB)

‣ The baryon acoustic peak (BAO)

‣ The Lyman-α forest

‣ Type Ia supernova

Large Scale Structures (LSS)★ The Sloan Digital Sky Survey (SDSS) has

observed almost 1 million galaxies and measured the red-shifts, which results in the most detailed map of the Universe

★ Another important survey is the 2dFGRS (2 degrees Field Galaxy Redshift Survey)

★ The analysis of large scale structures (comparison of models to observational data by constructing the power spectrum of density fluctuation P(k), i.e. the variance of the fluctuations in Fourier space) allows to study the properties of dark matter and the spectrum of primordial density fluctuations and in particular to put more stringent bounds on the neutrino mass

SDSS Galaxy map

The Lyman-α forest★ Lyman-α forest: absorption lines in the spectrum of distant quasars

due to intermediate hydrogen clouds which absorb the Lyman-α line (λα=1215.67 Å)

★ If the quasar is at redshift zq, the Lyman -α emission line has observed wavelength λαq=(1+zq)λα

★ Since the absorption clouds are at different redshifts, smaller than zq, it is common to observe a “forest” of absorption lines, below the Lyman -α emission line of the quasar

★ From the characteristics of the absorption lines it is possible to infer the power spectrum of density fluctuations for relatively small λ

Flux

Baryon Density

Position along line of Sight

Global fits of cosmological data★ It is assumed that galaxies

trace dark matter up to a constant factor b (the “bias”) which implies that the galaxy power spectrum Pg(k) is related to the dark matter power spectrum PDM(k) by:

★ The bias is extracted from fits to the data in the framework of specific cosmological models

pg(k) = b2PDM (k)

hep-ph/0503257

★ The CMB shows that the Universe used to be uniform, with density variations ~ 10−5

★ Gravitational instability caused these tiny density fluctuations to grow in amplitude into the galaxies and the large-scale structures

★ If all the matter can cluster (e.g. dark matter) then fluctuations grow as the cosmic scale factor a:

★ If a fraction of the matter density is unable to cluster, the fluctuation growth will be slower: if only a density fraction Ω* can cluster, then:

★ The length below which neutrino clustering is suppressed (“neutrino free-streaming”) roughly corresponds to the distance neutrinos have time to travel while the Universe expands by a factor 2

★ What matters is the density, so to translate limits from density fluctuations into a mass limit we need assumptions on the neutrino density (~112/cm3 per flavor)

Even for a small neutrino mass, get large impact on structure: power spectrum is excellent probe of

neutrino mass!

Mass limitsCMB★ WMAP CMB-measurements

alone do not constraint mν but other cosmological parameters that enable to measure the small-scale P(k)-suppression that massive neutrinos cause

★ Combining WMAP SDSS gives the 95% CL upper mν < 1.7eV

★ Including information about SDSS or 2dFGRS galaxy bias tightens this bound to mν < 0.6 − 0.7 eV

★ Including SDSS measurements of the Lyman forest further tightens the bound to mν < 0.42 eV (95%)

CMB+

LSS

Limit on number of ν’s from cosmological data

★ The effective number of additional neutrinos ΔNν can be constrained using CMBR data: since the Universe is not completely matter dominated at the recombination, additional neutrinos would increase the expansion rate, enhancing the first acoustic peak. The peaks are also shifted to larger values of l

★ A fit to WMAP data gives ΔNν=-0.9+6.7-2.2 (95% CL)

★ The formation of large scale structure is also sensitive to ΔNν by affecting the power spectrum (shift to the right). Combining WMAP data with LSS and 2dFGRS gives ΔNν=0.1+3.9-2.8 (95% CL)

★ If data from Hubble on h and from the Supernova Cosmology Project on Ω0M=0.28±0.14, gives ΔNν=1.0+3.0-2.1 (95% CL)

Discussion

★ The cosmological mass upper limits are complemented by the limits from neutrino oscillation experiments

‣ Atmospheric neutrino oscillations show that there is at least one neutrino whose mass exceeds a lower limit around 0.05 eV

‣ Solar neutrino oscillations occur at smaller mass scale, perhaps around 0.008 eV

★ Sterile neutrinos are currently disfavored, it can be assumed that only three neutrino flavors are present in the neutrino background; this means that none of the three neutrinos can weigh more than about 0.42/3 = 0.14 eV

★ The mass of the heaviest neutrino would therefore be in the range 0.05 − 0.14 eV

Summary neutrino masses

Phenomenon Measure Sensitivity Dirac vs Majorana

Flavor oscillations No

β-decay 0.2 eV No

Cosmology 0.1 eV No0νββ 0.01eV Yes