nonequilibrium neutrino in the early universe plasma daniela kirilova institute of astronomy...

Nonequilibrium neutrino in the early Universe plasma

Daniela Kirilova

Institute of Astronomy Bulgarian Academy of Sciences

Sofia, Bulgaria

2 Sept’08 Sozopol

OUTLINE

• Neutrinos in the standard cosmological model and in reality

• Deviations from the equilibrium Fermi-Dirac neutrino spectrum caused by different processes

• Cosmological influence of nonequilibrium processes involving neutrino on neutrino decoupling, BBN, CMB and CNB

• The effect of active-sterile oscillations on the Universe dynamics and on the nucleon kinetics during BBN

• The effect of particle decays on BBN

• Cosmological constraints on the basis of primordial abundance measurements, BBN, CMB and LSS results

• According to SCM our Universe is filled with massless non-oscillating neutrinos (an assumption).

• There exist three neutrino flavours - e, , (confirmed for weakly interacting species lighter than M/2 by LEP , however sterile neutrino is allowed).

• The lepton asymmetry is zero (an assumption).

• Neutrino spectra have the equilibrium Fermi-Dirac distribution (an assumption).

Neutrino in the standard cosmological model

exp( / ) /(1 exp( / ))eqn E T E T

• After the electron-positron annihilation neutrino temperature becomed lower than the temperature of the photons T=(4/11)1/3 Tcmb. The cosmological neutrino background (CNB) today is expected with an extremely low temperature ~ 1.9 K, i.e. less than the temperature of the CMB Tcmb~2.7 K.

Precise calculation of decoupling accounts for partial heating of neutrinos. Pastor et al, 2008

Dolgov, Hansen & Semikoz, NPB 503 (1997) 426; Mangano et al, PLB 534 (2002) 8

• Today relic neutrino (CNB) is expected to be the most numerous particle after the CMB photons.

n =3/11ncmb n =112 cm-3 ncmb =411 cm-3

Relic Neutrino Background

At high temperatures neutrinos were in equilibrium due to weak interactions with the particles of the high temperature plasma. Around 3 MeV muon and tau and at 2 MeV electron neutrino decoupled and since then they were free streaming, i.e. cosmological neutrino background.

Though numerous, the detection of CNB is very difficult: first, because it is an extremely elusive particle due to its weak interactions and second, because neutrinos are expected to have today extremely low energy ~ T.

Standard model description does not match reality:

• Flavour neutrino oscillations exist (solar, atmospheric, terrestrial expts data).

Not all neutrino species are massless!

Lepton flavor is violated.

Mixing in the lepton sector exists.

CPV?

• Sterile neutrino is allowed (not constrained by LEP, predicted by GUT models, wellcomed by cosmology LSS, DM, baryogenesis, astrophysics)

• LA may be non-zero: L < 0.1 in all neutrino sectors

• Neutrino spectra n(E) may differ from the equilibrium ones

Neutrino from experimental and observational data

Neutrino Oscillations

Positive indications for oscillations of neutrino were obtained at the greatest neutrino experiments. Solar neutrino problem, atmospheric neutrino anomaly and the positive results of terrestrial experiments can be resolved by the phenomenon of neutrino oscillations.

Neutrino oscillations: Mass eigenstates are distinct from the flavor eigenstates:

m = Umf f, (f =e, , )Transitions b/n different flavors are possible - flavor composition changes with time.

Neutrino oscillations imply non-zero mass differences and mixing: m2 0, at least 2 neutrino have m 0.

Observational evidence for oscillations:Solar neutrino anomaly: Homestake, Kamiokande, SuperKamioKa, Gallex, SAGE, SNO experiments

е LMA: m2 7.2-9.2 .10-5 eV2 sin22 ~0.3

Atmospheric neutrino anomaly: SuperKamioKa, Macro, Soudan 2, IMB

, m2 2.6.10-3eV2 maximal

Terrestrial experiments: KamLAND, MINOS, K2K, LSND e, m2 O(1eV2) и sin22=O(0.003)

alternative models with s give better agreement with Homestake and explain the variation of the flux with B.

Though neutrino anomalies are well described in terms of flavor neutrino oscillations, sub-leading sterile oscillations may provide better fit (3 plus 2 schemes).

Neutrino Oscillations in the early Universe

Cosmological influence of oscillations:

Active sterile oscillations corresponding to the regions favored by the atmospheric and solar neutrino data establish an equilibrium between active neutrino species before BBN epoch. No considerable influence on BBN, CMB, CNB.

Neutrino active-sterile oscillations may excite additional light particles into equilibrium, i.e change the expantion rate distort the neutrino energy spectrum from the equilibrium FD form affect neutrino-antineutrino asymmetry of the medium (suppress / enhance).

All these may play crucial role for neutrino involved processes in theearly Universe during BBN, CMB, LSS, CNB.

Cosmological constraints on oscillationsFrom the allowed range of the observables of the early Universe, likebaryonic density, light elements abundances, expansion rate, BBN, CMB spectrum, structure characteristics of the Universe, etc., it is possible to constrain the parameters of neutrino oscillations.


• Flavour neutrino oscillations exist (solar, atmospheric, terrestrial expts data).




CPV?





Sterile Neutrinos • Predicted by GUT models• Required for producing non-zero neutrino masses by most models• Welcomed by oscillations data: as a subdominant channel for better fit

(subdominant sterile oscillations channel required by Homestake data, Holanda, Smirnov, 2004, Chauhan, Pulido, 2004, variation of the flux with B,

Caldwell D, Sturrock P.,2005 ); required for explanation of LSND in combination with other expts

• Wellcomed by cosmology: * may be DM particle (accounting for all (?) DM: m<3.5 KeV if not MSM

produced, Shaposhnikov et al., 2007 ; may play subdominant role as DM component if in eV or KeV range and MSM)

* may play a role in LSS formation when constituting few % of the DM by suppressing the small scale power in the matter power spectrum and thus providing better fits to the observational data from SDSS, cluster abundance, weak lensing, Lyman Alpha forest, CMB Tegmark et al., 2004

*plays major role in natural baryogenesis through leptogenesis• Wellcomed by astrophysics: * may explain pulser kicks Kusenko 2007

*explain r-process nucleosynthesis *explain supermasive black holes *The X ray photons from sterile neutrino decays may catalize the production of

molecular H and speed up the star formation, causing earlier reionization

Sterile Neutrinos Constraints • Sterile neutrino is constrained by BBN, because it increases the expansion rate and

hence dynamically influences He production, Shvartzman 1969

recent analysis of He-4 do not favour Ns> 1, Peimbert et al,07, Izotov et al.07

Ns<0.6 for 3% He-4 uncertainty

• CMB feels the increase in the relativistic density due to sterile neutrino through its effect on the growth of cosmological perturbations (additional relativistic particles delay the epoch of matter-radiation equality, the power at small scales is suppressed)

WMAP: Ns< 6 Barger at al., 03, Ichikawa et al,07 ; PLANCK forcast Ns~0.2

• LSS tells Ns in more recent universe, T<100 eV, t>10 y, when formation of the smallest observable scale (Mpc) begins

galaxy clustering (SDSS, 2dF, ..) Ly Alpha +CMB provide looser bounds than BBN 1<Ns<5 preferring Ns~2, Seljak et al,06

• In case of oscillations with active neutrino it exerts major effect on nucleons kinetics during pre-BBN and its mixing parameters are constrained by BBN+CMB DK,1988

• In case of fast oscillations the sterile state can be brought into equilibrium during pre-BBN, causing H increase and overpoduction of He, hence mixing parameters are constrained, Dolgov,1981

• In case of radiative decays its decay products may distort CMB• In case of non-raditive decays, the decay poducts may influence nucleons kinetics

and hence BBN constraints on its decay time, mass and number densities exist Dolgov, DK, 1986

X ray photons from sterile neutrino decays provide observational feature Lyman alpha bound m>10 KeV (of MSM Dodelson-Widrow mechanism of

production)

Et cetera…..


• Flavor neutrino oscillations exist (solar, atmospheric, terrestrial expts data).




CPV?





Processes leading to deviations from the

equilibrium FD neutrino distribution • Electron-positron annihilation – negligible effect Dolgov et al. 1997

The equivalent number of neutrino species – 3.046 instead of 3.

Account for flavor oscillations Mangano et al, 2005:

number density of one neutrino species 113 per cubic cm instead 112 in SCM.

Flavor oscillations with parameters favored by the atmospheric and solar neutrino data establish an equilibrium between active neutrino species before BBN epoch. • Neutrino-antineutrino asymmetry – strongly constrained by BBN in all sectors because

of flavor oscillations <0.07 , Dolgov et al. 2002

• Active-sterile oscillations before neutrino decoupling slightly influence active neutrino disrbutions, because the states are refilled due to interactions with the plasma Barbieri&Dolgov, 1991, and may bring sterile neutrino into equilibrium.

• Active-sterile oscillations proceeding after decoupling may strongly distort neutrino energy spectrum DK, 1988, not introducing additional degrees of freedom, since sterile neutrino is filled for the sake of the active ones.

• Particles decays into neutrinos

Neutrino Oscillations in the early Universe

Cosmological influence of oscillations:

Mixing b/n active neutrinos influence neutrino spectra and BBN negligibly Dolgov 1981

Flavor oscillations corresponding to the regions favored by the atmospheric and solar neutrino data establish an equilibrium between active neutrino species before BBN epoch. No considerable influence of flavor oscillations on BBN, CMB, CNB should be expected.

Neutrino active-sterile oscillations may excite additional light particles into equilibrium, distort the neutrino energy spectrum from the equilibrium FD form affect neutrino-antineutrino asymmetry of the medium (suppress / enhance).

All these may play crucial role for neutrino involved processes in the Universe during BBN, CMB, LSS, CNB.

Oscillations effects a s

Dynamical effect – production of additional neutrino species. Additional degree of freedom enhances the energy density

and drives expansion faster. Dolgov ,1981

Tf ~ geff

1/6 4Не overproduction

Yd ~0.013 Ns

( 1 additional Yp/Yp = 5 %) oscillations dynamical effect

2~ TGgH eff7

10.75 34eff ssg N N N

Kinetic effect : a energy spectrum distortion,

e depletion, D.K.,1988; Barbieri,Dolgov, 1990;

Enqvist et.al., 92; DK M.Chizhov, PLB ,1997

energy threshold effect pre-BBN kinetics neutrino-antineutrino asymmetry growth Foot, Volkas,1996; DK M.Chizhov,1996 Dolgov et al., 2002

In case of oscillations effective after decoupling and provided that the sterile state is not in equilibrium (Ns<1), the spectrum distortion effect on BBN is the major one. Expressed in terms of effective number of neutrinos: Nk,0 6 for resonant oscillations Nk,0 3 for non-resonant oscillations DK , Astrop.Phys.,2003

Spectrum distortion should be precisely accounted for.

Effects of nonequilibrium a s 2

~osc

mГ

E

)(~ 2 EndEEN

2 2~ F EГ NG

Oscillations –medium influence

• Medium suppresses the oscillations amplitude

• Medium may enhance them

• Negligible spectrum

distortion ? (work with particle densities and T shift;

one momentum approximations.)

-Fast oscillations equilize pre-existing asymmetries

- Oscillations cause great spectrum distortion, asymmetry growth

Persists, and is often the leading effect , hence it

should be precisely described !

Evolution of neutrinos in the presence of oscillations

Approach: follow the evolution of neutrino for each momentum;

account for oscillations, expansion and interactions with the medium

simultaneously

20 2

( ) ( ), ( ) 2 , ( )F F

W

t t QHp i t i G L N t O G

t p M

H

*

3

, ,

~ ~ 2 ~ /

1 0exp / 1 exp /

0 0

e e

ie je i il l

LL LL

in eq in eqLL

U U U l e s

is free neutrino Hamiltonian

Q E T L L L L L d p N

n E T E T n

0H

2 2 2 2 3 30

20

1 2 2 cos / ,

where 0.1 / and =3 MeV.

e e

eqLL v

Pl eff

n c s c s BT E T T n

B M m g T

Analytical solution for vacuum neutrino oscillations (post BBN epoch):

The evolution of spectrum distortion

The distortion concerns first the low energetic

part of the spectrum because the oscillations

become effective first to low energy neutrinos

Soon after, the whole spectrum is distorted

from its equilibrium FD form.

The non-equilibrium initial condition

leads to considerable and continuous

deviations from the equilibrium.

2

~osc

mГ

E

Numerical solutions for matter neutrino oscillations

Evolution of the distortion

The spectrum distortion of the active neutrino for a wide range of

oscillation parameters persists during the nucleons freezing period.

Energy spectrum distortion evolution

Sterile neutrinos may be present at the onset of BBN epoch -- may be produced in GUT models, in models with large extra dimensions, Manyfold Universe models, mirror matter models, or by oscillations in 4-neutrino mixing schemes, etc. The degree of population may be different depending on the production model.

The distortion of the neutrino spectrum due to active-sterile oscillations and the kinetic effect caused

Nk depends on the degree of initial population of s.

The biggest effect is Nk,0 at Ns=0, the effect decreases with Ns . DK,Int.J.M.P.D,2004, 2007

Nk ~ Nk,0- Nk,0 Ns

Spectrum distortion for different initial population of s.: Ns=0 – the lowest curve, Ns=0,5 and Ns=0,8 – the upper

curve. The dashed curve shows the equilibrium spectrum.

Spectrum distortion

for non zero initial popuation of s

Spectrum distortion for different initial population of s. Ns=0 – the lowest curve, Ns=0,5 and Ns=0,8 – the upper curve. The dashed curve shows the equilibrium spectrum. DK , IJMPD2007

Distortion evolution for non zero Ns

The depletion of active neutrinos(an integral effect of the distortion)

CNB

2 neutrino mixing:

4 neutrino mixing:

Nk,4 < Nk,2

Sterile state is filled for the sake of e CNB flavor neutrinos may have the equilibrium number

density or be depleted depending on the type of oscillations to sterile neutrinos and their parameters.

0.5 neq < ne < neq

Energy spectum strongly distorted from the equilibrium Fermi-Dirac one.

Sterile state filles from e , while e is partially refilled for the sake of muon and tau neutrino

Flavor oscillations reestablish the equilibrium between the different neutrino flavors. Then CNB electron neutrinos will have slightly depleted number densities, as low as 3/4 of the SCM value, i.e

112*3/4=84 per cubic cm.

Flavor mixing decreases the depletion and spectrum distortion

BBN - one of the most precision probes of new neutrino physics

Cosmological constraintsFrom the allowed range of the observables of the early Universe, likebaryonic density, light elements abundances, expansion rate, BBN, CMB spectrum, structure characteristics of the Universe, etc., it is possible to constrain the parameters of neutrino oscillations and deviations from FD in the neutrino spectrum.

BBN provides a unique information about the physical conditions of ourUniverse at the very early epoch (t ~ 1 sec) and thus allows to constrainphysics beyond the standard cosmological model and the standard electroweak model.

We explored a modification of the standard Big Bang Nucleosynthesis with neutrino oscillations e s

effective after electron neutrino decoupling.

BBN with oscillationsHe-4 mass fraction is a strong function of the effective number of light stable

particles at BBN epoch

It depends also on the e characteristics decrease n/p freezes earlier 4Не is overproduced

BBN with fast a s : increaseeffective before a decoupling

BBN with a s e spectrum distortion effective after a decoupling and Ns<1

We explored a modification of the standard Big Bang Nucleosynthesis

with neutrino oscillations e s effective after electron neutrino decoupling.

2~ TGgH eff

710.75 3

4eff ssg N N N

2 2~ F EГ NG

sN

2 4 7sin 2 10m

Evolution of nucleons in the presence of е s

the numerical approach

)()(),,(2

LLnpen

nn

p nnnnpeApedp

nHp

t

n

)()~()~,,(2

LLpnennnpneAped

2 7 210 0 1

2 0.3sm eV all mixing angles N

MeV T MeV

The interplay b/n effects

dynamic effect increases

kinetic effect decreases

total effect decreases

N= Nk,0- Nk,0 Ns +NsNk,0 >1

Nk,0 Ns >Ns

m2 = 10-7 eV2 sin22 = 1

The role of additional light s

total effect increases

kinetic effect decreases

dynamic effect increases

Nk,0 < 1 N= Nk,0- Nk,0 Ns +Ns

Nk,0 Ns < Ns

Dependence of maximum overproduction on the mixing

0Y/Y 32% for resonant case

0Y/Y 14 % for non-resonant

Expressed in terms of effective number of neutrinos the kinetic effect due to e spectrum distortion: Nk,0 6 for resonant osc Nk,0 3 for non-resonant osc

2 7 210m eV

2 8 210m eV

Maximum He-4 overproduction in BBN with oscillations due to spectrum distortion

Maximal overproduction dependence on mass difference

BBN is very sensitive to neutrino spectrum distortion

BBN constraints do exist if He-4 uncertainty is over 5% but

for non-equilibrium oscillations.

BBN with nonequilibrium es

allows to constrain oscillation parameters for He-4 uncertainty up to32% (14%) in resonant (non-resonant) case.

Maximum He-4 overproduction in BBN with oscillations due to spectrum distortion

4Не – the preferred elementBBN - the most early and precision probe for physical conditions in the early Universe, and for constraining new physics, relevant at this E.For a precise analysis of the oscillations effect on BBN, He-4 is used because the most reliable and abundant data now available are for that element.- Observed in НІІ low metalicity regions of dwarf galaxies- Extrapolated towards zero metalicityYp=0,2421 0,0021 Izotov, Thuan 2000

Yp=0,2429 0,009 Izotov, Thuan 2004

Yp=0,2472 0,0012 Izotov, Thuan 20007 (93 spectra of 86 low-metalicity HII regions)

dispersion of the determinationsYp=0,245 0,013 Olive, Skillman 2004

Yp=0,2491 0,0091 Olive, Skillman 2004

Yp=0,2384 0,0025 Peimbert et al 2002

Yp=0,2474 0,0028 Peimbert, Luridiana. Peimbert 2007, new atomic data

Determinations indicate 3-5% uncertainty (systematic errors). Sasselov, 95Possibly it is related with the evaluation of ionization level, stellar absorption, .. Luridiana, 2002

The primordial abundance Yp, predicted from SBBN, is calculated withgreat precision: the theoretical uncertainty is less than 0.1% within a wide

range of baryon density. Yp=0,2482 0,0007

BBN constraints on oscillations

BBN with neutrino oscillations between initially empty s

and e

Observational data on primordial He- 4 was used to put stringent limits on the allowed oscillation parameters.

BBN constraints on е s :

Barbieri, Dolgov 91 – depletion accountDolgov 2000 – dashed curve; DK, Enqvist et al. 92 – one p approx. DK.,Chizhov 2001 – distortion and asymmetry growth accountDolgov, Villante, 2003 - spectrum distortion

Spectrum distortion reflected in neutrino oscillations constraints from BBN

The distortion leads to a decrease of the

weak rates to an increase of the n/p

freezing T and He overproduction.

Correspondingly the account of spectrum

distortion leads to strengthening of BBN

constraints at large mixings.

The account of the neutrino-antineutrino

asymmetry growth caused by resonant

oscillations leads to relaxation of the

constraints for small mixings.

42 2 9 2 2

2 10 2 2

sin 2 1.5 10 0

8.2 10 large , 0

m eV m

m eV m

Ns=0

Role of the initial population of s BBN constraints relaxed or strengthened?

Additional s population may lead to stonger or weaker BBN constraints on oscillation parameters.

There exist an interplay b/n the effects of

non-zero initial population of s on BBN: in case the dynamical effect dominates, He-4 overproduction is enhanced and BBN constraints strengthen, in case the kinetic effect dominates He-4 overproduction decreases and BBN constraints relax.

The dotted blue (red) contour presents

Yp/Yp=3% (Yp/Yp=5.2% ) for Ns=0,

the solid blue (red) contour presents

Yp/Yp=3% (Yp/Yp=5%) for Ns=0,5. DK, Panayotova, 2006, ; DK, 2007

BBN, CMB and CNB bounds • For oscillations parameters corresponding to the regions favored by the atmospheric and solar neutrino

data flavor equilibrium between active neutrino species is established before BBN epoch. No constraints.

• The case of fast active-sterile oscillations before decoupling of active neutrinos, leading to additional species into equilibrium:

Due to oscillations the sterile state is filled by the active neutrino, which is refilled by the plasma. Thus the net effect is energy density in neutrino sector increase. CMB anisotropy spectrum feels energy density increase caused by additional particles, hence it may constrain the fast active-sterile oscillations before decouling. CMB+LSS constraints (sensitive to the total energy density) can be obtained 1<Ns<5 .

BBN constraints on oscillation prameters exist, based mainly on the dynamical effect :

Dolgov A., F.Villante ,2003 m2>10-6 eV2, for non-resonant case:

• In case of the oscillations effective after active neutrino decoupling ,

the total energy density of neutrinos remains unchanged. Hence CMB constraints cannot be obtained. Stringent BBN constraints exist in 2-neutrino oscillations case:

For nonequilibrium oscillations the constraints are strengthened by orders of magnitude:

• :

22 4 5 2

22 4 5 2

sin 2 3.16 10

sin 2 1.74 10

es es

s s

m eV N

m eV N

42 2 9 2 2

2 10 2 2

sin 2 1.5 10 0

8.2 10 large , 0

m eV m

m eV m

2 4 7sin 2 10m

• In 2 neutrino oscillation case CNB is considerably influenced: relic electron neutrino can be expected in the extreme about twice less numerous than in SCM,

and considerably less energetic with an energy spectum strongly distorted from the equilibrium Fermi-Dirac one.

• In 4 neutrino case it should be expected that when flavor oscillations are taking place they will tend to reestablish the equilibrium between the different neutrino flavors. Then CNB electron neutrinos will have slightly depleted number densities, around 3/4 of the SCM value, i.e 112*3/4=84 per cubic cm. The depletion of the electron neutrino and, correspondingly, the kinetic effect of oscillations is reduced. Hence, the cosmological constraints on oscillation parameters, corresponding to 4-neutrino mixing case will be less stringent than the ones calculated for 2-neutrino case.

BBN - one of the most precision probes of new neutrino physics

BBN with decaying particles We explored a modification of the standard Big Bang

with decaying particles X during or before n-p feezing. The presence of such particles increases the expansion rate and their decay products change nucleons kinetics.

Depending on its mass, density and life time, n-p ratio

may be shifted in either direction. This allows relaxation of the BBN bounds on the number of relativistic species.

At high E (big X masses)

increase of n density, while

at low E (small masses) n decreases

The precise analysis of BBN with decaying X reveals the possibility to achieve either overproduction or underproduction

of primordial He-4

( ) ( )

( ) ( )

nn p

p e ne

nn n n pe n n p e n

t

n n pe n n n ne p

expn

p

n

n T

( ) ( )n pe p e n

( ) ( )p e n n pe

X X XX

X

n n mHp n

t p E

min

2

2 2

min

( ),

4 where .

4

X

X XX X X

E

X X

n n mHp dE n p

t p E

m EE

E

Evolution equations of decaying X and its decay products:

22

2 5 23

0

1 32

Fn n p

GN T N N d n

t T

standard

Xn nn

N NN

t t t

Evolution of neutrons:

4 He ~ 2Fn

pnucl

NY

N

BBN may provide a sensitive probe to additional non-radiatively decaying particles, due to the kinetic effect of the decay products on n/p freezing, and hence on He production.

For m < 7 MeV and in case

He underproduction is possible

The possibility for a considerable underproduction allows to weaken the BBN bounds on neutrino species BBN with decaying MeV particles may resolve the discrepancy b/n LSS+CMB constraints on N (based on WMAP3 power spectrum and galaxy clustering power spectrum of SDSS, Ly alpha data, 2dF, HST, SN, etc.) pointing to a higher than 3 value of N (in case it persists). Seljak et al., JCAP, 2006

In a non-standard BBN with more than 3 neutrino species, the dynamical effect of these may be compensated by the effect of the decay products on nucleons kinetics during BBN.

1XX

F

m

T

Conclusions • In case of fast active-sterile oscillations before decoupling of active neutrinos, leading

to additional species into equilibrium, CBM and BBN constraints hold. • In case of active-sterile oscillations after decoupling of active neutrinos only BBN

constraints hold. The number densities of CNB neutrinos are reduced, the spectrum differs from FD. CNB neutrinos may have the equilibrium number density or be depleted depending on the type of oscillations and their parameters.

• BBN provides stringent constraints on nonequilibrium decays of additional particles, leading to deviations of the electron neutrino spectrum, which reflects in changes in the kinetics of nucleons.

• BBN with decaying MeV particles may resolve the discrepancy b/n LSS+CMB constraints on N

• In case neutrino masses are in the eV range they can constitute several % of the DM, they can influence matter clustering (suppressing small-scale power of the matter power spectrum) providing better correspondence between models and observational data (from SDSS, cluster abundance, weak lensing, Lyman Alpha forest, CMB). Tegmark et al., 2004; Takada, 2008

nonequilibrium neutrino in the early universe plasma daniela kirilova institute of astronomy...

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