Efectos de las oscilaciones de sabor sobre el desacoplamiento de neutrinos c ó smicos Teguayco...

Efectos de las oscilaciones de sabor sobre el desacoplamiento

de neutrinos cósmicos

Efectos de las oscilaciones de sabor sobre el desacoplamiento

de neutrinos cósmicos

Teguayco Pinto Cejas

9-9-2005AHEP - IFIC

Teguayco Pinto Cejas

9-9-2005AHEP - IFIC

Trabajo de investigaciónTrabajo de investigación

hep-ph/0506164

OutlineOutlineMotivation Basic concepts

Early Universe

Thermal equilibrium

Neutrino decouplingQED corrections

Non instantaneous

First results

Effects of medium Final results

Flavour neutrino oscillations

MotivationMotivation

Neutrinos are abundant in the Universe Radiation component Hot dark matter Neutrino spectrum depends on decoupling Decoupling is flavour dependent

Neutrinos are abundant in the Universe Radiation component Hot dark matter Neutrino spectrum depends on decoupling Decoupling is flavour dependent

Hannestad 2003

Early UniverseEarly Universe Primordial Plasma: initial state of high

temperature The plasma is kept in thermal equilibrium

by different reactions whose rate is larger than the expansion rate

Primordial Plasma: initial state of high temperature

The plasma is kept in thermal equilibrium by different reactions whose rate is larger than the expansion rate

Neutrinos coupled by weak interactions

Decoupled neutrinos(CNB)

T~MeVt~sec

Primordial

Nucleosynthesis

Thermal equilibriumThermal equilibrium

After quark-hadron transition neutrinos are kept in thermal equilibrium by:

Equilibrium implies:

After quark-hadron transition neutrinos are kept in thermal equilibrium by:

Equilibrium implies:

€

Γ=σn >> H ~ 1/ t€

ναν β → ναν β

ναν α → ναν αναν α → e+e−

ν α e± →ν α e±

€

Te = Tγ = Tν

Neutrino decouplingNeutrino decoupling It occurs in the Radiation Dominated Era, where

the radiation component of is dominant

Since neutrinos have mν << MeV they were relativistic at decoupling

Relic neutrinos present a FD distribution with Tν

At T~me e-annihilation processes heat up the ’s and the temperature ratio is:

It occurs in the Radiation Dominated Era, where the radiation component of is dominant

Since neutrinos have mν << MeV they were relativistic at decoupling

Relic neutrinos present a FD distribution with Tν

At T~me e-annihilation processes heat up the ’s and the temperature ratio is:

€

Γν ~ H ⇒ GFT 5 ~ 8πρ

3M Pl2 ⇒ Tdec

ν ≈1 MeV

€

Tγ /Tν = 11/4( )1/ 3

≈1.40102

mν only could be important throughoscillations

Entropy conservation

It depends on momentum because ofthe smallness of neutrino mass

€

fν ( p,t)∝1

ep / T +1The present day temperatures

€

Tγ = 2.73 K

Tν =1.95 K

p and T scale with a

Non-instantaneous decouplingNon-instantaneous decoupling

Below Tdec there are residual interactions between ν and e± that distort the neutrino spectrum

Non-thermal effect


Non-thermal effect

€

(∂t − Hp∂p ) f ν α (t, p) = Icollν α (t, p)

Boltzmann equation

Icoll is basically proportional to:

Statistical factorWeak interaction amplitude

Icoll is basically proportional to:

Statistical factorWeak interaction amplitude

€

1

2E

d3 p

(2π)32E i

⎛

⎝ ⎜

⎞

⎠ ⎟

i= 2

4

∏∫ (2π)4δ 4 (p1 + p2 − p3 − p4 ) A2F

Since Tdec isclose to me

Non-instantaneous decouplingNon-instantaneous decoupling


Non-thermal effect

with the comoving variables


Non-thermal effect

with the comoving variables

€

(∂t − Hp∂p ) f ν α (t, p) = Icollν α (t, p)

€

x = ma(t)

y = pa(t)

Boltzmann equation

€

df ν α (x, y)

dx=

1

HxIcoll

ν α (x, y)

We also use the energy conservation to set a complete system of integro-differential equations

where The system has no analytical solution There have been several works dedicated to this

problem

We also use the energy conservation to set a complete system of integro-differential equations

where The system has no analytical solution There have been several works dedicated to this

problem

Dicus et al 1982Hannestad & Madsen 1995Dolgov et al 1997

Dicus et al 1982Hannestad & Madsen 1995Dolgov et al 1997

€

˙ ρ = −3H(ρ + P)⇒dρ

dx=

1

x(ρ − 3P )

€

=(x /m)4

P = P(x /m)4

€

dz

dx= ... where z = Tγ a(t)

First results (No oscillations)First results (No oscillations)

The next results have been obtained solving numerically the previous set of integrodifferential equations with a fortran code.

We use a grid on neutrino momenta

where i=1,…,100

The next results have been obtained solving numerically the previous set of integrodifferential equations with a fortran code.

We use a grid on neutrino momenta

where i=1,…,100

€

fν α(y i,x) → f i

ν α (x)

At x < 0.2 (T~2.5 MeV) there is nodecoupling

Between 0.2 < x < 4distortions are effective

At low temperatures(x > 4) the distortions freeze out due to:- decoupling- few e±

Also

€

δfν e> δfν μ

€

δνα=να

−να

eq

The distortion has beenMultiplied by 10 to see theeffect

z (Temp. ratio) Neff

Instant. 1.4010 0 0 3

No QED 1.3990 0.95 0.43 3.035

QED 1.3978 0.94 0.43 3.046

Take into account that this definitionis only correct when the neutrino are in equilibrium

Corrections at finite temperature

to the and e± plasma equation

of state

€

δν e/ρν

0 (%)

€

δν x/ρν

0 (%)


Instant. 1.4010 0 0 3

No QED 1.3990 0.95 0.43 3.035

QED 1.3978 0.94 0.43 3.046

€

Neff =ρ r − ργ

ρν0

ργ0

ργ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Two reasons for the variation of Neff

Distortion on neutrino spectrum Variation of T

Two reasons for the variation of Neff

Distortion on neutrino spectrum Variation of T

€

∝T 4 ∝ z4 ⇒ργ

0

ργ

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟=

z0

z

⎛

⎝ ⎜

⎞

⎠ ⎟

4

It is a usual way to parametrize the energy density in function of

€

r = 1+7

8

4

11

⎛

⎝ ⎜

⎞

⎠ ⎟4 / 3

Neff

⎡

⎣ ⎢

⎤

⎦ ⎥ργ

€

δν x/ρν

0 (%)

€

δν e/ρν

0 (%)


Instant. 1.4010 0 0 3

No QED 1.3990 0.95 0.43 3.035

QED 1.3978 0.94 0.43 3.046

This difference is one of the motivationsof this work

€

δν e/ρν

0 (%)

€

δν x/ρν

0 (%)

Flavour neutrino oscillationsFlavour neutrino oscillations

Experimental evidences of neutrino oscillations Experimental evidences of neutrino oscillations

€

Δm122

10−5= 8.1 eV2,

Δm312

10−3= 2.2 eV2

sin2 θ12 = 0.3, sin2 θ23 = 0.5, sin2 θ13 = 0

M. Maltoni et al 2004

Flavour neutrino oscillationsFlavour neutrino oscillations

Experimental evidences of neutrino oscillations They could play a role if they are effective at the

epoch of neutrino decoupling Density matrix

Evolution

Experimental evidences of neutrino oscillations They could play a role if they are effective at the

epoch of neutrino decoupling Density matrix

Evolution€

=ee ρ ex

ρ ex ρ xx

⎛

⎝ ⎜

⎞

⎠ ⎟=

1

2P0 + σP[ ] =

1

2

P0 + Pz Px − iPy

Px + iPy P0 − Pz

⎛

⎝ ⎜

⎞

⎠ ⎟

We have to take into account the presence of the medium.

It blocks the oscillations and its effect arises later.

Remember that it is composedby: e±, and ν

€

dP

dx= Vvac ×P

spin up signifies νe

spin down signifies ν

Effect of medium (I)Effect of medium (I)

The equation of motion

The equation of motion

€

dP

dx= Vvac + Vmed

( ) ×P − DPT

Two different effects induced by the medium

Change the axis and speed of

precession but not the length

Shrinkage of P which destroy the coherence of the evolution

MSW effect

Pz gives the excess of νe

over ν

Assuming collisions flavour conserving

€

V med ∝ ˆ u z

€

PT ∝ ˆ u x,y

Effect of medium (II)Effect of medium (II)

Medium potential Vmed Medium potential Vmed

Modifies the dispersion relation

€

Vαmed ∝ 2GF Nα + 2ºorder

Asymmetry of particle α

€

Nα = nα − nα

Neglect neutrino asymmetry Nν~0

For e± background we have to take into account the second order

Neglect neutrino asymmetry Nν~0

For e± background we have to take into account the second order

€

Vemed = −

8 2GF p

3mw2 ρ e

Nl~NB~10-10

At high T Ne<<e

At T~1 MeV Ne << Δm2/2p

Effect of medium (III)Effect of medium (III)Oscillations start to be effective immediately

At high T Ve >Δm2/2p

Oscillations start to be effective around 1MeV

Dolgov et al 2002

Final resultsFinal results

Two independent calculations are carried out in the x range from me/10 to 35

Two independent calculations are carried out in the x range from me/10 to 35

One evaluates the creation of distortions in the neutrinodistribution function dividing the evolution in 1000 steps

Then the evolution equations involving the components of neutrino matrix are solveddividing each step x0x1 by 100 or more

Oscillations smooth the flavour dependence of the distortion

Around T~1 MeV the oscillationsstart to modifythe distortion

The variationis larger for νe

The differencebetween differentflavors is reduced

zfin δνe δν δν Neff

QED 1.3978 0.94 0.43 -- 3.046

3ν-mixing 1.3978 0.73 0.52 0.52 3.046

s2 13=0.047 1.3978 0.70 0.56 0.52 3.046

Bimaximal 1.3978 0.69 0.54 0.54 3.046

Small correction

The effect on the energy density is higher

Fitting the distorsionsFitting the distorsions

For numerical calculations

If neutrino masses are relevant

For numerical calculations

If neutrino masses are relevant

CMBFAST or CAMB

€

fν e(y) = feq (y)[1+10−4 (1− 2.2y + 4.1y 2 − 0.047y 3)]

fν μ ,τ(y) = feq (y)[1+10−4 (−4 + 2.1y + 2.4 y 2 − 0.019y 3)]

€

fν 1(y) = 0.7 fν e

(y) + 0.3 fν x(y)

fν 2(y) = 0.3 fν e

(y) + 0.7 fν x(y)

fν 3(y) = fν x

(y)

€

fν i(y) = Uαi

2fν α

(y)α = e,μ ,τ

∑

Some examplesSome examples

Contribution of neutrinos to total energy density

Neutrino number density

Contribution of neutrinos to total energy density

Neutrino number density€

Ων =ν

c

=3m0

94.12h2 eV2

€

Ων =3m0

93.14h2 eV2

Effect of distorsion

€

nν = 335.7 cm-3

Effect of distorsion

€

nν = 339.3 cm-3

Big Bang nucleosynthesisBig Bang nucleosynthesis

Production of the primordial abundances of light elements (4He, H…)

The relevant temperature interval is from 1 MeV to 50 keV

Neutrinos have a double influence on nucleosynthesis

Production of the primordial abundances of light elements (4He, H…)

The relevant temperature interval is from 1 MeV to 50 keV

Neutrinos have a double influence on nucleosynthesisAs a component of the background radiationThe processes np are directly affected by neutrino spectrum

As a component of the background radiationThe processes np are directly affected by neutrino spectrum €

n + ν e → p + e−

n + e+ → p + ν e

n → p + e− + ν e


As a component of the background radiationThe processes np are directly affected by neutrino spectrum

As a component of the background radiationThe processes np are directly affected by neutrino spectrum

As a component of the background radiationAs a component of the background radiation

€

δem = −δρν

€

δXnem = −0.1

δTγ

Tγ

=0.1

4

δρν

ρν

Energy conservation

Change in tBBN

Energy conservation

Change in tBBN


As a component of the background radiationEnergy conservation

Change in tBBN

As a component of the background radiationEnergy conservation

Change in tBBN

The processes np are directly affected by neutrino spectrumThe processes np are directly affected by neutrino spectrum

€

δem = −δρν

€

δXnem = −0.1

δTγ

Tγ

=0.1

4

δρν

ρν

€

δtBBN

tBBN

≈ −δρν

ρ tot

≈ −1

2

δρν

ρν

€

δXnν = −0.1

δTν e

Tν e

= −0.1

4

δρν e

ρν e

€

δXnt ∝

δρν

ρν


ΔYp

QED 1.71 10-4

3ν-mixing 2.07 10-4

s2 13=0.047 2.12 10-4

Bimaximal 2.13 10-4

Yp~ 0.24

ConclusionsConclusions

We are able to calculate the neutrino spectrum considering the oscillations

We have calculated the frozen values of neutrino distributions

The effects on Neff are not yet detectable but could be measured in future experiments

The effect on BBN is not important over the total correction

We are able to calculate the neutrino spectrum considering the oscillations

We have calculated the frozen values of neutrino distributions

The effects on Neff are not yet detectable but could be measured in future experiments

The effect on BBN is not important over the total correction

CMBFASTCAMB

not PLANCKmaybe CMBPOL

FinFin

Efectos de las oscilaciones de sabor sobre el desacoplamiento de neutrinos c ó smicos Teguayco...

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