Neutrinos from cosmological cosmic rays:

A parameter space analysis

Diego González-Díaz, Ricardo Vázquez, Enrique ZasDepartment of Particle Physics, Santiago de Compostela University, 15706, Santiago, Spain

July 2003

Index of contents

July 2003

I. Energy loss processes and propagation equations.

II. Parameters involved in calculation. Predicted fluxes.

III. Main features of the resulting fluxes. Analytical approximation.

IV. The normalization problem.

V. An extremely constrained model. Cosmological CR dominating above the ankle.

VI. Limits for neutrinos. General bottom-up scenario.

VII. Limits for AGN’s.

VIII. Conclusions.

July 2003

July 2003

I. Energy loss processes and propagation equations.

July 2003

July 2003

I. Energy loss processes and propagation equations.

Interaction length: λint(E)=∫n(ε) n(θ) σ(E,ε,θ) dε dθ

Inelasticity: K(E) = < (E-E’)/E >E’

Attenuation length: λatt(E)= λint(E) / K(E) ≈ [1/E (dE/dx)]-1 (c.e.l.)

p, E

Δ(1232)

π, Eπ’

p(n), E’

γ, ε

θdN(E)/dE

n(ε)·n(θ)

dσ(E,ε,θ)/dE’

Limits of validity of c.e.l. approximation:

-λint(E) / K(E) slow varying function of E.

-Fluxes with spectral index γ larger than 1.

-λatt(E) << propagation distance.

July 2003

July 2003

I. Energy loss processes and propagation equations.

Good estimates of the main ν observables Total number(I) and Total energy(II).

Good estimate of propagated nucleon flux (I & II).

Advantages:I. Proper asymptotic limit at energies close to resonance (s~mΔ

2).II. Proper asymptotic limit for nucleon inelasticity at high energy:

K(E) →0.5.

Helpful assumption:

-For the distribution of recoiling protons and produced pions we assume a 2→2 body process with isotropy in the center of mass frame.

π-production

July 2003

For γ→1, processes far from resonances can contribute significantly to the shape of neutrino spectrum.

July 2003

I. Energy loss processes and propagation equations.

July 2003

Total cross-section for π-production for protons:

July 2003

I. Energy loss processes and propagation equations.

Pair production

Assumption:

-Parameterization of cross section and inelasticity for the regime

E>>εcmb (Born approximation), according to Chodorowsky et.al.

Redshift

In general:

231

,1

, )1)(1()1(1

)(),( zzc

H

dx

dE

EzzE MM

ozattzatt

July 2003

July 2003

I. Energy loss processes and propagation equations.

July 2003

Mean free paths

July 2003

I. Energy loss processes and propagation equations.

The propagation equations:

),('),'()()(),(

)(

1),(

int

xEdx

dE

EdExE

dE

dnnddxE

EE

xENNN

N

),('),'()()(

),(xE

dx

dE

EdExE

dE

dnndd

E

xEN

),(),(

),(xE

dx

dE

EdExE

dE

d

E

xE

Numerical method:Runge-Kutta. 200x200x150 bins in (E,E’,z).Running step Δx=0.5(1+z)-3 Mpc.Computation time less than 0.5 hs for z<8.

Consistency conditions:

dEEdExE NN )0,(),(

dEEEdExExEE NN )0,(),(),(

July 2003

July 2003

Cosmic ray flux from isotropic and homogeneous sources:

2

230

221

)(4)()()(

4

),()(

min kr

drrtRtt

R

tR

rR

zEE

roo

NN

dzzEzz

z

H

cE

z

N

MM

m

oooN

max

0 23

1

),()1)(1()1(

)1()(

activity [N/t]3)1()()(

)1()()(

zzt

zzt

o

mo

density

July 2003

I. Energy loss processes and propagation equations.

July 2003

II. Parameters involved in calculation. Predicted fluxes.

July 2003

July 2003

γ spectral injection index

m activity evolution index

zmax z of formation of sources

ΩM density of matter

Ho Hubble constant

B intergalactic magnetic field

ηLρL/ηoρo local enhancement

Emax maximum acceleration energy in source

Parameters Orientative order of magnitude

[1-3]

[3-5]

[1-4]

[0.2-1]

[50-80 Km/s/Mpc]

[B≤1nG]

[?]

[E max >4 1020 eV]

July 2003

II. Parameters involved in calculation. Predicted fluxes.

July 2003July 2003

CR flux from a source located at redshift z:

γ=2 Emax=1022 eV

II. Parameters involved in calculation. Predicted fluxes.

July 2003July 2003

Neutrino flux from a source located at fixed z:

γ=2 Emax= 1022 eV

II. Parameters involved in calculation. Predicted fluxes.

July 2003July 2003

CR and ν fluxes for different models

II. Parameters involved in calculation. Predicted fluxes.

July 2003July 2003

III. Main features of the resulting fluxes. Analytical approximation.

July 2003

III. Main features of the resulting fluxes. Analytical approximation.

According to Berezensky & Grigorieva, for low CR energies the high energy photon tail dominates, following:

EEco

att e λ∞=12Mpc

Eco=mpmπ(1+mπ/mp)/2KBTcmb =3 1020 eV

Features

I. The CR spectrum is sharply suppresed at an energy around 1020 eV for distances larger than ~10Mpc.

)ln()(

x

ExE co

c

July 2003

Features

II. Decoupling of redshift losses. Significative energy losses due to redshift occur within a distance much larger than the π-production scale.

u=3/2 Einstein-DeSitter

u=0 ΩM=01)1( 3

,

, u

Hzatt

att zR

)ln()1()( 1

x

EzzE co

c

Redshift losses occur after π-production, which takes place at constant z.

III. Main features of the resulting fluxes. Analytical approximation.

July 2003

III. Main features of the resulting fluxes. Analytical approximation.

July 2003

Idea of the analytical approximation:

Asumptions of the analytical approach:

Total number and flux shape

A. Due to the power-law character of the injected CR spectrum, the recoiling nucleons do not contribute significantly to the bulk of neutrinos if γ≥1.5. The effect of such nucleons can be absorbed in a constant factor of order unity.

III. Main features of the resulting fluxes. Analytical approximation.

B. The main part of the interactions occur close to the Δ-resonance.

July 2003

C. For low γ, injected energy in pions is well approximated by the total energy of the bulk of interacting protons.

Total energy

III. Main features of the resulting fluxes. Analytical approximation.

July 2003

Emax=1022 eV

Analytical approximation:

III. Main features of the resulting fluxes. Analytical approximation.

Features of cosmological flux:

I. Due to the pair suppression, the sources contributing to theobserved CR flux beyond E=1019 eV (Usually used for normalizationcondition) are placed within a distance Dee~λee=500 Mpc (z=0.15).

The last contributing source in this range fixes the GZK cutoff:

eVDE

DzDzEEee

coeeeecGZK

192 106)ln(

))(1())((

Normalization condition over E=1019 eV is dependent mainly on γ and (possibly) on the local enhancement ηLρL/ηoρo.

July 2003

III. Main features of the resulting fluxes. Analytical approximation.

Features of cosmological flux:

Dependence on cosmological parameters:

II. In the case of m>γ+1/2, unless ΩM<<1 the cosmological dependence is similar to an E-dS model. The depence is roughly absorbed in a factor 1/(HoΩM

1/2).

dzzEz

H

cE

z

N

M

m

oooN

max

0

2/5

),()1(

)(

III. The main contributors to the neutrino flux are placed at z≈zmax if m>γ+1/2.

In the extreme limiting case ΩM=0 (ΩΛ=1) the cosmology can affect strongly, changing m->m+3/2.

July 2003

III. Main features of the resulting fluxes. Analytical approximation.

CR flux

July 2003

CR cosmological flux

III. Main features of the resulting fluxes. Analytical approximation.

Neutrino cosmological flux

July 2003

III. Main features of the resulting fluxes. Analytical approximation.

July 2003

cnorm

c

m

EE

E

m

zQ

12

9

max

29

1)1(

normc

m

EE

E

m

zQ

max2

5

max ln

25

1)1(

max

1

max2

5

max

25

1)1(E

E

E

m

zQ

c

m

norm

c

m

E

E

m

zE

21

)1( 2

5

maxmax coEz

E2

max )1(

022.0(max)

Total energy injected in neutrinos:

Total number (if m>γ+1/2):

γ>2

γ=2

γ<2

July 2003July 2003

IV. The normalization problem.

July 2003

IV. The normalization problem.

I. Requiring normalization above a crossover energy E~1019eV < EGZK . Due to pair suppresion the main contributions are placed within Dee= 500 Mpc. Requiring number conservation the normalization constant can be approximated by:

localooLLeeoE

E

E

ooo DDH

c

dEE

dEE

H

c

norm

norm

1/

1)(

max

exp

II. Requiring normalization over the ankle E~3 1018 eV (Shape concordance). The resulting parameter space is highly constrained.

July 2003

July 2003

IV. The normalization problem.

July 2003

Posible ways of getting normalization

July 2003July 2003

V. An extremely constrained model. Cosmological CR dominating above the ankle.

July 2003July 2003

V. An extremely constrained model. Cosmological CR dominating above the ankle.

In order to fit the two slopes of the observed CR spectrum in the regime [3 1018- 5 1019 eV] the parameter m and γ must be related in some way.

Usual range for ordinary bottom-up sources (AGNs,QSOs)

July 2003July 2003

EGRET: Constraining the electromagnetic component it provides the harder limit for neutrinos at the moment.

Example of maximal fluxes compatible with EGRET

V. An extremely constrained model. Cosmological CR dominating above the ankle.

July 2003July 2003

Range of values for Emax and zmax compatible with EGRET.

V. An extremely constrained model. Cosmological CR dominating above the ankle.

July 2003July 2003

VI. Limits for neutrinos. General bottom-up scenario.

July 2003July 2003

VI. Limits for neutrinos. General bottom-up scenario.

Energy in neutrinos vs energy in photonsγ=1.5 γ=2.5

Qem/Qν is a slow varying function of the injection redshift –z-. We show that roughly the ratio Qem/Qν is dependent mainly on γ.

July 2003July 2003

VI. Limits for neutrinos. General bottom-up scenario.

Model with free m Model with free zmax

July 2003July 2003

VII. Limits for AGN’s.

July 2003July 2003

VII. Limits for AGN’s.

(If not specified, Emax is 1022 eV)

1

(1+z)3 z<1.9

(1+1.9)3 1.9<z<2.7

(1+1.9)3e(2.7-z)/z z>2.7

(1+z)4 z<1.9

(1+1.9)4 z>1.9

3

2 Like 3 but with Emax=1023 eV

models for QSOs

4(1+z)3.4 z<1.9

(1+z)0 1.9<z<2.9

0 z>2.7

July 2003July 2003

VIII. Conclusions.

July 2003July 2003

VIII. Conclusions.

-Main features of cosmological CR and ν fluxes are presented together with useful scalings for total energy and number in ν.

-Similar behaviour of Qem and Qν with z of injection allows to set the EGRET limit as an approximate function of γ.

-Requiring normalization above the ankle has the following implications:

1>γ>2.4 High m needed, [Emax , zmax] very constrained. 2.4<γ<2.6 Values of m compatible with evolution of QSOs. 2.6<γ<2.8 Small evolution of sources with z needed. γ>2.8 Not possible to fit AGASA data over the ankle.

-Unless extreme values are assumed, ν fluxes are highly independent on cosmological parameters.

July 2003July 2003

VIII. Conclusions.

-Generic bottom-up scenario requiring a normalization over Enorm~1019 eV implies:

EGRET limit constrains the neutrino fluxes at high γ through pair-production, limiting strongly the allowed range of parameters [m, zmax] but not Emax . At low γ the energy injected is very sensitive to Emax:

max

1

max2

5

max

25

1)1(E

E

E

m

zQ

c

m

For Emax=1022 eV then [m,zmax] are severely constrained.

Intermediate ranges 1.5<γ<2.5 allow a wider range for the parameters.

-Current models for AGN inspired on evolution of QSOs are close to the EGRET limit for high γ. For low γ the fluxes are strongly dependent on Emax; Emax≥1023 eV is at odds with EGRET.