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Neutrinos from cosmological cosmic rays: A parameter space analysis Diego González-Díaz, Ricardo Vázquez, Enrique Zas Department of Particle Physics, Santiago de Compostela University, 15706, Santiago, Spain July 2003

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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

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I. Energy loss processes and propagation equations. July 2003

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I. Energy loss processes and propagation equations. Interaction length: λ int (E)=∫n(ε) n(θ) σ(E,ε,θ) dε dθ Inelasticity: K(E) = E’ Attenuation length: λ att (E)= λ int (E) / K(E) ≈ [1/E (dE/dx)] -1 (c.e.l.) p, E Δ(1232) π, E π ’ 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: - λ int (E) / K(E) slow varying function of E. -Fluxes with spectral index γ larger than 1. - λ att (E) << propagation distance. July 2003

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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~m Δ 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.

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July 2003 I. Energy loss processes and propagation equations. July 2003 Total cross-section for π -production for protons:

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July 2003 I. Energy loss processes and propagation equations. Pair production Assumption: -Parameterization of cross section and inelasticity for the regime E>> ε cmb (Born approximation), according to Chodorowsky et.al. Redshift In general: July 2003

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I. Energy loss processes and propagation equations. July 2003 Mean free paths

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July 2003 I. Energy loss processes and propagation equations. The propagation equations: 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: July 2003

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Cosmic ray flux from isotropic and homogeneous sources: activity [N/t] density July 2003 I. Energy loss processes and propagation equations.

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July 2003 II. Parameters involved in calculation. Predicted fluxes. July 2003

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γ spectral injection index m activity evolution index z max z of formation of sources Ω M density of matter H o Hubble constant B intergalactic magnetic field η L ρ L /η o ρ o local enhancement E max 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 > eV] July 2003 II. Parameters involved in calculation. Predicted fluxes.

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July 2003 CR flux from a source located at redshift z: γ=2 E max =10 22 eV II. Parameters involved in calculation. Predicted fluxes.

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July 2003 Neutrino flux from a source located at fixed z: γ =2 E max = eV II. Parameters involved in calculation. Predicted fluxes.

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July 2003 CR and ν fluxes for different models II. Parameters involved in calculation. Predicted fluxes.

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July 2003 III. Main features of the resulting fluxes. Analytical approximation.

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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: λ ∞ =12Mpc E co =m p m π (1+m π /m p )/2K B T cmb = eV Features I. The CR spectrum is sharply suppresed at an energy around eV for distances larger than ~10Mpc. July 2003

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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 Ω M =0 Redshift losses occur after π -production, which takes place at constant z. III. Main features of the resulting fluxes. Analytical approximation. July 2003

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III. Main features of the resulting fluxes. Analytical approximation. July 2003 Idea of the analytical approximation:

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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

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III. Main features of the resulting fluxes. Analytical approximation. July 2003 E max =10 22 eV Analytical approximation:

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III. Main features of the resulting fluxes. Analytical approximation. Features of cosmological flux: I. Due to the pair suppression, the sources contributing to the observed CR flux beyond E=10 19 eV (Usually used for normalization condition) are placed within a distance D ee ~ λ ee =500 Mpc (z=0.15). The last contributing source in this range fixes the GZK cutoff: Normalization condition over E=10 19 eV is dependent mainly on γ and (possibly) on the local enhancement η L ρ L /η o ρ o. July 2003

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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 Ω M <<1 the cosmological dependence is similar to an E-dS model. The depence is roughly absorbed in a factor 1/(H o Ω M 1/2 ). III. The main contributors to the neutrino flux are placed at z≈zmax if m>γ+1/2. In the extreme limiting case Ω M =0 (Ω Λ =1) the cosmology can affect strongly, changing m->m+3/2. July 2003

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III. Main features of the resulting fluxes. Analytical approximation. CR flux July 2003 CR cosmological flux

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III. Main features of the resulting fluxes. Analytical approximation. Neutrino cosmological flux July 2003

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III. Main features of the resulting fluxes. Analytical approximation. July 2003 Total energy injected in neutrinos: Total number (if m> γ +1/2): γ>2 γ=2 γ<2

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July 2003 IV. The normalization problem.

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July 2003 IV. The normalization problem. I. Requiring normalization above a crossover energy E~10 19 eV < E GZK. Due to pair suppresion the main contributions are placed within D ee = 500 Mpc. Requiring number conservation the normalization constant can be approximated by: II.Requiring normalization over the ankle E~ eV (Shape concordance). The resulting parameter space is highly constrained. July 2003

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IV. The normalization problem. July 2003 Posible ways of getting normalization

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July 2003 V. An extremely constrained model. Cosmological CR dominating above the ankle.

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July 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 [ eV] the parameter m and γ must be related in some way. Usual range for ordinary bottom- up sources (AGNs,QSOs)

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July 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.

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July 2003 Range of values for E max and z max compatible with EGRET. V. An extremely constrained model. Cosmological CR dominating above the ankle.

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July 2003 VI. Limits for neutrinos. General bottom-up scenario.

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July 2003 VI. Limits for neutrinos. General bottom-up scenario. Energy in neutrinos vs energy in photons γ=1.5γ=2.5 Q em /Q ν is a slow varying function of the injection redshift –z-. We show that roughly the ratio Q em /Q ν is dependent mainly on γ.

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July 2003 VI. Limits for neutrinos. General bottom-up scenario. Model with free mModel with free z max

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July 2003 VII. Limits for AGN’s.

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July 2003 VII. Limits for AGN’s. (If not specified, E max is eV) 1 (1+z) 3 z<1.9 (1+1.9) 3 1.9

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July 2003 VIII. Conclusions.

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July 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 Q em and Q ν 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, [E max, z max ] very constrained Not possible to fit AGASA data over the ankle. -Unless extreme values are assumed, ν fluxes are highly independent on cosmological parameters.

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July 2003 VIII. Conclusions. -Generic bottom-up scenario requiring a normalization over E norm ~10 19 eV implies: EGRET limit constrains the neutrino fluxes at high γ through pair- production, limiting strongly the allowed range of parameters [m, z max ] but not E max. At low γ the energy injected is very sensitive to E max : For E max =10 22 eV then [m,z max ] 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 E max ; E max ≥10 23 eV is at odds with EGRET.

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