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Neutrinos from cosmological cosmic rays: A parameter space analysis

July 2003. 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. Index of contents.

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Neutrinos from cosmological cosmic rays: A parameter space analysis

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  1. July 2003 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

  2. July 2003 Index of contents 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

  3. July 2003 I. Energy loss processes and propagation equations. July 2003

  4. July 2003 I. Energy loss processes and propagation equations. Interaction length: λint(E)=∫n(ε) n(θ) σ(E,ε,θ) dε dθInelasticity: K(E) = < (E-E’)/E >E’Attenuation length: λatt(E)= λint(E) / K(E) ≈ [1/E (dE/dx)]-1 (c.e.l.) γ , ε n(ε)·n(θ) π, Eπ’ p, E dN(E)/dE θ Δ(1232) p(n), E’ 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 γlarger than 1. -λatt(E) << propagation distance. July 2003

  5. July 2003 I. Energy loss processes and propagation equations. π-production 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. Advantages: I. Proper asymptotic limit at energies close to resonance (s~mΔ2). II. Proper asymptotic limit for nucleon inelasticity at high energy: K(E) →0.5. Good estimates of the main ν observablesTotal number(I)andTotal energy(II). Good estimate of propagated nucleon flux (I & II). For γ→1, processes far from resonances can contribute significantly to the shape of neutrino spectrum. July 2003

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

  7. Redshift In general: 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. July 2003

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

  9. Consistency conditions: 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. July 2003

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

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

  12. 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. Parameters γspectral injection index m activity evolution index zmaxz of formation of sources ΩMdensity of matter HoHubble constant Bintergalactic magnetic field ηLρL/ηoρolocal enhancement Emaxmaximum acceleration energy in source July 2003

  13. July 2003 II. Parameters involved in calculation. Predicted fluxes. CR flux from a source located at redshift z: γ=2 Emax=1022 eV July 2003

  14. July 2003 II. Parameters involved in calculation. Predicted fluxes. Neutrino flux from a source located at fixed z: γ=2 Emax= 1022 eV July 2003

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

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

  17. July 2003 III. Main features of the resulting fluxes. Analytical approximation. Features I.The CR spectrum is sharply suppresed at an energy around 1020 eV for distances larger than ~10Mpc. According to Berezensky & Grigorieva, for low CR energies the high energy photon tail dominates, following: λ∞=12Mpc Eco=mpmπ(1+mπ/mp)/2KBTcmb =3 1020 eV July 2003

  18. III. Main features of the resulting fluxes. Analytical approximation. 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=3/2Einstein-DeSitter u=0 ΩM=0 Redshift losses occur afterπ-production, which takes place at constant z.

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

  20. III. Main features of the resulting fluxes. Analytical approximation. July 2003 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. B. The main part of the interactions occur close to the Δ-resonance. Total energy C. For low γ, injected energy in pions is well approximated by the total energy of the bulk of interacting protons.

  21. III. Main features of the resulting fluxes. Analytical approximation. July 2003 Analytical approximation: Emax=1022 eV

  22. III. Main features of the resulting fluxes. Analytical approximation. Normalization condition over E=1019 eV is dependent mainly on γand (possibly) on the local enhancement ηLρL/ηoρo. July 2003 Features of cosmological flux: I. Due to the pair suppression, the sources contributing to the observed CR flux beyond E=1019 eV (Usually used for normalization condition) are placed within a distance Dee~λee=500 Mpc (z=0.15). The last contributing source in this range fixes the GZK cutoff:

  23. III. Main features of the resulting fluxes. Analytical approximation. July 2003 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/(HoΩM1/2). In the extreme limiting case ΩM=0 (ΩΛ=1) the cosmology can affect strongly, changing m->m+3/2. III. The main contributors to the neutrino flux are placed at z≈zmax if m>γ+1/2.

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

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

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

  27. July 2003 IV. The normalization problem. July 2003

  28. 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: • Requiring normalization over the ankle E~3 1018 eV (Shape concordance). The resulting parameter space is highly constrained. July 2003

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

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

  31. V. An extremely constrained model. Cosmological CR dominating above the ankle. Usual range for ordinary bottom-up sources (AGNs,QSOs) July 2003 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. July 2003

  32. V. An extremely constrained model. Cosmological CR dominating above the ankle. July 2003 EGRET: Constraining the electromagnetic component it provides the harder limit for neutrinos at the moment. Example of maximal fluxes compatible with EGRET July 2003

  33. V. An extremely constrained model. Cosmological CR dominating above the ankle. July 2003 Range of values for Emax and zmaxcompatible with EGRET. July 2003

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

  35. 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 2003 VI. Limits for neutrinos. General bottom-up scenario. Energy in neutrinos vs energy in photons γ=1.5 γ=2.5 July 2003

  36. July 2003 VI. Limits for neutrinos. General bottom-up scenario. Model with free m Model with free zmax July 2003

  37. July 2003 VII. Limits for AGN’s. July 2003

  38. models for QSOs (1+z)4 z<1.9 (1+1.9)4 z>1.9 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 3 2 Like 3 but with Emax=1023 eV (1+z)3.4 z<1.9 (1+z)0 1.9<z<2.9 0 z>2.7 4 July 2003 VII. Limits for AGN’s. (If not specified, Emax is 1022 eV) July 2003

  39. July 2003 VIII. Conclusions. July 2003

  40. July 2003 VIII. Conclusions. -Main features of cosmological CR and νfluxes are presented together with useful scalings for total energy and number in ν. -Unless extreme values are assumed, νfluxes are highly independent on cosmological parameters. -Similar behaviour of Qem and Qνwith 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 Highm 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. July 2003

  41. July 2003 VIII. Conclusions. -Generic bottom-up scenario requiring a normalization over Enorm~1019 eV implies: EGRET limit constrains the neutrino fluxes at high γthrough pair-production, limiting strongly the allowed range of parameters [m, zmax] but not Emax . At low γthe energy injected is very sensitive to Emax: 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. July 2003

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