Flavor composition and energy spectrum of astrophysical neutrinos

The measurement of the flavor composition of the neutrino fluxes from astrophysical sources has been proposed as a method to study not only the nature of their emission mechanisms, but also the fundamental neutrino properties. It is however problematic to reconcile these two goals, since a sufficiently accurate understanding of the neutrino fluxes at the source is needed to extract information about the physics of neutrino propagation. In this work we discuss critically the expectations for the flavor composition and energy spectrum from different types of astrophysical sources, and comment on the theoretical uncertainties connected to our limited knowledge of their structure.

Figure 1

Distribution of the observable flavor ratio $R_{e\mu }=({\nu }_e/{\nu }_{\mu }{)}_{\mathrm{obs}}$ for different models of the $\nu$ source and different neutrino properties. A source model is defined by the relative intensity (summing over $\nu$ and $\overline{\nu }$) of the emission for the three flavors: $[{\nu }_e,{\nu }_{\mu },{\nu }_{\tau }{]}_{\mathrm{source}}$. The distribution of $R_{e\mu }$ is entirely due to the present uncertainties in the determination of the $\nu$ mixing parameters. The source models [0, 1, 0] and [1, 0, 0] correspond to pure ${\nu }_{\mu }$ or ${\nu }_e$ emission; the model [1, 1.86, 0] corresponds to the pion dominated emission from a thin source with a power law spectrum of slope 2. The two dashed curves show the distributions of $R_{e\mu }$ for a $\nu$-decay model [16] where only the lowest $\nu$ mass eigenstate (in the direct and inverse hierarchy) is stable.

Figure 2

Distribution of the observable flavor ratio $R_{\mu \tau }=({\nu }_{\mu }/{\nu }_{\tau }{)}_{\mathrm{obs}}$ (see caption of Fig. 1). Some of the curves have been renormalized for an easier observation of their shape.

Figure 3

Neutrino flavor ratio $({\nu }_{\mu }+{\overline{\nu }}_{\mu })/({\nu }_e+{\overline{\nu }}_e)$ at the source obtained from the chain decay of charged pions with a power law energy spectrum $\propto E_{\pi }^{-\alpha }$. Energy losses are assumed to be negligible. The dashed line neglects the effect of muon polarization in pion decay.

Figure 4

Neutrino flavor ratio $({\nu }_{\mu }+{\overline{\nu }}_{\mu })/({\nu }_e+{\overline{\nu }}_e)$ at the source for the chain decay of different mesons having a power law spectrum $\propto E^{-\alpha }$. The different lines refer to the decay of ${\pi }^{\pm }$, $K^{\pm }$, and $K_L$. The dashed (dot-dashed) line shows the effect of neglecting muon polarization in the chain decay of charged pions (kaons).

Figure 5

Characteristic times for a muon plotted as a function of its energy. The different lines correspond to (i) the decay time $t_{\mathrm{dec}}$; (ii) the time $t_{\mathrm{matter}}$ for energy loss in ordinary matter, assuming a gaseous medium with the average composition of the interstellar medium (ISM) and the density $\rho ={10}^{-10}\mathrm{g}{\mathrm{cm}}^{-3}$ (this time scales as $t_{\mathrm{matter}}\propto {\rho }^{-1}$); (iii) the time $t_{\mathrm{syn}}$ for synchrotron losses in a magnetic field $B={10}^4$ Gauss ($t_{\mathrm{syn}}\propto B^{-1}$), and (iv) the time $t_{\mathrm{IC}}$ for inverse Compton losses. The target radiation field is assumed isotropic, with energy spectrum of GRB form (63) with ${\epsilon }_b=1\mathrm{KeV}$, and energy density ${\rho }_{\gamma }={\rho }_B$ (with $B={10}^4$ Gauss).

Figure 6

Characteristic times for charged pions. The different lines correspond to: (i) the decay time $t_{\mathrm{dec}}$; (ii) the time $t_{\mathrm{matter}}$ for hadronic interactions with a gaseous medium with the ISM composition and density $\rho ={10}^{-10}\mathrm{g}{\mathrm{cm}}^{-2}$; (iii) the time $t_{\mathrm{syn}}$ for synchrotron losses (in a randomly oriented magnetic field $B={10}^4$ Gauss); (iv) the time $t_{\mathrm{IC}}$ for inverse Compton losses; (v) the time $t_{\pi \gamma }$ for photohadronic interactions. The last 2 times are calculated for a radiation field of GRB form (63) with ${\epsilon }_b=1\mathrm{KeV}$, and energy density ${\rho }_{\gamma }={\rho }_B$ (with $B={10}^4$ Gauss). Point a (b) is the intersection of the $t_{\mathrm{dec}}$ and $t_{\mathrm{syn}}$ ($t_{\mathrm{matter}}$) curves. The energy that corresponds to the point is the critical energy above which synchrotron losses (the hadronic interaction probability) are (is) important before decay. For the values of $B$ and $\rho$ chosen in this illustration the presence of a gas of ordinary matter is negligible.

Figure 7

Decay energy distribution $d{p}_{\mathrm{dec}}/dE(E,E_i)$ for particles losing energy as $-dE/dt=a{E}^2$. The critical energy $ϵ=\sqrt{m/(a\tau )}$ is the energy where the decay time and the energy loss time are equal.

Figure 8

Proton interaction rate in a radiation field of the form (63). The energy $E^{*}$ (defined in Eq. (72)) corresponds to the threshold proton energy for interaction with photons with energy ${\epsilon }_b$. The dashed line neglects the interactions with all photons with energy $\epsilon >{\epsilon }_b$.

Figure 9

Distribution of ${ϵ}_r$ (the photon energy in the proton rest frame) for the $p\gamma$ interactions of protons of different energy $E_p$. The isotropic photon field has the energy spectrum of GRB form (63). The dashed line neglects the contribution of photons above the break energy ${\epsilon }_b$.

Figure 10

Energy distributions (in the form $Q_j(E)E^2$ versus $E$) for different secondary particles $({\pi }^{\pm },K^{\pm },K_L)$ produced by the $p\gamma$ interactions of relativistic protons in a GRB radiation field. The shape of the primary proton spectrum is also shown (in the form $N_p(E)E^2$ versus $E$) as a thin solid line. The energy spectrum of the isotropic target photons has the form (63). All energies are in units of the characteristic energy $E^{*}$ (defined in Eq. (72)). The dashed lines are the distributions of the meson energy at decay assuming the presence of a magnetic field with value $B=2.78\times {10}^3({\epsilon }_b/\mathrm{KeV})$ Gauss.

Figure 11

Neutrino energy spectrum (summed over all flavors) from a GRB source, according to the model [25]. The calculation is performed assuming a proton spectrum $N_p(E)\propto E^{-\alpha }$, traveling in an isotropic radiation field of form (63). The different curves are obtained varying the exponent $\alpha$ of the proton spectrum, or the exponents ${\beta }_1$ and ${\beta }_2$ of the radiation field. All energies are measured as a fraction of $E^{*}=m_p{ϵ}_{\mathrm{th}}/(2{\epsilon }_b)$. The energy losses of all secondary particles are assumed to be negligible.

Figure 12

Neutrino energy spectrum (summed over all flavors) from a GRB source (see text). The primary proton spectrum has slope $\alpha =2$, while the target photons have a broken power law energy distribution (see Eq. (63)) with slopes ${\beta }_1=1$ and ${\beta }_2=2$ below and above the break energy ${\epsilon }_b$. The curves are calculated assuming different values for the magnetic field, corresponding to values of the critical energy for muons ${ϵ}_{\mu }=\infty$, 30, and 3 (see Eq. (74)).

Figure 13

Energy spectra for ${\nu }_e$ and ${\overline{\nu }}_e$ from a GRB source (proton slope $\alpha =2$, target photon slopes ${\beta }_1=1$ and ${\beta }_2=2$, no magnetic field). For the ${\overline{\nu }}_e$ flux also the contributions from meson and neutron decay are separately shown. Neutron decay dominates at low energy.

Figure 14

Ratio $({\nu }_{\mu }+{\overline{\nu }}_{\mu })/({\nu }_e+{\overline{\nu }}_e)$ for neutrinos from a GRB source. The ratio corresponds to the spectra shown in Fig. 11. The rapid increase at low energy is due to the effect of ${\overline{\nu }}_e$ from $n$ decay.

Figure 15

Ratio $({\nu }_{\mu }+{\overline{\nu }}_{\mu })/({\nu }_e+{\overline{\nu }}_e)$ for neutrinos from a GRB source. The ratio corresponds to the spectra shown in Fig. 12. The three curves are calculated assuming different values of the magnetic field in the source (and proton slope $\alpha =2$, target photon slopes ${\beta }_1=1$ and ${\beta }_2=2$). The decrease of the flavor ratio at high energy for large $B$ is associated to the important synchrotron losses of high-energy muons.

Figure 16

Ratio (${\nu }_e/{\overline{\nu }}_e$) at the source for neutrinos from a GRB source. The ratio corresponds to the spectra shown in Fig. 12. At low energy the ratio is low because of the importance of ${\overline{\nu }}_e$ from $n$ decay. At higher energy the ratio decreases because of the increased importance of ${\pi }^{-}$ production. For a high magnetic field, at high energy the dominant source of ${\nu }_e$ and ${\overline{\nu }}_e$ becomes $K_L$ decay, and the (${\nu }_e/{\overline{\nu }}_e$) ratio becomes unity.

Figure 17

Ratio (${\nu }_{\mu }/{\overline{\nu }}_{\mu }$) at the source for neutrinos from a GRB source. The ratio corresponds to the spectra shown in Fig. 12. The three curves correspond to three different values of the magnetic field in the source. At low energy (and at all energies for small $B$) the ratio is unity, reflecting the fact that the chain decay of a charged pion produces a ${\nu }_{\mu }$ and a ${\overline{\nu }}_{\mu }$. In the presence of a large magnetic field, the neutrinos from muon decay are suppressed, and the flavor ratio is determined by the ${\pi }^{+}/{\pi }^{-}$ ratio. At the highest energies charged kaon decay becomes the dominant source of ${\nu }_{\mu }$ and ${\overline{\nu }}_{\mu }$ and the flavor ratio reflects the $K^{+}/K^{-}$ ratio.

Figure 18

Observable ratio $({\nu }_{\mu }+{\overline{\nu }}_{\mu })/({\nu }_e+{\overline{\nu }}_e)$ after propagation for the neutrinos from a GRB source. The solid lines correspond to the source models shown in Figs. 12, 15. The three curves differ for the value of the magnetic field that corresponds to ${ϵ}_{\mu }=3$, 30, and $\infty$ (no field). The neutrino oscillation parameters have been chosen as ${\theta }_{23}={45}^{\circ }$, ${\theta }_{12}={34}^{\circ }$, and ${\theta }_{13}=0$. The thin dotted line calculates for the naive assumption of a source composition $[{\nu }_e,{\nu }_{\mu },{\nu }_{\tau }]=[1,2,0]$. The dotted line assumes the same mixing parameters as above. The dashed (dot-dashed) line has ${\theta }_{23}=50°$ (40°).