Proton and neutrino extragalactic astronomy

The study of extragalactic sources of high energy radiation via the direct measurement of the proton and neutrino fluxes that they are likely to emit is one of the main goals for the future observations of the recently developed air showers detectors and neutrino telescopes. In this work we discuss the relation between the inclusive proton and neutrino signals from the ensemble of all sources in the universe and the resolved signals from the closest and brightest objects. We also compare the sensitivities of proton and neutrino telescopes and comment on the relation between these two new astronomies.

Figure 1

Redshift evolution of the proton energy. The different energies correspond to protons observed at $z=0$ with energy $E_{\mathrm{now}}={10}^{22-0.1k}\mathrm{eV}$ (with $k=0,1,\dots ,25$).

Figure 2

Redshift horizon as a function of the observed proton energy $E$. The different curves are calculated using Eq. (18) for different values of the maximum acceleration energy $E_{\max }$.

Figure 3

Plot of the function $F_{\varphi }(z)=F_{\Phi }(z)$ that gives the redshift dependence of the flux from a $\nu$ source. The source spectrum is a power law of slope $\alpha$. The different curves are calculated for two different choices of the cosmological parameters (${\Omega }_{\mathrm{m}}$ and ${\Omega }_{\Lambda }$) and two different slopes ($\alpha =2$ and $\alpha =2.5$).

Figure 4

Plot of the function $F_{\Phi }(z,E_{\min })$ (in the form $z^2{F}_{\Phi }(z)$ versus $z$) for protons. The function $F_{\Phi }(z)$ gives the “dimming” of the integral proton flux above the minimum energy $E_{\min }$ with increasing redshift $z$. The solid (dashed) curves are calculated for a proton spectrum slope with slope $\alpha =2.2$ ($\alpha =2.6$) and different values of the threshold energy $E_{\min }$.

Figure 5

Shape of the luminosity function that fits the x-ray observations of AGN in the [10, 20] keV band (from 20).

Figure 6

Redshift evolution of the x-ray luminosity density of AGN in the [10, 20] keV band according to the fit of 20.

Figure 7

Plot of the redshift response (“Kepler-Olbers”) function $K_{\varphi }(z)=K_{\Phi }(z)$ for neutrinos assuming a power law emission of slope $\alpha$. In three of the calculations the neutrino emission is considered constant in time. In a fourth calculation the neutrino emission changes in time with the redshift dependence shown in Fig. 6.

Figure 8

Plot of the redshift response (“Kepler-Olbers”) function $K_{\varphi }(z,E)$ for protons assuming a power law emission of slope $\alpha =2.3$. The different curves are calculated (in the absence of source evolution) for different values of the observed proton energy. The thick (red) curve is the curve calculated for neutrinos (considering only redshift energy losses).

Figure 9

Shape factor ${\zeta }_{\varphi }(E)$ for protons, calculated for a power injection with slope $\alpha =2.0$, 2.2, 2.4, and 2.6. The solid lines are calculated for a constant value of the source power density, while the dashed lines assume for ${\mathcal{L}}_p(z)$ the redshift dependence shown in Fig. 6.

Figure 10

Redshift dependence of the Kepler-Olbers functions $K_{\Phi }(z,E_{\min })$ for $E_{\min }=6\times {10}^{19}\mathrm{eV}$ (thick curves) and $E_{\min }={10}^{20}\mathrm{eV}$ (thin curves). The solid (dashed) curves assume a power law injection spectrum with a slope $\alpha =2.0$ ($\alpha =2.4$). Integrating over $z$ one obtains the values ${\zeta }_{\Phi }=0.037$ (0.034) for $E_{\min }=6\times {10}^{19}\mathrm{eV}$ and ${\zeta }_{\Phi }=0.0107$ (0.0093) for $E_{\min }={10}^{20}\mathrm{eV}$.

Figure 11

The thick curves show the redshift horizon as a function of the proton energy. The horizon is defined as the value of $z$ within which 90% of the observed flux is emitted. The solid (dashed) line is calculated for a power law emission with slope $\alpha =2.7$ (2.0). The dotted curves indicate the selection criteria used by the Auger collaboration in their correlation analysis [13]. The redshift that corresponds to the Cen A position is also indicated. The curves labeled $B_1$ and $B_2$ correspond to an expected deviation of 3 degrees with the estimates of the extragalactic magnetic field of Eqs. (24, 26).

Figure 12

Plot of the function $G_{\varphi }(x)=G_{\Phi }(x)$ for neutrinos observations. The neutrino emission has a power law form with slope $\alpha$, and all neutrino sources have identical luminosity. The different curves are calculated for different cosmological parameters and for two different values of the slope $\alpha$. The dotted curve shows the function $x^{-5/2}/2$.

Figure 13

Plot of the function $G_{\Phi }(x)$ (in the form $x^2{G}_{\Phi }(x)$ versus $x$) for the proton integral flux. The calculation is performed for a power law energy spectrum with $\alpha =2.4$ and two energy thresholds $E_{\min }=6\times {10}^{19}\mathrm{eV}$ and $E_{\min }={10}^{20}\mathrm{eV}$. The dashed lines are calculated assuming that all sources have the same luminosity, while the solid line assumes the luminosity function of Eq. (47) shown also in Fig. 5. The dotted curves show the functions $x^{-5/2}/2$ and $x^{-5/2}⟨y^{3/2}⟩/2$.

Figure 14

Number of sources with an expected signal $m$ (plotted in the form $mdN/dm$ versus $m$). The distributions are calculated for the same, power density ${\mathcal{L}}_{\nu }$, assuming that all sources have the same luminosity $L$ and emit with a power law spectrum of slope $\alpha =2$. The different curves differ only because of the value of $L$.

Figure 15

Same as in Fig. 14 but plotted in the form $m^{2}dN/dm$ versus $m$. The area under each curve is proportional to the total number of detected neutrino events.

Figure 16

Expected total number of detected sources (solid lines) and number of sources with multiplicity $k\ge 2$ (dashed lines) plotted as a function of the median luminosity of the sources. The calculation assumes a threshold energy $E_{\min }\simeq 6\times {10}^{19}\mathrm{eV}$ and a total exposure of $9000({\mathrm{km}}^2\mathrm{yr}\mathrm{sr})$ for the Auger detector. The source power density is set to ${\mathcal{L}}_p\simeq 1.6\times {10}^{36}\mathrm{erg}/(\mathrm{s}{\mathrm{Mpc}}^3)$ to reproduce the observed event rate. The thin (thick) curves assumed a fixed luminosity (the luminosity function of Eq. (47)). The dotted line is $\propto \sqrt{L_0}$.

Figure 17

Expected number of proton sources detectable in Auger with multiplicity $n\ge 2$ (for $E_{\min }=6\times {10}^{19}\mathrm{eV}$) plotted as a function of detector live time. The power density of the sources is ${\mathcal{L}}_p=1.6\times {10}^{36}\mathrm{erg}/(\mathrm{s}{\mathrm{Mpc}}^3)$. To reproduce the observed inclusive event rate (the dotted line gives the total number of events as a function of time). The different curves are calculated assuming that all sources have the same spectral shape (a power law of slope $\alpha =2.0$) and three possible values of the median luminosity: ($L_0={10}^{38}$, ${10}^{40}$, and ${10}^{42}\mathrm{erg}/\mathrm{s}$). The dashed line is calculated with a fixed luminosity, while the solid lines are the result using the luminosity function of Eq. (47) and shown in Fig. 5.

Figure 18

The lines describe the curves in the {$z$, $L$} (luminosity, distance) plane that correspond to a rate of one event/year from a source. The luminosity must be understood as luminosity per energy decade. The different curves are for the detection of proton showers with threshold energy $E_{\min }=60$ and 100 EeV in Auger (assuming a source declination $\delta \simeq -30°$) and for the number of neutrino induced muons in IceCube (for a source with $\delta >0$). The dotted lines are $\propto z^2$.

Figure 19

Expected number of neutrino event clusters with $k\ge 2$ plotted as a function of the median luminosity $L_0$ of the neutrino sources. The calculation is performed with a neutrino power density ${\mathcal{L}}_{\nu }={10}^{37}\mathrm{erg}/(\mathrm{s}{\mathrm{Mpc}}^3)$ and assuming no cosmological evolution of the sources. The solid lines assume the luminosity function of Eq. (47) and the dashed lines a fixed value $L=L_0$.

Figure 20

Expected number of extragalactic neutrino sources detectable in IceCube with a multiplicity of upgoing muons $n\ge 2$ plotted as a function of detector live time. The neutrino sources are all identical and emit with power law spectra with slope $\alpha =2$ and a luminosity per energy decade $L_0$. The power density of the ensemble of all neutrino sources is $\mathcal{L}={10}^{37}\mathrm{erg}/(\mathrm{s}{\mathrm{Mpc}}^3)$. The different curves are calculated for different values $L_0={10}^{38}$, ${10}^{40}$, and ${10}^{42}\mathrm{erg}/\mathrm{s}$. The dashed line gives the inclusive number of detected upgoing muons that are produced by extragalactic neutrinos. For a detector, places in the Mediterranean Sea at a latitude of 36° the number of sources should be reduced by a factor 0.787.

Figure 21

Plot of the detector effective area $A(\Omega )$ as a function of the celestial declination $\delta$. In the case of the Auger-South detector, the effective area is divided by the surface area of the array, the detector latitude is ${\phi }_{\mathrm{lat}}=-24.8°$, and the projected area is averaged over a sidereal day considering only the time when a source has zenith angle $\theta <60°$. For the neutrino telescopes, the detection method is the observation of $\nu$ induced muons (with $E_{\min }\ge 1\mathrm{GeV}$), the detector projected area in the direction $\theta$ is approximated as constant, and $\nu$ detection is considered possible for all zenith angles $\theta >90°$.

Figure 22

Neutrino energy response for the upgoing muon signal. The area under the curves is proportional to the signal of $\nu$ induced muons (with threshold $E_{\mu }=1\mathrm{GeV}$) assuming a power law neutrino flux with slope $\alpha$. The thick (thin) curves are calculated for a slope $\alpha =2$ ($\alpha =2.4$); the solid (dashed) curves are calculated for a nadir angle $\theta =90°$ ($\theta =0°$) and negligible (maximum) $\nu$ absorption in the Earth. The relative normalization of the curves is correct if the spectra have the same integrated flux above 1 TeV. The $\nu$ induced signal is roughly proportional to the integrated neutrino flux in the approximate energy range $[{10}^{12},{10}^{16}]\mathrm{eV}$ in the case of a hard spectrum or $[{10}^{11},{10}^{15}]\mathrm{eV}$ for a soft spectrum.

Figure 23

Neutrino to muon conversion efficiency. The different curves correspond to different choices for the muon detection threshold and different celestial declination of a source for a neutrino detector at the South Pole.