Constraints on the origin of the ultrahigh energy cosmic rays using cosmic diffuse neutrino flux limits: An analytical approach

Astrophysical neutrinos are expected to be produced in the interactions of ultrahigh energy cosmic rays with surrounding photons. The fluxes of the astrophysical neutrinos are highly dependent on the characteristics of the cosmic-ray sources, such as their cosmological distributions. We study possible constraints on the properties of cosmic-ray sources in a model-independent way using experimentally obtained diffuse neutrino flux above 100 PeV. The semianalytic formula is derived to estimate the cosmogenic neutrino fluxes as functions of source evolution parameter and source extension in redshift. The obtained formula converts the upper limits on the neutrino fluxes into the constraints on the cosmic-ray sources. It is found that the recently obtained upper limit on the cosmogenic neutrinos by IceCube constrains the scenarios with strongly evolving ultrahigh energy cosmic-ray sources, and the future limits from a $1{\mathrm{km}}^3$ scale detector are able to further constrain the ultrahigh energy cosmic-rays sources with evolutions comparable to the cosmic star formation rate.

Figure 1

Integral neutrino fluxes, $J[{\mathrm{cm}}^{-2}{sec}^{-1}{\mathrm{sr}}^{-1}]$, as a function of neutrino energy. Bold lines represent the present analytical estimates and thin lines represent corresponding predictions by the full numerical calculations [9] or the Monte-Carlo simulations [6, 15].

Figure 2

Integral neutrino fluxes with energy above 1 EeV, $J[{\mathrm{cm}}^{-2}{sec}^{-1}{\mathrm{sr}}^{-1}]$, on the plane of the source evolution parameters, $m$ and $z_{\max }$.

Figure 3

Constraints on the UHECR source evolution $m$ and $z_{\max }$ with the IceCube 2008–2009 flux limit [11] (left) and with the full IceCube five-year sensitivity [12] (middle). The areas above the solid lines are excluded by null detection of $\nu$ events. The shaded belts represent uncertainties in the present analytical estimation. The right panel shows the full IceCube five-year constraint when the emission rate per comoving volume becomes constant above $z_s$ of 1.0. Excluded region at 68% confidence level (corresponding to $\simeq 1.1$ events assuming the same background rate of the IceCube 2008–2009 measurement [11]) and 90% confidence level ($\simeq 2.2$ events) is displayed.