Strong lensing of high-energy neutrinos

We consider the effects of strong gravitational lensing by galaxy-scale deflectors on the observations of high-energy ($\mathrm{E}\gg \mathrm{GeV}$) neutrinos (HENs). For HENs at cosmological distances, the optical depth for multiple imaging is $\sim {10}^{-3}$, implying that while we do not expect any multiply imaged HENs with present samples, next-generation experiments should be able to detect the first such event. We then present the distribution of expected time delays to aid in the identification of such events, in combination with directional and energy information. In order to assist in the evaluation of HEN production mechanisms, we illustrate how lensing affects the observed number counts for a variety of intrinsic luminosity functions of the source population. Finally, we see that the lensing effects on the cosmic neutrino background flux calculation would be negligible by taking kiloparsec-scale jets as an example.

Figure 1

VDFs for various redshifts. Red, yellow, green, blue, and violet solid lines correspond to the VDFs at $z=0$, 1, 3, 6, and 10, respectively, following the evolution discussed in Sec. 2a. The gray dashed line is for the local VDF from SDSS [23].

Figure 2

Redshift dependence of optical depths for various scenarios. The green solid line denotes the optical depth for a nonevolving VDF, and the blue solid and orange dashed lines correspond to the optical depths for the VDF evolution described in Sec. 2a applied to the full redshift range and up to $z=4$, respectively.

Figure 3

Observed LFs at $z_{\mathrm{s}}=20$ for various optical depths and bright-end slopes. LFs with bright-end slopes of $\alpha =-1.5,-2,-2.5,-3.6,-5.0$, and a Schechter function are shown in red, yellow, green, blue, indigo, and violet, respectively. The curves are shifted in the $x$ direction for each of the slopes for clarity, and the choice of ${\mathrm{\Phi }}^{*},L^{*}$, and $\beta$ are arbitrary. The intrinsic LFs are shown with thick lines, and the three observed LFs with different optical depths (with ${\tau }_{\mathrm{orig}}$ indicating the optical depth at $z_{\mathrm{s}}=20$ from Sec. 2b) are shown with three corresponding lines for each intrinsic LF with increasing transparency.

Figure 4

Observed $\nu \mathrm{LF}\mathrm{s}$ for the PDE model described in Sec. 3c1. Dashed lines indicate the intrinsic LFs, while solid lines represent the observed LFs. Red, yellow, green, blue, and violet lines are for LFs at $z=0$, 1, 3, 6, and 10, respectively. The right panel is simply a magnified version.