How bright can the brightest neutrino source be?

After the discovery of extraterrestrial high-energy neutrinos, the next major goal of neutrino telescopes will be identifying astrophysical objects that produce them. The flux of the brightest source $F_{\max }$, however, cannot be probed by studying the diffuse neutrino intensity. We aim at constraining $F_{\max }$ by adopting a broken power-law flux distribution, a hypothesis supported by observed properties of any generic astrophysical sources. The first estimate of $F_{\max }$ comes from the fact that we can only observe one universe and, hence, the expected number of sources above $F_{\max }$ cannot be too small compared with one. For abundant source classes such as starburst galaxies, this one-source constraint yields a value of $F_{\max }$ that is an order of magnitude lower than the current upper limits from point-source searches. Then we derive upper limits on $F_{\max }$ assuming that the angular power spectrum is still consistent with neutrino shot noise. We find that the limits obtained with upgoing muon neutrinos in IceCube can already be quite competitive, especially for rare but bright source populations such as blazars. The limits will improve nearly quadratically with exposure and, therefore, be even more powerful for the next generation of neutrino telescopes.

Figure 1

The source flux distribution $d{N}_s/dF$ multiplied by $F$ (top), $F^2$ (middle), and $F^3$ (bottom), for $2<\alpha <3$ and $1<\beta <2$. Both horizontal and vertical axes are in logarithmic scales. The shaded regions in the middle and bottom panels represent that areas below these broken lines correspond to the intensity $I_{\nu }$ [Eq. (7)] and the Poisson angular power spectrum $C_{\nu }^P$ [Eq. (8)], respectively; i.e., $I_{\nu }$ and $C_{\nu }^P$ are dominated by sources near $F_{*}$ and $F_{\max }$, respectively.

Figure 2

One-source upper limits (90% CL) on the neutrino flux per flavor from the brightest neutrino source, as a function of the characteristic source number $N_{*}$, for various values of $\alpha$ and $\beta .F_{\max }^{1\mathrm{s}}$ is defined from Eq. (10) as the flux for which there is a 90% probability of not finding any brighter source (solid and dotted). The blue band represents the region where the brightest source is located at 90% CL for given $N_{*}$, in the Euclidean case with $(\alpha ,\beta )=(2.5,1.5)$. The dashed horizontal line represents the upper limit from the search for pointlike source in Ref. [4] toward the South Pole (see also Appendix pp1). The orange band shows the characteristic flux $F_{*}$ of a single source required for the population from which it is drawn to explain the observed intensity $I_{\nu }$ according to Eq. (7).

Figure 3

Upper limits (90% CL) on the flux (per flavor) of the brightest source from the angular power spectrum, $F_{\max }^{\mathrm{APS}}$, as a function of the characteristic source number $N_{*}$ by using the HESE data set. The color code and line style are the same as in Fig. 2. Only the regions where $F_{\max }^{\mathrm{APS}}>F_{*}$ are valid as upper limits.

Figure 4

Upper limits (90% CL) on the flux (per flavor) of the brightest source from the angular power spectrum, $F_{\max }^{\mathrm{APS}}$, as a function of the characteristic source number $N_{*}$ by using the upgoing ${\nu }_{\mu }$ events above 300 TeV and assuming the current IceCube exposure [6]. The color code and line style are the same as in Fig. 2. Only the regions where $F_{\max }^{\mathrm{APS}}>F_{*}$ are valid as upper limits. The pink bank represents the region where the brightest source is located at 90% CL for given $N_{*}$, in the Euclidean case with $(\alpha ,\beta )=(2.5,1.5)$. The purple square, blue diamond, and green star are located at the expected neutrino flux for Mkn 412, Cen A and M82 or NGC 253, for values of $N_{*}$ typical of blazars, radio galaxies, and starburst galaxies, respectively (see text for details).

Figure 5

Projected 90% CL upper limits from angular power spectrum (solid) and one-source limits (dotted) as a function of $N_{*}$, for contained track events, assuming $(\alpha ,\beta )=(2.5,1.5)$. Projections for both KM3NeT and IceCube-Gen2, being coincidently the same, are shown as a solid line. The dashed horizontal line represents the upper limit from the search for pointlike sources [4] after scaled down by a factor of $1/\sqrt{10}$. The dotted line represents the 90% CL one-source upper limits, and the red region shows where the flux of the brightest source is located at 90% CL in the case of Euclidean sources.

Figure 6

Projected 90% CL upper limits from angular power spectrum as a function of $N_{*}$, if the high-energy neutrino sky remains isotropic after using detectors similar to KM3NeT (solid) and IceCube-Gen2 (dot-dashed), assuming $(\alpha ,\beta )=(2.5,1.5)$. The dotted is for the current IceCube configuration as in Fig. 4. See Table 1 for detector configurations. The dashed horizontal line represents the upper limit from the search for pointlike sources [4] after scaled down by a factor of $1/\sqrt{10}$, and the red region shows where the flux of the brightest source is located at 90% CL in the case of Euclidean sources.

Figure 7

Flux distribution $F_{\nu }d{N}_s/d{F}_{\nu }$ for the BL Lac model with $L_{\nu }\propto L_{\gamma }^2$ [38] and the FSRQ model with $L_{\nu }\propto L_{\gamma }^{1.5}$ [39], compared with the 2FHL distribution [37] assuming linear scaling, $L_{\nu }\propto L_{\gamma }$.

Figure 8

The same as the middle and bottom panels of Fig. 1, but for $\alpha <2$.

Figure 9

Lower limits on $N_{F_{\max }}$ (top) and upper limits on $F_{\max }$ (bottom) as a function of exposure normalized to that of 4-year IceCube ${\mathcal{E}}_{\mathrm{IC}4\mathrm{y}}$, from the angular power spectrum measurements, in the case of $\alpha =1.5$. Solid and dotted lines correspond to angular resolutions of ${\theta }_{\mathrm{psf}}=1°$ and 0.5°, respectively. The dashed line in the bottom panel is the upper limits from the point-source searches (Appendix pp1), extrapolated as ${\mathcal{E}}^{-1/2}$.

Figure 10

The same as Fig. 9 but for upgoing ${\nu }_{\mu }$ tracks above 300 TeV, for which ${\theta }_{\mathrm{psf}}=0.5°$ (solid) and 0.3° (dotted).