Cosmic tau neutrino detection via Cherenkov signals from air showers from Earth-emerging taus

We perform a new, detailed calculation of the flux and energy spectrum of Earth-emerging $\tau$-leptons generated from the interactions of tau neutrinos and antineutrinos in the Earth. A layered model of the Earth is used to describe the variable density profile of the Earth. Different assumptions regarding the neutrino charged- and neutral-current cross sections as well as the $\tau$-lepton energy loss models are used to quantify their contributions to the systematic uncertainty. A baseline simulation is then used to generate the optical Cherenkov signal from upward-moving extensive air showers generated by the $\tau$-lepton decay in the atmosphere, applicable to a range of space-based instruments. We use this simulation to determine the neutrino sensitivity for $E_{\nu }\gtrsim 10\mathrm{PeV}$ for a space-based experiment with performance similar to that for the Probe of Extreme MultiMessenger Astrophysics (POEMMA) mission currently under study.

Figure 1

Geometry for detecting an EAS from an upward-moving tau neutrino. The angles ${\beta }_v$ and ${\theta }_v$ label the elevation angle and local zenith angle for the point along the line of sight. Angles ${\beta }_{\mathrm{tr}}$ and ${\theta }_{\mathrm{tr}}$ describe the emerging tau trajectory.

Figure 2

Upper panel: The column depth as a function of ${\beta }_{\mathrm{tr}}$ following the PREM parametrization (black dashed) and using the seven-shell density model (solid red) for trajectory angles $0°\le {\beta }_{\mathrm{tr}}\le 20°$. For reference, the column depths for a sphere of radius $R_E$ composed entirely of water (solid blue line) and one composed of water and rock (dotted-dashed green) are also plotted. Lower panel: The column depth for the PREM and seven-shell density models, for trajectory angles ${\beta }_{\mathrm{tr}}\le 50°$.

Figure 3

As a function of incident neutrino energy, the charged-current cross sections evaluated using ALLM and BDHM $\mathrm{small}\text{−}x$ extrapolations and using next-to-leading order QCD with a power law extrapolation at $\mathrm{small}\text{−}x$ [76].

Figure 4

As a function of incident neutrino energy, at fixed neutrino angle relative to the horizon ${\beta }_{\mathrm{tr}}=1°$, 5°, 10° and 20°, the ratio of the column depth $X({\beta }_{\mathrm{tr}})$ to the neutrino charged-current interaction length evaluated using the nCTEQ15-1 PDFs.

Figure 5

The energy loss parameters $b_{\tau }^{\mathrm{nuc}}$ and $b^{\mathrm{pair}}$ for tau energy loss in $⟨dE/dX⟩$ as a function of initial tau energy $E$. The dotted and dashed ($b_{\tau }^{\mathrm{nuc}}$), and dotted-dashed ($b_{\tau }^{\mathrm{pair}}$) lines are for standard rock, and the solid ($b_{\tau }^{\text{pair}}$) line is for water. Two separate curves are shown for photonuclear energy loss, labeled with ALLM and BDHM. The photonuclear energy loss parameters for water are approximately equal to those for rock.

Figure 6

Tau decay distribution [Eq. (7)] normalized to 1 for $\tau \to {\nu }_{\tau }X$ from pythia (histogram) and $\tau \to {\nu }_{\tau }\ell {\nu }_{\ell }$ (solid curve). The neutrino energy fraction is $y=E_{{\nu }_{\tau }}/E_{\tau }$.

Figure 7

Tau exit probability as a function of ${\beta }_{\mathrm{tr}}$ for incident $E_{{\nu }_{\tau }}={10}^7$, $10^8$, $10^9$, and $10^{10}\mathrm{GeV}$ evaluated using the ALLM photonuclear energy loss and the nCTEQ-1 neutrino cross section (${\sigma }_{\mathrm{SM}}$). The dashed lines are without regeneration, and the solid lines include regeneration.

Figure 8

Upper panel: Relative probability of a tau emerging on a trajectory at an angle of ${\beta }_{\mathrm{tr}}=1°$, 5°, 10°, and 20° through the Earth for $E_{\nu }={10}^9\mathrm{GeV}$. Lower panel: As above, with $E_{\nu }={10}^{10}\mathrm{GeV}$. The figures are plotted as a function of $z_{\tau }=E_{\tau }/E_{\nu }$.

Figure 9

Neutrino plus antineutrino flux summed over flavors, scaled by the neutrino energy squared, from Ref. [18]. Curve 1: Uniform evolution, mixed composition, $E_{p\max }={10}^{11}\mathrm{GeV}$. Curve 2: SFR1 evolution, mixed composition, $E_{p\max }={10}^{11}\mathrm{GeV}$. Curve 3: SFR1 and GRB2 evolution, protons, dip model, $E_{p\max }=10^{12.5}\mathrm{GeV}$. Curve 4: FR II evolution, protons, $E_{p\max }=10^{12.5}\mathrm{GeV}$. Curve 5: Uniform evolution, an iron rich composition, $E_{p\max }={10}^{10}\mathrm{GeV}$. Curve 6: Uniform evolution, iron, $E_{p\max }={10}^{11}\mathrm{GeV}$. See text for details.

Figure 10

Upper panel: The ratio of the outgoing tau flux to the incident neutrino flux, at the same energies, for fixed values of the angle of the trajectory relative to the horizon ${\beta }_{\mathrm{tr}}$ for cosmogenic flux 1 [18]. The ALLM tau energy loss model is used, along with the standard model neutrino cross section. The solid histograms include regeneration, while the dashed histograms do not. Lower panel: As in the upper plot, for flux 4.

Figure 11

The five lower histograms show the exiting tau flux scaled by energy as a function of tau energy for cosmogenic neutrino flux 1 [18] and for fixed values of the angle of the trajectory relative to the horizon ${\beta }_{\mathrm{tr}}$. The ALLM tau energy loss model is used, along with the standard model neutrino cross section. The uppermost histogram shows the incident tau neutrino flux scaled by a factor of $1/10$.

Figure 12

The exiting tau flux scaled by energy as a function of tau energy for flux 1 [18], for fixed values of the angle of the trajectory relative to the horizon ${\beta }_{\mathrm{tr}}$. The ALLM tau energy loss model is shown with the solid histograms, while the BDHM energy loss model is shown with the dashed histograms, in both cases with the neutrino cross section taken to be ${\sigma }_{\mathrm{SM}}$. The band shows the minimum and maximum values of the energy-scaled flux when the BDHM energy loss and neutrino cross section, as well as the ALLM energy loss and neutrino cross sections, are considered.

Figure 13

Path length $s$ at altitude $a$ given a trajectory that emerges from the surface of the Earth at angle ${\beta }_{\mathrm{tr}}$ relative to the horizon.

Figure 14

Upper panel: The tau survival probability as a function of altitude $a$ for $E_{\tau }={10}^8\mathrm{GeV}$. Lower panel: As above, for $E_{\tau }={10}^{10}\mathrm{GeV}$.

Figure 15

Upper panel: The spatial profile of the Cherenkov signal ($\mathrm{photons}/{\mathrm{m}}^2$) at 525 km altitude for a 100 PeV upward EAS with a 10° Earth-emergence angle initiated at sea level. Lower panel: The simulated Cherenkov spectrum observed at 525 km altitude for the EAS.

Figure 16

The intensity and wavelength dependence of the Cherenkov signal for 100 PeV upward-moving EASs for a 5° Earth-emergence angle as a function of EAS starting altitude.

Figure 17

The fraction of taus that decay at an altitude larger than 33 km, as a function of ${\beta }_{\mathrm{tr}}$ and ${\log }_{10}(E_{\tau }/\mathrm{GeV})$. The colored bands show 0.1 increments of the tau decay fraction.

Figure 18

Upper panel: The Cherenkov angle ${\theta }_{\mathrm{Ch}}^0$ as a function of starting altitude for a 100 PeV air shower from a tau decay from the one-dimensional Cherenkov EAS model. Lower panel: Cherenkov cone photon distribution as a function of starting altitude and Earth-emergence angle for a 100 PeV air shower from the one-dimensional Cherenkov EAS model.

Figure 19

As in Fig. 1, with an exaggerated difference between ${\theta }_v$ and ${\theta }_{\mathrm{tr}}$. Tau decays in the atmosphere outside the observation window of the detector (outside the dashed red lines) cannot be detected, as discussed in Sec. 4a.

Figure 20

Aperture as a function of tau neutrino energy for the POEMMA 360° (solid) and 30° (dashed) configurations from Monte Carlo integration of Eq. (13) with $N_{\mathrm{PE}}^{\min }=10$. The black curves show the aperture for altitudes of decay in the range 0–20 km, while the red curves restrict the altitudes of decay to 5–20 km.

Figure 21

All-flavor sensitivity scaled by neutrino energy squared, as a function of neutrino energy, assuming an operating time of 5 years and a duty cycle of 20%, for showers produced at all altitudes (black curves and markers). The solid (dashed) curves follow from Eq. (19) for $\mathrm{\Delta }\varphi =30°(360°)$. The closed markers follow from Eq. (21) with $\mathrm{\Delta }\varphi =360°$. The 90% CL upper limits from Auger [45] (scaled for sliding decade-wide neutrino energy bins), IceCube [136], and ANITA [137] are shown along with projected sensitivities of ARIANNA [138], ARA-37 [139] and GRAND10k [41], for the all-flavor limits.

Figure 22

All-flavor sensitivity scaled by neutrino energy squared, as a function of neutrino energy, assuming an operating time of 5 years and a duty cycle of 20%, for showers produced at all altitudes (black curves), as in Fig. 21. The solid (dashed) black curves follow from Eq. (19) for $\mathrm{\Delta }\varphi =30°(360°)$. Curves and bands for diffuse all-flavor neutrino fluxes are shown for newborn pulsar sources [8], AGNs [9], galactic clusters with central sources [10, 11], late flares and prompt emission from GRBs [7] and from UHECR photodisintegration within a source (labeled UFA) [12]. Observational sensitivities are shown as in Fig. 21.

Figure 23

Three-flavor sensitivity for the standard Earth density model of Table 1 for the ALLM (solid black curve) and BDHM (solid blue curve) tau energy loss with the standard model neutrino nucleon cross section. The ALLM energy loss with the outermost shell density is set to ${\rho }_{\mathrm{rock}}=2.65\mathrm{g}/{\mathrm{cm}}^3$ and is shown with the black dashed curve. The black (blue) dotted-dashed curve shows $E_{\nu }^2{F}_{\mathrm{sens}}$ using ALLM (BDHM) for both the $\tau$-lepton energy loss and ${\sigma }_{\nu N}$.

Figure 24

Comparison of the geometric aperture from the Earth cap (upper blue curve) versus that for the Earth zone (lower black curve, defined by $\mathrm{\Delta }\alpha =7°$) for ${\theta }_d=1.5°$, as a function of altitude. The inset shows the calculation on a linear scale from 0 to 35 km altitude.

Figure 25

Ratio of tau exit probability with regeneration (up to five charged-current interactions) to no regeneration (one charged-current interaction) as a function of ${\beta }_{\mathrm{tr}}$ for incident $E_{{\nu }_{\tau }}={10}^7$, $10^8$, $10^9$ and $10^{10}\mathrm{GeV}$ evaluated using the ALLM photonuclear energy loss and the nCTEQ-1 neutrino cross section (${\sigma }_{\mathrm{SM}}$).

Figure 26

For angles ${\beta }_{\mathrm{th}}=1°$, 5°, 10° and 20°, the cumulative distribution function for the relative tau exit probability for $E_{{\nu }_{\tau }}={10}^7\mathrm{GeV}$, as a function of $z=E_{\tau }/E_{{\nu }_{\tau }}$. The ALLM $\mathrm{small}\text{−}x$ extrapolation of the electromagnetic structure function in $b_{\tau }^{\mathrm{nuc}}$ has been used.

Figure 27

For angles ${\beta }_{\mathrm{th}}=1°$, 5°, 10° and 20°, the cumulative distribution function for the relative tau exit probability for $E_{{\nu }_{\tau }}={10}^8\mathrm{GeV}$, as a function of $z=E_{\tau }/E_{{\nu }_{\tau }}$. The ALLM $\mathrm{small}\text{−}x$ extrapolation of the electromagnetic structure function in $b_{\tau }^{\mathrm{nuc}}$ has been used.

Figure 28

For angles ${\beta }_{\mathrm{th}}=1°$, 5°, 10° and 20°, the cumulative distribution function for the relative tau exit probability for $E_{{\nu }_{\tau }}={10}^9\mathrm{GeV}$, as a function of $z=E_{\tau }/E_{{\nu }_{\tau }}$. The ALLM $\mathrm{small}\text{−}x$ extrapolation of the electromagnetic structure function in $b_{\tau }^{\mathrm{nuc}}$ has been used.

Figure 29

For angles ${\beta }_{\mathrm{th}}=1°$, 5°, 10° and 20°, the cumulative distribution function for the relative tau exit probability for $E_{{\nu }_{\tau }}={10}^{10}\mathrm{GeV}$, as a function of $z=E_{\tau }/E_{{\nu }_{\tau }}$. The ALLM $\mathrm{small}\text{−}x$ extrapolation of the electromagnetic structure function in $b_{\tau }^{\mathrm{nuc}}$ has been used.

Figure 30

The frequency as a function of the ratio of $E_{\mathrm{shr}}/E_{\tau }$ for tau decays from a pythia 8 simulation.