Memorandum on the relation between MC weight, the effective area, and the sensitivity

Shigeru Yoshida

Expected event rate $N_{ev}$ from a primary neutrino flux $J_\nu$ at the earth surface can be obtained by

$${dN_{ev}\over d\log{E_{l}}d\Omega} = A^{eff}(\log{E_l}, \O... ...mega) {dJ_\nu \over d\log{E_\nu}d\Omega}(\log{E_\nu}, \Omega),$$ (1)

where $E_l$ is energy of the secondary lepton such as $\mu$, $E_{\nu}$ is primary energy of neutrinos, $N_{\nu\to l}$ is number of secondary leptons produced inside the earth and reaching to the IceCube volume, and $A^{eff}$ is the effective area of the IceCube observatory. The integral in the equation above accounts for the propagation effect in the earth and obtained by resolving the relevant transport equation [1]. This is equivalent to the secondary lepton flux at the IceCube depth and pre-calculated by JULIeT class like PropagatingNeutrinoFlux.java or PropagatingAtmMuonFlux.java. An example is shown in Figure 1.

Figure 1: Fluxes of the EHE particles at the IceCube depth for a scenario of the neutrino production by the GZK mechanism. Two cases in the nadir angle are shown in the figure.

The effective area $A^{eff}$ in Eq. 1 can be estimated by either the semi-analytical way [1] or the full-brown MC. The semi-analytical method gives $A^{eff}$ as

$$A^{eff}(\log{E_l}, \Omega) = \int\limits_{\log{E_{dep}^{th}... ...ep}\over d\log{E_{dep}}}(\log{E_l}, \log{E_{dep}}, \Omega) A_0$$ (2)

where $E_{dep}$ is energy deposit of the secondary lepton propagating over 1km inside the IceCube volume. In EHE the energy deposit takes place mostly in form of a bunch of cascades. $A_0$ is typically 1 km$^2$ for the IceCube.

By the full MC, the effective area will be given by

$$A^{eff}(\log{E_l}, \Omega) = \ A_0 {N^{detected}\over \ {dN_l^{MC}\over d\log{E_l}}(\log{E_l},\Omega)\Delta\log{E_l}}$$

$$= A_0 \sum\limits^{detected} \ {1\over {dN_l^{MC}\over d\log{E_l}}(\log{E_l},\Omega)\Delta\log{E_l}},$$ (3)

where $N^{detected}$ is number of MC events passing your criteria, $dN_l^{MC}/d\log{E_l}$ is the MC primary particle spectrum of the secondary leptons (mostly $\mu$ and $\tau$ for EHE). $A_0$ is the area of throwing primary particles in MC. $\Delta\log{E_l}$ is a bin width. Putting Equations [1] and [3] together will lead to the proper event weight as

$$w = A_0 {1\over {dN_l^{MC}\over d\log{E_l}}(\log{E_l},\Omega... ...E_\nu}, \log{E_l}, \Omega) {dJ_\nu \over d\log{E_\nu}d\Omega}.$$ (4)

Note that $J_\nu$ could be given by the GZK neutrino model, or any other EHE neutrino models as well as by the atmospheric muon/neutrino background prediction. So the weight given above could be not unique but various depending on what models you consider.

In this context the IceCube ``sensitivity'' for EHE neutrinos can be obtained from a quasi-differential event rate in neutrino model independent way. This approach has been widely used in many other experiments (for example, see [2]). The neutrino flux upper-bound with energy of $E_0$ from non-existence of signals is evaluated by putting

$${dJ_\nu \over d\log{E_\nu}d\Omega}(\log{E_\nu}, \Omega) =\le... ...mega}(\log{E_0}, \Omega)}\right] \delta(\log{E_\nu}-\log{E_0})$$ (5)

into Eq. 1. We get

$$\left[{{dJ_\nu \over d\log{E_\nu}d\Omega}(\log{E_0}, \Omega)... ...{dN_{\nu\to l}\over d\log{E_l}}(\log{E_0}, \log{E_l}, \Omega)}$$ (6)

where $E_l^{th}$ is threshold energy of the secondary leptons, $N^{ev}(\geq \log{E_l^{th}})^{95 \%}$ is number of upper bound events with 95 % C.L. The Poisson distribution gives 2.3 events for example.