Memorandum on the relation between MC
weight, the effective area, and the sensitivity
Shigeru Yoshida
September 1, 2005

log(dN/dE 2E[GeV cm−2sec−1sr−1 ])85 Degree
70 Degree−8ντ−10

νe−12νµτµ−14106

1081010

1012

Energy [GeV]
Figure 1: Fluxes of the EHE particles at the IceCube depth for a scenarioof the neutrino production by the GZK mechanism. Two cases in the nadir
angle are shown in the figure.
Expected event rate Nev from a primary neutrino flux Jν at the earth
surface can be obtained bydNev= Aef f (log El , Ω)d log El dΩZd log Eν

log EldNν→ldJν
(log Eν , log El , Ω)(log Eν , Ω),d log El
d log Eν dΩ(1)where El is energy of the secondary lepton such as µ, Eν is primary energy of neutrinos, Nν→l is number of secondary leptons produced inside theearth and reaching to the IceCube volume, and Aef f is the effective area of
the IceCube observatory. The integral in the equation above accounts forthe propagation effect in the earth and obtained by resolving the relevanttransport 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.
The effective area Aef f in Eq. 1 can be estimated by either the semi1

analytical way [1] or the full-brown MC. The semi-analytical method gives
Aef f asAef f (log El , Ω) =Z

d log Edepth '10PeVlog EdepdNdep(log El , log Edep , Ω)A0 (2)d log Edepwhere Edep is energy deposit of the secondary lepton propagating over 1km
inside the IceCube volume. In EHE the energy deposit takes place mostly inform of a bunch of cascades. A0 is typically 1 km2 for the IceCube.
By the full MC, the effective area will be given by
Aef f (log El , Ω) =A0 dN M C
ld log El= A0detected
XN detected(log El , Ω)∆ log El
1dNlM C

(log El , Ω)∆ log El
d log El,(3)where N detected is number of MC events passing your criteria, dNlM C /d log Elis the MC primary particle spectrum of the secondary leptons (mostly µ andτ for EHE). A0 is the area of throwing primary particles in MC. ∆ log El isa bin width. Putting Equations [1] and [3] together will lead to the proper
event weight asw = A01

ZdNlM Cd log El (log El , Ω)∆ log El log El

d log EνdJνdNν→l(log Eν , log El , Ω).
d log Eld log Eν dΩ(4)Note that Jν could be given by the GZK neutrino model, or any other EHEneutrino 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 independentway. This approach has been widely used in many other experiments (forexample, see [2]). The neutrino flux upper-bound with energy of E0 from
non-existence of signals is evaluated by putting"#dJνdJν
(log Eν , Ω) =(log E0 , Ω) δ(log Eν − log E0 )d log Eν dΩd log Eν dΩ2(5)into Eq. 1. We getdJν(log E0 , Ω) = Rd log Eν dΩdΩRlog ElthN ev (≥ log Elth )95%
ν→ld log El Aef f (log El , Ω) ddN
log El (log E0 , log El , Ω)
(6)where Elth is threshold energy of the secondary leptons, N ev (≥ log Elth )95%
is number of upper bound events with 95 % C.L. The Poisson distribution
gives 2.3 events for example.

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Bibliography[1] S. Yoshida, R. Ishibashi, H. Miyamoto Phys. Rev. D 69 103004 (2004).
[2] X. Bertou et.al., Astroparticle Physics 17 183-193 (2002).

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