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\begin{document}

\title{Search for ultra-high-energy cosmic neutrinos with the IceCube neutrino observatory}

\classification{98.70.Sa, 95.85.Ry, 95.55.Vj}

\keywords      {cosmic-rays, neutrinos}

\author{Shigeru Yoshida}{
  address={Department of Physics, Graduate School of Science, Chiba University, Chiba 263-8522, Japan}
}

%\author{&lt;author2&gt;}{
%  address={&lt;common address for author2 and author3&gt;}
%}

%\author{&lt;author3&gt;}{
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%  ,altaddress={&lt;author1 address&gt;} % additional visiting address
%}


\begin{abstract}
A search for extremely-high-energy (EHE) neutrinos with energies greater than $10^6$~GeV
has been performed using the data taken with the IceCube detector at the South Pole
between May 2010 and May 2012. Two neutrino-induced cascade events are 
observed passing all the signal search criteria
over an expected background of 0.06 events. Energies of the observed two events are estimated to 
be in the range of $10^{6} \sim 10^7$~GeV.
We find no events with energies above $10^9$~GeV in the sample of accumulated effective 
all-flavor-sum neutrino detection exposure of $\geq 4.8\times 10^{16}$ cm$^2$ sec sr. 
An upper limit on the neutrino flux in the energy range above $10^6$~GeV is obtained from the observations.
The present IceCube data disfavors strong cosmological evolutions of 
ultra-high-energy cosmic-ray sources such as AGNs associated with radio-loud jets.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% MAINMATTER
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%===============================================================
\section{Introduction}
\label{Introduction}%
%===============================================================

The origin of the ultra-high-energy cosmic-rays (UHECRs) has been a long-standing
important question in astrophysics.
%
Neutrinos, secondaries produced by UHECR nucleons, 
are expected to provide the direct information on the UHECR origin, since a neutrino 
penetrates over cosmological distance without being deflected
by cosmic magnetic field nor absorbed by the photon field.

In addition to neutrinos directly produced from cosmic-ray sources, 
secondary produced neutrinos in the 
propagation of ultra-high-energy cosmic-rays (UHECRs) with energies reaching up
to $\sim 10^{20}\ {\rm eV}$ are highly expected. 
These ``cosmogenic'' neutrinos are produced 
by the Greisen-Zatsepin-Kuzimin (GZK) mechanism via
interactions of UHECRs with the CMB and extragalactic background light 
(infra-red, optical, and ultra violet)~~\cite{greisen,zatsepin,Beresinsky:1969qj}. 
A measurement of the cosmogenic neutrinos probes the yet-unknown UHECR origin
because the spectral shapes and flux levels are sensitive to the redshift 
dependence of UHECR source distributions and 
cosmic-ray primary compositions~\cite{yoshida2012,ahlers2009}. 
The main energy-emission range of the GZK cosmogenic neutrinos
is predicted to be around EeV ($10^{18}$~eV)~\cite{yoshida93,ESS}. 
The cosmogenic production has been considered as a main source to emit extremely-high-energy (EHE)
cosmic neutrinos.

A measurement of these EHE neutrinos requires a huge detection volume
as their fluxes are expected to be extremely low. 
The IceCube Neutrino Observatory~\cite{icecube} at the geographical South Pole
is the first cubic-kilometer scale Cherenkov neutrino 
detector now in full operation. Its large instrumentation volume
as well as the omni-directional neutrino-detection capability
have extended the sensitivity into energy regimes even beyond EeV
by implementing the diffuse neutrino search technique.
We present the EHE neutrino search which is extended to the data obtained from 
May 2010 to May 2012. The basic search strategies are similar 
to the previous searches~\cite{icecubeEHE2010, icecubeEHE2011}
and established to be sensitive to all three flavors of neutrinos.
The main improvement comes from the detector enlargement, 
statistical enhancement of the data as well as the improvement 
in the understanding of the deep glacial ice
and resultant background and signal Monte Carlo simulations. 
This improvement allows refined
geometrical reconstruction of the events and thus a better background rejection is established.

%===============================================================
\section{The IceCube Detector}\label{sec:detector}
%===============================================================

IceCube is a detector to record Cherenkov light from relativistic 
charged particles induced in neutrino interactions with an array 
of Digital Optical Modules (DOMs).
Each DOM encloses a 10'' R7081-02 photomultiplier tube (PMT)~\cite{pmt} 
in a transparent pressure sphere along with the high 
voltage system, a LED flasher board for optical calibration in ice, 
and a digital readout board~\cite{digitizer}. 
These DOMs are deployed along electrical cable bundles that carry power and information 
between the DOMs and surface electronics.
The cable assemblies, often called strings, were lowered into holes drilled to a depth of 
2450~m with the horizontal spacing at approximately 125~m.
%
The DOMs occupy the depth from 1450~m to 2450~m at intervals of 17~m where the glacial 
ice is transparent.
%The PMT waveforms are recorded when satisfies the hard local coincident (HLC) condition 
%of the 
%nearest or next-to-nearest DOMs 
%observed photon hits within 1~$\mu$s. 
Then the events are considered to be triggered when 
eight DOMs recorded the signals 
within 5~$\mu$s (Simple Majority Trigger).

%===============================================================
\section{Experimental Data and Simulation}\label{sec:data}
%===============================================================
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
From May 31, 2010 to May 12, 2011, IceCube with 79 strings (IC79) was in operation, 
and the first year IceCube full configuration with 86 strings (IC86) took data from 
May 13, 2011 to May 15, 2012.
In the present analysis, we use the data in these periods.
The corresponding effective livetimes for 2010-2011 and 2011-2012 runs are 
319.18~days and 350.91~days respectively excluding the period of detector calibration 
and unstable operation.

%Online filtering
The total number of photo-electrons (NPE) in each event is calculated for the event 
fulfilling the trigger condition at the surface computer farm at the IceCube site.
The extremely-high energy (EHE) on-line filter sample used in the current analysis is
sent to the North via 
satellite when the NPE of the event is larger than 1000. A total of 4.5$\times 10^7$ and 6.3$\times 10^7$ 
EHE sample 
events were collected with IC79 and IC86 respectively. 
%blinding sample
For a blind analysis we used approximately 10\% of the total experimental sample 
randomly selected throughout the data-taking periods
for 
the simulation verification. These test samples correspond to 33.4 and 20.8~days for IC79 and IC86 
respectively. 
Final selection criteria are applied on the retained signal search sample of 285.8~days for IC79 and 
330.1~days for IC86 only after the selection is fixed to minimize a selection bias.

There are two classes of major background in this analysis. 
The dominant class by event number is that containing muon bundles, made up of a large 
number of muons produced by high energy cosmic-ray interactions in the atmosphere. 
The other class is that containing atmospheric neutrinos. The muon bundle was simulated with the CORSIKA 
air-shower simulation~\cite{Corsika} with the SIBYLL 2.1 hadronic interaction models~\cite{Ahn:2009wx} 
with the pure-iron primary-cosmic-ray composition.
For the atmospheric neutrinos, the All Neutrino Interaction 
Simulation (ANIS) package~\cite{anis} is used to simulate each neutrino flavor 
separately between
$10^{1.7}$ and $10^9$~GeV. 
Cosmogenic neutrino signals were also similarly simulated with the JULIeT package~\cite{juliet} 
in the energy range of $10^5$ and $10^{11}$~GeV
as with previous analyses~\cite{icecubeEHE2010, icecubeEHE2011}
except the improved optical ice properties.

%===============================================================
\section{Data process}\label{sec:process}
%===============================================================

After the initial hit cleaning to remove the coincident atmospheric muon
background, 
events were first required to have at least 300 hit DOMs and 3200~p.e.
In this analysis-level sample, a total of $5.0\times 10^5$ and $6.3\times 10^5$ 
events are retained for IC79 and IC86 respectively. 
At this analysis level, 
hit informations from DeepCore strings are also discarded to keep 
DOM separation uniformity across $\sim$1~km$^3$ detector volume.

%%%%%%%%%%%%                                                                  
%  TABLE1  %                                                                  
%%%%%%%%%%%%                                                                  
\begin{table}[tb]
\caption{
  Number of events passing cuts at on-line filtering, off-line analysis, and final level with
  350.9~days of total detector livetime for IC86. 
  The signal rates correspond to simulations of the GZK neutrino
  model~\cite{yoshida93}. Analysis sample requests the number of hit DOMs $\geq$ 300, 
  log$_{10}$ (NPE) $\geq$ 3.5.
  Preliminary systematic uncertainties in the expected
  event rates at the final selection level are given as asymmetric errors.
  The IC79 samples exhibited the similar numbers
  except no event found at the final level.
  \label{table:level2}}
{\small
  \begin{tabular}{lrrr}
    {\footnotesize Contributions}
      &amp;  {\footnotesize Experimantal}
      &amp;  {\footnotesize Bg MC}
      &amp;  {\footnotesize Sig MC~\cite{yoshida93}} \\

   \hline
   {\footnotesize {\it EHE filter level }}
   &amp; $6.3\times10^7$  &amp;  $9.5\times10^7$  &amp;  2.6  \\
   {\footnotesize {\it Analysis level}} &amp; $6.3\times10^5$  &amp;  $1.3 \times10^6$  &amp; 1.9 \\
   {\footnotesize {\it Final level}} &amp;  2 &amp; {\footnotesize 0.018$^{+0.013}_{-0.013}$ }
    &amp; {\footnotesize 1.109$^{+0.154}_{-0.133}$}\\
  \end{tabular}
}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[tb]
  \caption{Event number distributions in the plane of NPE and energies of neutrino-induced muons (left) 
    and neutrinos which induce cascades (right) of the analysis level of IC86 signal Monte Carlo simulations.
    The particle-energy distribution is assumed to follow $E^{-1}$ in these plots for illustrative purposes.
    The muon and neutrino energies are given when a particle intersects a radius of 880~m from the IceCube center. 
    Cascade events
    include all-flavor neutral current and $\nu_e$ charged current interactions.} \label{fig:energyNPE}
  \includegraphics[width=1.50in]{figure/NPEVsEnergyMuE1.pdf}
  \includegraphics[width=1.50in]{figure/NPEVsEnergyCascadeE1.pdf}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%T
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure*}[tbh]
  \includegraphics[width=1.85in]{figure/AnalysisLevelIC86NPE.pdf}
  \includegraphics[width=1.85in]{figure/AnalysisLevelIC86CosTheta.pdf}
  \caption{Event-number distributions are shown as functions of NPE 
    (left panels) and reconstructed zenith angle (right panels). 
    The experimental test sample, expected backgrounds from atmospheric neutrinos and muons, 
    and signals from various cosmogenic (GZK) neutrino models~\cite{yoshida93, ahlers2010, kotera2010} 
    are shown for the IC86 data sample. Event numbers presented here are 
    for the livetime of the test samples of the experimental data (20.8 days).
    The signal distributions are adding all three flavors of neutrinos. 
    The background sum includes atmospheric muons and conventional atmospheric neutrinos. 
    Each component of the background (single and coincident atmospheric muons, conventional 
    and prompt atmospheric neutrinos) is also shown separately.}
  \label{fig:analysislevel}
\end{figure*}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%T

As the energy spectrum of background atmospheric muons and neutrinos falls steeply with energy,
cosmogenic neutrino fluxes dominate over background in the high-energy region. 
%
Energy is inferred from NPE in the current analysis.
%The events induced by these extremely-high energy neutrinos, therefore, stand out
%against the background of cosmic-ray induced muon events due to the much higher Cherenkov light
%deposition measured by IceCube, because the amount of energy deposit, {\it i.e.}, visible energy 
%within/near the instrumentation volume is highly correlated with the energy of the incoming particles. 
%
Figure~\ref{fig:energyNPE} presents the simulated analysis-level distribution of NPE recorded by IC86 as a 
function of the Monte-Carlo-truth energies of in-coming particles.
A clear correlation between NPE and the energy of particles measured near IceCube
is observed for muon track events. 
The energies are sampled at a radius of 880~m from the IceCube center.
For the cascade events, events with the cascade vertex position
outside the IceCube detector volume weaken the correlation.

Figure~\ref{fig:analysislevel} displays the NPE and reconstructed zenith-angle distributions
from data and simulations at the analysis level.
It can be seen that the signal cosmogenic neutrino distributions dominate over the 
atmospheric $\mu$ and $\nu$ background distributions in the high NPE region.
The zenith angles are reconstructed by 
fitting of the first photon hit time by the muon-track hypothesis.
Although the detailed algorithms are different for the analysis between
IC79 and IC86, the resultant anglar resolution is approximately $\sim 1$ degree
or less, 
which is sufficient to remove atmospheric muon-bundle background events in the current analysis.
This is clearly visible in the right panel of FIG.~\ref{fig:analysislevel}.
Table~\ref{table:level2} summarizes the number of events retained in the on-line EHE filter, analysis level sample and the final level selections.

%===============================================================
\section{Analysis}\label{sec:analysis}
%===============================================================
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[bt]
%\begin{center}
  \includegraphics[width=1.7in]{figure/AnalysisLevel2D_IC86Data.pdf}
  \includegraphics[width=1.7in]{figure/AnalysisLevel2D_IC86AtmMuSingleCoinc.pdf}
  \includegraphics[width=1.7in]{figure/AnalysisLevel2D_IC86AtmNu.pdf}
  \includegraphics[width=1.7in]{figure/AnalysisLevel2D_IC86AllSignal.pdf}
%\end{center}
  \caption{Event-number distributions on the plane of NPE and reconstructed zenith angle 
for the IC86 run.  The experimental 
test samples are shown in left panels. The atmospheric muon (middle left panels) and the 
conventional atmospheric neutrino background (middle right panels), and signal 
cosmogenic neutrino model \cite{yoshida93} Monte Carlo simulations (right panels) are 
also shown. The colors indicate event numbers per 
livetime of 20.8~days for the IC86 test sample.
The signal distributions sum all three flavors of neutrinos. The background 
distributions include atmospheric muons from the CORSIKA package with SIBYLL high-energy 
interaction model and conventional atmospheric neutrinos. The solid lines in each panel 
indicate the final selection criteria.}
\label{fig:final}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[bth]
%  \begin{center}
    \includegraphics[height=2.1in, width=2.1in]{figure/EffectiveAreaEachFlavor_IC79.pdf}
    \includegraphics[height=2.1in, width=2.1in]{figure/EffectiveAreaEachFlavor_IC86.pdf}
    \includegraphics[height=2.1in, width=2.1in]{figure/EffectiveArea3FlavorSum_3years.pdf}
%  \end{center}
  \caption{The IceCube neutrino effective areas with different 
    string configurations, IC79 and IC86 (middle panel) for each 
    neutrino flavor, averaged over 4$\pi$ solid angle. Three-flavor sums of the effective areas 
    are are shown in the right panel.
    The effective area from the previous search~\cite{icecubeEHE2011} with 40 string configuration of IceCube 
    (IC40) are also shown for comparison. 
    Exposure of the sample used in this analysis is obtained by multiplying 
    the effective area with the livetime (369.3~days, 319.2~days and 350.9~days for 
    IC40, IC79, and IC86 respectively) and 4$\pi$ solid angle. 
    The sharp peaked 
    structure at 6.3~PeV for electron neutrinos is due to the Glashow resonance \cite{glashow}.
}
  \label{fig:effarea}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The analysis was optimized based on simulation samples of background and signal after 
simulation verification using the test sample.
We have performed a blind analysis which is constructed without 
looking at $\sim$90\% of the total sample where the signal event candidate is searched. 
Selection criteria are obtained by optimizing the NPE threshold values in each of the IC79 and IC86 samples
to minimize the model discovery factor \cite{icecubeEHE2011, MRF}.
In Fig.~\ref{fig:final}, the event-number distributions of the Monte Carlo simulations and 
experimental data corresponding to the 10\% test sample livetime of 
20.8~days for the IC86 run are shown. 
The signal distributions add all three flavors of neutrinos.
The solid lines in Fig.~\ref{fig:final} indicates the final criteria for each sample. 
The events above the criteria lines are considered to be the final sample.
A similar criteria is also applied to the IC79 data.
A large NPE threshold is required to eliminate 
the dominant atmospheric muon background for the downward-going region.
On the other hand, the NPE threshold value is given as a constant value
in the zenith angle region of cos $\theta$ $\leq$ ~$\sim$0.1. 
This implies that zenith angle reconstruction of events is mainly needed for the rejection of atmospheric muon background.
As one can see from the middle panels in Fig.~\ref{fig:analysislevel}, atmospheric and GZK neutrinos 
can be separated with a zenith-angle-independent NPE threshold value.
After removing the atmospheric muon background, 
a majority of signal neutrinos is thus selected by the fixed NPE thresholds
regardless of the reconstructed zenith angles. The criteria do not depend
on event topologies, either.
These features ensure stability of signal selection performance against mis-reconstruction of 
neutrino-induced event geometries.
%
The preliminary final level background event rate of atmospheric muons and conventional atmospheric 
neutrinos is 0.06~events in the 615.9~day effective period of signal search sample.
The background with prompt atmospheric neutrinos is 0.19 events 
assuming the nominal prompt neutrino flux from Ref.~\cite{prompt_enberg}. 

The effective neutrino-detection areas of the analysis 
with the different IceCube detector configurations are shown in 
Fig.~\ref{fig:effarea}.
The effective areas are given for each neutrino flavor, averaged over 
4$\pi$ solid angle for IC79 and IC86, assuming an equal flux of 
neutrinos and anti-neutrinos.
Below 5~PeV, the effective area of electron neutrinos exceeds 
that of muon or tau neutrinos. 
This reflects the fact that muon and tau tracks from 
muon/tau neutrinos partially deposit their energies into the detector 
volume while the cascades induced by charged current interactions of electron neutrinos 
deposit 100\% of their energy. Therefore, even though tracks have a longer propagation 
length, they less-frequently satisfy the selection criteria (Fig.~\ref{fig:final}) 
in the energy region, in contrast to the cascade events induced within the detector.
%
The right panel in Fig.~\ref{fig:effarea} shows the three-neutrino-flavor sum of 
effective areas from IC79 and IC86 as well as with IC40 from 
the previous analysis \cite{icecubeEHE2011}. The current analysis has approximately a factor 
of two enlarged effective area from IC40.
The difference between IC79 and IC86 effective 
areas below $3 \times 10^{7}$~GeV comes from the NPE threshold difference for 
each sample. 
%The slight difference above $3 \times 10^{9}$~GeV is due to rLLH cut in IC79.

%=====================================================
%=====================================================
\section{\label{sec:Resuts} Results}
%=====================================================
%=====================================================

Two events are observed in the final data sample
passing the final selection criteria.
One event (Event~118545) is recorded in August 8th, 2011
with the NPE value of $7.0\times 10^4$~p.e.
The second event (Event~119316) is from January 3rd, 2012 with
the NPE value of $9.6\times 10^4$~p.e.
The expected number of conventional background events in the period is $0.060\pm0.004$(stat.)$^{+0.040}_{-0.050}$ (syst.),
and $0.191\pm0.004$(stat.)$^{+0.078}_{-0.098}$ (syst.) with the nominal value prompt atmospheric neutrinos~\cite{prompt_enberg}.
%
The {\it preliminary} p-value of observing two events for
the conventional background-only hypotheses is 
$1.75\times10^{-3}$, equivalent to a 2.9$\sigma$ deviation,
and $1.54\times10^{-2}$ (2.2$\sigma$ equivalence)
when including the nominal prompt atmospheric neutrinos.
The systematic errors arise mainly from uncertainties
on the absolute detection efficiency, and the composition and intensity
of cosmic-rays.
Event displays of the observed two events are shown in Fig.~\ref{fig:eventview}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
\begin{figure}[tb]
%\begin{center}
	\includegraphics[width=1.55in, height=1.55in]{figure/Aug_EventView.pdf}
	\includegraphics[width=1.55in, height=1.55in]{figure/EventViewJan2012_3.pdf}
	\caption{The event displays of the two observed events from August 2011 (left panel) and January 2012 (right panel). Each spheres represent a DOM. Colors indicate the arrival timing of the photon (red$=$earliest, blue$=$latest). The size of the sphere indicates the measured amount of photons in each DOM.}
	\label{fig:eventview}
%\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5
%
Both events are from the IC86 sample and have a characteristic spherical photon distribution.
These distributions are consistent with Cherenkov photons from particle cascades induced 
by neutrino interactions within the IceCube detector without outgoing muon or tau tracks. 
%
It is likely that these events are induced by either electron neutrinos of any interaction 
or neutral-current interactions of muon or tau neutrinos. 

The two events are then reconstructed using maximum-likelihood fits with the cascade hypothesis, 
while only the track-event hypothesis was used in the analysis level.
The reconstructed energy deposits of these two
cascade events are 1.1 and 1.3~PeV with the reconstruction uncertainties of $\sim$35\% 
including statistical uncertainty and systematic uncertainties associated with the ice properties 
and the absolute DOM sensitivity. 
These energy deposits correspond to the incoming neutrino energies
if each cascade is the result of a charged-current 
interaction of an electron neutrino where 100\% of the neutrino energy is deposited near the interaction vertex. 
%
Including the hypothesis of neutral-current interaction of three flavors of neutrinos, 
the statistical error, and 
the systematic error, the 90\% most probable neutrino energies of two events 
at the earth surface correspond to 
780~TeV-5.6~PeV and 890~TeV-8.5~PeV, respectively, 
assuming that the surface neutrino spectrum follows an $E^{-2}$ power law.

Figure~\ref{fig:npe} shows the NPE distributions from two years of data 
passing the final selection criteria
for experimental data, signal and background Monte Carlo simulations.  
Signal models include three 
cosmogenic neutrino models with different cosmic-ray source distributions or spectra. 
%==============================================================
\begin{figure}[bt]
%\begin{center}
	\includegraphics[height=3.0in]{figure/FinalLevelIC86IC79NPE.pdf}
	\caption{
          Event distributions from the 2010-2012 sample of 615.9~days of livetime 
          passing the final selection criteria as a function of ${\rm log_{10} NPE}$. 
          Black marks indicate experimental data. 
          Purple (Kotera {\it et al.}~FRII ~\cite{kotera2010}), 
          red (Yoshida {\it et al.}~\cite{yoshida93}) 
          and orange (Ahlers {\it et al.}~\cite{ahlers2010}) 
          solid lines indicate cosmogenic neutrino models with different assumed parameters.
          The solid green line is the nominal prompt atmospheric neutrino distribution~\cite{prompt_enberg},
          and the solid blue line is the conventional atmospheric muon and neutrino background sum. Dotted green and blue lines indicates the contribution from the conventional atmospheric neutrinos and muons respectively.
        }\label{fig:npe}
%\end{center}
\end{figure}
%
%==============================================================
While some of the cosmogenic neutrino models predict 
consistent total event rates with the experimental observation of two 
neutrino events, 
they contradict the null observation of events above 100~PeV.

%=====================================================
\section{\label{sec:ModelTests} The tests on the cosmogenic neutrino models}
%=====================================================

We examine models predicting ultra-high-energy neutrino productions
to see whether it accounts the present observations.
This allows us to survey the GZK cosmogenic neutrino models that give us an insight 
into UHECR origins.  
The approach presented here is based on
the event rates above 100 PeV, where
the main population of cosmogenic neutrinos originating in UHECRs is expected.
This is more relevant to probing UHECR sources
with the cosmogenic neutrino productions. This can be considered as a view 
that a neutrino generation mechanism other than the GZK interactions
is responsible for the PeV neutrino sky including the observed two neutrino events.
The statistical significance of these tests is limited by our observational
exposure. To obtain the best limit, we combine the exposure 
of the previously published results obtained by
the half-completed IceCube detector with 40 string configuration (IC40)~\cite{icecubeEHE2011}
to the present results hereafter. 
The IC40 exposure increases the observation exposure by $\sim$30\% from the IC79 and IC86 only exposure, 
depending on the neutrino energy, as one can see in Fig.~\ref{fig:effarea}.

A prospective event rate in the energy region above 100 PeV 
is compared to the observational upper limit.
We calculate the p-values for several cosmic neutrino models
based on the rate.
Table~\ref{table:CL} summarizes the p-values for several neutrino models.
The hypothesis of stronger cosmological evolutions is ruled out at around 90\% C.L.
The maximal flux allowed by the constraints from the diffuse photon flux
(labeled as ``the maximal flux'' in the table) is also excluded. It demonstrates that
the present constraints from the limit on the ultra-high-energy neutrino flux are compatible 
with those from the photon-flux measurement by Fermi in the 10 GeV region~\cite{fermilimit}.

%==========================================
\begin{table}[bt]
\begin{tabular}{lcl}
\hline
\hline
$\nu$ Model &amp; Event rate &amp; p-value \\
\hline
Yoshida and Teshima~\cite{yoshida93} &amp;    &amp; \\
$m = 4.0,z_{max}=4.0$&amp;  2.6 &amp;  0.11 \\
\hline 
Kalashev {\it et al.}~\cite{kalashev02} &amp;   &amp;  \\
$m = 5.0,z_{max}=3.0$&amp; 4.0 &amp;  0.029 \\
\hline
Yoshida and Ishihara~\cite{yoshida2012} &amp;    &amp; \\
$m = 5.0,z_{max}=2.0$&amp;  2.0 &amp;  0.21 \\
\hline
Ahlers {\it et al.}~\cite{ahlers2010} &amp;    &amp; \\
$m = 4.6,z_{max}=2.0$&amp;  2.0  &amp; 0.21 \\
("the best fit") &amp;  &amp; \\
\hline
Ahlers {\it et al.}~\cite{ahlers2010} &amp;    &amp; \\
("the maximal flux") &amp; 4.1  &amp; 0.026 \\
\hline
Kotera {\it et al.}~\cite{kotera2010} &amp;    &amp; \\
GRB &amp; 0.63 &amp; 0.53 \\ 
\hline
%Kotera {\it et al.}~\cite{kotera2010} &amp;    &amp; \\
%SFR &amp; 0.60 &amp; 0.64 \\ 
%\hline
Kotera {\it et al.}~\cite{kotera2010} &amp;    &amp; \\
Fanaroff-Riley type II  &amp; 3.8 &amp;  0.022 \\ 
\hline
Top-down 1~\cite{sigl99} &amp;  &amp; \\
SUSY    &amp; 21 &amp; $\ll 0.001$ \\
\hline
Top-down 2~\cite{sigl99} &amp;  &amp; \\
GUT    &amp; 5.0 &amp; 0.011 \\
\hline
\hline
\end{tabular}
\caption{Expected number of events from several neutrino models
and the p-value for consistency with the present observation in the energy range
above 100 PeV.}
\label{table:CL}
\end{table}
%=================================================================

%==========================================
\begin{figure}[tb]
  \includegraphics[width=0.45\textwidth]{./figure/EvolutionContourIC40-79-86_1yr.pdf}
  \caption{Constraints on the UHECR source evolution $m$ and $z_{max}$ 
    with the present analysis. The semianalytic formulation~\cite{yoshida2012}
    estimates the neutrino flux for calculating the limit shown here.
    The area above the solid lines are excluded at the confidence level
    labeled in the figure.
 \label{fig:gzk_constraint}}
\end{figure}
%==========================================

%==========================================
\begin{figure}[bt]
  \includegraphics[width=0.45\textwidth]{./figure/UpdatedUpperlimt_withoutGR_withSys_v2.pdf}
  \caption{All-flavor neutrino flux differential upper limit
from the present IceCube EHE analysis including the IceCube exposure from the previously published result (IC40)~\cite{icecubeEHE2011} (bold solid lines).
%
The systematic errors are
included. Various model predictions (assuming primary protons)
are shown for comparison; Engel {\it et al.}~\cite{ESS},
Kotera {\it et al.}~\cite{kotera2010}, 
Ahlers {\it et al.}~\cite{ahlers2010},Yoshida {\it et al.}~\cite{yoshida93}.
The model-independent differential
upper limits by other experiments are also shown for Auger
Auger(PAO)~\cite{auger12}, RICE~\cite{rice11}, ANITA~\cite{anita2,anita2err}. 
 \label{fig:UL}} 
\end{figure}
%==========================================

The obtained limits on the characteristics of the UHECR sources are displayed
in a comprehensive manner using the formula to estimate the cosmogenic 
neutrino fluxes as functions of the source evolution parameter and source extension
in redshift $z$~\cite{yoshida2012}. The event rate in the  energy region beyond 100 PeV 
is calculated for a given $m$, and $z_{\rm max}$, 
the parameterization often used in the literature
where the spectral emission rate per comoving volume is expressed as $(1+z)^m$ for
$z\leq z_{\rm max}$. The calculated rates are converted into the constraints on
the parameter space of $m$ and $z_{\rm max}$.
Figure~\ref{fig:gzk_constraint} presents the resultant constraints.
It indicates that strongly evolved sources such as $m\gg 4$ 
are not responsible for the main bulk of UHECRs.
Among sources categorized in this class is
the Fanaroff-Riley type II (FR-II) radio galaxies, the long-standing favorite
as a candidate of the UHECR emitter~\cite{biermann87}. 
We should note, however, that the obtained bound is not valid 
if the mass composition of UHECRs is not dominated by
proton primaries.  The dominance of proton primaries is 
widely assumed in the models mentioned here while a predominance of heavier nuclei 
such as iron generally provides much smaller fluxes of neutrinos. The present
analysis is not sensitive to scan the flux levels in this scenario.

The quasidifferential model-independent 90\% C.L. limit
on neutrino fluxes normalized by energy decade is shown in Fig.~\ref{fig:UL} 
using the same method implemented in
our previous EHE neutrino searches~\cite{icecubeEHE2010,icecubeEHE2011}.
Full mixing in the standard
neutrino flavor oscillation scenario is assumed.
A difference from the calculation of limit shown in our previous publications
arises from the existence of two events in the final sample.
The 90\% event upper limit used in the calculation takes into account
the energy probability density function of each of the observed events.
This is the strongest constraint in the EeV regime obtained to date.

The constraint is weaker in the PeV range due to the detection
of two events. A flux of $E_{\nu}^2\phi \sim O(10^{-8})$ 
GeV/cm$^2\ \sec $ sr at PeV is consistent
with the observation.

\section{\label{sec:summary} Summary}

%                                                                                                                                   
We analyzed the 2010--12 data sample collected by the 79 and 86-string IceCube 
detector to search for extremely-high-energy neutrinos with energies exceeding $10^6$~GeV\null.
We observed two neutrino-induced cascade events passing the final selection criteria.
The energy profiles of the two events indicate that these events are cascades of which the energy deposited was $\sim$1~PeV.
An upper limit on the neutrino rate in the energy region above 100 PeV
puts constraints on the distribution of UHECR emitters in redshift space.
The observational constraints started to reach, for the first time,  
the parameter region where some leading UHECR-source candidates are expected to be distributed.
The bound---significantly upgraded from our previous publication~\cite{icecubeEHE2011}---was 
brought about by both the enlarged instrumentation volume and
the refined Monte Carlo simulations with improved agreements with the experimental data.
The ongoing data taking with the fully completed detector
continues to further enhance the IceCube sensitivity to cosmological EHE neutrinos.

\begin{theacknowledgments}
The author wishes to thank the conference organizers, especially Robert Lahmann, 
for their very warm hospitality. All the results presented here could not have been
accomplished without my colleagues, Aya Ishihara, and Keiichi Mase, who have been
driving the IceCube program to search for extremely-high-energy cosmic neutrinos.
I also deeply acknowledge the entire IceCube collaboration for their enthusiastic support
in the data analysis, simulations, as well as the stable operations of the detector.
A part of this work was supported by
the Grants-in-Aid in Scientific Research from the JSPS in Japan.
\end{theacknowledgments}

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