We have proposed a novel approach for studying
oscillations with extragalactic
neutrinos. Active Galactic Nuclei and Gamma Ray Bursts
are believed to
be sources of ultrahigh energy muon neutrinos. With distances of
100 Mpc or more, they provide an unusually long baseline for
possible detection of
with mixing parameters
down to
eV
, many orders of magnitude
below the current accelerator experiments. By solving the coupled
transport equations, we show that high-energy
's, as they propagate through the earth, cascade down in
energy,
producing the enhancement of the incoming
flux in the low energy
region, in
contrast to the
high-energy
's, which get absorbed.
For an AGN quasar model we find the
flux
to be a factor of
to
larger than the incoming flux
in the energy range
between
GeV and
GeV, while for a GRB fireball
model, the enhancement is
-
in
the same energy range and for zero nadir angle.
This enhancement decreases with larger nadir
angle, thus providing a novel way to search for
appearance
by measuring the angular dependence of the muons.
To illustrate
how the cascade effect and the
final flux depend on
the steepness of the incoming
,
we show the energy and angular distributions
for several generic cases of the
incoming tau neutrino flux,
for
and
.
We show that for the incoming flux that is not too steep,
the signal for the
appearance of high-energy
is the enhanced production of lower energy
and their distinctive angular dependence,
due to the contribution from the
decay into
just below the
detector.