We propose a novel approach for studying \nu _ { \mu } \rightarrow \nu _ { \tau } 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 \nu _ { \mu } \rightarrow \nu _ { \tau } with mixing parameters \Delta m ^ { 2 } down to 10 ^ { -17 } eV ^ { 2 } , many orders of magnitude below the current accelerator experiments .
By solving the coupled transport equations , we show that high-energy \nu _ { \tau } ’ s , as they propagate through the earth , cascade down in energy , producing the enhancement of the incoming \nu _ { \tau } flux in the low energy region , in contrast to the high-energy \nu _ { \mu } ’ s , which get absorbed .
For an AGN quasar model we find the \nu _ { \tau } flux to be a factor of 2 to 2.5 larger than the incoming flux in the energy range between 10 ^ { 2 } GeV and 10 ^ { 4 } GeV , while for a GRB fireball model , the enhancement is 10 \% - 27 \% 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 \nu _ { \tau } appearance by measuring the angular dependence of the muons .
To illustrate how the cascade effect and the \nu _ { \tau } final flux depend on the steepness of the incoming \nu _ { \tau } , we show the energy and angular distributions for several generic cases of the incoming tau neutrino flux , F _ { \nu } ^ { 0 } \sim E ^ { - n } for n = 1 , 2 and 3.6 .
We show that for the incoming flux that is not too steep , the signal for the appearance of high-energy \nu _ { \tau } is the enhanced production of lower energy \mu and their distinctive angular dependence , due to the contribution from the \tau decay into \mu just below the detector .