Several cosmologically distant astrophysical sources may produce high-energy cosmic neutrinos $(E>~{10}^6$ GeV) of all flavors above the atmospheric neutrino background. We study the effects of vacuum neutrino mixing in the three flavor framework on this cosmic neutrino flux. We also consider the effects of possible mixing between the three active neutrinos and the (fourth) sterile neutrino with or without big-bang nucleosynthesis constraints and estimate the resulting final high-energy cosmic neutrino flux ratios on Earth compatible with currently existing different neutrino oscillation hints in a model independent way. Further, we discuss the case where the intrinsic cosmic neutrino flux does not have the standard ratio.

The possibility of measuring ${\sin }^{2}2{\theta }_{13}$ using reactor neutrinos is examined in detail. It is shown that the sensitivity ${\sin }^{2}2{\theta }_{13}>0.02\mathrm{}$ can be reached with 40 ton yr data by placing identical CHOOZ-like detectors at near and far distances from a giant nuclear power plant whose total thermal energy is $24.3{\mathrm{GW}}_{\mathrm{th}}.$ It is emphasized that this measurement is free from the parameter degeneracies that occur in accelerator appearance experiments, and therefore the reactor measurement is complementary to accelerator experiments. It is also shown that the reactor measurement may be able to resolve the degeneracy in ${\theta }_{23}$ if ${\sin }^{2}2{\theta }_{13}$ and ${\cos }^{2}2{\theta }_{23}$ are relatively large.

If the flavor-dependent nonstandard interactions (NSIs) in neutrino propagation exist, then the matter effect is modified, and the modification is parametrized by the dimensionless parameter ${\epsilon }_{\alpha \beta }$ ($\alpha$, $\beta =e$, $\mu$, $\tau$). In this paper, we discuss the sensitivity of the T2HKK experiment, the possibility of which is now seriously discussed as a future extension of the T2K experiment, to such NSIs. On the assumption that ${\epsilon }_{\alpha \mu }=0$ ($\alpha =e$, $\mu$, $\tau$) and ${\epsilon }_{\tau \tau }=|{\epsilon }_{e\tau }|/(1+{\epsilon }_{ee})$, which are satisfied by other experiments to a good approximation, we find that, among the possible off-axis flux configurations of 1.3°, 1.5°, 2.0°, and 2.5°, the case of the off-axis angle 1.3° gives the highest sensitivity to ${\epsilon }_{ee}$ and $|{\epsilon }_{e\tau }|$. Our results show that the 1.3° off-axis configuration can exclude NSIs for $|{\epsilon }_{ee}|\gtrsim 1$ or $|{\epsilon }_{e\tau }|\gtrsim 0.2$ at $3\sigma$. We also find that in the presence of NSIs T2HKK (for the off-axis angle 1.3°) has better sensitivity to the two $CP$ phases [${\delta }_{CP}$ and arg(${\epsilon }_{e\tau }$)] than DUNE. This is because of the synergy between the two detectors, i.e., one in Kamioka and one in Korea. T2HKK has better sensitivity to the $CP$ phases than the atmospheric neutrino experiment at Hyper-Kamiokande in inverted hierarchy, but in normal hierarchy, the atmospheric neutrino experiment has the best sensitivity to the $CP$ phases.