Would someone with some knowledge of physics care to break down the ramifications of this (if it's not some sort of measurement error)?
Please?
Special Relativity holds only for space with static and continuous metric (to preview the complexity of the theory otherwise, try to play with Maxwell equations where integration would happen over non-static volumes/surfaces :) . The space expansion doesn't fit the premises of Special Relativity and thus there is no contradiction. Special Relativity is pretty artificial theory which applies to the true world of our expanding (ie. non static metric) Universe only by approximation - this approximation works only as much as the speed of expansion is slow and either time period under consideration is small or distances are small (but not very small as the space continuity premise of SR seems to get broken as we get to Planck distances)
In this particular case of neutrinos, i think, if they integrate Earth gravity time dilation variations along the neutrinos path, it would be correction on the scale around this billionth of second.
Isnt that effect in the reference frame of the neutrinos and not the measuring instrument(I'm not too sure about this)?....And also wouldnt it mean that the neutrinos take more time instead of less time?
Consider what we know about the Universe but can't fully explain so far. We know that space-time is dominated by a "dark energy" that may be due to background quantum vibrations or some as yet completely unknown phenomenon. We know that most of the mass of the Universe is bound up in dark matter which at present seems most likely to be a weakly interacting massive particle, specifically neutralinos (a supersymmetric particle). Neutralinos are thought to be their own anti-particles, they could have been responsible for bizarre "dark stars" in the early Universe and for an ongoing gamma ray flux due to neutralino-neutralino annihilations. Moreover, most of the dark matter in the Universe would essentially be the cosmic neutralino background, an echo from the big bang analogous to the cosmic neutrino background and cosmic microwave background. There is already tantalizing experimental evidence that such dark matter particles exist, from multiple sources.
Additionally, recent evidence seems to hint strongly of the existence of strange-matter. Unlike ordinary quark-based matter such as protons, neutrons, other baryons and hadrons strange-matter would consist of up, down, and strange quarks in a more-or-less liquid like state instead of individual quarks being bound up in 2 or 3-quark composite particles. It is possible that some of the so-called "neutron-stars" out there are in fact strange-matter stars. It's also possible that such matter is stable in effectively arbitrarily sized fragments, down to individual tons, grams, or even atomic nuclei scales. Who knows what kind of post-nuclear physics we could perform if we got our hands on such particles. One possibility would be to use them to create energy via fusion. Tossing particles at a strangelet with sufficient speed should cause them to dissolve and fuse with the strangelet, releasing more energy in the process than conventional fusion.
And that hardly touches the surface. There are even more bizarre things out there like micro black holes, the matter vs anti-matter chirality, the possibility that superconductors serve as mirrors for gravitational waves, and so much more. There's plenty of star trek out there waiting for us.