How is it possible to discover neutrinos from the big bang? Were the neutrinos emitted just before the big bang? Or at the same moment? If they moved outward in all directions, basically unhindered by matter or gravity, how can we hope to ever detect them? Unless something caused some of them to turn back and travel in a different direction?
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Issue 1: The Expansion Misconception
Forget everything you thought you visualized about the Big Bang. Let's start from scratch.
First, picture a sheet of rubber with a grid marked on it. This sheet represents space. It is not curved, nor will it ever curve in this example. It might stretch, but that's not particularly important here either. Attach a light bulb at each grid point, representing, say, a galaxy.
Now the conceptual leap: imagine you start off with an infinite sheet. There are galaxies going off as far as you can imagine. This is a pretty good qualitative description of our universe, with the caveat of being 2D rather than 3D. (Picture an infinite blueberry muffin if you want 3D.)
Let's say the light bulbs start all off. At some time $t_0$ turn them all on. The light bulb one meter away from you will have its first light reach you a few nanoseconds after $t_0$. The light bulb $300{,}000\ \mathrm{km}$ away from you will seem to turn on one full second after $t_0$. And so on.
At any given time $t > t_0$, light bulbs a distance $c(t-t_0)$ away will be seen just turning on. Such light bulbs can be found in all directions. Similarly, photons that decoupled from the primordial plasma at recombination, or neutrinos that did the same even earlier in the universe, are always arriving at Earth. The later they arrive, the further away they must have originated, but remember the universe is infinite -- there is always a "further away" for them to have originated from. The fact that the universe is expanding changes this simple model in quantitative detail, but not qualitatively.
Issue 2: "Unhindered by Matter or Gravity"
The fact that neutrinos don't interact much with matter is good. It means they won't be stopped in their multi-billion-year journey to reach us.
As for gravity, it's worth pointing out that any nonzero neutrino mass means they can in theory be captured by a gravitational potential well, even one that doesn't correspond to a black hole.
Issue 3: Detection
The only actual problem with detecting relic neutrinos, though a very big one indeed, is that neutrinos are incredibly difficult to detect at all. On average only one out of a very large number will interact with your detector. And the chances for interaction drop with neutrino speed. Because relic neutrinos are expected to be quite cold (for the same reason as CMB photons are cold -- they cool as the universe expands), the neutrino detectors we have have no hope of seeing them. All we can see are the much more energetic neutrinos that come from things like nuclear fusion in the Sun.
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Addressing part of your question:
Neutrinos were not emitted prior to (or at the start of) the big bang. "Before the big bang" is a phrase that drives the philosophers nuts, but because the big bang was the beginning of space and time, it is thought that there was no "before". Also, neutrinos of any sort could not have formed until a few fractions of a nanosecond after the big bang. Why? In the early universe (i.e. orders of magnitude smaller than that), there wasn't anything resembling what we know today. Neutrinos, electrons, quarks, and other elementary particles formed later - within a nanosecond or two.
Neutrino detectors are certainly feasible to make, and there are many in operation, such as the Super Kamiokande detector in Japan, but it is very hard to detect neutrinos, and I see no reason why it should be any easier (in fact, I would think it would be a lot harder) to find these ones. Any of these "relic neutrinos" (as @CuriousOne put it) would not be emitted from any single direction, unlike, say solar neutrinos, which come from one direction: the Sun. Finally, I should add that I have no idea what to make of your last sentence, so I won't try to address that.
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