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It's mostly well accepted today that the Big Bang was the reason for today's universe, mostly due to the expansion, the cosmic microwave background etc. and mathematically the FLRW metric supports it. Is there/could there be anything at all, existing or modeled through specific experiments, in the present universe that can give us a better picture as to what exactly happened at t = 0 or what could have caused the Big Bang?

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    $\begingroup$ There are, at least, two more important pieces of evidence, the cosmic gravitational wave background and the cosmic neutrino background. The latter is probably not measurable (at least I don't know of any practical experimental way), but the gravitational waves could go back way further, still, and tell us something important about the very early universe at $10^{-32}s$ and earlier. $\endgroup$ – CuriousOne Dec 30 '15 at 3:17
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As astrophysicists continue their research, we're pushing our understanding closer and closer to time $T=0$. However, our progress seems to be asymptotic to that final goal of understanding the beginning of time. In addition, our concept of "time" may not be up to the task of describing the true beginning of the Universe. For instance, if you review the Very early universe section of the Wikipedia Chronology of the universe article, you'll find that in some inflation models, the time when inflation ended seems to be shorter than the duration of inflation.

So, unless there is some sort of (perhaps even literal) quantum leap in our understanding, we may never know what happened at time $T=0$.

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    $\begingroup$ +1. Indeed, cosmic censorship may very well set a limit to what we can know about where "we" came from. $\endgroup$ – CuriousOne Dec 30 '15 at 4:22

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