Eventually stars will form and nucleosynthesis will begin again, 'cooking' these initial elements into heavier elements such as carbon, oxygen and others (Wikipedia). We've detected a great deal of evidence for this process. Older, lower mass (and therefore longer lived) stars tend to have lower amounts of the elements heavier than helium (low metallicity).
But this element formation process will not be uniformly distributed. Regions with stars will form these heavier elements at higher rates, and supernovae and other explosive events will expel them back into the interstellar medium. We expect regions such as galaxies and clusters of galaxies to have a higher fraction of these heavier elements leaving the regions between galaxies with a lower fraction of heavy elements. This non-uniformity in mixing means that there might still be some regions of the cosmos where unprocessed 'pristine' material from the original nucleosynthesis might be found.
We now have news that some of this left-over material may have indeed been found.
Here's a press report:
Discovery.com: Pristine Big Bang Gas Found
And here is a link to the paper published in Science magazine:
Science: Detection of Pristine Gas Two Billion Years After the Big Bang
Most cosmological alternatives to the Big Bang are either much younger (such as those advocated by young-Earth creationists) or of infinite age (advocated by Plasma Cosmology and Electric Universe, among others).
But infinite age cosmologies have a host of problems, rarely acknowledged by their advocates.
- In a universe of infinite age, everything has had enough time to reach its equilibrium configuration, which means there is no free energy left which can be liberated. Basically, an infinite age universe is a dead universe and there is no energy available for even life to function.
- Most infinite age cosmologies try to avoid the problem described above by advocating some mechanism for matter to be spontaneously created. While such a process would violate conservation of energy (COE), there are known quantum processes which can violate COE for very short intervals, therefore, we can't rule this possibility out at 100% confidence (but the difference with 100% would very, very, VERY tiny). Conservation of energy could be violated on a level below our current ability to detect it.
Here's some references that address some of the questions that get raised by energy issues in Big Bang cosmology: Machines Like Us: Does the Big Bang violate Conservation of Energy?
These discussions always generate confusion because General Relativity does define conserved quantities that we equate to energy defined on a cosmological time scale, but the definition does not map cleanly to small times scales, such as looking at the entire cosmos vs. energy conservation on the Earth. Sean Carroll gives a variation of the explanation here: Cosmic Variance: Energy is not Conserved.
This confusion has allowed various cranks to exploit the terminology to promote bad science.
Machines Like Us: Does the Big Bang violate the 2nd Law of Thermodynamics?
For a description of how we learned that symmetry principles are behind the conservation laws, see this chapter from the online text "Conceptual Physics".