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If the Cosmic microwave background happened when the universe was approximately 379,000 years old and it's still detectable today, is it probable/possible that other kinds of "radiation" from before that might be out there?

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There would not be so much radiation in the sense of electromagnetic radiation. However, with the breaking of electroweak symmetry, about 3 seconds into the big bang, lepton doublets acquired mass and the very tiny massed neutrino was "liberated" from the rest of the matter fields. There then exists a cosmic neutrino background (CNB) that should carry information from that time period.

Can these be detected? Neutrinos are notoriously tough to detect. Detecting neutrinos that are ancient neutrinos that carry information from behind the CMB is very difficult. We can identify neutrinos with a source such as a supernova, but that does not mean we can identify a source for every neutrino.

There should also exist primordial gravitational radiation. This gravitational radiation would be long wavelength gravitons (soft gravitons) that form another background. These would occur from the time quantum gravitation separated from gauge fields in a symmetry breaking in the first $10^{-40}$ to $10^{-35}$ seconds of the universe. Related to that is the production of gravitational radiation or gravitons in the inflationary period of the universe.

Can these be detected? At best maybe only indirectly. The BICEP2 team announced the detection of B-modes plasma in the CMB. Gravitational radiation with it polarization would induces the plasma of radiation dominated phase to emit radiation in certain polarization directions. This was announced in the spring of 2014. However, dust in the Milky Way galaxy can induce a similar polarization, and the certainty of detecting these early gravitational waves dropped from over $5$-$\sigma$ to $3$-$\sigma$.

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