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The CMBR or Cosmic Microwave Background Radiation, is not only consistent with the Big Bang, it was an expected outcome of the Big Bang! Like the neutrino and black holes, the CMBR was predicted by scientific and mathematical models before it was discovered. The explosion of energy that created the universe is still resonating in space even as the universe cools. This IS what the CMBR is, it's the leftover energy from the Big Bang itself.
Cosmic Microwave Background Radiation (CMBR) is radiation that fills most of the space in the universe. It is composed of very faintly-moving microwave radiation (other frequencies are almost too faint to detect) that can be detected by radio telescopes. The explanation for CMBR is that the universe is stil expanding, and so the energy inside it has been moving away from itself for billions of years; while the newly-born universe was extremely hot and agitated, the years of expansion has cooled the CMBR to a point where it can barely be detected. This is one of the major scientific proofs that the Big Bang happened; however, it also predicts the eventual Heat Death of the universe, where all the matter and energy particles in the universe are so far away from each other that no movement can be imparted, and so everything becomes the same, constant, dead temperature.
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