Your statement contains some correct elements, but it also has a few inaccuracies that can be clarified.
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Background Radiation: The radiation you are referring to is the Cosmic Microwave Background (CMB) radiation. It is indeed a critical piece of evidence supporting the Big Bang theory and is thought to be the remnant heat from the early universe, shortly after the Big Bang.
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Discovery: The CMB was discovered accidentally by Arno Penzias and Robert Wilson in 1965. Their discovery provided strong evidence for the Big Bang theory.
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Temperature and Uniformity: The temperature of the CMB is largely uniform, around 2.7 Kelvin, but there are tiny fluctuations in temperature that correspond to density variations in the early universe. These fluctuations are crucial for understanding the formation of large-scale structures in the universe.
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Age of the Universe: When you say we can see what the universe looked like "13,700,000,000 years ago," you are indeed touching on the fact that the CMB originated from a time approximately 380,000 years after the Big Bang when the universe cooled enough for photons to move freely. However, this is not exactly 13.7 billion years ago but rather reflects the time immediately after the Big Bang.
In summary, while your statement captures the essence of the CMB's significance, it could benefit from clarification regarding uniformity and age. The CMB provides insight into the early state of the universe, and the fluctuations seen in it are key to understanding cosmic evolution.