The Big Bang is the leading theory that almost all astrophysicists believe explains the origin of the universe. This is because all observations so far made support the Big Bang theory; there are four main lines of evidence that are most-often used.
The expansion of the universe. The universe is expanding now, so in the past it must have been smaller. If it were smaller in the past, then there probably was a time when it was infinitesimally small. One could ask why don't we think that it might be expanding now but it could have been shrinking before and we just don't know about it. The answer is that there is simply no mechanism that we know about that could accomplish this transition on a universal scale.
The second line of evidence is the Cosmic Microwave Background Radiation (CMB) that was discovered in 1965 by Arno Penzias and Robert Wilson from Bell Labs. They were working with a microwave receiver, but were getting noise from every direction they pointed the receiver. It was coming from all over the sky at what seemed to be exactly the same frequency. This was the first evidence for the CMB, and they later shared a Nobel Prize for this discovery.
The CMB is an "echo" left over from when the universe was approximately 300,000 years old, as predicted by the Big Bang model. As something becomes compressed, as matter was when the universe was young, it becomes hot. The actual "heat" comes from particles' movements - the faster they move, the more energetic they are, and so the more heat we see. The universe was so hot before it was 300,000 years old that atoms could not form. Because of this, photons - particles of light - could not move around, for they kept reacting with electrons.
Therefore, during this period, the universe was effectively opaque. Once the universe had reached 300,000 years old, atoms could form, and electrons were now bound to a nucleus. Once this happened, photons could move about freely. This "first light" is the CMB, and its existence is a very strong indication that the Big Bang occurred.
The third major pillar of the Big Bang theory lies in the abundance of the different elements of the universe. The theory predicts that certain amounts of hydrogen, helium, and other elements should be made. Observations have shown almost exactly the amounts that are predicted.
The fourth piece is that the Big Bang theory is the only one that comprehensively lays down a framework for the eventual evolution of the universe as we observe it today.
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