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Hmm.  After reading thru the previous two educators responses, I find it difficult to explicate the theory in a more cogent manner.

Or, to say it differently, for most readers, little needs to be added without resorting to Hawking, et. al.  We know that:

  1. The universe apparently had a beginning.  Astrophysicists have dated it to between 13 and 14 billion years ago.  Sources for this vary, but think of nuclear decay, for one.
  2. The Big Bang theory implies something similar, but not the same as, an explosion:  given this, the universe should be expanding and continue to expand.  Measurements of the velocities of galaxies and galactic groups support this.
  3. Anything similar to an explosion should produce heat.  There is a what appears to be a cosmic background radiation that is about 3 degrees above absolute zero.

Recent discoveries point to problems with the theory, but have yet to discredit it completely.

Big Bang's Gravitational Wave Goes Bust, Due to Dust

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The big bang theory is a widely accepted cosmological model of evolution of our universe. It explains how the universe was 'born' through a 'big bang' and how it has evolved over time. 

The theory states that our universe began from a singularity around 13.8 billion years ago and has been expanding ever since. The model provides comprehensive explanation for a number of observed phenomena. The discovery by Hubble (in 1929) that galaxies are drifting apart and the discovery of cosmic microwave background radiation (in 1964) support the big bang theory. Due to these and other discoveries in support of the Big Bang theory, it has become the widely accepted model for the birth and evolution of the universe and is favored over the Steady State Theory (another competing cosmological model for the universe's evolution).

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The Big Bang theory is an attempt to deal with the observed reality of an expanding universe. That is to say, in the wake of observations showing that all galaxies are moving away from one another, the "film" was run backward mentally, projecting back to a time when all the matter and energy in the universe was compressed into a single unit. In fact, this was a time before matter and energy had differentiated. This massive block of substance then exploded outward in all direction, differentiating into matter and energy, and into the different nuclear forces. That explosion is the Big Bang.

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