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The age of the universe has been subject to debate for a long time. Before the advent of science, the accepted theory was based in Judeo-Christian theology, which supports Young-Earth Creationism, or the theory that the universe was created in six days by God as stated in the Bible. Later, greater understanding of science began to create new theories on how the universe and the Earth came to exist.
Scientist and Catholic priest Georges Lemaître posited the first fledgling Big Bang Theory, which he called the "Hypothesis of the Primeval Atom." After using the Finite Universe Model in his initial paper, he refined his theory to include an infinitely expanding universe emanating from a single point. This theory allowed for a much greater age of the universe than the religious age of about 6000 years.
As technology increased in scope and ability, the methods used to determine the size and age of the universe became more accurate. By measuring light, radiation, and Red Shift (a Doppler Effect) it is now possible to estimate the age of the universe with greater accuracy.
The speed of light, famously codified by Albert Einstein in his Special Theory of Relativity, is 186,282 miles per second (299,792.458 kilometers per second). Knowing this, we can calculate the distance of far stars and galaxies based on what color their light is and how fast we know it to move in a vacuum. Red Shift seen in distant galaxies shows that they are receding from our point in space (if objects showed Blue Shift, they would be approaching our point in space). Background radiation, mostly Microwave Spectrum Radiation, is considered to be the leftover energy from the Big Bang, which could not be absorbed by solidifying atoms as the universe cooled; this radiation has been growing fainter as it moves through the universe.
The currently accepted age of the universe is 13.75 billion years, plus or minus 110 million years. This number is based on NASA's WMAP measurement of microwave background radiation. This spacecraft measured and removed the information of foreground radiation. By generating a map of the temperature of of the background radiation in the universe, and since the background radiation is very homogeneous, we can extrapolate the initial heat and activity of this radiation and calculate how old it is. The age of the background radiation corresponds with the age of the universe.
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