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The Big Bang theory is the most commonly accepted scientific theory for the formation and age of the universe. According to the most recent measurements, the age of the universe is just under 14 billion years old. The Big Bang theory came to be postulated and finally measured throughout the 20th century. Originally, people thought that the universe was static, or in essence a system in steady state, meaning that no net change was occurring. The thermodynamic concept of entropy, however, dictated that the universe must not be a static system. Astronomer and physicist Edwin Hubble in 1929 produced the first evidence that the universe is expanding. He measured a phenomenon known as red shift from light emanating from distant galaxies. He noticed that light absorbance values in the line spectra of these galaxies shifted to the red end of the visible spectrum. This could only happen if the galaxies were moving away from the Earth (sort of like a Doppler effect for visible light waves). So this was evidence that the universe was expanding. So scientists then took extremely accurate measurements of this expansion rate and then extrapolated it back in time to the zero point, or the start of the expansion of the universe at the Big Bang. This is how the age of the universe was measured. Writing down the actual mathematical steps would be beyond my skill or the scope of this forum.
So, they must know the size of the universe if our cosmos was formed 13.7 billion years ago.
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