The short answer: The big bang is how astronomers explain the way the universe began. It is the idea that the universe began as just a single point, then expanded and stretched to grow as large as it is right now (and it could still be stretching).
The long answer: For most of human history, observers of the sky assumed it eternal and unchanging. Edwin Hubble dealt this story an experimental blow in the 1920s when his observations showed both that galaxies outside the Milky Way existed, and that their light appeared stretched — a sign that they were rushing away from Earth.
George Lemaître, a contemporary Belgian physicist, interpreted data from Hubble and others as evidence of an expanding universe, a possibility permitted by Einstein's recently published field equations of general relativity. Thinking backwards, Lemaître inferred that today's separating galaxies must have started out together in what he called the "primeval atom."
The first public use of the modern term for Lemaître's idea actually came from a critic — English astronomer Fred Hoyle. On March 28, 1949, Hoyle coined the phrase during a defense of his preferred theory of an eternal universe that created matter to cancel out the dilution of expansion. Hoyle said the notion that "all matter of the universe was created in one big bang at a particular time in the remote past," was irrational. In later interviews, Hoyle denied intentionally inventing a slanderous name, but the moniker stuck, much to the frustration of some.
"The Big Bang is a really bad term," said Paul Steinhardt, a cosmologist at Princeton. "The Big Stretch would capture the right idea." The mental image of an explosion causes all kinds of confusion, according to Steinhardt. It implies a central point, an expanding frontier, and a scene where light shrapnel flies faster than heavier chunks. But an expanding universe looks nothing like that, he said. There's no center, no edge, and galaxies large and small all slide apart in the same way (although more distant galaxies move away faster under the cosmologically recent influence of dark energy).
Regardless of its name, the Big Bang theory found widespread acceptance for its unparalleled ability to explain what we see. The balance of light with particles like protons and neutrons during the first 3 minutes, for instance, let early elements form at a rate predicting the current amounts of helium and other light atoms.
"There was a small window in time where it was possible for nuclei to form," said Glennys Farrar, a cosmologist at New York University. "After that, the universe kept expanding and they couldn't find each other, and before [the window] it was too hot."
A cloudy plasma filled the universe for the next 378,000 years, until further cooling let electrons and protons form neutral hydrogen atoms, and the fog cleared. The light emitted during this process, which has since stretched into microwaves, is the earliest known object researchers can study directly. Known as the cosmic microwave background (CMB) radiation, many researchers consider it the strongest evidence for the Big Bang.
Misconseption: Beginning at the beginning, a common question asked by laymen and some physicists is what the universe expands into. The very terminology of the ``big bang'' suggests an explosion, which flings debris out into some void. Such a picture is strongly suggested by many semi-popular descriptions, which commonly include a description of the initial instant as one ``where all the matter in the universe is gathered at a single point'', or something to that effect. This phrase can probably be traced back to Lemaître's unfortunate term ``the primaeval atom''. Describing the origin of the expansion as an explosion is probably not a good idea in any case; it suggests some input of energy that moves matter from an initial state of rest. Classically, this is false: the expansion merely appears as an initial condition. This might reasonably seem to be evading the point, and it is one of the advantages of inflationary cosmology that it supplies an explicit mechanism for starting the expansion: the repulsive effect of vacuum energy. However, if the big bang is to be thought of explosively, then it is really many explosions that happen everywhere at once; it is not possible to be outside the explosion, since it fills all of space. Because the density rises without limit as t -> 0, the mass within any sphere today (even the size of our present horizon) was once packed into an arbitrarily small volume. Nevertheless, this does not justify the ``primaeval atom'' terminology unless the universe is closed. The mass of an open universe is infinite; however far back we run the clock, there is infinitely more mass outside a given volume than inside it.
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