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The big bang cosmology originally grew out of Einstein's general theory of relativity, which was published in 1915. In 1922, Alexander Friedmann, a Russian mathematician, derived equations from general relativity showing that the universe might be expanding, in contrast to the static universe that Einstein had assumed. Einstein originally posited a nonzero value for the cosmological constant, a term in his equations, but after the expansion of the universe was discovered, he lamented that this was his greatest blunder and set the constant to zero [Davies2007, pg. 58]. In 1924, American astronomer Edwin Hubble measured the distance nearby spiral nebulae and showed that these systems were actually other galaxies, not merely objects within the Milky Way. In 1927, Georges Lemaitre, a Belgian Roman Catholic priest, suggested that the recession of these nebulae was due to the expansion of the fabric of universe. In 1929, Hubble confirmed this hypothesis, by showing that the distances to these galaxies were roughly proportional to their outward velocities, as measured by the degree to which their light spectra was shifted to the red (this fact is now known as Hubble's Law). This inferred that the entire universe is expanding, not only away from us but also away from every other position in space -- much like dots on the surface of an expanding balloon all appear to be moving away from each other. In that case, there must have been a time when the universe was very much more dense than it is today.
At about the same time, theoretical calculations by several researchers concluded that the big bang cosmology would have produced a universe that is roughly 75% hydrogen, 25% helium, with traces of He-3 and Li-7 (heavier elements are produced by a different process in the explosion of stars). Careful measurements verified these abundance figures in impressive detail [Guth1997, pg. 101-103].
More recent astronomical measurements continue to confirm the big bang model. For example, in 1993, measurements of the cosmic microwave background using the Cosmic Microwave Background Explorer (COBE) satellite were found to perfectly fit a black body radiation curve with a characteristic temperature of 2.725 K, plus or minus 0.01 K. Data obtained from the Wilkinson Microwave Anisotropy Probe (WMAP) spacecraft, which was in operation from 2001 through 2010, showed even more spectacular agreement -- plus or minus 0.001 K. See the plot below (left). In this graph, the vertical error bars of the data, if shown, would be much too small to be distinguished from the curve itself! It should also be noted that careful measurements of the microwave background measurement show that this radiation is isotropic (equal in all spatial directions) to within one part in 100,000. Interestingly, though, if it were perfectly isotropic, this would be in conflict with theory requiring small early fluctuations in matter density to provide seeds for the subsequent formation of stars and galaxies. Fortunately, though, in the early 1990s fluctuations were found, at just the level predicted by theory [Tegmark2014, Ch. 5].
Below (right) is a map of these fluctuations, again based on WMAP data (both photos courtesy NASA):
The most recent data has been obtained by the Planck orbiting observatory, which began operation in 2010. Like the WMAP mission, the Planck satellite permits scientists to analyze the "angular power spectrum" of the data (the degree of correlation between the data in one direction with the data in different directions across the sky), but with even greater accuracy and resolution than WMAP. In March 2013, the Planck project released its latest results. Researchers concluded that the universe consists of 4.9% ordinary matter, 26.8% "dark matter" (which exhibits itself only through gravitational effects), and 68.3% "dark energy" (which is believed to be responsible for the accelerating expansion of the universe). This data also permitted the Planck team to pin down the age of the universe at 13.8 billion years, slightly older than the 13.75 billion year age estimated by WMAP [Cho2013].
A very interesting online tool, constructed by the WMAP team, permits one to "design" a universe by varying the composition of matter (atoms, cold dark matter and dark energy) together with the Hubble constant (which is related to the age), to obtain the best fit to the angular spectrum data [WMAP2009].
In March 2010, the age of the universe was accurately measured by a completely different approach. A team of researchers from several U.S. and European institutions employed a technique known as gravitational lensing (a phenomenon related to general relativity) to measure the distances traveled by light rays from a distant galaxy to the earth along different paths. By calculating the time it took to travel along each path and the effective speeds involved, these researchers were able to infer how far away the galaxy lies and also the overall scale of the universe, plus some details of its expansion. The researchers confirmed that the age of the universe is 13.75 billion years old, plus or minus 0.17 billion years, in agreement with the age found by the WMAP and, more recently, by the Planck team [SD2010i].
These results are further confirmed by measurements of the universe's expansion and acceleration, based on the observation of Type Ia supernovas. These results are now accurate enough to rule out one alternative theory to the accelerating universe, namely that the Milky way lies in a large void [SD2011f].
While inflation made sense according to the original mathematical formulation of the inflation process developed back in the 1970s and 1980s, it seems significantly less satisfactory today, and even some leading figures in the field now believe that this scenario needs to be rethought. More modern mathematical formulations have many "ad hoc" parameters and assumptions, many of which are increasingly dubious, or which, as with considerations of the "multiverse," must invoke the anthropic principle to explain the early evolution our universe. For example, highly improbable conditions are required for the inflation process to be initiated. Worse, it appears that the most natural course of events is for the inflation process to continue indefinitely, producing an infinite variety of outcomes, so that it is difficult to make any firm observational predictions from the theory.
These difficulties had led even long-time supporters of inflation to question the theory's viability (although not the viability of the overall big bang cosmology). Paul Steinhardt of Princeton University is one vocal detractor. In 2012 he declared, "We thought that inflation predicted a smooth, flat universe. Instead, it predicts every possibility an infinite number of times. We're back to square one." Similarly, Sean Carroll, a cosmologist at the California Institute of Technology, explains, "Inflation is still the dominant paradigm, but we've become a lot less convinced that it's obviously true. ... If you pick a universe out of a hat, it's not going to be one that starts with inflation." [Gefter2012].
An excellent summary of these controversies is given in a 2011 Scientific American article by cosmologist Paul J. Steinhardt [Steinhardt2011], and also in a 2012 New Scientist article by science writer Amanda Gefter [Gefter2012].
These controversies were re-ignited with the March 2014 discovery of gravitational wave ripples in the cosmic microwave background radiation, which were heralded as the first solid experimental confirmation of the inflationary theory of the big bang [Overbye2014a]. Proponents of the multiverse were quick to point out the implications. As Alan Guth, the MIT physicist who founded inflation theory, explains, "Most inflationary models, almost all, predict that inflation should become eternal." Frank Wilczek adds, "There doesn't seem to be anything unique about the event we call the big bang. It is a reproducible event that could and would happen again, and again, and again." [Moskowitz2014].
But critics of inflation and the multiverse were also quick to sow skepticism. As Peter Coles of the University of Sussex, UK, writes, "Perhaps there is a part of the multiverse in which the BICEP2 results for inflation provide evidence for a multiverse, but I don't think we live there." [Moskowitz2014]. What's more, the BICEP2 experimental evidence itself, on which the March 2014 discovery was based, has been called into question. A team of researchers at the University of California, Berkeley argues that the "twisting" effect cited in the gravitational wave study could just as easily be accounted for as a result of dust in the Milky Way [Cowen2014]. The latest data points even more strongly to dust as an explanation, so the current consensus is that the BICEP2 data can say nothing about inflation one way or the other [Wolchover2015].
But the overall outline of the evolution of the universe, as summarized above, is on very solid ground, both theoretically and, more importantly, empirically. In this regard, the status of big bang cosmology is analogous to that of the theory of biological evolution on earth -- while there are some questions as to what processes and laws governed the origin and very early epochs of the process, the overall history since the origin is hardly in doubt. See Origin.
The big bang and big bang cosmology have led to numerous additional scientific discoveries, some of them rather deep and far-reaching -- see
The theological implications of the big bang are discussed in the article
Big bang theology.