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First it is necessary to understand how the early universe evolved according to the big bang model.

Initially the universe was extremely small and extremely hot, comprised of energy rather than material. As the universe began to expand, it also began to cool (more space and less energy means less energy per unit of space). As the universe continued to cool, some of that energy began to condense into quarks which later formed protons and neutrons, and electrons (as well as a myriad other particles we need not concern ourselves with).

Eventually (after about 3 minutes) the universe was now at the temperature where nucleosynthesis could occur. Protons and Neutrons fused together to form heavier nuclei, first heavier hydrogen nuclei, then helium, then lithium. By about 20 minutes after the beginning of the universe, the temperature had fallen too much to allow nuclear fusion, and this process stopped.

For approximately the next 400000 years, the universe was a sea of nuclei and electrons, then the temperature finally dropped to the point that nuclei could capture those electrons, forming neutral atoms.

There is, of course, much more to it than this, but this is sufficient for our purposes.

The three most substantial (there are more) pieces of evidence for the big bang are the Hubble expansion, the cosmic microwave background and the relative abundances of lighter elements.

The Hubble expansion is named for the astronomer Edwin Hubble, who was the first to knowingly observe galaxies beyond the Milky Way. Not only did he discover these galaxies, he discovered that they were all moving away from us. This discovery was achieved by examining their spectra. If an object emitting light or sound is moving relative to an observer, that light or sound is observed at different wavelengths than it is emitted. If the emitter and observer are moving towards each other, the observed wavelength decreases (gets shorter). If the emitter and observer are moving away from each other, the observed wavelength increases (gets longer). You can observe this yourself by listening to the change in sound of vehicle's siren as it approaches, passes, and moves away from you.

This phenomenon is known as Doppler shift, named for Christian Doppler. In the case of light, if the observed wavelength increases, then the light has moved toward the red end of the spectrum and is described as "red shift". Conversely, if the observed wavelength decreases, moving towards the blue end of the spectrum, it is described as "blue shift".

However, Hubble's observations of the red shift of galactic spectra were too extreme be explained by mere Doppler shift. It was not simply that the light was red-shifting as it travelled the distance between the galaxies and Hubble's telescope, but that the actual space between them was expanding, creating a greater distance between the galaxies and us, increasing that red-shift.

Since the light from those galaxies had been travelling for billions of years, it follows that over those billions of years the universe has experienced expansion. This was the first observational evidence of the big bang.

The cosmic microwave background, first observed by Arno Penzias and Robert Wilson in 1964, is a sea of photons in the microwave band which fills the entire observable universe. In whichever direction Penzias and Wilson pointed their telescope they observed this phenomenon, which they initially believed to be caused by an equipment malfunction. Further observations measured this radiation as being a black body spectrum (emitted by an opaque body as a feature of its temperature) with a peak wavelength of 1.873 mm.

Since this radiation is observed everywhere, it must have been emitted everywhere; it could not be a local phenomenon. Logically, this requires the entire universe to have at one time been opaque and suddenly become transparent. When this happened, photons would now be able to move freely through space and - like the light Hubble observed from distance galaxies - red shift as the universe expanded.

There is only one event that could have caused this phenomenon; the spontaneous universal recombination of nuclei and electrons into neutral atoms. When this happened, the universe changed from being a soup of ions constantly interacting with photons to being a somewhat less thick soup of neutral atoms which did not. While neutral atoms can absorb photons, because the universe had cooled so much by this point, the energy of the background photons was several orders of magnitude too low to be absorbed, and so those photons continued on their merry way.

The third significant piece of evidence, the abundance of light elements, is a relic of the period of nucleosynthesis. Since we know what reactions can and therefore would have happened and how long those reactions take, and our model predicts how long the universe would have been at the appropriate temperature to allow those reactions, we can predict the relative abundances of everything up to and including lithium-7.

Our observations of the material composition of the Universe almost exactly match those predictions. At the end of primordial nucleosynthesis, by mass, the universe was approximately 75% hydrogen and 25% helium, with other nuclei making up about 0.01%. Since that time the composition has changed slightly due to stellar nucleosynthesis.

It is important to recognise that the big bang model is constantly being refined as more and better observations can be made. However, the model is extremely robust, and since these discoveries there has been no competing theory.

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Q: What do scientists beliene provides evidence for the big bang theory?
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