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We are taught that the universe began as a singularity - an infinitely small and infinitely dense point. At the beginning of time there was a 'Big Bang' or, more accurately, 'Inflation'.

The main evidence for this is the observation of the red shift of all of the galaxies. This shows us that as time increases, the universe becomes bigger. A logical outcome of this is that going back in time, the universe shrinks. This is then extrapolated back to the beginning of time, where the universe was infinitely small.

However, the only thing that I can see that we know for sure is that the universe used to be smaller. This does not necessarily imply that it used to be infinitely small. How do we know, for example, that the universe doesn't oscillate and that we are in a time where the universe is expanding (and accelerating) and will eventually contract again?

What other evidence is there that suggests we started from an singularity?

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A terminology note: the stage after Big Bang, where universe is getting bigger, is called expansion. Inflation is a very short time interval in the history of the universe when the expansion was exponentially fast and it is a theory used to explain some observed phenomena such as local fluctuation in the cosmic background radiation's temperature. –  Marek Dec 26 '10 at 19:28
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There are 3 observations that support the big bang theory, i.e. origin of the universe in a singularity:

  1. The redshift of galaxies, as you already mentioned.
  2. The cosmic background radiation.
  3. The amounts of different nuclei in the universe, notably the preponderance of light elements like hydrogen and helium.

Each of these alone would probably not be sufficient to support the big bang theory. The redshift of galaxies could be explained by some other theory, some have been suggested by Hoyle and Narlikar in the past. Probably the other two phenomena could be explained independently as well, but it is the conjunction that fits so well with the big bang hypothesis.

Does that settle the matter once and for all? Short answer is no. Since these 3 observations have been made and confirmed, more detailed observations have been added to the mix and this has complicated the story for the big bang model. But that would take us into a longer post. The current model which is the most widely accepted is the so-called Lambda-CDM model.

As for the problem of the universe starting in a real singularity, instead of a very dense state, this is still an open problem related to a yet to be invented (or completed) theory of quantum gravity. Our current understanding of singularities in General Relativity is going back to the Penrose-Hawking singularity theorems. They are of the kind "Here be dragons!" in that they delineate the conditions for singularities to form and point where our knowledge ends. More can not be done, because a singularity is basically a failure of the theory.

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more specifically, the CMB proves (at least "strongly supports") inflation theory, since points on the sky separated by more than the light-travel time since the big bang appear in thermal equilibrium, suggesting that they were once in close proximity. The form of the CMB power spectrum also suggestions the inflation of primordial quantum density perturbations. –  Jeremy Dec 26 '10 at 22:55
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If someone taught you that the expansion of the universe necessarily implies a singularity, then he was wrong. The past singularity is not the only possible initial condition. There are actually several models that predict different scenarios. For example there is the eternal inflation scenario, where there is no initial singularity. There is also Turoks ecpyrotic model, which has big bangs without singularities and there are many other models.

Regarding the oscillation, things are more straightforward. The dynamics of the universe depend on the mass content of the universe. The mass content is measured by the energy density parameter. If that parameter is over a critical value, then the universe is closed and could oscillate as you say. If the density is smaller than or equal to the critical, then the universe expands for ever. The density of the universe is measured in the cosmic microwave background and it is measured to be almost exactly the critical. So the universe is not going to recollapse. But that doesn't exclude something like the Turok oscillating model.

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I grew up with Hoyles steady state universe. The universe was expanding, but continuous creation of matter (light elements to match observed abundance) was postulated so that the universe had a (quasi) steady form (i.e. density of the universe was maintained throughout time). It sounds a bit magical to be sure, but I don't think it could be conclusively cast aside until the CMB was discovered.

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This is an excellent question. It indeed may be true that the universe oscillates, and that we are not experiencing just one of the phases of this oscillation. (This theory, which is plagued by some rather technical problems but is still viable, sometimes goes under the name of 'cyclic universe').

What we do know, mainly from the three pillars of cosmology that Raskolnikov explained, is that the universe was small and hot early on. But we do not factually know whether the Big Bang (ie. singularity) actually happened, and we do not know what happened before inflation (i.e. before about 10^{-35} seconds after the Big Bang).

So while the smart money is still betting on the existence of the singularity - being arguably the simplest theory that also passes all observational tests with flying colors - future observations will likely able to distinguish between it and the cyclic scenario.

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Until further clarification the answer is NO:
The claim that 'space expands' must be backed by the claim that 'the atom is invariant'.

A bigger atom in the past is observed as a redshift (a longer wavelength) because the standard rule we use to measure it is based in the actual atomic dimension.
In a figurative way: to measure wavelengths of the past we must use 'inches' and now we use 'cm'. The same amount of distance gives two different measures. Did the distance increased ? or the unit of measure decreased?

A claim that the atom can vary in size, using the same laws of physics, can be found here: A self-similar model of the Universe unveils the nature of dark energy.

Asserting that the atom is invariant is equal to say: 'Here is an absolute reference frame', against Einstein.

From now on a clarification is needed. imo.

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Actually, the "invariance" of the atom is a statement of the Einstein Equivalence Principle. There are experimental constrains on the variation with time of the physical constants (the fine structure constant in particular). –  Vagelford Jul 8 '11 at 13:54
    
How to read that Einstein EP equals 'invariance'? quoting the paper "we cannot rely on the apparent invariance of bodies’ based length unit, as stressed by Einstein[3] when he called “reference-mollusk” to the reference-body;" ; see Table I, page 10, Fine structure constant is not scalable (its really a constant); and the same with Force; EP (gravitational mass=inertial mass, the ratio = 1 watever the mass). –  Helder Velez Jul 8 '11 at 17:19
    
John Webb, 1998 "A Search for Time Variation of the Fine Structure Constant", and also more recently. But no one know how can $\alpha$ vary and keep the phy.laws. A self-similar model explains Webb's words (pag.4) "What other physical phenomena could give rise to the observational effect we report? The spacing of the MgII and FeII isotopes is such that a significant change in the isotopic ratios could explain the observations." When I look to the table of isotopes I think: the ratios can change thru time. This paper sheds light into Webb's measures. –  Helder Velez Jul 8 '11 at 17:46
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