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Limestones are formed mostly from skeletal fragments of marine organisms.

However, calcium carbonate dissolves naturally in water, so the living organisms are not the only source of it in water.

Is it possible for limestone to form without living organisms? Are such deposits observed? Or is the presence of limestone a clear indicator that life is, or was present on given planet?

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Oolites are limestones that are usually considered as in-organic, although they may have bits of shell/etc in them. These form by the precipitation of calcium carbonate around particles (sand, broken shell, etc) with a process comparable to that of an oyster.

Some oolite references:

Recently I saw a reference that limestones do not deposit when Fe2+ is present. Unfortunately I can't find the reference now. If this is the case, then for the Earth, limestones require the indirect presence of life. The evolution of cyanobacteria created free oxygen in the atmosphere and the dissolved in the oceans. Hence all dissolved iron was oxidised from Fe2+ (a green colour) to Fe3+ (orange colour).

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I would add to your answer that there is travertine, a non-biogenic calcium carbonate rock. – Michael Aug 1 '14 at 13:46

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