Am I correct in understanding that in string theory before the birth of the universe, fundamental strings did not interact with each other, and when the strings began to interact, space-time appeared?
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Your question cannot be answered unless you specify exactly what concrete scenario you have in mind.
We haven't reached a complete understanding of how spacetime emerge in the context of string theory; because of that, all standard string inspired cosmologies assume preexisting backgrounds over which strings can propagate.
Some scenarios such as the string gas scenario take as work hypothesis a large system of weakly coupled strings, but others, such as brane inflation provide configurations where strings can interact at finite coupling.
Further references:
- R. H. Brandenberger, "String gas cosmology: progress and problems", Class. Quantum Grav. 28 204005 (2011), arXiv:1105.3247.
- Book "The Universe Before the Big Bang: Cosmology and String Theory" by Maurizio Gasperini, Springer (2008).
- L. McAllister and E. Silverstein, "String cosmology: a review", Gen. Rel. Grav. 40, 565-605 (2008), arXiv:0710.2951.
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