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We analytically derive the expected number density distribution of Nambu-Gotocosmic string loops at any redshift soon after the time of string formation totoday. Our approach is based on the Polchinski-Rocha model of loop formationfrom long strings which we adjust to fit numerical simulations and complementby a phenomenological modelling of gravitational backreaction. Cosmologicalevolution drives the loop distribution towards scaling on all length scales inboth the radiation and matter era. Memory of any reasonable initial loopdistribution in the radiation era is shown to be erased well before Big BangNucleosynthesis. In the matter era, the loop distribution reaches full scaling,up to some residual loops from the radiation era which may be present forextremely low string tension. Finally, the number density of loops below thegravitational cutoff is shown to be scale independent, proportional to anegative power of the string tension and insensitive to the details of thebackreaction modelling. As an application, we show that the energy densityparameter of loops today cannot exceed 10^(-5) for currently allowed stringtension values, while the loop number density cannot be less than 10^(-6) perMpc^3. Our result should provide a more robust basis for studying thecosmological consequences of cosmic string loops.

Cosmic string loop distribution on all length scales and at any redshift