Cosmic string loop distribution on all length scales and at any redshift
  We analytically derive the expected number density distribution of Nambu-Goto
cosmic string loops at any redshift soon after the time of string formation to
today. Our approach is based on the Polchinski-Rocha model of loop formation
from long strings which we adjust to fit numerical simulations and complement
by a phenomenological modelling of gravitational backreaction. Cosmological
evolution drives the loop distribution towards scaling on all length scales in
both the radiation and matter era. Memory of any reasonable initial loop
distribution in the radiation era is shown to be erased well before Big Bang
Nucleosynthesis. In the matter era, the loop distribution reaches full scaling,
up to some residual loops from the radiation era which may be present for
extremely low string tension. Finally, the number density of loops below the
gravitational cutoff is shown to be scale independent, proportional to a
negative power of the string tension and insensitive to the details of the
backreaction modelling. As an application, we show that the energy density
parameter of loops today cannot exceed 10^(-5) for currently allowed string
tension values, while the loop number density cannot be less than 10^(-6) per
Mpc^3. Our result should provide a more robust basis for studying the
cosmological consequences of cosmic string loops.
