  Numerical simulations of Nambu-Goto cosmic strings in an expanding universe
show that the loop distribution relaxes to an universal configuration, the
so-called scaling regime, which is of power law shape on large scales. Precise
estimations of the power law exponent are, however, still matter of debate
while numerical simulations do not incorporate all the radiation and
backreaction effects expected to affect the network dynamics at small scales.
By using a Boltzmann approach, we show that the steepness of the loop
production function with respect to loops size is associated with drastic
changes in the cosmological loop distribution. For a scale factor varying as
a(t)~t^nu, we find that sub-critical loop production functions, having a
Polchinski-Rocha exponent chi < (3nu-1)/2, yield scaling loop distributions
which are mostly insensitive to infra-red (IR) and ultra-violet (UV)
assumptions about the cosmic string network. For those, cosmological
predictions are expected to be relatively robust, in accordance with previous
results. On the contrary, critical and super-critical loop production
functions, having chi >= (3nu-1)/2, are shown to be IR-physics dependent and
this generically prevents the loop distribution to relax towards scaling. In
the latter situation, we discuss the additional regularisations needed for
convergence and show that, although a scaling regime can still be reached, the
shape of the cosmological loop distribution is modified compared to the naive
expectation. Finally, we discuss the implications of our findings.
