# Cosmic string loop production functions

## TL;DR

This paper investigates how the shape of the loop production function influences the distribution of cosmic string loops in the universe, revealing conditions under which the distribution reaches a stable scaling regime or not.

## Contribution

It introduces a Boltzmann approach to analyze the impact of the loop production function's steepness on the cosmic string loop distribution, highlighting the importance of the Polchinski-Rocha exponent.

## Key findings

- Sub-critical production functions lead to IR-insensitive, robust scaling distributions.
- Critical and super-critical functions depend on IR physics, preventing stable scaling.
- Regularizations can enable scaling but alter the loop distribution shape.

## Abstract

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.