We have recently made public a rather technical paper in which we explicitly and unambiguously calculate the cosmic string loop number density distribution in our universe coming from various motivated loop production functions. Such a density distribution crucially determines the spectrum of their emitted gravitational waves, a smooking gun for their potential discovery.

Together with **Pierre Auclair**,
**Mairi Sakellariadou**
and **Danièle Steer**, we have
carefuly explored in Ref.[1] the consequences of
changing the slope of the Polchinski-Rocha loop production function to the actual
observable cosmic string loop distribution. This is the parameter
\(\chi\) in the next plot:

For the cases we have referred to as “sub-critical”, \(\chi < (3 \nu - 1)/2\), corresponding to steep slopes in the previous figure, we recover the results of Ref. [2]. The parameter \(\nu\) encodes the growing rate of the scale factor \(a(t)\), namely, we assume \(a(t) \propto t^\nu\).

For shallower slopes, the so-called “critical” and “super-critical” cases, \(\chi \ge (3\nu-1)/2\), we find that either the loop distribution incessantly grows, or, with some regularisation, reaches a stationnary distribution whose shape depends on what happens on the larger length scales. This is best illustrated by the following plot:

It compares the loop number density produced by assuming an infinitely
sharp loop production function peaked at \(\gamma = 0.1\) (green
curve) with a (regularised) super-critical Polchinski-Rocha
distribution having \(\chi > 0.25\) (purple curve), in the radiation
era (\(\nu=1/2\)). As this plot shows, the gravitational backreaction
scale, \(\gamma_\mathrm{c}\), at which the loop production function is
cut **always matters** and is responsible for the plateau on the
purple curve at small \(\gamma\) values. The existence of a second
plateau around \(\gamma \simeq 10^{-6}\) comes from the Infra-Red
sensitivity of all super-critical loop production functions.