What the tension of a cosmic string measures

You need powers of ten, the idea of a density and the Hubble rate $H$, the fractional rate at which the universe expands. From the next section on we use units with $c=1$, so the number is written simply as $G\mu$.

Mass per length and tension

A Nambu–Goto string is the idealisation of a string so thin that only its length matters. Its energy per unit length $\mu$ and its tension are equal in these units. If the string came from a field theory, $\mu$ is about $\eta^2$, where $\eta$ is the energy scale of the transition that made it, and its width is about $1/\eta$, far smaller than any scale we care about here. Auclair et al., 2020, Section 1

To restore units, divide the dimensionless number by $G/c^2$:

Why one dimensionless number controls gravity

Every gravitational effect of the string involves Newton's constant together with $\mu$, so the only combination that can appear in a dimensionless result is $G\mu$. A loop radiates gravitational waves with power $\Gamma G\mu^2$, where the pure number $\Gamma$ depends on the loop's shape and is about 50 for realistic loops. Dividing by the loop's rest energy $\mu\ell$ gives a fractional loss rate proportional to $G\mu$. Auclair et al., 2020, Section 3

The connection to particle physics is

A string from a grand-unification scale near $10^{16}\,\mathrm{GeV}$ has $G\mu$ near $10^{-6}$. Measurements of the cosmic microwave background already require $G\mu$ below about $10^{-7}$ for several string types. Auclair et al., 2020, eq. 1.1 and Section 1

A network that keeps its share of the energy

Long strings stretch across the visible universe and are separated by a typical distance $L$, defined by $\rho_\infty=\mu/L^2$, where $\rho_\infty$ is their energy density. Simulations and the velocity-dependent one-scale model agree that $L$ grows in proportion to cosmic time. Write $L=\xi t$. In the radiation era $\xi_{\mathrm r}=0.271$ and the root-mean-square string speed is $0.662$; in the matter era $\xi_{\mathrm m}=0.625$. This is called scaling. Auclair et al., 2020, Section 3.1

Scaling means the strings never come to dominate the universe. In the radiation era $H=1/(2t)$, so the critical density is $\rho_{\mathrm c}=3H^2/(8\pi G)=3/(32\pi Gt^2)$. The fraction in long strings is

which contains no time at all.

Stretching alone would make the strings longer and their share larger. The network stays in scaling because it keeps cutting string into loops. In the one-scale model the energy put into loops per unit volume and time is $\tilde c\,\bar v\,\rho_\infty/L$, with the chopping efficiency $\tilde c=0.23$ taken from simulations.

Where new loops come from

The simplest model, Model I of Auclair and colleagues, assumes every loop is born with the same fraction of the string separation, $\ell=\alpha_{\mathrm L}L$. In units of cosmic time the birth size is $\alpha=\alpha_{\mathrm L}\xi$. NANOGrav uses $\alpha_{\mathrm L}\simeq0.37$ and $\xi_{\mathrm r}\simeq0.27$, which gives $\alpha\simeq0.10$ deep in the radiation era. NANOGrav Collaboration, 2023, Section 5.4, eqs. 43 to 45

Loops then shrink at the steady rate $\mathrm d\ell/\mathrm dt=-\Gamma G\mu$. A loop born at time $t_{\mathrm i}$ with length $\alpha t_{\mathrm i}$ disappears after

Because the network scales, the number of loops per volume and per length takes the form $n(\ell,t)=t^{-4}\,n(x)$ with $x=\ell/t$. Within one era, a single function of the ratio $x$ describes the loops at every time. Auclair et al., 2020, eq. 3.1

Lab task

Open the cosmic-string Lab and keep the loop size at its default. Read the lifetime as you move the tension from $10^{-10}$ to $10^{-17}$, then set the tension to $10^{-12}$ and check the value against the formula for $\tau$.

Exercises

1. Find the mass per length of a string with $G\mu=10^{-12}$.
2. A loop with $G\mu=10^{-12}$ is born with $\alpha=0.1$. How many times the birth age does it live?
3. What fraction of the critical density do long strings with $G\mu=10^{-17}$ hold in the radiation era?
4. In the one-scale model, what fraction of the long-string energy goes into loops during one unit of cosmic time $t$ in the radiation era?