Reading pulsar-timing and LISA results on cosmic strings

This guide shows where each of those statements sits in the papers and how much weight it carries. It assumes you have read From a loop population to the background spectrum. The two sources read in full for it are NANOGrav Collaboration, 2023, arXiv version 1, Section 5.4 and figures 9 to 11, and Auclair et al., 2020, arXiv version 1, Sections 6.1 and 6.2 and figure 8. Section and equation numbers below refer to those versions.

Four kinds of statement

Sort every sentence about cosmic strings and gravitational waves into one of four kinds before weighing it.

The NANOGrav numbers are quoted from arXiv version 1, which this guide did not compare with the journal version. The Auclair paper appeared in JCAP 04 (2020) 034; the version read here is arXiv version 1.

What a Bayes factor and a K ratio say

NANOGrav compares each new-physics model with a reference model in which the background comes from inspiralling supermassive black-hole binaries. A Bayes factor $B$ above one favours the new model, below one the binaries. $B$ depends on the prior range given to each parameter: widening a prior that the data do not use lowers $B$ even when the fit is unchanged. NANOGrav Collaboration, 2023, Section 3.2

The K ratio of eq. 10 compares the likelihood at a parameter value with the likelihood where the new signal is negligible. Where $K$ falls below $1/10$ the predicted signal exceeds what the data allow. NANOGrav stresses that these excluded regions do not depend on where the observed signal comes from. That makes them the safest numbers to quote from the paper.

Stable strings: too weak or too flat

For four stable-string models, which differ in how the power is shared among harmonics (cusps, kinks, a single fundamental mode, or simulated loops), the Bayes factors against the binary model are $0.277$, $0.364$, $0.379$ and $0.307$. Adding a binary population to the strings gives $0.76$ to $0.89$. Stable strings are the only new-physics source in the paper that consistently favours the binaries. NANOGrav Collaboration, 2023, Section 5.4

The reason is spectral shape. A tension low enough to give a rising spectrum in the pulsar-timing band gives too little amplitude; a tension high enough for the amplitude gives a spectrum that is too flat. Figure 9 shows the resulting posteriors peaking near $\log_{10}G\mu\sim-10.5$ to $-10$ without binaries.

With binaries included, the paper gives two kinds of upper limit on $\log_{10}G\mu$ for the four models:

These are published limits for one set of loop-model assumptions. The paper uses Model I loops with $\alpha_{\mathrm L}\simeq0.37$, $F=0.1$ and harmonics up to $10^5$, close to the model in the Lab.

Superstrings and metastable strings fit better, with caveats

Superstrings reconnect with probability $P<1$. NANOGrav models their spectrum by dividing the ordinary spectrum by $P$, eq. 52, with $\log_{10}P$ between $-3$ and $0$. They obtain $B=46\pm2$ without binaries and $37\pm2$ with them, the largest for any string model. Figure 11 shows a strong correlation between $G\mu$ and $P$. The highest posterior density sits at $\log_{10}P\sim-3$, the edge of the prior, and $\log_{10}G\mu\sim-12$, a region in tension with the LIGO–Virgo–KAGRA background bound unless the early expansion history was non-standard.

Metastable strings, whose network decays by monopole nucleation, give $B=13.4$ and $21.3$ for two variants, falling to $11.1$ and $18.9$ with binaries. The preferred tensions, $\log_{10}G\mu\sim-7$ to $-4$, mostly conflict with the ground-based bound, which corresponds to about $G\mu\lesssim2\times10^{-7}$ in their model.

Three caveats come from the paper itself. The abstract says the comparisons with binaries strongly depend on modelling assumptions about the binary population and should not be regarded as evidence for new physics. The superstring section says their spectrum is not well understood and more work is needed for a reliable prediction. And the ground-based bounds rely on extrapolating the spectrum across ten decades in frequency through an unknown early expansion history.

A fitted spectrum is not a detection

A good Bayes factor means a model's spectral shape matches a measured amplitude and slope in one narrow band. Other sources can produce the same shape. Telling strings apart from binaries would need a signature that binaries cannot mimic: bursts from cusps, a spectrum confirmed in a second frequency band, or a consistent amplitude across bands. None of these has been reported in the sources read here.

How a second band breaks a degeneracy

In a single band, tension and reconnection probability trade off against each other. The Lab's model shows this. With $\alpha=0.1$ and cusps:

The two cases agree within $1.2\,\%$ at one cycle per year but differ by a factor of $7.7$ at $3\,\mathrm{mHz}$, in the LISA band. These are model predictions from the Lab's teaching model with the simple $1/P$ rescaling, not fits to data.

The rescaling itself is uncertain. Auclair and colleagues instead write the chopping efficiency as $\tilde c(P)=\tilde c\,P^{\gamma}$, eq. 6.10, citing simulations that give $\gamma=1/2$ or $1/3$ rather than the naive $\gamma=1$. Auclair et al., 2020, Section 6.2.2 and Table 1 Different choices move the same data to different tensions. Which scaling holds is still open.

What LISA is expected to reach

LISA is a planned space mission, so every LISA statement is a model prediction. Auclair and colleagues find that the trailing edge of the spectrum's bump is the last part to leave the LISA band as the tension falls, and that three loop models all give a reach of order $G\mu\sim10^{-17}$. Figure 8 compares the three models with the LISA power-law sensitivity at $G\mu=10^{-17}$. Changing the harmonic index has at most order-one effects on that reach. Auclair et al., 2020, Section 6.1

Two printed numbers in that paper are not reproduced by the same model: the plateau coefficient of eqs. 6.1 and 6.2, and the plateau of $6.04\times10^{-14}$ quoted after figure 8. The recomputation guide records both as unresolved.

Claims and where to check them

To see the degeneracy yourself, open the cosmic-string Lab at its default, then lower the tension to $10^{-12}$ and the reconnection probability to $10^{-1.8}$. The pulsar-timing readout returns almost to its starting value while the LISA readout grows about eightfold.