Constraints on cosmic string tension imposed by the limit on the stochastic gravitational wave background from the European Pulsar Timing Array

TL;DR

This work addresses constraining the cosmic string tension $G\mu/c^2$ via the stochastic gravitational wave background limits from pulsar timing arrays, while explicitly accounting for large uncertainties in loop production and GW emission. It implements a one-scale network model to compute the GW spectrum by tracking the loop number density $n(\ell,t)$, loop evolution, and harmonic emission with a cutoff $n_*$ and spectral index $q$, propagating through cosmic expansion to obtain $\Omega_{\rm gw}(f)$. The authors explore a wide parameter space, including variations in loop size $\alpha$, emission parameters $(q,n_*)$, intercommutation probability $p$, and extensions to two-scale and log-normal loop distributions, ultimately deriving a robust conservative bound $G\mu/c^2<5.3\times10^{-7}$ from the EPTA limit and discussing prospects for orders-of-magnitude improvements with LEAP and SKA. They show that the PTA-derived constraint is highly sensitive to the assumed emission physics and loop distribution, and present comparative constraints with LIGO, highlighting the complementary nature of PTA and ground-based detectors. The results indicate that future PTA capabilities could either detect a cosmic-string SGWB or tighten the tension constraints by several orders of magnitude, depending on the true loop production physics.

Abstract

We investigate the constraints that can be placed on the cosmic string tension by using the current Pulsar Timing Array limits on the stochastic gravitational wave background (SGWB). We have developed a code to compute the spectrum of gravitational waves (GWs) based on the widely accepted one-scale model. In its simplest form the one-scale model allows one to vary: (i) the string tension, Gμ/c^2; (ii) the size of cosmic string loops relative to the horizon at birth, α; (iii) the spectral index of the emission spectrum, q; (iv) the cut-off in the emission spectrum, n_*; and (v) the intercommutation probability, p. The amplitude and slope of the spectrum in the nHz frequency range is very sensitive to these unknown parameters. We have also investigated the impact of more complicated scenarios with multiple initial loop sizes, in particular the 2-αmodels proposed in the literature and a log-normal distribution for α. We have computed the constraint on Gμ/c^2 due to the limit on a SGWB imposed by data from the European Pulsar Timing Array. Taking into account all the possible uncertainties in the parameters we find a conservative upper limit of Gμ/c^2<5.3x 10^{-7} which typically occurs when the loop production scale is close to the gravitational backreaction scale, α\approxΓGμ/c^2. Stronger limits are possible for specific values of the parameters which typically correspond to the extremal cases α\ll ΓGμ/c^2 and α\gg ΓGμ/c^2. This limit is less stringent than the previously published limits which are based on cusp emission, an approach which does not necessarily model all the possible uncertainties. We discuss the prospects for lowering this limit by two orders of magnitude, or even a detection of the SGWB, in the very near future in the context of the Large European Array for Pulsars and the Square Kilometre Array.

Figures (7)

• Figure 1: The GW energy density per logarithmic frequency interval $\Omega_{\rm{gw}}(f)h^2$ of a cosmic string network with $G\mu/c^2=10^{-7}$, $\alpha=10^{-3}$ and $n_{*}=1$. The black (solid) line is the full spectrum from the network due to loops formed in both radiation and matter eras, whereas the red (dashed) line is that from the radiation-dominated era and the blue (dot-dashed) line is from the matter-dominated era. The grey shaded area shows the frequency window probed with the highest sensitivity by PTA experiments with duration between 5 and 10 years.
• Figure 2: The GW sensitivity curves for a 10-year (black thick line) and a 5-year (black dashed line) PTA experiment, with the 10-year experiment achieving slightly better maximum sensitivity. The frequencies where these experiments achieve maximum sensitivity are $3.2\,{\rm nHz}$ and $6.3\,{\rm nHz}$ respectively. The red thick line is the GW spectrum of a cosmic string network for $\alpha_1=5.7\times10^{-10}$ and the red dashed line is the spectrum for $\alpha_2=2.8\times10^{-10}$ network. While the 10-yr experiment has a greater overall sensitivity at its minimum frequency, it has a lower sensitivity at the frequencies to which the 5-year experiment is sensitive to (see text for details).
• Figure 3: Regions of the $\alpha -n$ parameter space which can be probed by PTA experiments. The dark gray region includes all the cosmic string network configurations which create a SGWB probed at maximum sensitivity by a 10-year PTA experiment. Additionally to this region, the light gray slice includes all the extra configurations which can be probed at maximum sensitivity by a 5-year PTA experiment. The white area includes all those configurations which are probed by the reduced sensitivity slope (see, Fig. \ref{['fig::ptasens']}) for both 5- and 10-year experiments. The hatched area includes the configurations which are inaccessible to PTAs.
• Figure 4: Plots of normalized gravitational wave energy density per logarithmic frequency interval, $\Omega_{\rm gw}h^2$, due to cosmic string networks with different tensions but the same fiducial values of $\alpha$, $n_*$, $q$ and $p$. The thick blue lines are for networks in the large loop regime and the thin red lines are from networks in small loop regime. The dashed black line signifies the network for which $\alpha =\Gamma G\mu/c^2$. The analytic approximations of the peak frequency are also shown: the approximation found in CA92 (red long dashed curve) and our improved approximation (short dashed green curve).
• Figure 5: $\Omega_{\rm gw}h^2$ for cosmic string networks with different values of $\alpha$ and the fiducial values of $G\mu/c^2$, $n_*$, $q$ and $p$. With thick blue lines we plot the networks in the regime of large loops and with thin red lines the networks in the regime of small loops. With dashed line we plot the network with $\alpha=\Gamma G\mu/c^2$ which signifies the critical point after which we have no amplitude decrease.