Relic Gravitational Waves from Cosmic Strings: Updated Constraints and Opportunities for Detection

TL;DR

This work refines the relic gravitational-wave spectrum from a cosmic-string network by introducing a phenomenological loop-emission model that incorporates gravitational back-reaction via a high-frequency cut-off $n_*$, leading to a robust pulsar-timing bound of $G\mu/c^2 < 5.4(\pm1.1)\times10^{-6}$. The analysis reveals a red-noise portion of the spectrum spanning $f \sim 10^{-8}$–$10^{10}$ Hz, whose amplitude is sensitive to the thermal history through the evolution of relativistic degrees of freedom $g(T)$, and a peaked portion shaped by the loop spectrum index $q$ and $n_*$. Analytic expressions for the spectrum clarify the parameter dependencies, while observational bounds from pulsar timing, nucleosynthesis, and CMB data constrain the viable cosmic-string parameter space. The paper also outlines detection prospects for advanced LIGO/VIRGO and LISA, showing that cross-correlation of a detector network could reveal the stochastic background from cosmic strings if the model holds, thereby enabling a probe of high-energy physics in the early universe.

Abstract

We examine the spectrum of gravitational radiation emitted by a network of cosmic strings, with emphasis on the observational constraints and the opportunities for detection. The analysis improves over past work, as we use a phenomenological model for the radiation spectrum emitted by a cosmic string loop. This model attempts to include the effect of the gravitational back-reaction on the radiation emission by an individual loop with a high frequency cut-off in the spectrum. Comparison of the total spectrum due to a network of strings with the recently improved bound on the amplitude of a stochastic gravitational wave background, due to measurements of noise in pulsar signal arrival times, allows us to exclude a range of values of $μ$, the cosmic string linear mass density, for certain values of cosmic string and cosmological parameters. We find the conservative bound $Gμ/c^2 < 5.4 (\pm 1.1) \times 10^{-6}$ which is consistent with all other limits. We consider variations of the standard cosmological scenario, finding that an under dense, $Ω_0 < 1$ universe has little effect on the spectrum, whereas the portion of the spectrum probed by gravitational wave detectors is strongly sensitive to the thermal history of the cosmological fluid. We discuss the opportunity for the observation of this stochastic background by resonant mass and laser interferometer gravitational wave detectors.

Figures (5)

• Figure 1: The effect of a non-standard thermal history of the cosmological fluid on the amplitude of the red noise portion of the gravitational wave spectrum is shown. The solid curve displays the spectrum produced using a minimal GUT with a maximum $g = 106.75$. The dashed curve shows the spectrum produced allowing for a hypothetical, non-standard evolution of $g(T)$, as might occur if there were a series of phase transitions, or a number of massive particle annihilations as the universe cooled. For temperatures $T > 10^9 \,{\rm GeV}$, the number of degrees of freedom is $g = 10^4$. For $10^5 \, {\rm GeV} < T < 10^9 \, {\rm GeV}$, $g = 10^3$. For $T<10^5 \, {\rm GeV}$, the standard thermal scenario is resumed.
• Figure 2: The effect of a cut-off in the radiation mode number on the spectrum of gravitational radiation is shown. Curves for the loop radiation spectral index $q=2,\, 4/3$ for various values of $n_{*}$ are shown. The vertical line shows the location of the frequency bin probed by pulsar timing measurements. For $n_{*} {{\lesssim}} 10^2$ the shape of the spectrum is insensitive to the value of $q$ for purposes of pulsar timing measurements. For increasing $n_{*}$, more radiation due to late-time cosmic string loops is emitted in the pulsar timing frequency band.
• Figure 3: The effect of a low density, $\Omega_0 < 1$ universe on the peaked portion of the gravitational wave spectrum. The solid, long- and short-dashed curves represent spectra for $\Omega_0 = 1, \,0.6, \,0.2$. The vertical line shows the location of the frequency bin probed by pulsar timing measurements. For the loop spectral index $q=2$, a low density universe dilutes only the lowest frequency waves, corresponding the radiation emitted by loops still present today.
• Figure 4: The effect of a low density, $\Omega_0 < 1$ universe on the peaked portion of the gravitational wave spectrum. The solid, long- and short-dashed curves represent spectra for $\Omega_0 = 1, \,0.6, \,0.2$. The vertical line shows the location of the frequency bin probed by pulsar timing measurements. For the loop spectral index $q=4/3$, a low density universe leads to a dilution of gravitational waves with wavelengths up to $f \sim 10^{-5}\, {\rm Hz}$.
• Figure 5: Curves of constant $\Omega_{\rm gr}$ in $(\alpha, G\mu/c^2)$ parameter space are shown. For a given value of $\alpha$, these figures give the observational bound on $G\mu/c^2$ in the case $h=0.5, \, 0.75$. In each figure, the constraining curves for $q=10,\, 2,\, 4/3$ are given by the solid, long-, and short-dashed curves. The light dashed lines show $\alpha = \Gamma G\mu/c^2$. The most conservative constraint is $G\mu/c^2 < 5.4 \times 10^{-6}$.