Stochastic gravitational wave background from light cosmic strings

Spectra of the stochastic gravitational wave backgrounds from cosmic strings are calculated and compared with present and future experimental limits. Motivated by theoretical expectations of light cosmic strings in superstring cosmology, improvements in experimental sensitivity, and recent demonstrations of large, stable loop formation from a primordial network, this study explores a new range of string parameters with masses lighter than previously investigated. A standard “one-scale” model for string loop formation is assumed. Background spectra are calculated numerically for dimensionless string tensions $G\mu /c^2$ between ${10}^{-7}$ and ${10}^{-18}$, and initial loop sizes as a fraction of the Hubble radius $\alpha$ from 0.1 to ${10}^{-6}$. The spectra show a low frequency power-law tail, a broad spectral peak due to loops decaying at the present epoch (including frequencies higher than their fundamental mode, and radiation associated with cusps), and a flat (constant energy density) spectrum at high frequencies due to radiation from loops that decayed during the radiation-dominated era. The string spectrum is distinctive and unlike any other known source. The peak of the spectrum for light strings appears at high frequencies, significantly affecting predicted signals. The spectra of the cosmic string backgrounds are compared with current millisecond pulsar limits and Laser Interferometer Space Antenna (LISA) sensitivity curves. For models with large stable loops ($\alpha =0.1$), current pulsar-timing limits exclude $G\mu /c^2>{10}^{-9}$, a much tighter limit on string tension than achievable with other techniques, and within the range of current models based on brane inflation. LISA may detect a background from strings as light as $G\mu /c^2\approx {10}^{-16}$, corresponding to field theory strings formed at roughly ${10}^{11}\mathrm{GeV}$.

Figure 1

The gravitational wave energy density per log frequency from cosmic strings is plotted as a function of frequency for various values of $G\mu$, with $\alpha =0.1$ and $\gamma =50$. Note that current millisecond pulsar limits have excluded string tensions $G\mu >{10}^{-9}$ and LISA is sensitive to string tensions $\sim {10}^{-16}$ using the broadband Sagnac technique, shown by the bars just below the main LISA sensitivity curve. The dotted line is the Galactic white dwarf binary background (GCWD-WD) from A. J. Farmer and E. S. Phinney, and G. Nelemans, et al., [70, 71]. The dashed lines are the optimistic (top) and pessimistic (bottom) plots for the extra-Galactic white dwarf binary backgrounds (XGCWD-WD) [71]. Note the GCWD-WD eliminates the low frequency Sagnac improvements. With the binary confusion limits included, the limit on detectability of $G\mu$ is estimated to be $>{10}^{-16}$.

Figure 2

The power per unit volume of gravitational wave energy per log frequency from cosmic strings is measured at different times in cosmic history as a function of frequency for $G\mu ={10}^{-12}$, $\alpha =0.1$, and $\gamma =50$. Note that early in the Universe the values were much larger, but the time radiating was short. The low frequency region is from current loops and the high frequency region is from loops that have decayed. This plot shows the origin of the peak in the spectrum at each epoch; the peaks from earlier epochs have smeared together to give the flat high frequency tail observed today.

Figure 3

The gravitational wave energy density per log frequency from cosmic strings is shown with varying $\alpha$ for $G\mu ={10}^{-12}$ and $\gamma =50$. Here $\alpha$ is given values of 0.1, ${10}^{-3}$, and ${10}^{-6}$ from top to bottom. Note the overall decrease in magnitude, but a slight increase in the relative height of the peak as $\alpha$ decreases; while the frequency remains unchanged. The density spectrum scales as ${\alpha }^{1/2}$, and larger loops dominate the spectrum over small loops.

Figure 4

The gravitational wave energy density per log frequency from cosmic strings for $\alpha ={10}^{-5}$ and $\gamma =50$. Smaller $\alpha$ reduces the background for equivalent string tensions, with a LISA sensitivity limit of $G\mu >{10}^{-14}$ if the Sagnac sensitivity limits are used. The bars below the main LISA sensitivity curve are the Sagnac improvements to the sensitivity, while the dotted line is the Galactic white dwarf binary background (GCWD-WD) from A. J. Farmer and E. S. Phinney, and G. Nelemans, et al., [70, 71]. The dashed lines are the optimistic (top) and pessimistic (bottom) plots for the extra-Galactic white dwarf binary backgrounds (XGCWD-WD) [71]. Note the GCWD-WD eliminates the low frequency Sagnac improvements and increases the minimum detectable $G\mu$ to $>{10}^{-12}$. $G\mu$ ranges from the top curve with a value ${10}^{-7}$ to the bottom curve with ${10}^{-15}$.

Figure 5

The energy density in gravitational waves per log frequency from cosmic strings with varying luminosity $\gamma$ for $G\mu ={10}^{-12}$ and $\alpha =0.1$. Here the $\gamma$’s are given as follows at high frequencies: 50 for top curve, 75 for the middle curve, and 100 for the bottom curve. The peak frequency, peak amplitude, and amplitude of flat spectrum all vary with the dimensionless gravitational wave luminosity $\gamma$.

Figure 6

The gravitational wave energy density per log frequency from cosmic strings for $\gamma =50$, $G\mu ={10}^{-12}$, and a combination of two lengths ($\alpha$’s). 90% of the loops shatter into loops of size $\alpha ={10}^{-5}$ and 10% remain at the larger size $\alpha =0.1$. The overall decrease is an order of magnitude compared to just large alpha alone. From bottom to top the values of $G\mu$ run from ${10}^{-17}$ to ${10}^{-8}$.

Figure 7

The gravitational wave energy density per log frequency for $\gamma =50$, $G\mu ={10}^{-12}$, $\alpha =0.1$; plotted in one case (dashed) with only the fundamental mode, while for another case (solid) with the first six modes. The dashed curve has all power radiated in the fundamental mode, while the solid curve combines the fundamental with the next five modes: $P_1=25.074$, $P_2=9.95$, $P_3=5.795$, $P_4=3.95$, $P_5=2.93$, $P_6=2.30$. Note the slight decrease in the peak and its shift to a higher frequency. The LISA sensitivity is added as a reference.