Updated constraints on the cosmic string tension

We reexamine the constraints on the cosmic string tension from cosmic microwave background (CMB) and matter power spectra, and also from limits on a stochastic background of gravitational waves provided by pulsar timing. We discuss the different approaches to modeling string evolution and radiation. In particular, we show that the unconnected segment model can describe CMB spectra expected from thin string (Nambu) and field theory (Abelian-Higgs) simulations using the computed values for the correlation length, rms string velocity and small-scale structure relevant to each variety of simulation. Applying the computed spectra in a fit to CMB and SDSS data we find that $G\mu /c^2<2.6\times {10}^{-7}$ ($2\sigma$) if the Nambu simulations are correct and $G\mu /c^2<6.4\times {10}^{-7}$ in the Abelian-Higgs case. The degeneracy between $G\mu /c^2$ and the power spectrum slope $n_{\mathrm{S}}$ is substantially reduced from previous work. Inclusion of constraints on the baryon density from big bang nucleosynthesis (BBN) imply that $n_{\mathrm{S}}<1$ at around the $4\sigma$ level for both the Nambu and Abelian-Higgs cases. As a by-product of our results, we find there is “moderate-to-strong” Bayesian evidence that the Harrison-Zel’dovich spectrum is excluded (odds ratio of $\sim 100:1$) by the combination of CMB, SDSS, and BBN when compared to the standard 6 parameter fit. Using the contribution to the gravitational wave background from radiation era loops as a conservative lower bound on the signal for specific values of $G\mu /c^2$ and loop production size, $\alpha$, we find that $G\mu /c^2<7\times {10}^{-7}$ for $\alpha c^2/(\Gamma G\mu )\ll 1$ and $G\mu /c^2<5\times {10}^{-11}/\alpha$ for $\alpha c^2/(\Gamma G\mu )\gg 1$.

Figure 1

(Left) Comparison of cosmic string power spectra computed with the Unconnected Segment Model (USM) using the Nambu—model A—(solid line) and AH mimic—model B—(dotted line) parameters. Spectra have been normalized to the WMAP value of $C_{10}$, giving $G\mu /{10}^{-6}{c}^2=1.18$ for the Nambu model and 1.91 for the AH mimic. (Right) Comparison of scalar, vector and tensor modes for the USM AH mimic (solid line) parameters and the actual AH spectra from simulations (dotted line) [25]. The former has $G\mu /{10}^{-6}{c}^2=1.91$, and the latter 2.04. From top-to-bottom at low $\ell$ the ordering of the curves is the total power, then the anisotropy from vectors, scalars and tensors, respectively.

Figure 2

A montage of plots which show the dependence of the string spectrum on parameters of the USM. In each case the solid and dotted lines are the Nambu and AH mimic models, respectively, from Fig. 1. In all but the top left plot we have varied the one parameter keeping all the others the same as in the AH mimic model: (top left) the dashed line is the USM spectrum with $\xi =0.13$, $v=0.65c$ and $\beta =1.9$ but no matter-radiation transition; (top right) the short-dashed line has $\xi =0.2$ and the long-dashed line has $\xi =0.5$; (bottom left) the short-dashed line has $v=0.0c$ and the long-dashed line has $v=0.8c$; (bottom right) the short-dashed line has $\beta =1.5$ and the long-dashed line $\beta =1.9$.

Figure 3

2D likelihoods of scalar spectral index $n_{\mathrm{S}}$ versus cosmic string amplitude parameter $q_{\mathrm{str}}$ for the Nambu (red, likelihood surface at front) and AH (blue, surface behind) string spectra. In the left panel we show constraints from $\mathrm{CMB}+\mathrm{SDSS}$ data, and on the right constraints from $\mathrm{CMB}+\mathrm{SDSS}+\mathrm{BBN}$.

Figure 4

2D likelihoods of $n_{\mathrm{S}}$ versus ${log}_{10}[G\mu /c^2]$. Labeling is the same as in Fig. 3.

Figure 5

Pulsar constraints on $G\mu /c^2$ as a function of dimensionless loop production size $\alpha$. The various lines show limits from different observational constraints on the gravitational wave background—the solid line is derived from ${\Omega }_g{h}^2<2\times {10}^{-8}$ [65], the dotted line from ${\Omega }_g{h}^2<2\times {10}^{-9}$ [66] and the short-dashed line from ${\Omega }_g{h}^2<9.3\times {10}^{-8}$ [67]. We also show the relation $\alpha =60G\mu /c^2$ (long-dashed).

Figure 6

B-mode polarization power spectra from strings, tensors and gravitational lensing. For the string spectra, we use values of $G\mu /c^2$ corresponding to the $2\sigma$ upper limit from $\mathrm{CMB}+\mathrm{SDSS}$ data, that is. $G\mu /c^2=2.6\times {10}^{-7}$ for the USM Nambu model (solid line) and $6.4\times {10}^{-7}$ for the AH case (dotted line). We also show the inflationary primordial tensor spectrum with $r=0.1$ at $k=0.05{\mathrm{Mpc}}^{-1}$ (short-dashed line) and $r=0.01$ (long-dashed line). Finally, we show the gravitational lensing spectrum generated from E-mode mixing (dot-dashed line) expected in the inflationary model.