CMB power spectrum contribution from cosmic strings using field-evolution simulations of the Abelian Higgs model

We present the first field-theoretic calculations of the contribution made by cosmic strings to the temperature power spectrum of the cosmic microwave background (CMB). Unlike previous work, in which strings were modeled as idealized one-dimensional objects, we evolve the simplest example of an underlying field theory containing local U(1) strings, the Abelian Higgs model. Limitations imposed by finite computational volumes are overcome using the scaling property of string networks and a further extrapolation related to the lessening of the string width in comoving coordinates. The strings and their decay products, which are automatically included in the field theory approach, source metric perturbations via their energy-momentum tensor, the unequal-time correlation functions of which are used as input into the CMB calculation phase. These calculations involve the use of a modified version of CMBEASY, with results provided over the full range of relevant scales. We find that the string tension $\mu$ required to normalize to the WMAP 3-year data at multipole $\ell =10$ is $G\mu =[2.04\pm 0.06(\mathrm{stat}.)\pm 0.12(\mathrm{sys}.)]\times {10}^{-6}$, where we have quoted statistical and systematic errors separately, and $G$ is Newton’s constant. This is a factor 2–3 higher than values in current circulation.

Figure 1

Slices through a ${512}^3$ simulation in the radiation era, showing the Abelian Higgs analogue of magnetic flux density. Magnetic flux tubes run along the cosmic strings, which appear as dark regions, with a shape dependent upon the nature of the string intersection with the slice. Varying left to right, the horizon size (measured as $2\tau$) is 0.35, 0.63, and 1.00 times the boxside while the string width decays inversely with the scale factor $a$ and hence as ${\tau }^{-1}$ ($a\propto \tau$). Note the decay products visible in these images.

Figure 2

The variation of the string width $r$ and coupling parameters with conformal time $\tau$ (in units of ${\varphi }_0^{-1}$) for the $s=0.3$ simulations described in Sec. 2c. The subscript 0 indicates the value at the end of the simulation.

Figure 3

Results for the Lagrangian measure of $\xi$ from 5 simulations in the radiation era with $s=1$ (top) and $s=0$ (lower). The shaded regions show the $1-\sigma$ and $2-\sigma$ variations in the mean (error bars have not been used since correlations extend across most of the plot and would tempt the reader into believing individual points were independent). The best-fit straight lines over the region $80{\varphi }_0^{-1}<\tau <128{\varphi }_0^{-1}$ ($s=1$) and $64{\varphi }_0^{-1}<\tau <128{\varphi }_0^{-1}$ ($s=0$) are also shown. Note that for the later case, this excludes the acausal final period.

Figure 4

The raw equal-time scaling function ${\tilde{C}}_{11}^{\mathrm{S}}(k\tau ,k\tau )$ as averaged over 5 realizations for $s=0.3$ in the radiation era. Results are plotted at roughly uniformly-spaced $\tau$ values in the range $64{\varphi }_0^{-1}<\tau <128{\varphi }_0^{-1}$. The lower lines correspond to increasingly early times (dashed), with the $1-\sigma$ and $2-\sigma$ uncertainties in the mean indicated for the latest time (solid).

Figure 5

The equal-time scaling function ${\tilde{C}}_{11}^{\mathrm{S}}(k\tau ,k\tau )$ as in the previous figure, but with the time offset taken into account. For the present plot, the mean offset across the 5 realizations is used to adjust results for each one, whereas the actual CMB calculations use independent offsets for each realization.

Figure 6

The temporal dependence of the offset corrected ${\tilde{C}}_{11}^{\mathrm{S}}(k\tau ,k\tau )$ at four fixed values of $k\tau$: 17, 50, 110 and 220 (which appear $\mathrm{\text{top}}\to \mathrm{\text{bottom}}$ in the plot). Results are from 5 realizations with $s=0.3$ in the radiation era, using a common time offset for each realization.

Figure 7

The dependence of the $\tilde{C}(k\tau ,k\tau )$ upon $s$ in the radiation era at ${\tau }_{\mathrm{sim}}=128{\varphi }_0^{-1}$. The shaded regions show the $1-\sigma$ and $2-\sigma$ uncertainties for the $s=1$ case, while the lines indicate the $s=0.3$ (solid), $s=0.2$ (dashed) $s=0.1$ (dashed-dotted) results.

Figure 8

The unequal-time scaling functions ${\tilde{C}}_{11}^{\mathrm{S}}$ in the radiation era (left) and matter era (right). Results are from 5 realizations with $s=0.3$. Each realization is given the average offset time, hence there is no interpolation involved which might otherwise smooth these plots. The data is then raw, albeit it has been extrapolated as a constant for low $k\sqrt{\tau {\tau }^{\prime }}$ and the relative time symmetries have been used since the simulations output only for $\tau >{\tau }^{\prime }$. Note there are large statistical uncertainties which cannot be shown here.

Figure 9

The unequal time scaling functions from a global texture model, in order to provide a comparison case for the previous figure. The results shown are from 10 realizations in the radiation era (left) and matter era (right), using a ${256}^3$ lattice. Note the difference in the axis scales between this and the string case.

Figure 10

The remaining unequal-time correlation functions from $s=0.3$ string simulations. For the cross-correlation function ${\tilde{C}}_{12}^{\mathrm{S}}$ only the magnitude is shown and by virtue of Eq. (38) this is almost entirely anticorrelated. Note that its form can vary significantly from realization-to-realization and hence it suffers from very large statistical uncertainties. The vertical axes are constant within each scalar-vector-tensor class (including Fig. 8) so as to aid size comparisons.

Figure 11

The convergence of the eigen-contribution sum for the temperature power spectrum due to cosmic strings. Results are for $s=0.3$ with (dashed) 1, 2, 4, 8, 16, 32, 64, and (solid) 128 terms included.

Figure 12

The CMB power spectrum contribution from cosmic strings simulated with $s=0.3$ (solid), 0.2 (dashed), and 0.0 (dashed-dotted). The estimated $1-\sigma$ and $2-\sigma$ uncertainties (in the mean) are indicated in the $s=0.3$ case by the shaded regions.

Figure 13

The decomposition of the CMB power spectrum (solid) into scalar (dashed), vector (dashed-dotted), and tensor (solid-gray) modes. Results are shown for the $s=0.3$ case.

Figure 14

A comparison of the form of the $s=0.3$ string contribution (solid) to the WMAP results [62], enabled by an excessively large normalization to give a match to the data at $\ell =10$. Also shown are results from our global texture simulations (dashed). Note that the binned WMAP 3-year data are plotted which does not include output for $\ell =10$ and that $T$ is the mean CMB temperature.