CMB power spectra from cosmic strings: Predictions for the Planck satellite and beyond

We present a significant improvement over our previous calculations of the cosmic string contribution to cosmic microwave background (CMB) power spectra, with particular focus on sub-WMAP angular scales. These smaller scales are relevant for the now-operational Planck satellite and additional suborbital CMB projects that have even finer resolutions. We employ larger Abelian Higgs string simulations than before and we additionally model and extrapolate the statistical measures from our simulations to smaller length scales. We then use an efficient means of including the extrapolations into our Einstein-Boltzmann calculations in order to yield accurate results over the multipole range $2\le \ell \le 4000$. Our results suggest that power-law behavior cuts in for $\ell \gtrsim 3000$ in the case of the temperature power spectrum, which then allows cautious extrapolation to even smaller scales. We find that a string contribution to the temperature power spectrum making up 10% of power at $\ell =10$ would be larger than the Silk-damped primary adiabatic contribution for $\ell \gtrsim 3500$. Astrophysical contributions such as the Sunyaev-Zeldovich effect also become important at these scales and will reduce the sensitivity to strings, but these are potentially distinguishable by their frequency-dependence.

Figure 1

Results for $\xi$ from simulations using our new initial conditions compared to those using BHKU initial conditions, which yielded an offset scaling law. Statistical uncertainties determined from 3 realizations are denoted using the shaded regions ($1\mathrm{\text{−}}\sigma$ dark, $2\mathrm{\text{−}}\sigma$ light) and the linear best-fits over the period between the short vertical lines are indicated by the dashed lines. The results come from ${1024}^3$, $s=0$, $\Delta x=0.5{\varphi }_0^{-1}$ simulations in the case of the new initial conditions and ${768}^3$, $s=0.3$, $\Delta x=0.4295{\varphi }_0^{-1}$ for the BHKU ones.

Figure 2

The equal-time scaling functions $\tilde{C}(k\tau ,1)$ from ${1024}^3$, $s=0$ simulations with a matter-dominated FRW background for 5 times, as indicated in the legend in units of ${\varphi }_0^{-1}$. Results shown are the average of 3 realizations, with the estimated statistical uncertainties for $\tau =296{\varphi }_0^{-1}$ indicated by the shaded regions.

Figure 3

The equal-time scaling functions $\tilde{C}(k\tau ,1)$ from simulations with a matter-dominated FRW background at three values of $s$. Results shown are the average of 3 realizations, with statistical uncertainties for the $s=0.5$ case indicated by the shaded regions. Results are plotted for $\tau =150{\varphi }_0^{-1}$, which is the start of the period when we take UETC data for the $s=0$ case.

Figure 4

The UETC scaling functions $\tilde{C}(k\sqrt{\tau \tau },\tau /{\tau }^{\prime })$ in the matter era from ${1024}^3$ simulations with $s=0$, averaged over 3 realizations. The raw data is highlighted by the lighter central region, with extrapolations to more extreme $\tau /{\tau }^{\prime }$ values and to lower $k\sqrt{\tau {\tau }^{\prime }}$ indicated by the darker regions. The vertical axis indicates $|\tilde{C}|$ and it should be noted that the cross-correlator is negative near its peak.

Figure 5

The decoherence functions $\tilde{D}(k\sqrt{\tau \tau },\tau /{\tau }^{\prime })$ in the matter era from ${1024}^3$ simulations with $s=0$, averaged over 3 realizations. The raw data is highlighted by the lighter central region, with extrapolations to more extreme $\tau /{\tau }^{\prime }$ values and to lower $k\sqrt{\tau {\tau }^{\prime }}$ are indicated by the darker regions.

Figure 6

Slices through the decoherence functions $\tilde{D}$ at small scales for the matter era. In order to demonstrate the approximately constant form at these scales when plotted as a function of $k(\tau -{\tau }^{\prime })$, results are shown for the three values $k(\tau +{\tau }^{\prime })/2$ indicated by the legend. The latter quantity is perpendicular to $k(\tau -{\tau }^{\prime })$ in the $k\tau -k{\tau }^{\prime }$ plane (see also Fig. 9) and simplifies to $k\tau$ for the equal-time case.

Figure 7

Results for $k\tau {\tilde{C}}_{11}^{\mathrm{S}}$ from $s=0$ simulations, plotted against $k\tau$ (upper pane) and $k$ (lower pane). The 5 lines correspond to equally spaced times between $\tau =150$ and $300{\varphi }_0^{-1}$. In the upper pane the 5 lines are indistinguishable at low $k\tau$ due to the observed scaling, but scaling is broken on small scales where lines progressively move to the right. In the lower pane the lines are indistinguishable at high $k$, showing that on such scales $k\tau {\tilde{C}}_{11}^{\mathrm{S}}$ is then simply a function of $k$, i.e. the comoving length scale, and indicating that the attenuation of the signal occurs on scales a few times the comoving string width, which is constant in this $s=0$ case.

Figure 8

Results for all five equal-time scaling functions multiplied by $k\tau$ [or $(k\tau {)}^3$ in the vector case], with power-law fits over the range $k\tau =50\to 100$ (between the two vertical lines). The uppermost line is for ${\tilde{C}}_{11}^{\mathrm{S}}$, then of the middle pair of lines the black one is ${\tilde{C}}_{22}^{\mathrm{S}}$ and the gray one is $|{\tilde{C}}_{12}^{\mathrm{S}}|$, and finally of the lower pair the black one is ${\tilde{C}}^{\mathrm{V}}$ and the gray one if ${\tilde{C}}^{\mathrm{T}}$. A horizontal fit line would indicate that $\tilde{C}$ (or $(k\tau {)}^2{\tilde{C}}^{\mathrm{V}}$ in the vector case) varies as $1/k\tau$.

Figure 9

The blockwise coverage of the $k\tau -k{\tau }^{\prime }$ plane. A large primary block contains the dominant UETC data at near- and superhorizon scales ($k\tau <100$). Then smaller blocks of side $\Delta (k\tau )=20$ cover the diagonal, with overlapping to aid coverage. The gray-scale image shows the coherence function ${\tilde{D}}_{11}^{\mathrm{S}}$ to illustrate that these patches cover the region over which the UETC is significant, with light colors indicating higher values. The dark outer region indicates the region for which we extrapolate the coherence function, as in Fig. 5.

Figure 10

Contributions to the vector component of the temperature power spectrum from discrete $k\tau$ ranges. The highest line shows the dominant contribution from horizon and superhorizon scales. The contribution from first subhorizon matrix block is shown in the middle of the plot, while the approximate scaling law is indicated by the high $k\tau$ lines, which are lowest in the figure.

Figure 11

The build up of the high $\ell$ scalar mode temperature power spectrum as the maximum included $k\tau$ value is increased. From bottom to top in the plot these correspond to the $k\tau \lesssim 150$, $k\tau \lesssim 300$, $k\tau \lesssim 550$ and finally $k\tau \lesssim 5000$. Results shown by thin solid grey lines are from full CMB calculations, while the thick grey dashed and thick black solid lines correspond to results obtained using the rescaling property for $k\tau \gtrsim 150$. The power-law extrapolation is shown by the thin dashed line and is a reasonably accurate description of the final $k\tau \lesssim 5000$ result for $\ell \gtrsim 3000$.

Figure 12

The CMB temperature power spectrum determined from $s=0$ simulations and incorporating estimated UETC power for $k\tau \lesssim 5000$. The plot shows the total (thick line) plus the decomposition (thin lines) into scalar (S), vector (V), and tensor (T) modes.

Figure 13

Results for ${\ell }^3{\mathcal{C}}_{\ell }$, highlighting the form of the power spectrum at high $\ell$. The plot shows the total (thick line) plus the decomposition (thin lines) into scalar (S), vector (V), and tensor (T) modes. Additionally, the right of the figure shows the results of our tentative power-law extrapolations to $\ell >4000$.

Figure 14

The string contribution to the temperature power spectrum when including the string sources only after recombination (thin) or at all times (thick), with the difference between the two additionally shown (dashed). Recombination is not instantaneous and additionally the string source functions must be temporally differentiable and thus must be gradually switched on, hence this decomposition is ambiguous and there is an artificial reduction in the amplitude of the power law at high $\ell$ for the post-recombination results.

Figure 15

A comparison of three contributions to the CMB power spectrum: the inflationary scalar contribution (grey, solid line), the string contribution (black line) and the SZ effect (grey, dashed line). The inflationary scalar mode and $f_{10}\approx 1$ string contribution are both normalized to match the WMAP data [1] at $\ell =10$, while the $f_{10}\approx 0.1$ string contribution is set to yield 10% of the observed power at this multipole (the approximation signs are because $f_{10}$ is defined relative to the total theoretical power spectrum, not the observations). Two normalizations are shown for the SZ contribution, one for each of the QUAD bands: 100 and 150 GHz.

Figure 16

A comparison between the string contribution (thick line) to the CMB temperature and polarization power spectra with the adiabatic scalar contribution from inflation (thin line). The normalization of the string component is set at $f_{10}\approx 0.1$, while the inflationary scalar contribution uses the same settings as in Fig. 15. For the BB spectrum we also plot the possible contribution from the inflationary tensor mode (dot-dashed line), normalized at its 95 upper limit [82].