CMB power spectrum of Nambu-Goto cosmic strings

We improve predictions of the cosmic microwave background (CMB) power spectrum induced by cosmic strings by using source terms obtained from Nambu–Goto network simulations in an expanding universe. We use three high-resolution cosmic string simulations that cover the entire period from recombination until late-time $\mathrm{\Lambda }$ domination to calculate unequal time correlators (UETCs) for scalar, vector, and tensor components of the cosmic string energy-momentum tensor. We calculate the CMB angular power spectrum from strings in two ways. First, to aid comparison with previous work, we fit our simulated UETCs to those obtained from different parameter combinations from the unconnected segment model and then calculate the CMB power spectra using these parameters to represent the string network. Second and more accurately, we decompose the UETCs into their corresponding eigenvalues and eigenvectors and input them directly into an Einstein–Boltzmann solver to calculate the power spectrum for each of the three simulation time periods. We combine the three simulations together, using each of them in its relevant redshift range, and we obtain overall power spectra in temperature and polarization channels. Finally, we use the power spectra obtained with the latest Planck and BICEP2 likelihoods to obtain constraints on the cosmic string tension.

Figure 1

Evolution of the string network in the simulation covering the matter era (redshift range 945 to 37.2).

Figure 2

Energy density component of the string network in real space evaluated at time 384 out of 1536 for the simulation in the radiation era for resolutions of $128^3$, $512^3$, and $1536^3$.

Figure 3

Scalar UETCs obtained from a grid resolution of 1024: the figures of the left represent oblique 3D views of the three scalar UETCs ($⟨{\mathrm{\Theta }}_{00}{\mathrm{\Theta }}_{00}⟩$—top, $⟨{\mathrm{\Theta }}^S{\mathrm{\Theta }}^S⟩$—middle, and $⟨{\mathrm{\Theta }}_{00}{\mathrm{\Theta }}^S⟩$—bottom), the top right plot represents a contour plot of the 00-00 UETC in linear scale, and two bottom right plots represent the three scalar UETCs in linear and logarithmic scales.

Figure 4

Vector and tensor UETC components obtained from a grid resolution of 1024: oblique 3D views (left) and diagonal sections in linear scale (right).

Figure 5

Network correlation in the initial conditions from the simulation covering the matter era: left—oblique 3D view; right—front view.

Figure 6

$⟨{\mathrm{\Theta }}_{00}{\mathrm{\Theta }}_{00}⟩$ UETCs exhibiting scale invariance. In the bottom two rhs plots, the two plotted curves are indistinguishable. Scaling can be observed between the figures despite the correlation time used.

Figure 7

$⟨{\mathrm{\Theta }}_{00}{\mathrm{\Theta }}_{00}⟩$ equal time correlators at different resolutions.

Figure 8

Evolution of the averaged shape and amplitude correlators.

Figure 9

Comparison between the TT power spectrum for USM, Abelian–Higgs, and Nambu–Goto simulated strings. Simulation 1 covers the radiation era, Simulation 2 the matter era, and Simulation 3 matter and cosmological constant eras. The USM and Abelian–Higgs power spectra are the standard results used in the Planck cosmic defects paper [17].

Figure 10

Power spectra of the cosmic strings obtained from the simulations in the radiation era assuming scale invariance. From left to right: scalar, vector, and tensor power spectra; from top to bottom: TT, EE, TE, and BB power spectra ($G\mu =1.5\times 10^{-7}$). The numbers in the legend represent the number of eigenvectors used. The colors in the tensor spectra plots represent different numbers of eigenvectors used compared to the scalar and vector spectra.

Figure 11

Comparison between the TT power spectra obtained through the best fit method and using eigenvectors.

Figure 12

Power spectra of the cosmic strings obtained by using each of the three sets of UETCs and assuming scaling for the whole history of the Universe. The red, green, and blue show the power spectra considering the extrapolation of the results obtained in the radiation, matter, and $\text{matter}+\mathrm{\Lambda }$ epochs. The contributions from the UETCs from just the time interval where they are valid are plotted in the yellow, cyan, and magenta curves, and their sum is in black. The black curve represents the final overall power spectrum obtained. From left to right: The scalar, vector, and tensor power spectra; from top to bottom: the TT, EE, TE, and BB power spectra ($G\mu =1.5\times 10^{-7}$).

Figure 13

Comparison between the TT power spectra obtained using the three simulations and the USM, Abelian–Higgs (standard results), and CMBACT version 4. The string tension is taken $G\mu =2.07\times 10^{-6}$. On the left the results are represented showing that USM and Nambu–Goto have different amplitudes to Abelian–Higgs cosmic strings, while the right shows similar shapes (normalized at $l=10$).