CMB temperature bispectrum induced by cosmic strings

The cosmic microwave background (CMB) bispectrum of the temperature anisotropies induced by a network of cosmic strings is derived for small angular scales, under the assumption that the principal cause of temperature fluctuations is the Gott-Kaiser-Stebbins effect. We provide analytical expressions for all isosceles triangle configurations in Fourier space. Their overall amplitude is amplified as the inverse cube of the angle and diverges for flat triangles. The isosceles configurations generically lead to a negative bispectrum with a power-law decay ${\ell }^{-6}$ for large multipole $\ell$. However, collapsed triangles are found to be associated with a positive bispectrum whereas the squeezed triangles still exhibit negative values. We then compare our analytical estimates to a direct computation of the bispectrum from a set of 300 statistically independent temperature maps obtained from Nambu-Goto cosmic string simulations in a Friedmann-Lemaître-Robertson-Walker universe. We find good agreement for the overall amplitude, the power-law behavior, and the angle dependency of the various triangle configurations. At $\ell \sim 500$ the cosmic string Gott-Kaiser-Stebbins effect contributes approximately the same equilateral CMB bispectrum amplitude as an inflationary model with $|f_{\mathrm{NL}}^{\mathrm{loc}}|\simeq {10}^3$, if the strings contribute about 10% of the temperature power spectrum at $\ell =10$. Current bounds on $f_{\mathrm{NL}}$ are not derived using cosmic string bispectrum templates, and so our $f_{\mathrm{NL}}$ estimate cannot be used to derive bounds on strings. However it does suggest that string bispectrum templates should be included in the search of CMB non-Gaussianities.

Figure 1

Sketches of string correlation functions $V(\sigma )$ (velocity-velocity), $T(\sigma )$ (tangent-tangent), and $M_1(\sigma )$ (velocity-tangent), as a function of the string world sheet spacelike separation $\sigma$, defined in Eqs. (10, 11, 12).

Figure 2

Angular dependency of the isosceles bispectrum as a function of the angle $\theta$ in between the wave vectors ${\mathit{k}}_1$ and ${\mathit{k}}_2$. The particular values ${\theta }_{\mathrm{eq}}=\pi /3$ corresponding to the equilateral configuration as ${\theta }_0$ at which the bispectrum vanishes are represented. Notice the divergences for flat triangle configurations at $\theta \to 0$ (squeezed) and $\theta \to \pi$ (collapsed).

Figure 3

Typical CMB temperature map on a 7.2° field (resolution of ${\theta }_{\mathrm{res}}={0.42}^{\prime }$, $n_{\mathrm{pix}}^2={1024}^2$) obtained from Nambu-Goto cosmic string simulations (left panel). The right panel traces back the strings projected on our past light cone.

Figure 4

Exact versus approximated geometrical $S_{\ell \ell \ell }^{(w)}$ factors for a map with ${1024}^2$ pixels and various choices of the window function with $w$ ($w=2\pi$ corresponds to one Fourier mode per bandwidth). For large values of $w$, the deviations for the small multipoles come from the averaging effect. For the small angles we are interested in, Eq. (72) provides a good approximation to the exact expression (70).

Figure 5

The reduced bispectrum computed by the scale convolution method and averaged over 100 of non-Gaussian synthetic $\Lambda \mathrm{CDM}$ temperature maps whose probability distribution function is given by Eq. (73). The red dashed line is the expected non-Gaussian signal. The missing power for low multipoles comes from the cutoff at ${\ell }_w\simeq w/{\theta }_{\mathrm{fov}}\simeq 1050$ introduced by the choice of large $w$ values for the window functions.

Figure 6

Squeezed angle dependency of the mean squeezed bispectrum $[\ell (\ell +1)/(2\pi ){]}^{3/2}{b}_{\ell \ell \theta }$. The spurious plateau (dotted lines) for the lower multipoles comes from the cutoff associated with a given squeezed angle $\theta$ together with the window function width $w$ and occurs at ${\ell }_{\min }=w/({\theta }_{\mathrm{fov}}\theta )$.

Figure 7

Mean value and standard deviation of the squeezed bispectrum $[\ell (\ell +1)/(2\pi ){]}^{3/2}{b}_{\ell \ell \theta }$ for $\theta =0.2\mathrm{rad}$. The window function width has been increased to $w=30\pi$ to reduce the variance pushing up the lower multipole cutoff to ${\ell }_{\min }=w/({\theta }_{\mathrm{fov}}\theta )\simeq 3750$. The dashed line is the best power-law fit whose power has been set to the one obtained from the equilateral configuration [see Eq. (84)].

Figure 8

Rescaled mean squeezed bispectrum ${\theta }^3[\ell (\ell +1)/(2\pi ){]}^{3/2}{b}_{\ell \ell \theta }$ showing the $1/{\theta }^3$ dependency. Notice the large window function width $w=30\pi$ chosen to reduce the statistical errors and pushing up the numerical multipole cutoff $l_{\min }$.

Figure 9

Mean collapsed bispectrum $[\ell (\ell +1)/(2\pi ){]}^{3/2}{b}_{\ell \phi \phi }$. This configuration is extremely sensitive to the window function properties and statistical errors dominate. The change of sign for small values of $\phi$ is nevertheless observable.

Figure 10

Mean and standard deviation of the equilateral bispectrum $[\ell (\ell +1)/(2\pi ){]}^{3/2}{b}_{\ell \ell \ell }$ over our 300 string temperature maps and obtained for window function width $w=30\pi$. The blue dotted line is the power-law fit of Eq. (84).

Figure 11

Ratio of the mean bispectra $b_{\ell \ell \ell }/({\theta }^3{b}_{\ell \ell \theta })$ obtained for $\theta =0.2\mathrm{rad}$. Up to the statistical errors, the expected value $4/\sqrt{3}$ is recovered for $\ell >{\ell }_{\min }$.

Figure 12

Standard deviation of the bispectrum estimator over the 300 string temperature maps (black solid curve) compared with the nearly Gaussian random expected standard deviation obtained from Eq. (88).