Signatures From Topological Defects

Each type of topological defect produces a characteristic signature in the CMB - for example, cosmic strings would produce line-like discontinuities on the sky (Kaiser & Stebbins 1984), while cosmic textures produce distinct hot and cold spots (Coulson et al. 1994). This is illustrated in Figure 1.11 which compares the CMB anisotropies expected in a cosmic string model with the inflationary cold dark matter described in the previous section. Notice the step-like discontinuities in the string model - generally the fluctuations produced by topological defects are non-Gaussian, unlike the Gaussian noise predicted by inflationary models. The non-Gaussianity can be measured in a number of different ways, e.g. by testing for correlations between the spherical harmonic coefficients $a_\ell^m$ (Magueijo 1996), by calculating higher moments of the temperature maps such as the skewness $\langle \Delta T^3 \rangle /\langle \Delta T^2 \rangle^{3/2}$ and kurtosis $(\langle \Delta T^4\rangle - 3 \langle \Delta T^2 \rangle^2)/ \langle \Delta T^2\rangle^2$, which are zero for a Gaussian distribution, or by designing statistics to test for specific features such as discontinuities. However, detailed numerical calculations (Coulson et al. 1994) show that high sensitivities $\Delta T/T \sim 10^{-6}$, large sky coverage, and angular resolutions of much less than a degree are required to detect the non-Gaussianities arising from topological defects. High resolution is particularly crucial - for example, the line-like discontinuities of the cosmic string model shown in Figure 1.11 would be smeared out by a beam of width $\lower.5ex\hbox{$\; \buildrel \gt \over \sim \;$}1^\circ$ and so the anisotropy pattern would be difficult to distinguish from the Gaussian fluctuations predicted by inflation. For this reason, statistical tests of Gaussianity applied to the COBE CMB maps (Kogut et al.1995 ) do not provide strong constraints on topological defect models.

Figure 1.11: The picture on the left shows a simulated map of the sky of area 12.8° x 12.8° for an inflationary CDM model. The picture to the right shows the anisotropy pattern in a 12.8° x 12.8° patch of sky in a universe seeded by cosmic strings.

Figure 1.12: The dot-dashed line shows the temperature power spectrum of a cosmic string model with _b = 0.05 and h = 0.5. The solid curve shows a scale invariant inflationary CDM model with _b = 0.1 and h = 0.4, which matches the height of the first Doppler peak predicted in the string model, but has additional structure at high multipoles.

The temperature power spectrum can also be used to distinguish between topological defect models and inflation. Figure 1.12 shows a calculation of the power spectrum in a cosmic string model compared to an inflationary cold dark matter model (Magueijo et al.1996). At low multipoles the curves are quite similar - topological defect models predict a nearly scale invariant fluctuation spectrum as expected in the simplest models of inflation. At multipoles $\ell \gt 100$, the power spectra differ significantly. The Doppler peak is much broader in the string model and the secondary peaks characteristic of the inflationary models are absent in the string model. Similar conclusions apply to the CMB power spectra in cosmic texture theories (Crittenden & Turok, 1995). Recent developments in cosmic strings studies (see e.g., Allen et al.1997) suggest that there is a lack of large scale power in the matter power spectrum derived from string models with respect to what is obtained from inflationary CDM models. This discrepancy may be overcome to a large extent by introducing a cosmological constant component in the energy density (see e.g., Battye, Robinson & Albrecht 1998) or allowing for an open universe (see e.g., Avellino, Caldwell & Martins 1997).

In summary, the non-Gaussian signatures in the CMB arising from topological defects can distinguish between these models and the inflationary theories described above. Furthermore, the locations and shapes of the Doppler peaks in the temperature power spectrum might allow a distinction between these two classes of theories. High resolution and high sensitivities, such as those provided by PLANCK are required to apply these tests.

In the previous section, we showed how cosmological parameters could be determined with high accuracy if the CMB anisotropies were generated by adiabatic perturbations. Would the same be true if the CMB fluctuations arose from topological defects (i.e. isocurvature perturbations)? Certainly, the CMB power spectrum in topological defect theories at high multipoles is sensitive to cosmological parameters such as $\Omega_0$, $\Omega_b$, H₀ etc in a similar way to the adiabatic fluctuations described above. However, theoretical computations of the CMB anisotropies in defect theories are inherently much more complex than for adiabatic fluctuations and calculations with the precision achievable by PLANCK have not yet been made. It is therefore, not yet possible to compute the accuracies of cosmological parameters expected in defect theories, though we expect that comparable accuracies to those presented in the previous section are probably achievable.

[last update: 1 August 1999 by P. Fosalba]