Cosmic Structure Formation with Topological Defects

TL;DR

This work surveys how topological defects formed during symmetry breaking could seed cosmic structure, providing a complete framework from defect formation (Kibble mechanism) through linear perturbation theory with seed sources to predictions for CMB anisotropies and matter clustering. It develops both the theoretical apparatus (seed energy–momentum tensors, unequal-time correlators, and the Boltzmann–Einstein system) and the numerical machinery (σ-model and large-N limits for global defects, and Nambu–Goto treatments for strings) needed to compute power spectra. The analysis finds that pure global defects and local strings typically fail to reproduce observed acoustic peaks and large-scale structure unless one allows idealized, highly coherent scaling seeds or admixtures with inflation; decoherence and vector/tensor contributions suppress peaks and modify polarization signals. The study further shows that generic, causally scaling seeds can mimic inflationally generated spectra in principle, and mixed models can remain compatible with data, but definitive discrimination requires polarization measurements and a clear physical origin for the seeds. Overall, defects remain a compelling concept with rich phenomenology, but current data strongly constrain their dominant role in structure formation, favoring either inflation-dominated scenarios or carefully constructed mixed/alternative seed models.

Abstract

Topological defects are ubiquitous in physics. Whenever a symmetry breaking phase transition occurs, topological defects may form. The best known examples are vortex lines in type II super conductors or in liquid Helium, and declination lines in liquid crystals. In an adiabatically expanding universe which cools down from a very hot initial state, it is quite natural to postulate that topological defects may have emerged during a phase transition in the early universe and that they may have played the role of initial inhomogeneities seeding the formation of cosmic structure. This basic idea goes back to Kibble (1976). In this report we summarize the progress made in the investigation of Kibble's idea during the last 25 years. Our understanding of the formation and evolution of topological defects is reported almost completely in the beautiful book by Vilenkin & Shellard or the excellent Review by Hindmarsh & Kibble, and we shall hence be rather short on that topic. Nevertheless, in order to be self contained, we have included a short chapter on spontaneous symmetry breaking and defect formation. Our main topic is however the calculation of structure formation with defects, results which are not included in the above references.

Figures (27)

• Figure 1: The ETCs $C_{11}(z,1)=\langle|\Phi|^2\rangle(k\tau)$ (panel a) and $C_{22}(z,1)=\langle|\Psi|^2\rangle(k\tau)$ (panel b) are shown for different times. In grid units the times are $\tau=4$ (dashed), $\tau=8$ (dotted), $\tau=12$ (long dashed), $\tau=16,~20$ (dash dotted, long dash dotted) and $\tau=24$ (solid). Clearly $C_{22}$ scales much sooner than $C_{11}$. To safely arrive in the scaling regime one has to wait until $\tau\sim 16$ and $C_{ij}(k\tau=0)$ is best determined at $\tau\ge 20$ but $k\tau< 1$.
• Figure 2: Energy momentum conservation of numerical simulations is shown. The lines represent the sum of the terms which has to vanish if energy (solid) respectively momentum (dashed) is conserved, divided by the sum of the absolute values of these terms. The abscissa indicates the wavelength of the perturbation as fraction of the size of the entire grid. (from DKM
• Figure 3: The $C_\ell$ power spectrum is shown for the large-$N$ limit (bold line) and for the texture model. The main difference is clearly that the large-$N$ curve shows some acoustic oscillations which are nearly entirely washed out in the texture case.
• Figure 4: The two point correlation functions $C_{11}(z,r) = k^4\sqrt{\tau\tau'}\langle\Phi_s(\hbox{\boldmath $k$},t)\Phi_s^*(\hbox{\boldmath $k$},\tau')\rangle$ (left), $C_{22}(z,r) = k^4\sqrt{\tau\tau'}\langle\Psi_s(\hbox{\boldmath $k$},\tau)\Psi_s^*(\hbox{\boldmath $k$},\tau')\rangle$ (center) and $|C_{12}(z,r)| = k^4\sqrt{\tau\tau'}|\langle\Phi_s(\hbox{\boldmath $k$},\tau)\Psi_s^*(\hbox{\boldmath $k$},\tau')\rangle|$ (right). Panels (a) represent the result from numerical simulations of the texture model; panels (b) show the large-$N$ limit. For fixed $r$ the correlator is constant for $z<1$ and then decays. Note also the symmetry under $r\rightarrow 1/r$ for $C_{11}$ and $C_{22}$ which is lost for $C_{12}$ (from DKM).
• Figure 5: On the left the correlators $C_{ij}(z,1)$ are shown, while the right figure depicts $C_{ij}(0,r)$ with $r=\tau'/\tau$. The solid, dashed and dotted lines represent $C_{22}~,~C_{11}$ and $|C_{12}|$ respectively. Panels (a) are obtained from numerical simulations of the texture model and panels (b) show the large-$N$ limit. A striking difference is that the large-$N$ value for $|C_{12}|$ is relatively well approximated by the perfectly coherent result $\sqrt{|C_{11}C_{22}|}$ while the texture curve for $|C_{12}|$ lies nearly a factor 10 lower (from DKM).

Theorems & Definitions (1)

• Theorem 1