Advances in Astronomy

Volume 2010 (2010), Article ID 380507, 28 pages

http://dx.doi.org/10.1155/2010/380507

## Cosmic Strings and Their Induced Non-Gaussianities in the Cosmic Microwave Background

Institute of Mathematics and Physics, Centre for Cosmology, Particle Physics and Phenomenology, Louvain University, 2 Chemin du Cyclotron, 348 Louvain-la-Neuve, Belgium

Received 15 January 2010; Accepted 27 May 2010

Academic Editor: Eiichiro Komatsu

Copyright © 2010 Christophe Ringeval. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Motivated by the fact that cosmological perturbations of inflationary quantum origin were born Gaussian, the search for non-Gaussianities in the cosmic microwave background (CMB) anisotropies is considered as the privileged probe of nonlinear physics in the early universe. Cosmic strings are active sources of gravitational perturbations and incessantly produce non-Gaussian distortions in the CMB. Even if, on the currently observed angular scales, they can only contribute a small fraction of the CMB angular power spectrum, cosmic strings could actually be the main source of its non-Gaussianities. In this paper, after having reviewed the basic cosmological properties of a string network, we present the signatures Nambu-Goto cosmic strings would induce in various observables ranging from the one-point function of the temperature anisotropies to the bispectrum and trispectrum. It is shown that string imprints are significantly different than those expected from the primordial type of non-Gaussianity and could therefore be easily distinguished.

#### 1. Motivations

The origin of cosmic strings dates back to the discovery that cosmological phase transitions triggered by the spontaneous breakdown of the fundamental interaction symmetries may form topological defects [1–3]. Cosmic strings belong to the class of line-like topological defects, as opposed to point-like monopoles and the membrane shaped domain walls. As shown by Kibble, the appearance of defects in any field theory is related to the topology of the vacuum manifold [3]. If the ground state of a field theory experiences a spontaneous breakdown from a symmetry group to a subgroup , Kibble showed that cosmic strings will be formed if the first homotopy group is nontrivial. In other words, if noncontractile loops can be found in the manifold of equivalent vacua. Similarly, the other homotopy groups and determine the formation of domain walls and monopoles, respectively. Once formed and cooled, these defects cannot be unfolded, precisely due to their nontrivial topological configuration over the vacuum manifold of the theory. This simple statement suggests that cosmic strings, and topological defects in general, are a natural outcome of the unification of the fundamental interactions in the context of Cosmology. As remnants of unified forces, their discovery would be an incredible opportunity to probe extremely high-energy physics with “a telescope”.

In the last thirty years, many works have been devoted to the cosmological consequences, signatures, and searches for topological defects [4–7]. They have pushed cosmic strings to the privileged place to be generically compatible with observations. Indeed, domain walls and monopoles are prone to suffer from the cosmological catastrophe problem; their formation is sufficiently efficient (or their annihilation sufficiently inefficient) to either overclose the universe or spoil the Big-Bang Nucleosynthesis (BBN) predictions [8, 9]. For domain walls, this implies that either they should be extremely light, that is, formed at an energy scale less than a few , or no discrete symmetry should have been broken during the cooling of the universe. There is not so much choice for the monopoles; if interactions were unified, monopoles would have been formed. The homotopy group of with containing the of electroweak interactions is indeed nontrivial.As often with topological defects, sensitivity to the underlying model is such that one can often find a counter-example of any result. Both of these statements, on walls and monopoles, can be evaded in some particular models or with some amount of fine-tuning, as for instance if cosmic strings can be attached to them and catalyse annihilations [10, 11].Cosmic inflation was originally designed to solve the monopole problem. If a phase of accelerated expansion of the universe occurs, then any defects will be diluted enough to no longer have any (dramatic) consequences on cosmology [12–15]. Meanwhile, Inflationary Cosmology solves the flatness and homogeneity problem of the standard Big-Bang model, explains the origin and spectrum of the cosmic microwave background (CMB) anisotropies, as the formation of the large-scale structures [16–18, 17]. Inflation provides a priori an easy solution to the topological defects problem by diluting them to at most one per Hubble radius. However, one has to keep in mind that this mechanism works only if the defects were formed before inflation, and even in that case some may survive [20]. This has to be the case for monopoles and heavy walls, but not for local strings. On the contrary, exhaustive analysis of particle physics motivated inflationary models, embedding the Standard Model , has shown that strings are generically produced at the end of inflation [21]. In this picture, our universe should contain cosmic strings whose properties are closely related to those of the inflation [22–24]. String Theory provides an alternative framework to Field Theories: brane inflationary models propose that the accelerated expansion of the universe is induced by the motion of branes in warped and compact extradimensions [25–28]. Inflation ends when two branes collide and such a mechanism again triggers the formation of one-dimensional cosmological extended objects, dubbed cosmic superstrings [29–32]. These objects may be cosmologically stretched fundamental strings or one-dimensional D-brane [33, 34]. Although cosmic superstrings are of a different nature than their topological analogue, they produce the same gravitational effects and share similar cosmological signatures [35, 36].

Among the expected signatures, cosmic strings induce temperature anisotropies in the CMB with an amplitude typically given by , where is the string energy per unit length and the Newton constant [37]. (To avoid any confusion with Greek tensor indices, we will use the Carter’s notations and for the string energy density and tension [38].) For the Grand Unified Theory (GUT) energy scale, one has , which precisely corresponds to the observed amplitude of the CMB temperature fluctuations [39]. However, the power spectra do not match; topological defects are active sources of gravitational perturbations, that is, they produce perturbations all along the universe history, and cannot produce the characteristic coherent patterns of the acoustic peaks [40–44]. Current CMB data analyses including a string contribution suggest that they can only contribute to at most of the overall anisotropies on the observed angular scales [45, 46]. For Abelian cosmic strings (see Section 2), numerical simulations in Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes show that this corresponds to an upper two-sigma bound [47]. Direct detection searches provide less stringent limits but are applicable to all cosmic string models: [48–50]. Detecting cosmic strings in the CMB certainly requires one to go further than the power spectrum [51, 52] (see, however, Section 4.5). In fact, strings induce line-like discontinuities in the CMB temperature through the so-called Gott-Kaiser-Stebbins effect, which are intrinsically of non-Gaussian nature [53, 54]. In the inflationary picture, cosmological perturbations find their origin in the quantum fluctuations of the field-metric system, and therefore were born generically Gaussian. Non-Gaussianities can nevertheless appear from non-linear effects during inflation or from couplings to other fields (see the other articles in this issue). These non-Gaussianities are of the primordial type, that is, they exist before the cosmological perturbations reenter the Hubble radius. On the other hand, cosmic strings are a source of non-Gaussianity at all times and, as we will see, produce different signals from the CMB point of view.Notice that second-order perturbations, being non-linear, actively generate non-Gaussianities but at a relatively small amplitude [55–58].

In this paper, we review the non-Gaussian features a cosmological network of cosmic strings produce in the CMB anisotropies. In a first section, we briefly scan various cosmic string models and emphasize their similarities and differences for cosmology. Making observable predictions for cosmic strings faces the problem of understanding their cosmological evolution. Not only one has to solve the local dynamics in curved space, but as extended objects, cosmic strings follow a globally nonlocal evolution: the fate of one string depends on its interactions with the others. The cosmological evolution of a network of cosmic strings is a nontrivial problem which can be overcome by means of numerical simulations. These simulations permit an estimation of the various statistical properties affecting the observational signatures, such as the number of strings per Hubble radius, their shapes, velocities, or the loop density distribution. Latest results in this area, for the Nambu-Goto (NG) type of cosmic strings, are presented in Section 3. Once the statistical properties of a cosmological cosmic strings network are known, it is possible to extract meaningful observables depending only on the unique model parameter . (If no currents are flowing along the string, Lorentz invariance implies that the string tension equals the energy density .) In Section 4, we recap the expected CMB temperature anisotropies induced by cosmic strings, derived from various methods. Particular attention is paid to small angle CMB maps which preserve all of the projected statistical information. We then derive the cosmic string signals expected in various non-Gaussian estimators ranging from the one-point function of the CMB temperature fluctuations to the bispectrum and trispectrum. We conclude in Section 5 and discuss various non-Gaussian aspects which still have to be explored.

#### 2. Cosmic Strings of Various Origins

Cosmic strings of cosmological interest can be of various kinds depending on the microscopic model they stem from. As mentioned in the introduction, they can either be nontrivial stable, or metastable, field configurations or more fundamental objects in String Theory. From a gravitational point of view, they all are, however, line-like energy density and pressure distributions. In the following, we briefly review the different kinds of string having a cosmological interest and we emphasize their similarities and differences.

##### 2.1. Abelian Vortices

The simplest example of cosmic string illustrating the Kibble mechanism is the Abelian Higgs model. The theory is invariant under a local gauge group and the Higgs potential assumes its standard Mexican hat renormalisable formwhere is the self-coupling constant and the vacuum expectation value of the Higgs field . In Minkowski space, the Lagrangian readswhere is the field strength tensor associated with the vector gauge boson and At high enough temperature, loop corrections from the thermal bath restore the symmetry and the effective potential has an overall minimum at [1, 59]. Starting from high enough temperature, one therefore expects the symmetry to be spontaneously broken during the expansion and cooling of the universe. During the phase transition, the Higgs field reaches its new vacuum expectation value . At each spacetime location, the phase will have a given value, all of them being uncorrelated on distances larger than the typical correlation length of the phase transition. As pointed by Kibble, this is at most the horizon size although one expects it to be much smaller [3, 60–63]. As a result, there exists closed paths in space along which varies from to (or a multiple of ). Such phase configurations necessarily encompass a point at which (see Figure 1); the old vacuum has been trapped into a nontrivial configuration of the new vacuum, and this prevents its decay. Such a structure is invariant by translations along the third spatial dimension and is string shaped.