The current status of cosmic strings

Patrick Peter, GR$\varepsilon$CO - Institut d'Astrophysique de Paris - Université Pierre et Marie Curie peter-at-iap.fr

The interest in cosmic string (CS) research has seen a renewal recently when it was observed that fundamental (super)strings could act as actual topological defects, although with possibly rather different properties. It was even said, obviously in an excess of enthusiasm, that the observation of a single cosmic string in the sky would be ``a proof'' of the existence of superstring theory!

Most of the original idea dates back to the work of Kibble in 1976 who realized that if the Higgs mechanism were to take place in the early stages of the Universe's expansion, with a very small Hubble radius (and thus Horizon size)--recall it was a time during which inflation had not yet acquired its status of a paradigm, as it is now part of the standard model of cosmology--then even a simple causality argument led to the existence of linear topologically stable configurations, called cosmic strings, that should have been formed during the symmetry breaking phase transition. The evolution of the thus-formed network of cosmic strings was found to have the ability to provide, in the long run, a spectrum of scale invariant cosmological perturbations that could then have acted as seeds for large scale structure formation. Moreover, the order of magnitude of the temperature fluctuations in the CMB is $\Delta T/T \sim G_{\mathrm{N}} U$, with $U \sim E_{_{\mathrm{GUT}}}^2$ the energy per unit string length, and $G_{\mathrm{N}} = M_{\mathrm{P}}^{-2}$ the Newton constant, i.e., the inverse of the Planck mass squared. Based on these assumptions, and provided the transition was that expected at the GUT scale ( $E_{_{\mathrm{GUT}}} \sim 10^{16}$GeV), then the theoretical number fits the observation, with no fine-tuning involved. General references on these topics are Vilenkin & Shellard (2000) and Peter & Uzan (2008).

CS are predicted in almost any conceivable GUT (Jeannerot et.al. 2003), and they can be used to impose constraints on the high energy parameters, e.g. those stemming from supergravity (Rocher & Sakellariadou 2004). If endowed by a current, which happens to be often the case (although it is very much model-dependent), a CS network evolution (Ringeval et.al. 2007) ends up producing vorton states which overclose the Universe, thereby leading to a cosmological catastrophe [see, e.g. Cordero-Cid et.al. (2002) and Postma & Hartmann (2007)].

Models in which Nambu-Goto strings can form a network and evolve are expected to reach scaling, as shown numerically (Fig. 1), in the sense that the energy density contained in the network eventually behaves as the inverse of the square of the horizon scale. This point was subject to some polemics (see Vanchurin, Olum & Vilenkin (2006) and Martins & Shellard (2006)) but seems now to have been settled by Ringeval et.al. (2007).

More precise data have shown, however, that the scale invariant spectrum of primordial perturbations on large scales is not the whole story, and the recent WMAP sky observation has revealed that the most important contribution of the CMB fluctuations originates with an adiabatic coherent source such as predicted by the inflation models (see Lemoine et.al. 2007 for a summary on inflation and Riazuelo et.al. 2000 for CS calculations). In the best case, cosmic strings represent a small fraction of the CMB fluctuation. The current constraints (which are not only model-dependent, but also quite unclear because the actual CS perturbation spectrum is not known with certainty) are around 10 % or less (Wyman et.al. 2005). Accordingly, CS research seems to be marginalized.

There was a renewal of interest with the realization that, contrary to all expectations, fundamental superstrings might actually act as CS, although with some differences. Of course, these strings should have, from the outset, almost Planckian energy per unit length, thereby producing far too high an amplitude in the perturbations. But moreover, Witten (1985), based on perturbation arguments, had shown that long fundamental BPS strings in heterotic theory were cursed with instabilities and had therefore no chance to ever be observed. Non-BPS states were also believed to be unstable.

Figure 1: xxxx

Figure: Left panel: A typical string loop distribution according to the latest simulation by Ringeval et.al. (2007) during the matter area. The observable Universe occupies one eighth of the box, whose edge is $100 \ell_{\mathrm{c}}$, with $\ell_{\mathrm{c}}$ being the correlation length of the Vachaspati-Vilenkin (1984) initial conditions. The right panel shows, for the same era, the evolution of the energy density associated with long strings (top) and loops (bottom) of physical sizes $\ell_{\mathrm{phys}} = \alpha d_{\mathrm{h}}$, $d_{\mathrm{h}}$ being the horizon size. The time variable is the rescaled conformal time $\eta/\ell_{\mathrm{c}}$. These results show the network reaching scaling, as required.

Figure 3: String intercommutation

All this changed around 2004, when it was found that because of the large number of possible geometries for the compact dimensions and with the existence of localized branes, the string energy per unit length could be much lower than originally thought. Besides, new solutions were found, called F, D, and $(p,q)$ strings, that not only resemble the usual local topological defects, but also have sufficiently different properties to be discriminated. Moreover, those solutions could be stabilized over cosmologically relevant time scales. Polchinski (2005) and Majumdar (2005) provide nice overviews of these topics.

The most important difference, to date, between fundamental (or cosmological size string solutions in supertring theories) and ordinary CS is the so-called reconnection probability. Indeed, in a 4D space-time, when two topological defects collide, they exchange ends with a probability $P\sim 1$ (see Fig. 2). For superstring strings however, because of the extra-dimensions in which the strings (or branes) are actually evolving, this probability could be drastically reduced $P\ll 1$, leading to a completely different network evolution. The same reason implies that the string energy per unit length can be much lower than the Planck scale, once the Calabi-Yau volume is taken into account. All this leads to observationally compatible effects.

This topic is currently seeing some serious development, for these strings can easily be embedded in scenarios of brane inflation, where they are produced after the inflation epoch. They can produce observable amounts of gravitational radiation with particular spectral properties, be involved in the (p)reheating mechanism, form structures, help reionization and baryogenesis ... in short, the future for research in CS seems still quite bright!

References:
A. Cordero-Cid, X. Martin and P. Peter, Phys. Rev. D65, 083522 (2002) and references therein.
R. Jeannerot, J. Rocher and M. Sakellariadou, Phys.Rev. D68, 103514 (2003).
M. Lemoine, J. Martin and P. Peter (Eds.), Inflationary Cosmology, Lecture Notes in Physics, Springer (2007).
M. Majumdar, Lecture notes for COSLAB 2004, University of Leiden, Imperial College and Dhaka University, (2005). hep-th/0512062
C. J. A. P. Martins and E. P. S. Shellard, Phys. Rev. D73, 043515 (2006).
P. Peter and J.-P. Uzan, Primordial Cosmology, Oxford University Press, to appear (2008).
J. Polchinski, AIP Conf.Proc. 743, 331 (2005); Int. J. Mod. Phys. A20, 3413 (2005).
M. Potsma and B. Hartmann, (2007) arXiv:0706.0416
A. Riazuelo, N. Deruelle and P. Peter, Phys. Rev. D61, 123505 (2000) and references therein.
C. Ringeval, M. Sakellariadou, and F. Bouchet, JCAP 0702, 023 (2007).
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V. Vanchurin, K. D.Olum, and A. Vilenkin, Phys. Rev. D74, 063527 (2006).
A. Vilenkin and E. P. S. Shellard, Cosmic strings and other topological defects, Cambridge University Press (2000).
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M. Wyman, L. Pogosian, and I. Wasserman, Phys.Rev. D72, 023513 (2005); Erratum-ibid. D73, 089904 (2006).