## Abstract

Cosmic superstrings are expected to be formed at the end of brane inflation, within the context of brane-world cosmological models inspired from string theory. By studying the properties of cosmic superstring networks and comparing their phenomenological consequences against observational data, we aim to pin down the successful and natural inflationary model and get an insight into the stringy description of our Universe.

## 1. Introduction

Inflation was proposed in the 1980s as a simple and elegant solution to the shortcomings of the hot big bang model. Inflation provides a solution to the flatness and horizon problems within the framework of quantum field theory and general relativity, while it dilutes any undesired relics from possible extensions of the standard model. Inflation essentially consists of a phase of accelerated expansion that took place at a very high energy scale. Thus, by construction, inflation can liberate the standard model of cosmology from the requirement of special initial conditions. In addition, the amplification of the quantum fluctuations of the inflaton field, which drives inflation, offers a simple mechanism for the origin of the initial density fluctuations, which via gravitational instability could lead to the observed structure formation. The remarkable agreement of the inflationary-induced temperature fluctuations in the cosmic microwave background (CMB) radiation with all measurements is without doubt an element that strongly supports inflation. However, inflation still lacks a precise theoretical model. In this sense, it remains a paradigm in search of a model. An inflationary scenario should be considered as successful not only when it fits the data, but also when it can be accommodated within some fundamental theory.

In the context of supersymmetric grand unified theories (GUTs), the most natural inflationary model is that of hybrid inflation. In this context, a detailed study (Jeannerot *et al*. 2003) of all spontaneously symmetry-breaking (SSB) schemes that bring the initial symmetric state of the Universe (described by a large grand unified theory gauge group) down to a less symmetric state (described by the standard model gauge group) has shown that cosmic strings are generically formed at the end of hybrid inflation (Sakellariadou 2008). Thus, cosmic strings have to be considered as a *subdominant partner* of inflation, and they contribute (Bouchet *et al*. 2002) to the CMB temperature anisotropies.

Current CMB data impose strong constraints on the maximum allowed contribution of cosmic strings. Cosmic strings can contribute at most 11% (Bouchet *et al*. 2002; Wyman *et al*. 2005; Bevis *et al*. 2008) to the power spectrum of temperature anisotropies. This upper limit implies constraints on the string tension and therefore on the energy scale of string formation. Equivalently, one can impose (Rocher & Sakellariadou 2005*a*,*b*, 2006) constraints on the parameter space (couplings and mass scales) of the inflationary model.

Inflation must also prove itself generic, in the sense that the onset of inflation must be independent of initial conditions. The first studies (Calzetta & Sakellariadou 1992, 1993) addressing the onset of inflation were inconclusive, in the sense that no robust conclusions could be drawn as a quantum theory of gravity was missing. More recently, it has been shown (Germani *et al*. 2007) that successful single-field inflation with a polynomial potential is highly improbable within the semi-classical regime of loop quantum cosmology.

If string theory is the fundamental theory of all matter and forces, including a consistent quantum gravity sector, then there must exist a natural inflationary scenario within string theory. Such an approach will allow the identification of the inflaton and the determination of its properties, while at the same time cosmological measurements will provide an insight into the stringy description of our Universe. Since the *discovery* of Dirichlet (*D*) branes, a natural realization of our Universe in string theory is the brane-world scenario. In this context, a simple and well-motivated inflationary model is brane inflation, where inflation takes place while two branes move towards each other, and their annihilation releases the brane tension energy that heats up the Universe to start the hot big bang era. Typically, strings of all sizes and types may be produced during the collision. Large fundamental (*F*) strings and/or *D*1-branes (*D*-strings) that survive the cosmological evolution become cosmic superstrings. By observing strings in the sky, we will be able to test, for the first (and maybe only) time, string theory.

I highlight the most important properties of cosmic superstrings (there is a longer version with an *almost* complete list of references). First, I describe briefly the formation of cosmic superstrings at the end of brane inflation. Then, I discuss the differences between cosmic superstrings and their solitonic analogues, which arise in gauge theories. After a short review of the evolution of Nambu–Goto cosmic strings, I summarize the current *understanding* of the evolution of cosmic superstring networks. This issue is far from being resolved. The characteristics of the superstring network are strongly dependent on the specific brane inflationary model, while a realistic modelling through numerical simulations is particularly complex. Finally, I discuss the observational signatures of cosmic strings and superstrings. This is a very promising area of research, since it may give us the (*unique*) way of testing string theory as a realistic theory of nature. There is a large number of open directions that need to be thoroughly investigated and I briefly mention them. Note that we are working in type IIB string theory.

## 2. Cosmic superstring formation

The possible astrophysical role of fundamental strings was first advocated more than 20 years ago. It was proposed (Witten 1985) that superstrings of the O(32) and *E*_{8}×*E*_{8} string theories are likely to generate string-like stable vortex lines and flux tubes. However, in the context of perturbative string theory, the high tension of fundamental strings ruled them out as potential cosmic string candidates for the following three reasons. First, such heavy strings would produce CMB inhomogeneities far larger than the ones that have been measured. Second, such high-tension strings could not have been produced after inflation, since the scale of their tension is higher than the upper bound on the energy scale of the inflationary vacuum. However, any topological defects produced before inflation would have been diluted. Third, some instabilities were identified, implying that such strings would not be able to survive on cosmologically interesting time scales.

This picture has changed in the framework of brane-world cosmology, which offers an elegant realization of nature within string theory. Within the brane-world picture, all standard model particles are open string modes. Each end of an open string lies on a brane, implying that all standard model particles are stuck on a stack of *Dp*-branes, while the remaining *p*−3 of the dimensions are wrapping some cycles in the bulk. Closed string modes live in the high-dimensional bulk. In the brane-world context, the extra dimensions can even be infinite, if the geometry is non-trivial and they are warped. As a result of brane interactions, higher-dimensional *Dp*-branes unwind and evaporate so that we are left with *D*3-branes embedded in a (9+1)-dimensional bulk. One of these *D*3-branes could play the role of our Universe (Durrer *et al*. 2005). Cosmic superstrings are also left behind.

Inflation can be easily accommodated in a string theory-motivated cosmological model. String theory *lives* in a high-dimensional space, so compactification down to four space–time dimensions introduces many gravitationally coupled scalar fields (moduli) from the point of view of the four-dimensional theory. One of these fields could play the role of the inflaton, provided it does not roll quickly. Runaway or light moduli are extremely problematic in cosmology, so any realistic model should incorporate a mechanism for moduli stabilization. A correct brane inflation scenario will offer us valuable information on the early stages of our Universe, as well as on the particular compactification in string theory. String inflation models can be classified according to the origin of the inflaton field. If the inflaton is a scalar field arising from open strings ending on a *Dp*-brane (*p* stands for the dimensionality of the Dirichlet brane), then these open string models are called *D-brane* (or just *brane*) *inflation models*. If the inflaton field is a modulus (the most promising closed string modes), then these closed string models are called *moduli* or *modular inflation*.

Brane annihilations can also provide a natural mechanism for ending inflation. To illustrate the formation of cosmic superstrings at the end of brane inflation, let us consider a brane–anti-brane pair annihilation to form a *D*(*p*−2) brane. Each *parent* brane has a *U*(1) gauge symmetry and the gauge group of the pair is *U*(1)×*U*(1). The *daughter* brane possesses a *U*(1) gauge group, which is a linear combination, *U*(1)_{−}, of the original two *U*(1)'s. The branes move towards each other and, as their inter-brane separation decreases below a critical value, the tachyon field, which is an open string mode stretched between the two branes, develops an instability. The tachyon couples to the combination *U*(1)_{−}. The rolling of the tachyon field leads to the decay of the *parent* branes. Tachyon rolling leads to spontaneous symmetry breaking that supports defects with even co-dimension. Therefore, brane annihilation leads to vortices, *D*-strings; they are cosmologically produced via the Kibble mechanism. The other linear combination, *U*(1)_{+}, disappears, since only one brane remains after the brane collision. The *U*(1)_{+} combination is thought to disappear by having its fluxes confined by fundamental closed strings. Such strings are of cosmological size and they could play the role of cosmic strings (Sarangi & Tye 2002; Jones *et al*. 2003; Dvali & Vilenkin 2004); they are referred to in the literature as cosmic superstrings (Polchinski 2005).

## 3. Differences between cosmic strings and cosmic superstrings

Solitonic cosmic strings are classical objects that have been traditionally assumed to share the characteristics of type II Abrikosov–Nielsen–Olesen (ANO) vortices in the Abelian Higgs model. Cosmic superstrings, despite the fact that they are cosmologically extended, are quantum objects. Thus, one can expect differences in their properties, leading to a different behaviour, and distinct observational signatures.

Over distances that are large compared with the width of the string, but small compared with the horizon size, solitonic cosmic strings can be considered as one-dimensional objects and their motion can be well described by the Nambu–Goto action. However, this action cannot be used to describe what happens when two strings intercommute—a study that necessitates full field theory. When two strings of the same type collide, they may either pass simply through one another or reconnect (intercommute). A necessary, but not sufficient, condition for string reconnection is that the initial and final configurations be kinematically allowed in the infinitely thin-string approximation. Such a classical string solution for reconnection has been shown to exist, but the precise outcome of the string intersection depends on the internal structure of strings. Numerical simulations (and analytical estimates) of type II (and weakly type I) strings in the Abelian Higgs model suggest that the probability that a pair of strings will reconnect, after they intersect, is close to unity. The only exception for which the string reconnection probability was found to be different from 1 is the case of ANO strings with ultra-high collision speeds, in which case they just pass through each other.

The reconnection probability for cosmic superstrings is smaller (often much smaller) than unity. The corresponding intercommutation probabilities are calculated in string perturbation theory. The result depends on the type of strings and the details of compactification. For fundamental strings, reconnection is a quantum process and takes place with a probability of order (*g*_{s} is the string tension). It can thus be much less than 1, leading to an increased density of strings (Sakellariadou 2005), implying an enhancement of observational signatures. The reconnection probability is a function of the relative angle and velocity during the collision. One may think that strings can miss each other as a result of their motion in the compact space. Depending on the supersymmetric compactification, strings can wander over the compact dimensions, thus missing each other, effectively decreasing their reconnection probability. However, it was found (Jackson *et al*. 2005) that, in realistic compactification schemes, strings are always confined by a potential in the compact dimensions. The value of *g*_{s} and the scale of the confining potential will determine the reconnection probability. Even though these are not known, for a large number of models, it was found that the reconnection probability for *F*–*F* collisions lies between 10^{−3} and 1. The case of *D*–*D* collisions is more complicated; for the same models, the reconnection probability is anything between 0.1 and 1. The reconnection probability for *F*–*D* collisions can vary from 0 to 1.

Brane collisions not only lead to the formation of *F*- and *D*-strings; they also produce bound states, (*p*, *q*)-strings, that are composites of *p* *F*-strings and *q* *D*-strings (Copeland *et al*. 2004; Leblond & Tye 2004). The presence of stable bound states implies the existence of junctions, where two different types of string meet at a point and form a bound state leading away from that point. Thus, when cosmic superstrings of different types collide, they cannot intercommute; instead they exchange partners and form a junction at which there are three string segments. This is just a consequence of charge conservation at the junction of colliding (*p*, *q*)-strings. For *p*=*np*′ and *q*=*nq*′, the (*p*, *q*)-string is neutrally stable to splitting into *n* bound (*p*′, *q*′)-strings. The angles at which strings pointing into a vertex meet are fixed by the requirement that there be no force on the vertex. The formation of three-string junctions (Y-junctions) and kinematic constraints for their collisions have been investigated analytically (Copeland *et al*. 2006, 2007) under the assumption that each string evolves according to the Nambu–Goto action. Whether these results hold for cosmic superstrings, which carry fluxes of a gauge field and are therefore described by the Dirac–Born–Infeld action, remains to be shown.

The tension of solitonic strings is set from the energy scale of the phase transition followed by a spontaneously broken symmetry that left behind these defects as false vacuum remnants. Cosmic superstrings span a whole range of tensions, set from the particular brane inflation model. The tension of *F*-strings in 10 dimensions is *μ*_{F}=1/(2*πα*′) and the tension of *D*-strings is *μ*_{D}=1/(2*πα*′*g*_{s}). In 10 flat dimensions, supersymmetry dictates that the tension of the (*p*, *q*)-bound states reads (Schwarz 1995) Individually, the *F*- and *D*-strings are 1/2 Bogomol'nyi–Prasad–Sommerfield (BPS) objects, which, however, break a different half of the supersymmetry each. This equation represents the BPS bound for an object carrying the charges of *p* *F*-strings and *q D*-strings. In IIB string theory, our Universe can be described as a brane-world scenario with flux compactification. In this context, the standard model particles are light open string modes in a warped throat of the Calabi–Yau manifold. The string tension for strings at the bottom of a throat is different from the (simple) expression given above. The formula for tension depends on the choice of flux compactification. For example, for the Klebanov–Strassler throat (Klebanov & Strassler 2000), inside which the geometry is a shrinking *S*^{2} fibred over an *S*^{3}, the tension of the bound state of *p* *F*-strings and that of *q D*-strings reads (Gubser *et al*. 2004) and respectively, where *p* and *q* are integers, *h*_{A} is the warp factor at the bottom of the throat, *b*=0.93 is a number of the order of unity, and *M* denotes the number of fractional *D*3-branes. The tension formula for the (*p*, *q*)-bound states reads (Firouzjahi *et al*. 2006) .

Type I vortices in the Abelian Higgs model can also have three-vertex junctions and a range of tensions (Donaire & Rajantie 2006); in this way, they have more similarities with cosmic superstrings. However, cosmic superstrings are the only ones to have the integer-valued charges *p* and *q*. Thus, cosmic superstrings can, at least in principle, be distinguished from gauge theory strings.

## 4. Evolution of cosmic string and superstring networks

Let me first summarize our understanding of the evolution of cosmic string networks (Sakellariadou 2007). Knowing the differences between cosmic strings and cosmic superstrings, it is at least, in principle, *easy* to identify possible deviations between the evolution of the two networks. The first studies of the evolution of a cosmic string network were analytical. They have shown (Kibble 1985) the existence of *scaling*, where at least the basic properties of the string network can be characterized by a single length scale, roughly the persistence length or the inter-string distance *ξ*, which grows with the horizon. This is a key property for cosmic strings since it renders them cosmologically acceptable, a crucial difference between cosmic strings and monopoles or domain walls. The scaling solution was supported by subsequent numerical work (Albrecht & Turok 1985). However, further investigation revealed dynamical processes, including loop production, at scales much smaller than *ξ* (Bennett & Bouchet 1988; Sakellariadou & Vilenkin 1990).

The energy density of super-horizon (*infinite* or long) strings in the scaling regime is given (in the radiation-dominated era) by *ρ*_{long}=*κμt*^{−2}, where *κ* is a numerical coefficient (*κ*=20±10). The sub-horizon loops, their size distribution and the mechanism of their formation remained for years the least understood parts of the string evolution. Assuming that the super-horizon strings are characterized by a single length scale *ξ*(*t*), one obtains *ξ*(*t*)=(*ρ*_{long}/*μ*)^{−1/2}=*κ*^{−1/2}*t*. The typical distance between the nearest string segments and the typical curvature radius of the strings are both of the order of *ξ*. Early numerical simulations have shown that, indeed, the typical curvature radius of long strings and the characteristic distance between the strings are both comparable with the evolution time *t*. Clearly, these results agree with the picture of the scale-invariant evolution of the string network and with the one-scale hypothesis. However, the numerical simulations have also shown (Bennett & Bouchet 1988; Sakellariadou & Vilenkin 1990) that small-scale processes (such as the production of small sub-horizon loops) play an essential role in the energy balance of long strings. The existence of an important small-scale structure (*wiggliness*) superimposed on the super-horizon strings was also evident by analysing the string shapes (Sakellariadou & Vilenkin 1990).

In response to these findings, a three-scale model was developed (Austin *et al*. 1993), which describes the network in terms of three scales: the energy density scale *ξ*; a correlation length along the string; and a scale *ζ* relating to local structure on the string. The small-scale structure (wiggliness), which offers an explanation for the formation of the small sub-horizon-sized loops, is basically developed through intersections of long string segments. It seemed likely from the three-scale model that *ξ* and would scale, with *ζ* growing slowly, if at all, until gravitational radiation became important when *ζ*/*ξ*≈10^{−4} (Sakellariadou 1990). According to the three-scale model, the small length scale may reach scaling only if one considers the gravitational back-reaction effect. Aspects of the three-scale model have been checked (Vincent *et al*. 1997), evolving a cosmic string network in Minkowski space–time. These string simulations found that loops are produced with tiny sizes, which led the authors to suggest that the dominant mode of energy loss of a cosmic string network is particle production and not gravitational radiation as the loops collapse almost immediately.

Recently, the numerical simulations of the cosmic string evolution in a Friedmann–Lemaıˆtre–Robertson–Walker universe have found evidence (Ringeval *et al*. 2007) of a scaling regime for the cosmic string loops in the radiation- and matter-dominated eras down to 100th of the horizon time. The scaling was found without considering any gravitational back-reaction effect; it was just the result of string intercommutations. The scaling regime of string loops appears after a transient relaxation era, driven by a transient overproduction of string loops with length close to the initial correlation length of the string network. Calculating the amount of the energy–momentum tensor lost from the string network, it was found that a few per cent of the total string energy density disappears during a very brief process of formation of numerically unresolved loops, which takes place during the very first time steps of the string evolution. Analytical studies (Polchinski & Rocha 2006, 2007; Dubath *et al*. 2007) confirmed the numerical results.

The evolution of cosmic superstrings is clearly a more involved problem. Cosmic superstring networks not only have sub-horizon loops and super-horizon strings, but also have Y-junctions. In addition, one must consider a multi-tension spectrum and reconnection probabilities that can be much lower than unity. Certainly, computers are at present much more efficient than in the 1980s and 1990s when we performed the first numerical experiments with solitonic cosmic strings, and we obviously gained a lot of experience from those studies. Nevertheless, one must keep in mind that the evolution of cosmic strings has been studied almost exclusively in the (simple) case of the infinitely thin-string approximation. Therefore, even for the case of solitonic strings, the problem is so complex that all numerical experiments have been performed for the simplest (and, probably, less realistic) models. The evolution of a cosmic superstring network has very important consequences for the validity of the brane inflation model employed. The existence of Y-junctions may prevent a scaling solution. If such a network freezes, it may lead to undesirable cosmological consequences. A number of numerical experiments (Avgoustidis & Shellard 2005, 2006; Copeland & Saffin 2005; Saffin 2005; Sakellariadou 2005; Hindmarsh & Saffin 2006; Rajantie *et al*. 2007; Urrestilla & Vilenkin 2008) have addressed this issue, each of them at a different level of approximation.

I briefly describe the approach and findings of one of these numerical approaches (Rajantie *et al*. 2007). The aim of that study was to build a simple field-theory model of (*p*, *q*)-bound states, in analogy with the Abelian Higgs model used to investigate the properties of solitonic cosmic string networks, and to study the overall characteristics of the network using lattice simulations. The (*p*, *q*)-string network was modelled using two sets of Abelian Higgs fields. Two models were investigated, one in which both species of string have only short-range interactions and another one in which one species of string features long-range interactions. More precisely, we modelled the network with no long-range interactions using two sets of fields, complex scalars coupled to gauge fields, with a potential chosen such that the two types of strings will form bound states. In this way, junctions of three strings with different tensions were successfully modelled. In order to introduce long-range interactions, we considered a network in which one of the scalars forms global strings. This is important if the strings are of a non-BPS species. For example, for cosmic superstrings at the bottom of a Klebanov–Strassler throat, the *F*-string is not BPS while the *D*-string is. Thus, the different components of the (*p*, *q*) state are expected to exhibit different types of long-range interactions. The evolution of the string networks suggested that the long-range interactions have a much more important role in the network evolution than the formation of bound states. In the local–global networks, the bound states tend to split as a result of the long-range interactions, resulting in two networks that evolve almost independently. The formation of short-lived bound states and their subsequent splitting only increases the small-scale *wiggliness* of the local strings. In the case of a local–local network, the absence of long-range interactions allows the bound states to be much longer-lived and significantly influences the evolution of the network.

Even though preliminary studies indicate that the presence of junctions is not itself inconsistent with scaling, this issue is far from being resolved. Numerical experiments support a scaling solution, but this does not necessarily imply that realistic superstring networks formed at the end of a successful and natural brane inflationary era will reach scaling. Modelling a (*p*, *q*) network is a challenging task, and further investigations are necessary before this issue gets satisfactorily answered.

## 5. Observational consequences

Cosmic superstrings interact with the standard model particles via gravity, thus their detection involves gravitational interactions. By comparing the predictions against astrophysical data, one should be able to constrain the free parameters of the specific model. However, since the particular brane inflationary scenario remains unknown, the tensions of superstrings will be only loosely constrained.

I briefly discuss the observational signatures of cosmic strings, studied in the case of the Abelian Higgs model, and relate them to those of cosmic superstrings. All observational consequences of solitonic strings have been studied in the case of the Abelian Higgs model. The case of superconducting strings has almost always been neglected. Under these assumptions, cosmic strings can be well described by the Nambu–Goto action. In addition, the reconnection probability has always been considered as exactly equal to unity. Solitonic strings have been considered as having winding number equal to unity, thus junctions are not formed. Certainly, given the complexity of string evolution (nonlinear process), it is very difficult to go beyond this simple context in which solitonic strings have been studied. Nevertheless, one should be very careful when one deduces the observational consequences of strings; at least, the quantitative discussion will be invalid for more general string configurations. The complexity of cosmic superstrings and the uncertainty of the physical context (compactification, brane inflation model) of their formation render any discussion on their observational consequences even more uncertain.

Cosmic strings perturb the space–time around them so that a conical space–time is generated, leading to a unique gravitational lensing signature through the appearance of undistorted double images. The finding of even a single such gravitational lensing event would be seen as convincing evidence for the existence of cosmic strings. In the case of strings with junctions, the Y-shaped junctions give rise to lensing events that are qualitatively distinct from that produced in the case of *conventional* strings (Shlaer & Wyman 2005; Brandenberger *et al*. 2007). Identifying such a triple imaging event in the sky would provide a *smoking gun* for the existence of a string network with non-trivial interactions. Certainly, to generalize this study in the case of cosmic superstrings may not be straightforward.

Micro-lensing is very useful in detecting lensing when the image splitting is too small to resolve with astronomical measurements. The gravitational micro-lensing of distant quasars by solitonic cosmic strings has recently been investigated (Kuijken *et al*. 2008). The analysis seems to indicate rather pessimistic results for the detectability of such micro-lensing events generated by cosmic strings. These studies have not been extended for strings with non-trivial interactions.

Cosmic strings can also lead to weak gravitational lensing. A recent study (Dyda & Brandenberger 2007) on gauge cosmic strings has shown that, if such strings have a small-scale structure leading to a local gravitational attractive force towards them, then an elliptical distortion of the shape of background galaxies in the direction corresponding to the projection of the string onto the sky may be expected. Weak lensing has not been investigated in the case of cosmic superstrings (nor in the simpler case of strings with Y-junctions).

The CMB temperature anisotropies offer a powerful test for theoretical models aimed at describing the early Universe. The characteristics of the CMB multipole moments can be used to discriminate among theoretical models and to constrain the parameter space. According to our present understanding, the CMB temperature anisotropies originate mainly from the amplification of quantum fluctuations at the end of inflation, with a small contribution from the cosmic (super)string network. Given the small size of the observed CMB temperature anisotropies, the perturbations may be treated linearly. Thus, any coupling between perturbations induced by inflation and those seeded by cosmic strings can be neglected. Using the latest WMAP data and big bang nucleosynthesis (BBN) data, the fractional contribution from cosmic strings to the temperature power spectrum at the multipole moment *ℓ*=10 is at most 0.11 (Bevis *et al*. 2008). In other words, if the normalization of the string component has been set to match the data at multipole *ℓ*=10, the string contribution cannot exceed 11%. This translates into the upper limit on the dimensionless parameter *Gμ* (*G* is the gravitational constant and *μ* the string tension) given by (Bevis *et al*. 2008) *Gμ*<0.7×10^{−6}. This limit was derived for classical Abelian Higgs strings with equal vector and scalar particle masses; it is not expected to be valid for other types of strings.

The polarization of the CMB photons can give further information and constraints on the cosmological role of cosmic strings. The B-mode polarization spectrum provides an important window on cosmic strings, since the corresponding contribution from inflation is rather weak. Scalar modes may contribute to the B-mode only via the gravitational lensing of the E-mode signal, with a second inflationary contribution arising from the subdominant tensor modes. It is thus conceivable that the large vector contributions from cosmic strings enable the detection of their imprint through future B-mode measurements. In this way, current views on *natural* inflationary models may challenge the conventional thought that a detection of B-mode polarization in the CMB will show the existence of gravity waves (GWs) in the early Universe and determine the energy scale of inflation. To distinguish the cosmic string from the inflationary gravity wave signal, one should go to rather high energy resolution, since the signal from cosmic strings seems to be dominant at *ℓ*∼1000, while the gravity wave signal from inflation peaks at *ℓ*∼100 (Seljak & Slosar 2006). The prediction of a large cosmic string contribution to the B-mode polarization power spectrum has been confirmed even for small string contributions to the CMB. More precisely, it has been argued (Bevis *et al*. 2007) that data from future ground-based polarization detectors may bound the dimensionless string parameter to *Gμ*<0.12×10^{−6}. Cosmic strings can also become apparent through their contribution to the small-angle CMB temperature anisotropies. More precisely, at high multipoles *ℓ* (small angular resolution), the mean angular power spectrum of string-induced CMB temperature anisotropies can be described (Fraisse *et al*. 2007) by *ℓ*^{−α}, with *α*∼0.889. Thus, a non-vanishing string contribution to the overall CMB temperature anisotropies may dominate at high multipoles *ℓ* (small angular scales). In an arcminute resolution experiment, strings may be observable for *Gμ* down to 2×10^{−7} (Fraisse *et al*. 2007). Cosmic strings should also induce deviations from Gaussianity. On large angular scales, such deviations are washed out due to the low string contribution; however, on small angular scales, optimal non-Gaussian, string-devoted statistical estimators may impose severe constraints on a possible cosmic string contribution to the CMB temperature anisotropies.

One should keep in mind that all string-induced CMB temperature anisotropies were performed for Abelian strings in the zero-thickness limit, with reconnection probability equal to unity and winding number equal to 1. Even though, in any model where fluctuations are constantly induced by sources (*seeds*) having a nonlinear evolution, the perfect coherence that characterizes the inflationary-induced spectrum of perturbations gets destroyed (Durrer & Sakellariadou 1997; Durrer *et al*. 1997), there is still no reason to expect that, quantitatively, the results found for conventional cosmic string models will hold in more general cases.

Cosmic strings are expected to produce a stochastic background of GWs, which can be estimated by the incoherent superposition of GW bursts at the cusps and kinks of a network of oscillating string loops. This stochastic GW background may be detectable by pulsar timing observations. Gravity wave bursts emitted from the cusps of oscillating string loops may be detected by the LIGO/VIRGO and LISA interferometers. For *conventional* cosmic strings, it has been argued (Damour & Vilenkin 2000, 2001) that, even if only 10% of all string loops have cusps, the GW bursts might be detectable by the planned GW detectors for string tensions as small as *Gμ*∼10^{−13}. The result depends on the number of cusps, which is still not well known. Preliminary studies of gravity wave emission from cosmic superstring networks (Damour & Vilenkin 2005) seem to indicate that the smaller reconnection probability will enhance the observational signature of cosmic superstrings. The only difference that has been considered for those investigations is the reconnection probability. Analysing the BBN, CMB and pulsar timing bounds, it seems that the BBN and CMB bounds are consistent with, but somewhat weaker than, the pulsar bound. It is argued (Siemens *et al*. 2007) that, considering string networks with small reconnection probabilities, , strings with *Gμ*≥10^{−12} are ruled out when ∼10^{−3}. Increasing the reconnection probability, strings with *Gμ*≥10^{−10} are ruled out, while the bound becomes *Gμ*≥10^{−8} for ∼1. These results depend on the evolution of the string network, on the number of cusps/kinks, as well as on the reconnection probability. Since only the reconnection probability has been taken into account, it is rather immature to claim that these bounds correspond to realistic cosmic superstring networks. Further investigations are necessary. The energy scale of cosmic strings and superstrings can also be constrained from the emission of moduli. Preliminary studies have been performed; however, further studies with more realistic cosmic superstring networks are required.

## 6. Conclusions

For a number of years, inflation and topological defects have been considered as either two incompatible or two competing aspects of modern cosmology. Historically, one of the reasons for which inflation was proposed is to rescue the standard hot big bang model from the monopole problem. However, such a mechanism could also dilute cosmic strings unless they were produced at the end of or after inflation. Later on, inflation and topological defects competed as the two alternative mechanisms generating the density perturbations, which led to the observed large-scale structure and the anisotropies in the CMB. The plethora of data on the CMB has revealed an early Universe in striking agreement with the basic predictions of inflation, while there is a clear inconsistency between predictions from the topological defect models and the CMB data. This indicates a clear preference for inflation. However, despite its success, inflation still lacks a precise theoretical model.

In the context of a brane-world cosmological scenario within string theory, brane inflation can be easily accommodated. Brane inflation takes place while two branes move towards each other, and their annihilation releases the brane tension energy that heats up the Universe to start the radiation-dominated era of the hot big bang model. Typically, strings of all sizes and types may be produced during the collision. Large fundamental strings and/or *D*1-branes that survive the cosmological evolution become cosmic superstrings. The study of cosmic superstrings and their observational signatures will give us an understanding of the early stages of our Universe, and it will provide information on identifying the details of the string theory model that has relevance with our Universe. Thus, the issue of cosmic superstrings is gaining a lot of interest from the scientific community. Numerical experiments of cosmic superstring networks and analytical studies are trying to unravel the properties and characteristics of such networks. The problem is quite complex and our intuition from the *conventional* solitonic strings described by the Nambu–Goto action may turn out to be misleading. Many open questions are currently under investigation and we expect soon to have a better understanding of their properties and observational signatures that will allow us to detect them.

## Acknowledgments

It is a pleasure to thank Tom Kibble and George Pickett, the organizers of the Royal Society Discussion Meeting, who invited me to give this lecture. This work was supported in part by the European Union through the Marie Curie Research and Training Network UniverseNet (MRTN-CN-2006-035863).

## Footnotes

One contribution of 16 to a Discussion Meeting Issue ‘Cosmology meets condensed matter’.

- © 2008 The Royal Society