Cosmic string
Encyclopedia
Cosmic strings are hypothetical 1-dimensional (spatially) topological defect
s which may have formed during a symmetry breaking
phase transition
in the early universe when the topology
of the vacuum
manifold associated to this symmetry breaking is not simply connected
. It is expected that at least one string per Hubble volume
is formed. Their existence was first contemplated by the theoretical physicist Tom Kibble in the 1970s. The formation of cosmic strings is somewhat analogous to the imperfections that form between crystal grains in solidifying liquids, or the cracks that form when water freezes into ice. The phase transitions leading to the production of cosmic strings are likely to have occurred during the earliest moments of the universe's evolution just after cosmological inflation and are a fairly generic prediction in both Quantum field theory
and String theory
models of the Early universe. In string theory the role of cosmic strings can be played by the fundamental strings (or F-strings) themselves that define the theory perturbatively
, by D-strings which are related to the F-strings by weak-strong or so called S-duality
, or higher dimensional D-, NS- or M-branes that are partially wrapped on compact cycles associated to extra spacetime dimensions so that only one non-compact dimension remains, see the article by Copeland, Myers and Polchinski (pdf). The prototypical example of a quantum field theory with cosmic strings is the Abelian Higgs model. The quantum field theory and string theory cosmic strings are expected to have many properties in common, but more research is needed to determine the precise distinguishing features. The F-strings for instance are fully quantum-mechanical and do not have a classical definition, whereas the field theory cosmic strings are almost exclusively treated classically.
Cosmic strings, if they exist, would be extremely thin with diameters of the same order of magnitude as that of a proton, i.e. ~ 1 fm, or smaller. Given that this scale is much smaller than any cosmological scale these strings are often studied in the zero width, or Nambu-Goto approximation. Under this assumption strings behave as one-dimensional objects and obey the Nambu-Goto action
, which is classically equivalent to the Polyakov action
that defines the bosonic sector of superstring theory
. Even though cosmic strings are thought to be extremely thin they would have immense density and so would represent significant gravitational wave sources. In field theory, the string width is set by the scale of the symmetry breaking phase transition. In string theory, the string width is set (in the simplest cases) by the fundamental string scale, warp factors (associated to the spacetime curvature of an internal six-dimensional spacetime manifold) and/or the size of internal compact dimension
s. (In string theory, the universe is either 10- or 11-dimensional, depending on the strength of interactions and the curvature of spacetime.) A cosmic string about a kilometer in length may be more massive than the Earth. However general relativity
predicts that the gravitational potential of a straight string vanishes: there is no gravitational force on static surrounding matter. The only gravitational effect of a straight cosmic string is a relative deflection of matter (or light) passing the string on opposite sides (a purely topological effect). A closed cosmic string gravitates in a more conventional way. During the expansion of the universe, cosmic strings would form a network of loops, and in the past it was thought that their gravity could have been responsible for the original clumping of matter into galactic superclusters. It is now known that their contribution to the structure formation in the universe is less than 10%.
Other types of topological defects in spacetime are domain wall
s, monopole
s, and texture
s.
The violent oscillations of cosmic strings generically lead to the formation of cusps
and kinks. These in turn cause parts of the string to pinch off into isolated loops. These loops have a finite lifespan and decay (primarily) via gravitational radiation. This radiation which leads to the strongest signal from cosmic strings may in turn be detectable in gravitational wave experiments, such as LIGO and LISA. An important open question is to what extent do the pinched off loops backreact or change the initial state of the emitting cosmic string—such backreaction effects are almost always neglected in computations and are known to be important, even for order of magnitude estimates.
Gravitational lensing of a galaxy by a straight section of a cosmic string would produce two identical, undistorted images of the galaxy. In 2003 a group led by Mikhail Sazhin reported the accidental discovery of two seemingly identical galaxies very close together in the sky, leading to speculation that a cosmic string had been found. However, observations by the Hubble Space Telescope
in January 2005 showed them to be a pair of similar galaxies, not two images of the same galaxy. A cosmic string would produce a similar duplicate image of fluctuations in the cosmic microwave background, which might be detectable by the Planck Surveyor
mission.
A second piece of evidence supporting cosmic string theory is a phenomenon observed in observations of the "double quasar
" called Q0957+561A,B
. Originally discovered by Dennis Walsh
, Bob Carswell, and Ray Weymann in 1979, the double image of this quasar is caused by a galaxy positioned between it and the Earth. The gravitational lens
effect of this intermediate galaxy bends the quasar's light so that it follows two paths of different lengths to Earth. The result is that we see two images of the same quasar, one arriving a short time after the other (about 417.1 days later).
However, a team of astronomers at the Harvard-Smithsonian Center for Astrophysics
led by Rudolph Schild studied the quasar and found that during the period between September 1994 and July 1995 the two images appeared to have no time delay; changes in the brightness of the two images occurred simultaneously on four separate occasions. Schild and his team believe that the only explanation for this observation is that a cosmic string passed between the Earth and the quasar during that time period traveling at very high speed and oscillating with a period of about 100 days.
The earthbound Laser Interferometer Gravitational-Wave Observatory
(LIGO) and especially the space-based gravitational wave detector Laser Interferometer Space Antenna
(LISA) will search for gravitational waves and are likely to be sensitive enough to detect signals from cosmic strings, provided the relevant cosmic string tensions are not too small.
). The possibility of cosmic strings being produced in the early universe was first envisioned by quantum field theorist Tom Kibble in 1976, and this sprouted the first flurry of interest in the field. In 1985, during the first superstring revolution, Edward Witten
contemplated on the possibility of fundamental superstrings having been produced in the early universe and stretched to macroscopic scales, in which case (following the nomenclature of Tom Kibble) they would then be referred to as cosmic superstrings. He concluded that had they been produced they would have either disintegrated into smaller strings before ever reaching macroscopic scales (in the case of Type I superstring theory), they would always appear as boundaries of domain walls whose tension would force the strings to collapse rather than grow to cosmic scales (in the context of Heterotic superstring
theory), or having a characteristic energy scale close to the Planck energy they would be produced before cosmological inflation and hence be diluted away with the expansion of the universe and not be observable.
Much has changed since these early days, primarily due to the second superstring revolution. It is now known that string theory in addition to the fundamental strings which define the theory perturbatively also contains other one-dimensional objects, such as D-strings, and higher dimensional objects such as D-branes, NS-branes and M-branes partially wrapped on compact internal spacetime dimensions, while being spatially extended in one non-compact dimension. The possibility of large compact dimensions
and large warp factors allows strings with tension much lower than the Planck scale. Furthermore, various dualities that have been discovered point to the conclusion that actually all these apparently different types of string are just the same object as it appears in different regions of parameter space. These new developments have largely revived interest in cosmic strings, starting in the early 2000s.
In 2002, Henry Tye
and collaborators predicted the production of cosmic superstrings during the last stages of brane inflation
, a string theory construction of the early universe that gives leads to an expanding universe and cosmological inflation. It was subsequently realized by string theorist Joseph Polchinski
that the expanding Universe could have stretched a "fundamental" string (the sort which superstring theory considers) until it was of intergalactic size. Such a stretched string would exhibit many of the properties of the old "cosmic" string variety, making the older calculations useful again. As theorist Tom Kibble remarks, "string theory cosmologists have discovered cosmic strings lurking everywhere in the undergrowth". Older proposals for detecting cosmic strings could now be used to investigate superstring theory.
Superstrings, D-strings or the other stringy objects mentioned above stretched to intergalactic scales would radiate gravitational waves, which could be detected using experiments like LIGO and especially the space-based gravitational wave experiment LISA. They might also cause slight irregularities in the cosmic microwave background, too subtle to have been detected yet but possibly within the realm of future observability.
Note that most of these proposals depend, however, on the appropriate cosmological fundamentals (strings, branes, etc.), and no convincing experimental verification of these has been confirmed to date. Cosmic strings nevertheless provide a window into string theory. If cosmic strings are observed which is a real possibility for a wide range of cosmological string models this would provide the first experimental evidence of a string theory model underlying the structure of spacetime.
Topological defect
In mathematics and physics, a topological soliton or a topological defect is a solution of a system of partial differential equations or of a quantum field theory homotopically distinct from the vacuum solution; it can be proven to exist because the boundary conditions entail the existence of...
s which may have formed during a symmetry breaking
Symmetry breaking
Symmetry breaking in physics describes a phenomenon where small fluctuations acting on a system which is crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken. To an outside observer unaware of the fluctuations , the choice will appear arbitrary...
phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
in the early universe when the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
of the vacuum
Vacuum state
In quantum field theory, the vacuum state is the quantum state with the lowest possible energy. Generally, it contains no physical particles...
manifold associated to this symmetry breaking is not simply connected
Simply connected space
In topology, a topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question .If a space is not simply connected, it is convenient...
. It is expected that at least one string per Hubble volume
Hubble volume
In cosmology, the Hubble volume, or Hubble sphere, is the region of the Universe surrounding an observer beyond which objects recede from the observer at a rate greater than the speed of light, due to the expansion of the Universe....
is formed. Their existence was first contemplated by the theoretical physicist Tom Kibble in the 1970s. The formation of cosmic strings is somewhat analogous to the imperfections that form between crystal grains in solidifying liquids, or the cracks that form when water freezes into ice. The phase transitions leading to the production of cosmic strings are likely to have occurred during the earliest moments of the universe's evolution just after cosmological inflation and are a fairly generic prediction in both Quantum field theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
and String theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
models of the Early universe. In string theory the role of cosmic strings can be played by the fundamental strings (or F-strings) themselves that define the theory perturbatively
Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
, by D-strings which are related to the F-strings by weak-strong or so called S-duality
S-duality
In theoretical physics, S-duality is an equivalence of two quantum field theories or string theories. An S-duality transformation maps states and vacua with coupling constant g in one theory to states and vacua with coupling constant 1/g in the dual theory...
, or higher dimensional D-, NS- or M-branes that are partially wrapped on compact cycles associated to extra spacetime dimensions so that only one non-compact dimension remains, see the article by Copeland, Myers and Polchinski (pdf). The prototypical example of a quantum field theory with cosmic strings is the Abelian Higgs model. The quantum field theory and string theory cosmic strings are expected to have many properties in common, but more research is needed to determine the precise distinguishing features. The F-strings for instance are fully quantum-mechanical and do not have a classical definition, whereas the field theory cosmic strings are almost exclusively treated classically.
Cosmic strings, if they exist, would be extremely thin with diameters of the same order of magnitude as that of a proton, i.e. ~ 1 fm, or smaller. Given that this scale is much smaller than any cosmological scale these strings are often studied in the zero width, or Nambu-Goto approximation. Under this assumption strings behave as one-dimensional objects and obey the Nambu-Goto action
Nambu-Goto action
The Nambu–Goto action is the simplest invariant action in bosonic string theory, and is also used in other theories that investigate string-like objects . It is the starting point of the analysis of zero-thickness string behavior, using the principles of Lagrangian mechanics...
, which is classically equivalent to the Polyakov action
Polyakov action
In physics, the Polyakov action is the two-dimensional action of a conformal field theory describing the worldsheet of a string in string theory...
that defines the bosonic sector of superstring theory
Superstring theory
Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modelling them as vibrations of tiny supersymmetric strings...
. Even though cosmic strings are thought to be extremely thin they would have immense density and so would represent significant gravitational wave sources. In field theory, the string width is set by the scale of the symmetry breaking phase transition. In string theory, the string width is set (in the simplest cases) by the fundamental string scale, warp factors (associated to the spacetime curvature of an internal six-dimensional spacetime manifold) and/or the size of internal compact dimension
Compact dimension
In string theory, a model used in theoretical physics, a compact dimension is curled up in itself and very small . Anything moving along this dimension's direction would return to its starting point almost instantaneously, and the fact that the dimension is smaller than the smallest particle means...
s. (In string theory, the universe is either 10- or 11-dimensional, depending on the strength of interactions and the curvature of spacetime.) A cosmic string about a kilometer in length may be more massive than the Earth. However general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
predicts that the gravitational potential of a straight string vanishes: there is no gravitational force on static surrounding matter. The only gravitational effect of a straight cosmic string is a relative deflection of matter (or light) passing the string on opposite sides (a purely topological effect). A closed cosmic string gravitates in a more conventional way. During the expansion of the universe, cosmic strings would form a network of loops, and in the past it was thought that their gravity could have been responsible for the original clumping of matter into galactic superclusters. It is now known that their contribution to the structure formation in the universe is less than 10%.
Other types of topological defects in spacetime are domain wall
Domain wall
A domain wall is a term used in physics which can have one of two distinct but similar meanings in magnetism, optics, or string theory. These phenomena can all be generically described as topological solitons which occur whenever a discrete symmetry is spontaneously broken.-Magnetism:In magnetism,...
s, monopole
Magnetic monopole
A magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...
s, and texture
Texture (cosmology)
In cosmology, a texture is a type of topological defect in the structure of spacetime that forms when larger, more complicated symmetry groups are completely broken. They are not as localized as the other defects, and are unstable...
s.
Observational evidence
It was once thought that the gravitational influence of cosmic strings might contribute to the large-scale clumping of matter in the universe, but all that is known today through galaxy surveys and precision measurements of the cosmic microwave background (CMB) fits an evolution out of random, gaussian fluctuations. These precise observations therefore tend to rule out a significant role for cosmic strings and currently it is known that the contribution of cosmic strings to the CMB cannot be more than 10%.The violent oscillations of cosmic strings generically lead to the formation of cusps
Cusp (singularity)
In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
and kinks. These in turn cause parts of the string to pinch off into isolated loops. These loops have a finite lifespan and decay (primarily) via gravitational radiation. This radiation which leads to the strongest signal from cosmic strings may in turn be detectable in gravitational wave experiments, such as LIGO and LISA. An important open question is to what extent do the pinched off loops backreact or change the initial state of the emitting cosmic string—such backreaction effects are almost always neglected in computations and are known to be important, even for order of magnitude estimates.
Gravitational lensing of a galaxy by a straight section of a cosmic string would produce two identical, undistorted images of the galaxy. In 2003 a group led by Mikhail Sazhin reported the accidental discovery of two seemingly identical galaxies very close together in the sky, leading to speculation that a cosmic string had been found. However, observations by the Hubble Space Telescope
Hubble Space Telescope
The Hubble Space Telescope is a space telescope that was carried into orbit by a Space Shuttle in 1990 and remains in operation. A 2.4 meter aperture telescope in low Earth orbit, Hubble's four main instruments observe in the near ultraviolet, visible, and near infrared...
in January 2005 showed them to be a pair of similar galaxies, not two images of the same galaxy. A cosmic string would produce a similar duplicate image of fluctuations in the cosmic microwave background, which might be detectable by the Planck Surveyor
Planck Surveyor
Planck is a space observatory of the European Space Agency and designed to observe the anisotropies of the cosmic microwave background over the entire sky, at a high sensitivity and angular resolution...
mission.
A second piece of evidence supporting cosmic string theory is a phenomenon observed in observations of the "double quasar
Quasar
A quasi-stellar radio source is a very energetic and distant active galactic nucleus. Quasars are extremely luminous and were first identified as being high redshift sources of electromagnetic energy, including radio waves and visible light, that were point-like, similar to stars, rather than...
" called Q0957+561A,B
Twin Quasar
The Twin Quasar , also known as SBS 0957+561, or TXS 0957+561 , was the first identified gravitationally lensed object.-Quasar:...
. Originally discovered by Dennis Walsh
Dennis Walsh
Dennis Walsh was an English astronomer, born into a poor family in Manchester. He was best known for his discovery in 1979 of the first example of a gravitational lens which he made while studying quasars found in the Jodrell Bank 966MHz survey.He developed an early aptitude for mathematics and...
, Bob Carswell, and Ray Weymann in 1979, the double image of this quasar is caused by a galaxy positioned between it and the Earth. The gravitational lens
Gravitational lens
A gravitational lens refers to a distribution of matter between a distant source and an observer, that is capable of bending the light from the source, as it travels towards the observer...
effect of this intermediate galaxy bends the quasar's light so that it follows two paths of different lengths to Earth. The result is that we see two images of the same quasar, one arriving a short time after the other (about 417.1 days later).
However, a team of astronomers at the Harvard-Smithsonian Center for Astrophysics
Harvard-Smithsonian Center for Astrophysics
The Harvard–Smithsonian Center for Astrophysics is one of the largest and most diverse astrophysical institutions in the world, where scientists carry out a broad program of research in astronomy, astrophysics, earth and space sciences, and science education...
led by Rudolph Schild studied the quasar and found that during the period between September 1994 and July 1995 the two images appeared to have no time delay; changes in the brightness of the two images occurred simultaneously on four separate occasions. Schild and his team believe that the only explanation for this observation is that a cosmic string passed between the Earth and the quasar during that time period traveling at very high speed and oscillating with a period of about 100 days.
The earthbound Laser Interferometer Gravitational-Wave Observatory
LIGO
LIGO, which stands for the Laser Interferometer Gravitational-Wave Observatory, is a large-scale physics experiment aiming to directly detect gravitational waves. Cofounded in 1992 by Kip Thorne and Ronald Drever of Caltech and Rainer Weiss of MIT, LIGO is a joint project between scientists at MIT,...
(LIGO) and especially the space-based gravitational wave detector Laser Interferometer Space Antenna
Laser Interferometer Space Antenna
The Laser Interferometer Space Antenna is a planned space mission to detect and accurately measure gravitational waves from astronomical sources. LISA was originally conceived as a joint effort between the United States space agency NASA and the European Space Agency...
(LISA) will search for gravitational waves and are likely to be sensitive enough to detect signals from cosmic strings, provided the relevant cosmic string tensions are not too small.
String theory and cosmic strings
During the early days of string theory both string theorists and cosmic string theorists believed that there was no direct connection between superstrings and cosmic strings (the names were chosen independently by analogy with ordinary stringTwine
Twine is a light string or strong thread composed of two or more smaller strands or yarns twisted together. More generally, the term can be applied to any thin cord....
). The possibility of cosmic strings being produced in the early universe was first envisioned by quantum field theorist Tom Kibble in 1976, and this sprouted the first flurry of interest in the field. In 1985, during the first superstring revolution, Edward Witten
Edward Witten
Edward Witten is an American theoretical physicist with a focus on mathematical physics who is currently a professor of Mathematical Physics at the Institute for Advanced Study....
contemplated on the possibility of fundamental superstrings having been produced in the early universe and stretched to macroscopic scales, in which case (following the nomenclature of Tom Kibble) they would then be referred to as cosmic superstrings. He concluded that had they been produced they would have either disintegrated into smaller strings before ever reaching macroscopic scales (in the case of Type I superstring theory), they would always appear as boundaries of domain walls whose tension would force the strings to collapse rather than grow to cosmic scales (in the context of Heterotic superstring
Heterotic string
In physics, a heterotic string is a peculiar mixture of the bosonic string and the superstring...
theory), or having a characteristic energy scale close to the Planck energy they would be produced before cosmological inflation and hence be diluted away with the expansion of the universe and not be observable.
Much has changed since these early days, primarily due to the second superstring revolution. It is now known that string theory in addition to the fundamental strings which define the theory perturbatively also contains other one-dimensional objects, such as D-strings, and higher dimensional objects such as D-branes, NS-branes and M-branes partially wrapped on compact internal spacetime dimensions, while being spatially extended in one non-compact dimension. The possibility of large compact dimensions
Large extra dimension
In particle physics, models with universal extra dimensions propose that there are one or more additional dimensions beyond the three spatial dimensions and one temporal dimension that are observed...
and large warp factors allows strings with tension much lower than the Planck scale. Furthermore, various dualities that have been discovered point to the conclusion that actually all these apparently different types of string are just the same object as it appears in different regions of parameter space. These new developments have largely revived interest in cosmic strings, starting in the early 2000s.
In 2002, Henry Tye
Henry Tye
Sze-Hoi Henry Tye is a Chinese-American cosmologist and theoretical physicist most notable for proposing that a brane and an antibrane attraction and annihilation with one another, causes cosmic inflation and his work on superstring theory, brane cosmology and elementary particle physics. He...
and collaborators predicted the production of cosmic superstrings during the last stages of brane inflation
Brane cosmology
Brane cosmology refers to several theories in particle physics and cosmology motivated by, but not exclusively derived from, superstring theory and M-theory.-Brane and bulk:...
, a string theory construction of the early universe that gives leads to an expanding universe and cosmological inflation. It was subsequently realized by string theorist Joseph Polchinski
Joseph Polchinski
Joseph Polchinski is a physicist working on string theory. He graduated from Canyon del Oro High School in Tucson, Arizona in 1971, obtained his B.S. degree from Caltech in 1975, and his Ph.D. from the University of California, Berkeley in 1980 under the supervision of Stanley Mandelstam...
that the expanding Universe could have stretched a "fundamental" string (the sort which superstring theory considers) until it was of intergalactic size. Such a stretched string would exhibit many of the properties of the old "cosmic" string variety, making the older calculations useful again. As theorist Tom Kibble remarks, "string theory cosmologists have discovered cosmic strings lurking everywhere in the undergrowth". Older proposals for detecting cosmic strings could now be used to investigate superstring theory.
Superstrings, D-strings or the other stringy objects mentioned above stretched to intergalactic scales would radiate gravitational waves, which could be detected using experiments like LIGO and especially the space-based gravitational wave experiment LISA. They might also cause slight irregularities in the cosmic microwave background, too subtle to have been detected yet but possibly within the realm of future observability.
Note that most of these proposals depend, however, on the appropriate cosmological fundamentals (strings, branes, etc.), and no convincing experimental verification of these has been confirmed to date. Cosmic strings nevertheless provide a window into string theory. If cosmic strings are observed which is a real possibility for a wide range of cosmological string models this would provide the first experimental evidence of a string theory model underlying the structure of spacetime.
See also
- 0-dimensional topological defect: magnetic monopoleMagnetic monopoleA magnetic monopole is a hypothetical particle in particle physics that is a magnet with only one magnetic pole . In more technical terms, a magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring...
- 1-dimensional topological defect: cosmic string
- 2-dimensional topological defect: domain wallDomain wallA domain wall is a term used in physics which can have one of two distinct but similar meanings in magnetism, optics, or string theory. These phenomena can all be generically described as topological solitons which occur whenever a discrete symmetry is spontaneously broken.-Magnetism:In magnetism,...
- cosmic string loop stabilised by a fermionic supercurrent: vortonVortonA vorton is a circular cosmic string loop stabilized by the angular momentum of the charge and current trapped on the string.For a recent review article of cosmic strings see ....
External links
- http://www.damtp.cam.ac.uk/user/gr/public/cs_interact.html
- astro-ph/0302547 CSL-1: a chance projection effect or serendipitous discovery of a gravitational lens induced by a cosmic string?
- astro-ph/0406434 Anomalous Fluctuations in Observations of Q0957+561 A,B: Smoking Gun of a Cosmic String?
- astro-ph/0410073 Cosmic strings reborn?
- astro-ph/0503120 Signatures of Cosmic Strings in the Cosmic Microwave Background
- astro-ph/0506400 Further spectroscopic observations of the CSL-1 object
- astro-ph/0603838 Hubble Imaging Excludes Cosmic String Lens
- Cosmic strings and superstrings on arxiv.org