Cosmic Strings: Lovely Objects!Safa MotesharreiPhysics Department,University of Maryland,College Park, MD, USA,[email protected] 15, 2007AbstractCosmis strings are the most promising amongthe cosmological defects. They also consti-tute earliest possible observable strings, there-fore prove vital for string theorists. Here I reviewdifferent kinds of cosmological defects, then fo-cus on the cosmic strings. I give a short theoreti-cal derivation first and then discuss experimentalefforts to detect them.1 IntroductionTopological defects are stable configurations ofmatter formed at phase transitions in the veryearly universe. There are a number of possi-ble types of defects, such as domain walls, cos-mic strings, monopoles, textures and other ‘hy-brid’ creatures. The type of defect formed isdetermined by the symmetry properties of thematter and the nature of the phase transition.[1]The critical temperature Tcis determined by thesymmetry-breaking scale η.Domain walls are two-dimensional objectsthat form when a discrete symmetry is broken ata phase transition. A network of domain wallseffectively partitions the universe into various”cells”. Domain walls have some rather pecu-liar properties. For example, the gravitationalfield of a domain wall is repulsive rather thanattractive.A cosmic string is a hypothetical 1-dimensional topological defect in the fabricof spacetime. Cosmic strings are hypothesizedto form when different regions of spacetimeundergo phase changes, resulting in domainboundaries between the two regions when theymeet. This is somewhat analogous to theboundaries that form between crystal grainsin solidifying liquids, or the cracks that formwhen water freezes into ice. In the case of ouruniverse, such phase changes may have occurredin the early days as the universe formed. Thetension or energy per unit length µ of thesestrings is proportional to η1/2.Monopoles are zero-dimensional (point-like)objects which form when a spherical symmetryis broken. Monopoles are predicted to be su-permassive and carry magnetic charge. The ex-istence of monopoles is an inevitable predictionof grand unified theories (GUTs); this is one ofthe puzzles of the standard cosmology. However,1monopoles could be eliminated by introducingan inflationary period. This is only one of theadvantages of the inflation hypothesis.Textures form when larger, more complicatedsymmetry groups are completely broken. Tex-tures are delocalized topological defects whichare unstable to collapse.Most of these defects except cosmic strings,if exist, will cause drastic deviations from as-tronomical observations. This provides a goodground for arguing on their existence in reality.The idea of cosmic strings was given first byVilenkin[11]. Cosmic strings became very popu-lar during eighties and nineties, since they couldoffer an altenative to the inflation theory. Theywere aimed to explain the primordial densityperturbations which resulted in the early growthof the galaxies and clusters. However, data fromCOBE, WMAP, and BOOMERanG excludedcosmic strings as the main source for such per-turbations. However, they could still count par-tially for primordial density fluctuations.In the early 2000s, theorists of string the-ory revived interest in cosmic strings. It waspointed out by Joseph Polchinski that the ex-panding Universe could have stretched a ”funda-mental” string (the sort which superstring the-ory considers) until it was of intergalactic size.Such a stretched string would exhibit many ofthe properties of the old ”cosmic” string vari-ety, making the older calculations useful again.Furthermore, modern superstring theories of-fer other objects which could feasibly resemblecosmic strings, such as highly elongated one-dimensional D-branes (known as ”D-strings”).As theorist Tom Kibble[6] remarks, ”string the-ory cosmologists have discovered cosmic stringslurking everywhere in the undergrowth”. Olderproposals for detecting cosmic strings could nowbe used to investigate superstring theory.String theorists are in deep demand of cos-mic string, not only because if observed, pro-vides the first experimental evidence for theirtheory, but also because any symmetry breakingchain–which results in G321Group of the stan-dard model starting from M-theory–demands aGUT intermediate stage which in turn by it-self definitely predicts the formation of cosmicstrings at the end of the inflation era.2 Geometry of Cosmic StringsThe energy-momentum tensor of a cosmic stringcould be written in the following way[4]:Tµν= ρ(r)diag[1, 0, 0, −1], (1)where ρ(r) = ρ0for r < R and zero otherwise.We assume that the string is lying along the thirdaxis. R shows how thick the string is. Moreover,µ = πR2is the energy per unit length of thestring. Note that the effective gravitational massof the string vanishes, i.e. ρ +3p = ρ −ρ = 0.[12]Let us take the following ansatz for the metricds2= dt2− dr2− f2(r)dθ2− dz2. (2)The nontrivial components of the Einstein tensorbecomeG00= G33= −f00f= 8πρ(r) (3)The solution is thenf(r) =(k−1sin kr for r < RA + Br for r > R, (4)where k =√8πρ0. You can see that this solutionassures that everything remains regular near r =0. Furthermore, f (r) and its first derivative must2remain continuous at r = R, thereby we get thefull solution outside the stringf(r) = (r − R) cos(kR) +sin kRk. (5)Now it’s time to expand this for r  R andkR < 1f(r) = (1 − 4πρ0R2)r = (1 − 4πµ)r. (6)Therefore we finally obtain the line element forthe spacetime around a cosmic string:ds2= dt2− dr2− (1 − 8πµ)r2dθ2− dz2. (7)Of course, we may transform this metric lo-cally into the Minkowskian metric with φ =(1 − 4µ)θ:ds2= dt2− dr2− r2dφ2− dz2. (8)However, it does not represent a Euclideanspace, since φ only changes from 0 to (1−4µ)2π.For example, circumference of a circle of radiusr = a in a t=const., z=const. plane would beC = (2π − 8πµ)a, (9)which shows that we are in a conical space witha ”deficit angle” given by∆θ = 8πµ = 5.200µ10−6. (10)Indeed, as a light ray gets from very far awayclose to the string and then again to very faraway, φ changes by π and hence θ changes byπ(1+4µ). Therefore, light deflection is δθ = 4πµ,independent of the impact parameter. Thus cos-mic objects b ehind the string within δθ fromthe string will have two distinct images of thesame size. For a GUT-string, δθ ∼ 10−5,which is indeed well withing