Lensing and CMB Anisotropies by Cosmic Strings at a Junction
aa r X i v : . [ h e p - t h ] O c t Lensing and CMB Anisotropies by Cosmic Strings at a Junction
Robert Brandenberger, ∗ Hassan Firouzjahi, † and Johanna Karouby ‡ Physics Department, McGill University,3600 University Street, Montreal, Canada, H3A 2T8
Abstract
The metric around straight arbitrarily-oriented cosmic strings forming a stationary junction isobtained at the linearized level. It is shown that the geometry is flat. The sum rules for lensingby this configuration and the anisotropies of the CMB are obtained.Keywords : Cosmic strings, Cosmology
PACS numbers: ∗ Electronic address: [email protected] † Electronic address: fi[email protected] ‡ Electronic address: [email protected] . INTRODUCTION There has been a renewal of interests in cosmic strings [1]. This is partially due to therealization that in models of brane inflation cosmic strings, and not monopoles and domainwalls, are copiously produced [2, 3]. In these models of inflation, the inflaton is the distancebetween a D3-brane and an anti D3-brane [4] . There is an attractive force between thebrane and anti-brane. Inflation ends when they collide and annihilate each other. Theend product is a network of one-dimensional defects in the form of fundamental strings,F-strings, and D1-branes, D-strings. These can further combine to form a bound states ofp F-strings and q D-strings, (p,q)-strings, for integer p an q.In more developed models of brane inflation, inflation happens in a warped region insidethe string theory compactification [5]. This way, the scale of inflation as well as the effectivetension of cosmic strings, µ , are considerably smaller than the string scale [6]. One caneasily saturate the current bounds on Gµ , the dimensionless number corresponding to thecosmic string tension [7].When two (p,q) cosmic strings intersect generally a junction is formed. This is due tocharge conservation. This is in contrast to U(1) gauge cosmic strings. When two U(1) gaugecosmic strings intersect, they usually exchange partners and intercommute with probabilityclose to unity. In this view the formation of junctions may be considered a novel feature ofthe network of cosmic superstrings. Different theoretical aspects of (p,q) strings constructionwere studied in [8, 9, 10, 11] while the cosmological evolution of a network of strings withjunctions have been investigated in [12].The most important cosmological implications of cosmic strings are the gravitational oneswhich are controlled by Gµ . Among them are the lensing effects. The geometry around acosmic string is locally flat but globally it produces a deficit angle in the plane perpendicularto the string [13]. An observer looking at an object behind the string may see two identicalimages located on opposite sides of the string.In this paper we study the lensing and CMB anisotropies due to an arbitrary configurationof straight cosmic strings forming a stationary junction. We will provide the sum rule forthe formation of multiple images. The effects of cosmic string wakes on structure formationis briefly studied. 2 I. THE SETUP
We are interested in the metric of strings at a stationary junction. We assume the junctionis at rest. With an appropriate boost, one can also consider the case of a stationary junctionmoving with a constant velocity.The action of N semi-infinite strings joined at a point is S = − X i µ i Z d t d σ p −| γ i | , (1)where µ i = | ~µ i | is the tension of the i-th string (the vector pointing in direction of the string)and γ i mn is the metric induced on each string γ i mn = g µν ∂ m X µi ∂ n X νi . (2)Here m, n = { t, σ } are the coordinates along the string worldsheet, X µ are the space-timecoordinates and g µν is the space-time metric. We shall choose σ = 0 at the junction andincreasing away from this point. Furthermore, X = t while X i = x i , X i = y i and X i = z i .In order for the junction to be stationary, the vector sum of the tensions should vanishat the junction: P i ~µ i = 0. Suppose the junction is the origin of the coordinate system andthe unit vector along the i-th string is denoted by ~n i . The conditions for the junction to bestationary are translated into X i µ i n i x = X i µ i n i y = X i µ i n i z = 0 . (3)To solve the Einstein equations, we need to find the energy-momentum tensor T µν forthe string configuration, obtained by varying the string action with respect to the metric δ g S = − X i µ i Z d t d σ √− γ i γ mni ∂ m X µi ∂ n X νi δg µν = − Z d x T µν δg µν , (4)For each string, one can choose σ to represent the line element along the string d σ = dx i + dy i + dz i ≡ dl i , (5)which implies γ i = 1 , γ iσσ = − , γ i σ = 0 . (6)3sing these in Eq. (4) one obtains T µν ( x ) = X i µ i Z dl i γ mni ∂ m X µi ∂ n X νi δ ( x − x i ) , (7)where here and in the following, the boldface letters represent the spatial parts of the four-vectors X µ .We start from a flat background metric η µν = diag (1 , − , − , −
1) . Up to linearizedlevel, the metric is g µν = η µν + h µν , (8)where h µν is the perturbation due to cosmic strings, | h µν | << R µν = 8 πG ( T µν − η µν T ) ≡ πG S µν , (9)where R µν is the Ricci tensor and T = T µµ is the trace of the energy-momentum tensor.Furthermore, using (7) one obtains S µ = 0 while S ab ( x ) = X i µ i ( δ ab − n i a n i b ) Z dl i δ ( x − x i ) , (10)where a, b are the spatial indices.The Einstein equations as usual are subject to the choice of gauge. We use the harmonicgauge where h µν ,µ − h µµ,ν = 0 , (11)and R µν = − (cid:3) h µν , (12)where (cid:3) is the four-dimensional Laplacian.The general solution of (9) and (12) is (for example see [14] ) h µν ( x ) = − G Z d y S µν ( t − | y − x | , y ) | y − x | . (13)The term t − | y − x | stands for the retarded time. For our case of a static junction we areinterested in the time-independent solution.4sing (10) in (13) yields h µ = 0 while h ab ( x ) = − G X i µ i ( δ ab − n i a n i b ) Z ∞ dl i | x − x i | . (14)Knowing that along each string x i = l i ~n i , the above integral can be calculated, and ourfinal solution is h ab = 4 G X i µ i ( δ ab − n i a n i b ) ln (cid:18) r − ~r . ~n i r (cid:19) , (15)where r = | x | is measured from the point of the junction and r is a constant of integration.One can directly check that the solution (15) satisfies the harmonic gauge (11).Similar to what happens in the case of a single infinite string, at points on each stringwhere ~r = r ~n i , the metric is singular. This is because we have started with delta functionsources. In the realistic situation when the strings have finite width, this singularity issmoothed out [16]. On the other hand, at points on the opposite side of each string where ~r = − r ~n i , the metric is non-singular.Our solution in Eq. (15) is valid for any arbitrary configuration of straight cosmic stringsin a stationary junction. In order to understand its general applicability, let us consider thecase of a single infinite straight string, the case originally considered by Vilenkin [13]. Thisconfiguration in our formalism corresponds to two semi-infinite strings with equal tensionextended back to back [15] with ~n = − ~n . We may choose the strings to extended oppositelyalong the z -axis. We obtain h zz = h zx = h zy = h xy = 0, while h xx = h yy = 4 G µ ln (cid:18) r − ( r.~n ) r (cid:19) = 8 G µ ln (cid:18) r ⊥ r (cid:19) . (16)But r ⊥ is the normal distance to the string from the point of the observer at position x , andwe obtain Vilenkin’s solution [13]. III. THE FLATNESS OF THE GEOMETRY
The geometry around an infinite cosmic string is flat away from the string core. One mayask whether or not this is also true in our case of cosmic strings at a static junction. At firstsight, the metric given in (15) does not seem to be flat. To address this question we need to5alculate the components of the Riemann tensor. Since the time coordinate decouples fromour solution, effectively we are dealing with a three-dimensional spatial geometry. In threedimensions, both the Riemann tensor R abcd and the Ricci tensor R ab have 6 independentcomponents. This indicates that the components of Riemann tensor can be expressed interms of Ricci tensor. More explicitly R xzzy = R xy , R xyyz = R xz , R yxxz = R yz , R xyxy = 12 ( R zz − R xx − R yy ) , (17)while the remaining two components R xzxz and R yzyz are obtained by the appropriate per-mutations of the x, y and z coordinates.Our solution is a vacuum solution with the strings as sources. It is clear from (9) that R ab = 0 away from the strings. Using this in (17) we can immediately conclude that allcomponents of the Riemann tensor vanish and the geometry given by the metric (15) isindeed flat.The fact that the junction is stationary is the crucial requirement for the flatness of thegeometry. It is evident that for non-stationary junctions the space-time is curved. Theextent of the departure from a flat geometry is directly controlled by the extent of violationof the stationarity conditions. For example, using the metric (15) one can show that R xyxy = − Gr X i µ i ( n i x x + n i y y − n i z z ) . (18)This is zero due to the force balance conditions (3). IV. THE PROPAGATION OF LIGHT
One of the novel cosmological features of cosmic string is the lensing effect. The metricof a straight cosmic string is locally flat. But globally the geometry around the string hasa deficit angle given by ∆ = 8 πGµ . This results in the formation of two identical imagesof an object located behind the string. When looking at an object located behind a staticjunction of semi-infinite strings, one naturally expects multiple images to form.The lensing by three co-planar strings forming a Y -shaped junction was studied by Shlaerand Wyman [17]. An observer looking at an object, say a galaxy, located behind the planeof the strings will see three (identical) images; one image is the object itself and the othertwo are its lensing counterparts. 6 IG. 1: In this figure two parallel light rays emitted from infinity are deflected towards the pointG. The contour C is made of the point G, the light rays and the line connecting the two lightrays distance at infinity, denoted by the dashed line. The strings cross the plane spanned by thecontour at the points indicated by the cross signs. Only those strings which are enclosed by thecontour contribute in (19) and (25).
In the method used in [17], one starts with three infinite strings intersecting at thepoint of the junction. This will produce two Y -shaped junctions oriented oppositely at thejunction. One can “cut-and-paste” one junction and keep the remaining one. This methodwill correctly produce the lensing by the junction, as demonstrated in [17].In this section we would like to consider the general case of an arbitrary number ofstationary cosmic strings forming a junction, co-planar or not, and study the resultinglensing phenomena in cosmology.Suppose two parallel light rays are emitted from infinity towards the junction. Once theline connecting the light rays passes a string, the light rays are expected to be bent towardseach-other. Suppose the light rays stay co-planar and meet at a point, say G. The differencein the velocity vectors at the point of intersection G can be obtained by the method ofparallel transportation around a closed curved C. The curve C is composed of the point G,the two light rays from infinity and the line connecting these rays at infinity. For a schematicview see Fig. 1 . The difference in the velocity vectors at G is [18] δv α = − Z S R αβγλ v β dx γ ∧ dx λ , (19)where the integration is over the closed surface S bounded by C.Of course, in the empty regions away from strings the Riemann tensor vanishes as weshown before and the above integral is zero, as expected. However, when the surface in-tersects strings the Riemann tensor provides delta-function contributions and the integral7oes not vanish. This demonstrates that the relative change of the velocity vectors andconsequently the lensing effects are directly controlled by the number and the orientationof the cosmic strings which intersect the surface S. In the spirit, the method is analogous tothe residue theorem for the integration of an analytical function in the complex plane.Without loss of generality, we may suppose that the light rays are emitted along thenegative z-direction in the y-z plane. Using Eq. (10) one obtains S ab = X i µ i n i x ( δ ab − n i a n i b ) δ ( y − y i ) δ ( z − z i ) , (20)where y i and z i are the coordinates of the point of intersection of the i-th string with they-z plane. The factor n i x in the denominators originates from replacing dl i along the stringby dx via dx = n i x dl i .Using this in Eq. (19) one obtains δv x = Z dy dz R xzzy = Z dy dz R xy = 8 π G Z dy dz S xy = − π G X i µ i n i y , (21)where the sum is over those strings which intersect the plane of the light rays (the y-z planehere) and are enclosed by the surface S bounded by the light rays.Similarly, for the change of velocity in the y -direction one obtains δv y = Z dy dz R yzzy = 12 Z dy dz ( R yy + R zz − R xx )= 8 π G X i µ i n i x , (22)where to obtain the last equation, the identity n i x + n i y + n i x = 1 has been used.Finally, from (19) one can easily see that δv z = 0, which implies that there is no changeof velocity in the direction tangential to the initial light rays.Combining these results, one obtains the following coordinate independent representationof the change in velocity δ~v = − π G ~k × X i ~µ i , (23)where the unit vector ~k represents the direction of the light rays at infinity (line of sight).The angle between the two light rays at the point of intersection is∆ = | δ~v | = 8 π G µ eff , (24)8here µ eff = | ~k × X i ~µ i | . (25)As explained before, the sum is over those strings which intersect the plane S formed by thelight rays up to their point of intersection. For the case of a single string with tension µ enclosed between the light rays, µ eff = µ and we obtain the standard result for the deficitangle.Now let us try to apply Eq. (24) to some examples. The first interesting example is thelensing by a Y -shaped junction studied in [17]. The strings are in the x-y plane and the lightrays are emitted along the z-direction. A schematic view of this situation is presented in Fig.2 . Starting with the object located at point A and moving in counterclockwise direction, weenclose the string with tension µ and µ eff = µ . Consequently, there is an image at point A separated from A by the deficit angle 8 π G µ . Continuing further counterclockwise, weenclose a second string (with tension µ ) and µ eff = | ~µ + ~µ | . Because of the force balancecondition this is the same as µ . So the second image is at the point A which is separatedfrom the object A by a deficit angle 8 π G µ . Finally, continuing further to enclose the thirdstring, the effective tension vanishes due to force balance condition ~µ + ~µ + ~µ = 0 andthere is no other image. This means that the points A, A and A form a triangle. Thishas an interesting geometrical interpretation: Each image acts as a source for the other twoimages. The object A is the source for A through the string µ , the image A acts as sourcefor A via the string µ and the image A is the source for A via the string µ . This is inexact agreement with the prescription provided for a Y -shaped junction in [17].In this example the strings are between the observer and the object and the plane of thestrings is perpendicular to the line of sight. Whether or not the observer actually sees animage depends on the angular distances between each string and the object. If a distance islarger than the deficit angle, then the corresponding image is not observable.One can generalize the example of the Y -shaped junction to the case of N co-planarcosmic strings at junction. As before, the plane of the strings is between the object andthe observer and is perpendicular to the line of sight. Following the same steps as above,we obtain N − N points forms a closed loopdue to the force balance condition. For each image, the effective tension is the magnitudeof the vector sum of all strings enclosed between the object and the corresponding image.9 IG. 2: In this figures the multiple lensings by N co-planar strings at a junction is sketched. Theobject A and its lensing counterparts A i , i = 1 ..N −
1, form a closed loop. Each image is the sourcefor the nearby images via the enclosed string. For example, in the three-string junction in the leftfigure, the object A is the source for the image A via the string µ , A is the source for the image A via the string µ and A is the source for the object A via the string µ . Geometrically, as before, this means that each image is the source for the nearby images viathe enclosed string. Starting with the object A , the chain of object → image is given by A → A → A → .. → A N − → A . The schematic view of this case is presented in Fig. 2 .Another interesting case is when the light rays are parallel to the plane of the co-planarstrings. Now the strings are located in the y-z plane and two light rays are emitted alongthe z-direction. If the light rays are on the same side of the plane, then no string is enclosedby them and the light rays will stay parallel. Now suppose the light rays are emitted fromopposite sides of the plane. To simplify the analysis suppose the light rays are in the x-zplane so they are emitted at ( x , y , ∞ ) and ( x , y , ∞ ) with x x <
0. We already knowthat δv z = 0, i.e. there is no change of velocity in the direction parallel to the light rays.Also, from Eq. (22) one obtains δv y = 0 because the strings have no n x components. Onthe other hand, from Eq. (21) one obtains | δv x | = 8 π G | X i µ i n i y | , (26)where the sum is over the strings in the upper part of the y-z plane. From the force balancecondition, this is also equivalent to the sum over the strings in the lower part of the y-zplane. Interestingly enough, we see that the light rays which start out parallel to the plane10f strings are now deflected towards the plane. V. CMB ANISOTROPIES AND COSMIC STRING WAKES
The effects of moving string on CMB anisotropy were studied by Kaiser and Stebbins [19].These authors showed that as a consequences of the conical structure of space perpendicularto the string, a moving string will produce a line discontinuity in temperature anisotropymaps, with the amplitude of the discontinuity given by δTT = Gµγ ( v ) v , (27)where v is he string velocity perpendicular to our line of sight towards the string, and γ ( v )is the Lorentz factor. The effect is due to the relative Doppler shift in the photons passingon the two sides of the string. The Kaiser-Stebbins effect is a key distinctive observationalsignature for strings. Methods to search for this signature have recently been discussed in[20, 21] (see also [22] for older work).One may naturally ask how this can be generalized to the case of strings at a junctionwhen the junction is moving with a constant velocity v . It is understood that the junctionand the strings attached to it move as a solid object and the no force condition (3) stillholds.We are interested in change in the observed frequency between two parallel light rays(with the same initial frequency) in the presence of the moving junction. As before, we takethe light rays to be moving in the y-z plane, and initially along the z-direction. In terms ofEq. (19) instead of v µ we use the momentum four-vector p µ = ( E, p ), where E = | P | = ~ ω .This yields δω = ω Z R zzy dy dz . (28)Unlike in the case of a static junction, the Riemann tensor has non-zero components like R tabc and R tatb due to the string motion. In 4-D space-time the Riemann tensor has morecomponents than the Ricci tensor does. However, we can go to the junction’s rest framewhere the space-time is static and our results from the previous section can be used readily.Denote the coordinate system where the junction is static by x µ ′ while the coordinate system11sed by the observer and the light sources is represented by x µ . We have R zzy = ∂ x α ′ ∂ t ∂ x β ′ ∂ z ∂ x µ ′ ∂ z ∂ x ν ′ ∂ y R α ′ β ′ µ ′ ν ′ , (29)where now R α ′ β ′ µ ′ ν ′ has no time-like indices.In general the junction velocity vector ~v can have components in x, y and z-directions.Suppose the junction is moving along the x-direction. Then one obtains δωω = (cid:12)(cid:12)(cid:12)(cid:12)Z d y ′ d z ′ ∂ x ′ ∂ t R x ′ z ′ z ′ y ′ (cid:12)(cid:12)(cid:12)(cid:12) = 8 πGγ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i µ i n i y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (30)To obtain the final line, the same procedure as in Eq. (21) has been used. As before, in theabove the sum is over the strings which are enclosed by the light rays.Similarly, in the case when the junction is moving along the y-direction one obtains δωω = (cid:12)(cid:12)(cid:12)(cid:12)Z d y ′ d z ′ ∂ y ′ ∂ t R y ′ z ′ z ′ y ′ (cid:12)(cid:12)(cid:12)(cid:12) = 8 πGγ v (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X i µ i n i x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (31)where to obtain the final result Eq. (22) was used.Finally, if the junction is moving along the z-direction parallel to the light arrays one canshow that δω vanishes.Combining all these results, one can show that δωω = 8 πG γ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ~v . ( ~k × X i ~µ i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (32)where as before ~k represents the line of sight and the sum is over the strings which areenclosed by the light rays. Interestingly enough, this formula has the same functional formas that of the a single string [23].The implications of Eq. (32) for the CMB anisotropies are parallel to those of Eq. (24)for lensing. Each string at the junction produces its own CMB anisotropies. The change inthe CMB temperature across the i-th string is given by δTT = 8 πG γ | ~v. ( ~µ i × ~k ) | . (33)12nterestingly enough, the temperature anisotropy is different for different legs of the stringsat the junction, depending on the tensions and orientations of the strings.We conclude that the distinctive signature of strings with junctions for CMB anisotropiesis the possibility of line discontinuities joined at a point for an observer looking at the surfaceof last scattering. The details of this picture depends on the orientations of the strings andthe direction of its motion of the junction with respect to the line of sight. In recent work[20], the Canny algorithm [24], an edge detection algorithm, was shown to yield a sensitivestatistic with which CMB line discontinuities can be identified in the sky. Since stringjunctions are a prediction of cosmic superstrings (simple gauge theory strings do not admitjunctions), it would be interesting to develop modified edge detection algorithms which candifferentiate between strings with and without junctions in CMB anisotropy maps. Workon this issue is in progress.The lensing produced by the cosmic string deficit angle also produces distinctive signa-tures for structure formation, namely “wakes” [25]. For the rest of this section, we will brieflydiscuss the implications of string junctions for cosmic string wakes. Thus, we consider theaccretion of matter by a moving junction.Consider two massive non-relativistic objects at rest in the frame of the cosmic microwavebackground in the presence of a moving junction. As in the previous section, it is convenientto go to the rest frame of the junction. In this frame, the two massive objects are movingtowards the junction. When the line connecting the objects passes any string, the objectsare attracted towards each other behind the cosmic string. This can provide a mechanismfor structure formation by string wakes [25, 26, 27].In the case of a single infinite string, the lensing of matter behind a moving string leadsto a region of twice the background density in the wake of the moving string. The openingangle of the wake is given by the deficit angle. The question we wish to address in thissection is how the wake structure generalizes to strings with junctions.The results obtained in Section ( IV ) for the bending of two light rays can be used formassive objects too. The only change is the addition of γv factor due to the change of frame.The relative velocity between the massive objects at the intersection is δ~v = − πG γ v ~k × X i ~µ i . (34)On the other hand, the angle between the particles trajectories is ∆ m = | δ~v | /v , and one13btains ∆ m = 8 πG γ | ~k × P i ~µ i | . This means that around each leg of the string junctionthere is a wedge-shaped wake with opening angle ∆ m . In each of these regions, the matterdensity is doubled, i.e. δρ/ρ = 2.Since the cosmic strings cannot be the dominant source of structure formation, and sincestring wakes undergo non-trivial non-linear evolution, it will presumably be more difficult tofind distinctive signatures for string junctions in large-scale structure surveys than in CMBanisotropy maps. In principle, topological statistics such as Minkowski functionals [28] havethe power to find such signatures [29]. VI. CONCLUSIONS
Working at the level of linearized gravity, we have derived the metric of a static stringjunction, with an arbitrary number of strings joining. We have shown that away from theworld sheet of the strings, the metric is flat. Thus, the geometry generalizes that for a singlestraight infinitely long string. Each string segment produces a deficit angle, and thus deflectsboth light and matter. We have derived the lensing of light by a string junction, discussedthe CMB anisotropies produced by a string junction which is in uniform motion relative tothe frame of the microwave background, and commented on the formation of string wakes.We have identified a junction of line discontinuities in CMB anisotropy maps as a distinc-tive signature of strings with junctions, and proposed that one could look for these signaturesby a generalized edge detection algorithm. Finding positive evidence for such string seg-ments would provide a boost for superstring theory, since it is in the context of superstringtheory that the existence of strings with junctions first came to prominence. Strings withjunctions are quite generic in the context of superstring theory, but they do not appear insimple gauge field theory models of cosmic strings.
Acknowledgments
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