Cosmic String Loop Microlensing

Jolyon K. Bloomfield1 and David F. Chernoff2

1MIT Kavli Institute for Astrophysics and Space Research,Massachusetts Institute of Technology, Cambridge, MA 02139, USA∗

2Center for Radiophysics and Space Research,Cornell University, Ithaca, NY 14853, USA†

(Dated: November 28, 2013)

Cosmic superstring loops within the galaxy microlens background point sources lying close to theobserver-string line of sight. For suitable alignments, multiple paths coexist and the (achromatic)flux enhancement is a factor of two. We explore this unique type of lensing by numerically solving forgeodesics that extend from source to observer as they pass near an oscillating string. We characterizethe duration of the flux doubling and the scale of the image splitting. We probe and confirm theexistence of a variety of fundamental effects predicted from previous analyses of the static infinitestraight string: the deficit angle, the Kaiser-Stebbins effect, and the scale of the impact parameterrequired to produce microlensing. Our quantitative results for dynamical loops vary by O(1) factorswith respect to estimates based on infinite straight strings for a given impact parameter. A number ofnew features are identified in the computed microlensing solutions. Our results suggest that opticalmicrolensing can offer a new and potentially powerful methodology for searches for superstring looprelics of the inflationary era.

PACS numbers: 04.25.D-, 98.80.Cq

I. INTRODUCTION

Shortly after its birth, the universe is believed tohave grown exponentially in size via an inflationarymechanism. The almost scale-invariant density per-turbation spectrum predicted by inflation is stronglysupported by observations of the cosmic microwavebackground radiation carried out by the WMAP [1]and Planck satellites [2]. String theory, the best-developed tool to explore this epoch, suggests thebirth and survival of a network of one-dimensionalstructures on cosmological scales [3]. Numerous fos-sil remnants of these cosmic superstrings may existwithin the galaxy, which could be revealed throughthe optical lensing of background stars by way of thetechnique of microlensing [4]. This paper investigatesthe fundamental physical features of microlensing rel-evant to optical searches for string loops generated bythe network.

The primary parameter that controls string networkand cosmic loop properties and dynamics is the stringtension, µ. The first exploration of strings was inthe context of phase transitions in grand unified fieldtheories (GUTs), which generated horizon-crossingdefects with tension Gµ/c2 ∼ 10−6 (hereafter, c = 1)set by the characteristic grand unification energy [5].Such defects could seed the density of fluctuations forgalaxies and clusters, and would also create cosmicstring loops.

∗ jolyon@mit.edu† chernoff@astro.cornell.edu

The lifetime of a loop decaying by gravitationalevaporation scales as µ−1. With large string tensions,GUT strings decay quickly, and do not live many Hub-ble expansion times. This implies that for such loops,clustering is largely irrelevant: they move rapidly atbirth, become briefly damped by cosmic expansion,and are re-accelerated to relativistic velocities by themomentum recoil of anisotropic gravitational waveemission (known as the ‘rocket effect’) before fullyevaporating. As such, GUT loops were thought to behomogeneously distributed throughout space [6].

However, empirical upper bounds on µ from a num-ber of experiments in the past decade have essentiallyruled out GUT scale strings. Such experiments includenull results from lensing surveys [7], gravitational wavebackground [8] and burst searches [9], pulsar timingobservations [10], and observations of the cosmic mi-crowave background [11, 12] (see [13] for a generalreview). Roughly speaking, these upper bounds im-ply Gµ . 3 × 10−8 − 3 × 10−7, although all suchbounds are model-dependent and subject to a varietyof observational and astrophysical uncertainties, withmore stringent bounds typically invoking additionalassumptions. Future gravitational wave observato-ries (including Advanced LIGO [Laser InterferometerGravitational Wave Observatory], LISA [Laser In-terferometer Space Antenna] and Nanograv [NorthAmerican Nanohertz Observatory for GravitationalWaves]) may achieve limits as low as Gµ ∼ 10−12 [14].

The situation for strings in modern string theoryis somewhat different than it is for GUT strings. Inwell-studied models of string theory, the compacti-fication of extra dimensions involves manifolds pos-

arX

iv:1

311.

7132

v1 [

astr

o-ph

.CO

] 2

7 N

ov 2

013

2

sessing warped throat-like structures which redshiftall characteristic energy scales compared to those inthe bulk space. String theory contains multiple ef-fectively one-dimensional objects collectively referredto here as superstrings. Any superstring we observewill have tension µ exponentially diminished fromthat of the Planck scale by virtue of its location atthe bottom of the throat. There is no known lowerlimit for µ. The range of interest for microlensing is10−14 < Gµ < 3× 10−7, with the lower limit set bythe difficulty of observing optical microlensing of stars(finite source size versus lensing angle; finite durationversus lensing timescale) and the upper limit by thecollective empirical tension bounds.

The lowered tension of superstring loops qualita-tively alters their astrophysical fate compared to GUTstrings. With much smaller µ, superstring loops ofa given size live longer. This has two importantimplications: the loops that dominate the lensingprobability are small loops (sub-galactic rather thanhorizon-crossing), and were born before the matterera — they are old. Because of their age, cosmic ex-pansion has had sufficient time to dampen the initialpeculiar motion of the loops, which allows them tocluster as matter perturbations grow [4, 15]. As such,below a critical tension Gµ ∼ 10−9, all superstringloops accrete along with cold dark matter.

The bottom line is that low-tension loops withinthe galaxy are over-abundant with respect to loopswithin the universe as a whole by roughly the samefactor as cold dark matter is over-abundant withinthe Galaxy. By investigating a detailed model ofthe tension-dependent distribution of loops withinour galaxy [16], it was found that the local enhance-ment of string loops has implications for microlensingsearches, gravitational wave detection experimentsand for pulsar timing measurements. In this paper,we concentrate on microlensing.

Previous work on cosmic string lensing has mainlyassumed high string tensions, wherein multiple imagesof the same object may be resolved [17–20]. Numericalcomputations show that the lensing patterns of stringloops can become quite involved [21], with complicatedcaustics defining where multiple images may exist.

Based on the low string tensions required by re-cent cosmological observations, it is likely that onlya microlensing signature can be detected optically,as low tensions imply small angular deflections [4].In a microlensing event, multiple unresolved imageslead to an apparent doubling of flux over the durationof the event. The defining features of such an eventare an achromatic doubling of flux that repeats manytimes with a characteristic time-scale set by the loopperiod. Cosmic string microlensing is thus distinctfrom lensing in that it is a time domain measurementrather than an image signature, and is distinct from

other astrophysical microlensing events because of itsunique flux signature.

Stars are the ideal target for microlensing withinthe galaxy. They are numerous, and ever more ca-pable time-domain surveys are being planned andcarried out. Furthermore, the characteristic angularscale for microlensing is 8πGµ, which is well-matchedto the stellar size. For Gµ = 10−13 this angle is com-parable to the angular size of a solar mass star at 10kpc. For Gµ > 10−13 the stars are point-like targets.Microlensing of stars may be spatially correlated onthe sky in a manner similar to that of normal lensing[22].

Current descriptions of rate calculations for mi-crolensing events [23] neglect the galactic clusteringeffect, and updated results are in progress [24]. It islikely that constraints on the parameter space of cos-mic string loops can be determined from investigationsof microlensing in current observational data.

It is important to derive a full understanding of themicrolensing signature to be able to hunt for and iden-tify the rare but meaningful events that may occurduring a survey. In this paper, we begin a detailedexploration of the nature of microlensing for point-likesources, presenting the first numerical realizations ofcosmic string microlensing. The framework of this dy-namical calculation combines two elements that havenot hitherto been melded but are equally important:the space curvature of the mass component of thebent loop and the deficit angle of the static stringsource. This opens the possibility of considering addi-tional elements of interest, including substructure anddiscontinuities on the loop, and paves the way to mak-ing microlensing a viable search technique for currentand upcoming large scale surveys such as PanStarrs,LSST, Gaia and WFIRST, and deep bulge surveyssuch as OGLE and MOA.

This paper is organised as follows. We begin withan overview of string lensing from infinite straightstrings in Section II. In Section III, we present themathematical description of cosmic string loops anddetail the formalism we use to propagate geodesicsin the string spacetime. In Section IV, we describethe computational implementation of this formalism.We briefly describe the string configuration we usein Section V, before presenting the results of ournumerical investigation in Section VI and concludingin Section VII.

II. STRING LENSING BASICS

Lensing is a fundamental feature of general rela-tivity. Curved spacetime can create multiple imageswith varying magnification, shear and distortion. In

3

microlensing the individual images of the source areunresolved but the total brightness varies in timeas the geometry of observer, lens and source evolve.Refsdal [25] calculated microlensing for a gravitatingNewtonian point mass and concluded that significantbrightness amplification was possible. Paczynski [26]proposed utilizing microlensing to search for dark,massive objects contributing to the total mass densityof the Galactic halo.

Vilenkin [27] was the first to discuss lensing by acosmic string. Unlike a Newtonian point mass whichcurves spacetime, a straight string’s positive energydensity and negative pressure (along its length) con-spire to leave spacetime flat. Its presence inducesa deficit angle with size ∆ = 8πGµ and creates aconical geometry. There is no magnification, shear ordistortion of a particular image, but there are multipledistinct paths for photons to travel from source toobserver when the source, string and observer are allnearly aligned. For GUT strings with Gµ = 10−6 thetypical angular splitting of images is ∼ 5 arc seconds.Vilenkin noted that exact double images of cosmicobjects like galaxies located behind the string mightreveal the string’s presence. Since that suggestion,the observational bounds (from CMB, gravity wavesearches and pulsar timing) have tightened, constrain-ing Gµ . 3 × 10−8 − 3 × 10−7, and the expectedsplitting cannot be resolved. In addition, advancesin string theory naturally yield superstrings, string-like entities which can have much smaller µ. In thiscontext, Chernoff and Tye [4] suggested that one lookfor the transient change of the unresolved flux whena star passes behind the string. A search for super-strings by string microlensing is conceptually similarto the search for Newtonian masses by way of normalmicrolensing.

The lensing geometry for static, straight infinitestrings is illustrated in Figure 1. The observer-lensdistance is d1, the star-lens distance is d2 and theobserver line of sight to the star is n. Lensing requiresclose alignment of star, string and observer to order∆. We will work to lowest non-vanishing order in ∆.

The string unit vector s forms an angle θ withrespect to the line of sight (cos θ = n · s). The wedgespans the projected deficit angle ∆′ = ∆ sin θ, isremoved from the plane (green hatched region) andopposite sides identified. When the star lies anywherealong the blue strip on the left, two lines of sight existfor the observer to see the star. Let the displacementof the star from line of symmetry in the plane beh, and write d1 = d1/(d1 + d2), d2 = d2/(d1 + d2),

h = 2h/(d2∆′). The range of microlensing is |h| < 1.

The angle of separation between the images is

δφ = α+ β = d2∆′ +O(∆3), (1)

String

Observer

Observer

D'

Star

Α

Β

d2 d1

Figure 1. The black dot at the center is a straight stringwith tension µ and intrinsic deficit angle ∆ = 8πGµ.An angular section of Minkowski space has been alignedwith the observer and removed (green hatched region).This makes clear that certain points can transmit lightto the observer by both clockwise and counterclockwisecircumnavigation of the string. The red dotted line isthe symmetrical line from observer to string. The stringpierces the plane formed by the star on the left (bluedot) and the observer on the right (black dots) at angleθ (not drawn). The wedge has projected deficit angle∆′ = ∆ sin θ. When the star at distance d2 lies along theblue strip subtended by the dotted lines there exist twolines of sight for the observer to see the star. The totalflux is very nearly double the flux of a single image andthe angular separation of the pair of images is δφ = α+ β.

the path length difference is

δd

d1 + d2=

1

2d1d2h(∆′)2 +O(∆4) (2)

and the fractional change in energy for photons inthis geometry is

δν

ν= ∆γn · (~v × s) (3)

where γ is the relativistic factor for the string [28–31].During a microlensing event, geodesics must have adistance of closest approach to the string r in therange

|r| . d1d2d1 + d2

∆′ . (4)

The microlensing event will have a duration of

δt ≈ 2d1d2d1 + d2

∆′

|~v · (n× s)|&

2d1d2d1 + d2

∆′ (5)

as the string sweeps through the line of sight.

The most numerous string loops today are expectedto have a length ` ∼ ΓGµτ , where τ is the age of theuniverse, and Γ ∼ 50− 100. Such a loop has a typicalcurvature scale ∼ 1/` and a bounding box of sidelength ∼ `/4. Microlensing caused by the deficit angleis possible as long as the distance of closest approachsatisfies r `; otherwise the loop’s mass will lens (ormicrolens) in a traditional Newtonian fashion. The

4

first possibility requires that the source (observer)distance must be less than `/∆′, which is comparableto the Hubble scale today. The infinite straight stringresults should provide a valid approximation as longas the characteristic string scale is ∼ `. Of course, anindividual segment of string may have a much smallerradius of curvature, particularly near a cusp, wherethese results become more approximate. One of thepurposes of this paper is to compare these analyticalresults to numerical simulations for microlensing incosmic string loops.

III. GRAVITATIONAL WAVE EMISSIONAND GEODESIC PROPAGATION

In this section, we present the formalism used tocalculate the metric in the string spacetime. We alsodescribe the geodesic equation formulation that weemploy.

A. String Dynamics

We begin with a brief overview of string dynamicsfollowing Vilenkin and Shellard [5]. The string isdefined by its location xµ(ξa), where ξ0 = τ andξ1 = σ are the worldsheet coordinates. We work withthe Nambu-Goto string action

S = −µ∫d2ξ√−γ (6)

where µ is the string tension, and γab = gµν∂axµ∂bx

ν

is the induced metric on the worldsheet.

Working in flat space, the equation of motion forthe string is given by

xµ − xµ′′ = 0 (7)

where xµ = ∂xµ/∂τ and xµ′ = ∂xµ/∂σ, and we usethe conformal gauge

x · x′ = 0, x2 + x′2 = 0. (8)

We further simplify the gauge by choosing τ = t, thetime coordinate. This allows us to write the stringtrajectory as a three-vector ~x(σ, t), subject to theconditions

~x · ~x′ = 0, ~x2 + ~x′2 = 1, ~x− ~x′′ = 0. (9)

The general solution to these equations is

~x(σ, t) =1

2

[~a(σ − t) +~b(σ + t)

](10)

subject to the conditions

~a′2 = ~b′2 = 1 (11)

which imply that the tangent vectors live on a unitsphere. The energy of the string is simply E = µLfor invariant string length L, and the stress-energytensor for the string is given by

Tµν(~r, t) = µ

∫dσ(xµxν − xµ′xν′)δ(3)(~r − ~x(σ, t)).

(12)

B. Metric Perturbations

We work in linearized gravity, with gµν = ηµν +hµνand metric signature (−,+,+,+). The linearizedEinstein equation is given by

hµν = −16πGSµν (13)

where

Sµν = Tµν −1

2ηµνT

σσ (14)

and we work in harmonic gauge

∂νhνµ =

1

2∂µh

σσ. (15)

To invert the linearized Einstein equation, we usethe Green’s function method following Damour andBuonanno [32]. The inversion yields

hµν(~r, t) =

∫d2σ 8Gµ Fµνθ(t− x0)δ((r − x)2)

(16)

for the metric perturbation, where

Fµν = xµxν − x′µx′ν + ηµνxσ′x′σ. (17)

Here, r and x in the delta function refer to thefour-vectors, where rµ = (t, ~r). Defining Ωµ =rµ−xµ(σ, τ), the retarded time needed in integratingout the delta function is the retarded solution to

ηµνΩµΩν = 0 (18)

which yields

τ = t− |~r − ~x(σ, τ)|. (19)

We call the solution to this equation τret, the retardedtime. Integrating over the delta function, the metricperturbation becomes

hµν(~r, t) =

∫dσ 4Gµ

Fµν(σ, τret)

|Ωµxµ(σ, τret)|. (20)

5

Note that Ωµxµ is negative when evaluated on theretarded time, except on the string or on a line ofcusp radiation, where it vanishes and the integraldiverges logarithmically. If desired, one can avoidthis situation by implementing an over-retarded timewith τ = t−

√|~r − ~x(σ, τ)|2 + ε2, which regulates the

denominator.

Ωµxµ = (~r − ~x) · ~x−√|~r − ~x|2 + ε2 < 0 (21)

A reasonable size for ε is the distance of closest ap-proach at which the metric perturbation becomesnonlinear. In the numerical part of this work, ε wasset to zero for all practical purposes.

Calculating the average field a long way away fromthe loop, one finds

〈hµν(~r)〉 =2

L

∫ L/2

0

hµν(~r, t)dt

=2GM

|~r|diag(1, 1, 1, 1) (22)

which corresponds to the linearized Schwarzschildmetric for a loop of mass M = E = µL.

C. Metric Derivatives

The reason for using Damour and Buonanno’s ap-proach is that it allows for metric derivatives to bestraightforwardly calculated. A single partial deriva-tive acting on the metric perturbation is

∂αhµν(r) =

∫dσdτ 8Gµ Fµν(σ, τ)θ(r0 − τ)∂αδ(F )

(23)

where we let F = (r − x(σ, τ))2 = ηµνΩµΩν . Notethat the derivative acting on the step function yields

a product of delta functions at two separate locations,which vanishes (except on the string). The derivativeacting on the delta function yields

∂αδ(F ) = ∂αFδ′(F ) =

∂αF

∂F/∂τ

∂δ(F )

∂τ(24)

by repeated use of the chain rule. Integrating byparts and noting that derivatives of F can be simplyexpressed in terms of Ωµ as

∂F

∂τ= −2Ωµxµ , ∂αF = 2Ωα , (25)

leads to Damour and Buonanno’s Eq. (2.40),

∂αhµν(r) =

∫dσ

4Gµ

|Ωµxµ(σ, τ)|∂

∂τ

[Fµν(σ, τ)

ΩαΩµxµ

](26)

where all expressions should be evaluated at τret. Theτ derivative yields

∂

∂τ

[Fµν

ΩαΩµxµ

]= Fµν

ΩαΩµxµ

− Fµνxα

Ωµxµ

− FµνΩα

(Ωµxµ)2Λ (27)

where we define Λ = ∂τ (Ωµxµ) = Ωµxµ − xµxµ.

Extending this approach, we can compute the sec-ond derivative of the metric, which will be needed tocompute curvature invariants. Starting from

∂α∂βhµν(r) = ∂β

∫dσdτ 8Gµ Fµνθ(r

0 − τ)∂αδ(F ) ,

(28)

we follow the same idea as for the first derivative:evaluate all the derivatives before integrating out thedelta function. The resulting expression is as follows.

∂α∂βhµν(~r, t) =

∫dσ

4Gµ

|Ωµxµ|∂

∂τ

[Fµν

ηαβΩµxµ

− FµνxβΩα + xαΩβ

(Ωµxµ)2+ Fµν

ΩαΩβ(Ωµxµ)2

− FµνΩαΩβ

(Ωµxµ)3Λ

](29)

All expressions here should be evaluated at τret. Evaluating the derivatives, the final result is

∂α∂βhµν(~r, t) =

∫dσ

4Gµ Σµναβ|Ωµxµ(σ, τret)|3

(30)

where

Σµναβ = Fµν

[−2x(αΩβ) + 2xαxβ − ηαβΛ + 6

x(αΩβ)

ΩµxµΛ + 3

ΩαΩβ(Ωµxµ)2

Λ2 − ΩαΩβΩµxµ

Λ

]+ Fµν

[ηαβΩλxλ − 2(xβΩα + xαΩβ)− 3

ΩαΩβΩµxµ

Λ

]+ FµνΩαΩβ (31)

6

and Λ = ∂τΛ = Ωµ...xµ − 3xµxµ.

D. Geodesic Equation

We are interested in ray-tracing null geodesicsthrough the perturbed spacetime. The geodesic equa-tion is given by

∂2xµ

∂λ2+ Γµσλ

∂xσ

∂λ

∂xλ

∂λ= 0 (32)

where xµ describes the position of a photon in space-time, and λ is an affine parameter. This equationcan be integrated directly, but there exist better ap-proaches. Because of the mass shell constraint p2 = 0(where pµ = ∂xµ/∂λ, with affine parameter λ chosensuch that pµ is the four-momentum of the photon),only three components of the geodesic equation actu-ally need to be integrated. Furthermore, the affineparameter can be disposed of by using coordinatetime, as no horizons are present in the spacetimesunder consideration. An efficient method of integrat-ing the geodesic equation that takes advantage ofthese properties is described by Hughes et al. [33],which does not require time derivatives of the metriccomponents and thus saves computational time.

Consider the metric written in the ADM decompo-sition.

ds2 = −α2dt2 + γij(dxi + βidt)(dxj + βjdt) (33)

With the metric written in this manner, we can writethe geodesic equation as follows.

dxi

dt= γij

pjp0− βi (34a)

dpidt

= −ααip0 + βk,ipk −1

2γjk,i

pjpkp0

(34b)

p0 =1

α

√γijpipj (34c)

The variables being integrated are pi and xi; theenergy p0 is algebraically determined at each step,ensuring the mass-shell condition. The inverse spatialmetric γij is used to raise and lower indices on βi.Note that the affine parameter has vanished, and thatno time derivatives of the metric are needed. Thisformulation is completely general, but fails near hori-zons, where the time coordinate becomes problematic.This system of equations can be specialized to thelinear approximation if desired. However, the amountof extra computational time required to numericallyintegrate the full equations compared to the linearizedequations is negligible.

IV. COMPUTATIONAL TOOLS

In this section, we describe the computational ap-proach that we employ to calculate metric perturba-tions and their derivatives, to integrate geodesics, andto solve the geodesic equation as a boundary valueproblem. We also include a detailed analysis of theapproximations included in our calculations.

The formalism that we use to calculate geodesics inthe microlensing context is similar to that developedby de Laix and Vachaspati [21] to investigate the lens-ing properties of cosmic strings. However, there are anumber of differences. In particular, we work with fi-nite source and observer distance, rather than placingthem at infinite distances. Furthermore, we do notwork in a thin lens approximation. Finally, becausewe trace the full geodesic rather than just looking atthe deflection angle, we can investigate features alongthe geodesic that allow us to understand phenomenain the observables, and also compare theoretical pre-dictions to details of the geodesics such as distance ofclosest approach.

To calculate the metric perturbation and its deriva-tives at a given point in spacetime, we make use of Eqs.(20), (26) and (30). These formulas all include an in-tegral over the string loop. The required componentsare stored as a vector and integrated together.

At each point on the string loop, the retarded timeneeds to be evaluated. As the implicit equation de-scribing the retarded time is strictly monotonic (ex-cept for exactly on a cusp where it is stationary),this is reasonably straightforward. For points on thestring close to a cusp, the derivative of the implicitequation becomes very small near the root, and soa combination of Newton’s method and a bisectionmethod are employed. We use a tolerance δτ for theaccuracy of the retarded time calculation.

Once the retarded time has been evaluated, it isstraightforward to evaluate the integrands for the met-ric perturbations and their derivatives. The integral isthen summed using a psuedospectral method. As thestring loops we considered were smooth and periodic,it is highly efficient to evaluate the integral by a bisec-tion method, using a rectangular approximation. Webegin by evaluating the integral with 32 divisions, andcompare it to 64 divisions in order to estimate relativeand absolute accuracy. The number of bisections isdoubled until the relative and absolute tolerances aresatisfied. Due to the limits of long integers, we de-mand that at most 30 bisections be allowed, althoughsuch large numbers are only typically needed whenevaluating points very close to the string. While inte-

7

grating over the loop, the shortest distance to the loop(in the form t− τret) is recorded. Using this methodof integration precludes the investigation of stringswith kinks, which are not appropriately continuous.In principle, the method also breaks down if a cusp isencountered (due to the discontinuity), but as cuspsonly form for an instant in time, the string can alwaysbe taken to be smooth.

In order to understand how an object is microlensed,we want to know the null geodesic(s) that start at thesource and end at the observer at a given time. Thisspecifies a geodesic as a boundary value problem. Forthe string spacetimes we are considering, there willalways be at least one solution (a deviation from theEuclidean geodesic), and the possibility of microlens-ing suggests that there will sometimes be multiplesolutions.

We solve the geodesic equation as an initial valueproblem by selecting an observer position (threeboundary conditions) and photon arrival direction(two angles describing a unit vector). The third com-ponent of the momentum sets the arrival energy, whichwe normalise to one. Coordinate time is used as theintegration parameter, and so an initial time mustalso be selected. The initial value problem is thusspecified in terms of 5 boundary conditions and oneparameter. To integrate the geodesics, we make useof the GNU Scientific Library (GSL) [34] ODE inte-gration routines. We integrate the position and picomponents backwards in time for a predeterminedperiod of time, using the Runge-Kutta-Fehlberg (4,5) method. The accuracy on each integration stepis specified in terms of relative and absolute errortolerances, and an adaptive step size is used.

When searching for a geodesic that connects thesource and observer, there are three parameters thatcan be varied: two initial conditions which control theangle at which the photon arrives at the observer, andone parameter which describes the time the sourceemitted the photon (and thus how far back to inte-grate the geodesic). As we are working with linearizedgravity about a Minkowski background, the flat spacegeodesic that connects the source and observer pro-vides a good initial guess for these parameters. Inorder to improve a guess, we consider geodesics withangles that slightly vary in orthogonal directions andalso propagate the geodesic a little further backward intime, constructing three basis vectors. We then calcu-late the Euclidean deviation from the beginning of thegeodesic to the source, and decompose that deviationin terms of the three basis vectors. The correspondingmodifications are then made to the angles and tim-ing, and the process repeated until the geodesic landswithin some tolerance of the source. This methodis computationally intensive: each iteration requiresthree geodesics to be computed. Nonetheless, the

approach typically led to exponential convergence upto the point where the tolerances were insufficientlytight to allow further convergence.

Iterating this procedure for arrival times across oneperiod of the string loop obtains one geodesic perarrival time. For our initial searches, we broke thestring period up into 1000 timesteps. In order toobtain multiple geodesics at a given arrival time, wesearched for geodesics that arrived one time step apartand whose angle at the observer jumped significantly.Such pairs were always found to have local minima inthe distance of closest approach. We then used theangles for these geodesics modified by the appropri-ate angular velocity as initial guesses for time stepsmoving forward or backward in time appropriatelyin order to obtain multiple geodesics arriving at agiven time. Due to rapid changes occurring as pho-tons approach a string segment very closely, we usedincreasingly small timesteps to obtain new geodesics.This method of identifying microlensing solutions re-lies on each family of smoothly-connected geodesicsbeing the only solution at some point in the evolu-tion. For microlensing events where a microlensedimage appears and then disappears while the primaryimage is always visible, this method is unlikely toidentify the second set of solutions. A source mightbe microlensed twice by two different lengths of stringthat lie close to the observer-source line of sight, anintrinsically low probability event. Generally speak-ing, events with more than two images will not beidentified with this method.

A. Approximations

Our approach to calculating metric perturbationsis fundamentally limited by the linearized gravityapproximation. In our numerical implementation, anumber of approximations are made, mainly involvingaccuracy tolerances. Our approach to our numericalapproximations is to ensure that the numerical preci-sion is superior to the linear approximation. In thissection, we estimate the magnitude of each error, andcalculate appropriate tolerances.

The linearized gravity approximation drops termsof order h2, where h is the metric perturbation. FromEq. (20), the metric perturbation in the linear regimeis

hµν = 4Gµ

∫ L

0

dσFµν|Ωµxµ|

. (35)

Heuristically, the error from the linear approximationis roughly δh ∼ h2 ∼ (Gµ)2, and so we aim for thenumerical errors to also be of this order.

We would like to know when the linear perturba-tion becomes nonlinear. Very roughly, the numerator

8

Fµν ∼ O(1), while the denominator |Ωµxµ| ∼ |~r − ~x|.Let the distance of closest approach be rmin. Thesegment of string nearest this distance of closest ap-proach will dominate the metric perturbation integral.As a crude estimate, model that segment of stringas a straight segment of string of length αL, with itscenter offset from the point of interest by the distanceof closest approach. We have

h ∼ 4Gµ

∫ αL/2

−αL/2

dh√h2 + r2min

∼ 8Gµ ln

(αL

rmin

)+O

(r2minα2L2

), (36)

displaying the typical logarithmic divergence. Forh ∼ 1, we find

rmin ∼ αLe−1/8Gµ , (37)

showing that the linear approximation should be validup to distances very, very close to the string.

When calculating the metric perturbations, the inte-gration routine estimates relative and absolute errors.Writing h = htrue + δhintegration, the relative error isδhintegration/h and the absolute error is δhintegration.Comparing to the linear approximation, tolerancesof Gµ for the relative error and (Gµ)2 for the abso-lute error are reasonable. As we desire the numericalerrors to be subdominant compared to the linear ap-proximation, we set the tolerances to be two ordersof magnitude smaller again.

The metric perturbation is also sensitive to errorsin the retarded time calculation. Roughly speaking,the error in the metric perturbation can be estimatedas

δh ∼ h(δτ

L+δτ

r?

)(38)

where r? is the closest retarded distance to the stringfrom that point in spacetime. The first term comesfrom errors in the numerator Fµν , while the secondterm comes from errors in the denominator Ωµxµ.The appropriate conditions for the tolerance in theretarded time are then δτ < GµL, δτ < Gµr?. Inpractice, because root finding using Newton’s methodis very cheap when a good estimate for the root isknown (such as calculated from the previous segmentof loop considered), we demanded δτ < (Gµ)2L/100,which satisfies both requirements up to r? two ordersof magnitude closer than we expect is required formicrolensing to exist.

We now come to integrating the geodesic. At eachintegration step, two errors are calculated: the ab-solute and relative error. The total absolute erroraccumulated across the geodesic will be the numberof steps times the absolute error per step. As we do

not know a priori how many steps will be taken, wethus demand an absolute error of zero. In order to un-derstand the relative error, we look at Eq. (34a) overa finite difference step. The error in computing ∆xi

from the linear approximation is O(h2), and so thetolerance on the relative error should be Gµ. Again,we set this to be a couple of orders of magnitude lowerin order to make the numerical errors subdominantto the linear approximation.

The total error in the location at which the geodesiclands arising from the linear approximation is then∼ Gµ(d1+d2), where d1+d2 is the Euclidean distancefrom the source to the observer. Again, we demandthat the numerical precision is a couple of orders ofmagnitude better than this.

We see that essentially all tolerances scale with Gµ.As such, working at smaller Gµ becomes more chal-lenging, as satisfying the desired tolerances becomesmore computationally demanding.

V. STRINGY SPACETIMES

In this section, we introduce the string loop config-uration that we used, and investigate the spacetimethat it generates.

For our goal of investigating string microlensing,we desired the simplest loop configuration we couldfind. More complicated loop configurations would bemore computationally demanding when calculatingthe metric perturbations and their derivatives, andwould complicate the interpretation of the results.As such, we chose to work with the Burden loopconfiguration [35], defined as follows.

~a(ξ) =1

α

(sin(αξ), 0, cos(αξ)

)(39a)

~b(ξ) =1

β

(cos(ψ) sin(βξ), sin(ψ) sin(βξ), cos(βξ)

)(39b)

Here, α = 2πm/L and β = 2πn/L for relatively primeintegers m and n, and ψ is an arbitrary angle. Whenone of m or n is unity, this configuration has no self-intersections (for 0 < ψ < π) or kinks, but possessesa cusp. The period of the loop is T = L/2mn. Thetangent sphere representation is two great circles, withn and m revolutions each as σ varies from 0 to L. Forthis work, we used L = 2π in arbitrary units, m = 1,n = 2, and ψ = π/3. The tangent sphere for ourloop is shown in Figure 2, and some images of theloop configuration through its period are presented inFigure 4. The string loop sits roughly along the x-zplane.

Despite the simple nature of this particular loopconfiguration, the metric solution it sources is still

9

Figure 2. The tangent sphere representation of our string.One great circle is traversed twice for every time the otheris traversed once.

0.75Π

2

x

K

Figure 3. Plots of the Kretschmann scalar for the oscillat-ing string along the x axis for 0 < x < L/4, where L = 2π.The red dashed line indicates the Kretschmann scalar fora Schwarszchild black hole of equivalent mass. Plots aregiven at five different times, with a tenth of a period be-tween them. For reference, the string is contained in thebounds −0.75 ≤ x ≤ 0.75. It is evident that there is littlecurvature in the inner portion of the loop.

highly complicated. To demonstrate this, we plot themetric components along the x axis for different timesin Figure 5 (there is nothing special about the x axis;these plots are simply intended to be indicative ofbehavior).

The time-averaged metric produced by the stringyields a Schwarzschild-like metric when far from thestring, as shown in Eq. (22). However, for radius r <L, this approximation breaks down, and the dynamicsbecome far more complicated. To demonstrate howthe inner portion of the string spacetime behaves interms of curvature, we compute the Kretschmannscalar RµνγδRµνγδ along the positive x axis from the

origin at five different times in Figure 3. We alsoplot the Kretschmann scalar K = 48G2M2/c4r6 for aSchwarzschild black hole of mass µL for comparison.It is readily seen that the behavior of geodesics nearthe string in this spacetime is going to be far morecomplicated than for either black holes or the infinitestraight string.

VI. STRING MICROLENSING

In this section, we present the first demonstration ofcosmic string loop microlensing in the form of two nu-merical realizations of the phenomena. We present thegeometry of each situation, discuss features evidentin the simulations, and quantitatively compare phe-nomena with theoretical predictions from the infinitestraight string.

A. First Demonstration of Microlensing

For our first demonstration of microlensing, wechose a simple geometry where we strongly suspectedthat microlensing should occur. We chose sourceand observer positions on opposite sides of the stringsuch that the Euclidean path between the two pointsintersected the volume swept out by the string in asimple manner. This geometry is detailed in Figure6. It is straightforward to see that the Euclideanline joining the source and observer will lie outsidethe loop for part of the period, and inside for therest. This line crosses the string twice, once for ahorizontal segment of string, and once for a diagonalsection of string. Note that this line is nowhere nearthe cusps, and also avoids regions where multipleclose encounters with the string might occur. Thestring configuration is as described in Section V, theobserver is located at (1,−4,−3), and the source islocated at (−1.26, 3.73, 3.3). In order to make themicrolensing solutions as pronounced as possible, weused Gµ = 10−3.

Using the computational method described in Sec-tion IV, we relaxed Euclidean geodesics to obtaingeodesics connecting the source and observer, witharrival times spaced throughout a period. This alonecould not identify microlensing solutions, as it onlyproduced one geodesic arriving at any given time.From plots of the distance of closest approach as afunction of arrival time, it is straightforward to iden-tify when the solutions jump from geodesics passingoutside the loop to geodesics passing inside the loop(although which is which requires 3D visualizationtools). Solutions on either side of this jump couldthen be used as initial guesses from which to step for-

10

Figure 4. The loop considered in this paper in snapshots in time. The stills are equally spaced throughout the period,moving left to right, top to bottom. Note that these are plots of the position of the string, not the retarded position asseen by an observer from a string that “shines”. The colour-coding displays the string’s velocity at that point, with bluebeing slowest, and red approaching the speed of light. Images three and seven are at times near when a cusp forms.

-0.8 0.8 2 Π

10

20

30

40

50

htt

-0.8 0.8 2 Π

-10

-5

5

htx

-0.8 0.8 2 Π

-10

-5

5

hty

-0.8 0.8 2 Π

-40

-30

-20

-10

10

htz

-0.8 0.8 2 Π

5101520253035

hxx

-0.8 0.8 2 Π

-15

-10

-5

5

hxy

-0.8 0.8 2 Π

-5

5

10

15

hxz

-0.8 0.8 2 Π

10

20

30

40

hyy

-0.8 0.8 2 Π

-5

5

10

15

hyz

-0.8 0.8 2 Π

10

20

30

40

50

60

hzz

Figure 5. Plots of the metric perturbation components (divided by Gµ) along the x axis, ranging from −L to L (whereL = 2π). Five times have been plotted, with 1/10th of the period passing between each curve. For reference, the stringis contained in the bounds −0.75 ≤ x ≤ 0.75.

11

Figure 6. Plot of the retarded image of a loop throughoutits period (shown in 15 steps). The red dot is the positionof the source, and the loop is shown from the perspectiveof the observer. The colour scheme is as previously, withred indicating string velocities close to the speed of light,and blue indicating slower velocities, typically ∼ 0.1c.

T 8 2 T 8 3 T 8 4 T 8 5 T 8 6 T 8 7 T 8 T

0.05

0.10

0.15

r

Figure 7. Plot of distance of closest approach between ageodesic and the string loop as a function of time of arrivalat the observer. The distance is a Euclidean distance, andmeasured in the same distance units as the string (recallthat we use L = 2π). The period for this loop is T = L/4.The red shaded background is indicative of flux from thesource across the period, in arbitrary units.

wards or backwards in time appropriately, iterativelyextending the solutions until they hit the string (orin practice, got too close to the string to computefurther data points in a reasonable amount of time).

The results are plotted in Figure 7, which shows thedistance of closest approach of the geodesics arrivingat different points in time. The existence of multi-

ple solutions for some points in time is evident: wehave identified microlensing solutions. The red partof the curve corresponds to geodesics lying outsidethe string, while the dark blue curve corresponds togeodesics passing through the string. At points intime where multiple solutions are present, the curvesare coloured differently to highlight this. The blackline is drawn at the theoretical estimate of the dis-tance at which microlensing solutions exist for aninfinite straight string (see Section II). We see thatwith an O(1) correction factor, this estimate is correct,although it is an underestimate at the beginning ofthe microlensing event, and an overestimate at theend. One important observation from this plot is thatmicrolensing events begin and end when one geodesichas an impact parameter of zero.

Another quantity that can be extracted from thisplot is the duration of each microlensing event, whichwill be important for rate calculations and potentiallyobserving such events. The duration of these eventsare about 8% and 12% of the period. Given theestimate of Eq. 5, these durations are approximatelydouble (triple) the estimated minimum duration forthis loop. Such factors are very reasonably explainedby various angles and velocity-dependent coefficients.

The jagged peak near t = 7T/8 is explained bydividing the string loop into two halves. To the leftof this peak, one half is closer to the geodesic, but ismoving away from it. On the other side, the other halfof the string is closer to the geodesic, and continuesto move closer. At the discontinuity in the derivative,the two segments of the loop are equidistant from thegeodesic. This suggests that whenever the velocityof the string is towards a geodesic, the slope of thedistance of closest approach is negative, while theopposite holds when the string is moving away fromthe geodesic. This fact is important when consideringtime delay and redshift effects below.

We investigated the linear approximation by lookingat the metric perturbation at the distance of closestapproach for each geodesic. We computed the maxi-mum deviation of any component from the Minkowskimetric as an indication of the size of the perturbation.The typical values we found were h ∼ 0.01, spiking upto a maximum of h ∼ 0.07 as the distance of closestapproach became small. When applying Eq. (36)with rmin ∼ 0.03 and α ∼ 0.2, corresponding to themiddle of the microlensing event, we obtain h ∼ 0.03,and so these values are as expected. This confirmsthat the linear approximation works well, and showsthat it is sufficient for investigations of microlensing.

In Figure 8, we plot the angle of arrival of thegeodesics, choosing an arbitrary orientation. This cor-responds to the path that the position of the sourcewould track in the sky. The colours in this plot arechosen to be the same as in the previous plot, so that

12

-0.020 -0.015 -0.010 -0.005Dx

-0.005

0.005

0.010

0.015

Dy

Figure 8. Plot of the angle (in radians) at which thegeodesics arrive at the observer plotted in an x− y plane,where the −z direction points directly at the source. Thecolour scheme is chosen identically to the previous plot.

different curves can be matched; the locations of thesource where microlensing events are occurring areevident. The origin in this plot represents the Eu-clidean line between the source and the observer. Forsmall deflection angles, the difference between the twoseparate curves is approximately the deflection anglecaused by traversing the opposite side of the string,which is roughly 0.01− 0.015 radians. As discussedin Section II, the deflection from an infinite straightstring is 8πGµ sin θd2/(d1 + d2) ∼ 0.012 radians inthis case, where we assume θ ∼ π/2. We see thatwith O(20%) correction factors, this deflection angleis validated by the microlensing solutions found. Itis interesting to note that at the ends of both curves,where the geodesic is becoming very strained and pass-ing very close to the string, the angle of arrival takesa sharp hook in a new direction. Closer inspection ofthis hook shows that it points directly towards thelocation of the microlensed solution. Another interest-ing point to note is the “Newtonian” deflection angle,caused simply by the mass of the string. While itvaries depending on how the string is configured atany given point in time, it is of the same order as thedeflection from the deficit angle.

We can also investigate the time of flight of thegeodesics. These are plotted in Figure 9, where the yaxis plots the time delay compared to the Euclideandistance between the observer and the source. Weimmediately see that the time delay is always positive.The magnitude of the delay is due to a number offactors. The most obvious reason is that the geodesicsare bent, and are thus travelling a greater distance.In the lower figure, we plot the Euclidean distancetravelled by the geodesics (ignoring O(h) corrections).This plot shows that the extra distance traversedbecause of the bent geodesics incurs only 1− 2% ofthe observed extra travel time. It also shows that

T 8 2 T 8 3 T 8 4 T 8 5 T 8 6 T 8 7 T 8 T0.072

0.073

0.074

0.075

0.076

0.077

Dt

T 8 2 T 8 3 T 8 4 T 8 5 T 8 6 T 8 7 T 8 T0.0000

0.0005

0.0010

0.0015

Dx

Figure 9. Above: plot of the time delay of geodesics as afunction of arrival time, compared to the Euclidean traveltime.Below: plot of the increased Euclidean distance throughgeodesic bending as a function of arrival time.

the travel distance is roughly constant for each curve,and that geodesics passing through the loop have aslightly shorter journey. The dominant contributionto the time delay is from the Shapiro time delay [36].The time delay from this effect is proportional toGµL ∼ 6 × 10−3 with a coefficient depending uponthe geometry, which could easily be O(10). When thegeodesic is far from the string, we see that the traveltime is less. Similarly, the curve describing geodesicsthat travel through the string loop is significantly moredelayed. We attribute this effect to the coefficientdescribing the Shapiro time delay increasing as thegeodesics moves deeper into the potential well of thestring. As the geodesics approach the string moreclosely, they move into stronger wells, explaining theincreased time delay at the ends of the curves. Notethat the intersections between microlensing solutionsthat occur on this plot do not occur at the same timesas the intersections on the plot showing distance ofclosest approach. It is intriguing that these curvesintersect, as it is not clear that this is required.

Finally, we can plot the redshift of the geodesics(fobs − fsrc)/fsrc, as shown in Figure 10. A generaltrend is evident: when the string is moving away from

13

T 8 2 T 8 3 T 8 4 T 8 5 T 8 6 T 8 7 T 8 T

-0.005

0.000

0.005

0.010

0.015Df f

Figure 10. Plot of the redshift (fobs − fsrc)/fsrc forgeodesics as a function of arrival time.

Observer

1.005

1.010

1.015

1.020

1.025

Df f

Figure 11. Plot of redshift in a geodesic as a functionof position. The x-axis ranges from the source to theobserver, while the y-axis shows the redshift. Four curvesare plotted, with a time difference of a tenth of a periodbetween them. These curves are all from geodesics thatpass outside the loop.

the geodesic, photons become blueshifted, while whenthe string is moving towards the geodesic, the photonsbecome redshifted. The size of the shift red/blueshiftvaries between ∼ ±4πGµ, in good agreement withthe prediction from the Kaiser-Stebbins effect [28]described in Section II.

The final plot that we include in Figure 11 is theredshift along a geodesic as a function of position. Wesee that as a photon leaves the source and travels to-wards the string, it encounters waves of gravitationalradiation. In the middle of the loop, the photon isblueshifted by the potential well, and as it leaves, it“surfs” a wave at a redshift that decays as 1/r (thedecay of a gravitational wave from a cosmic stringloop). The final redshift is somewhat dependent uponwhether the photon was emitted at a peak or a troughof the gravitational waves. These observations coin-cide with observations about the computational timenecessary for integrating geodesics: the path of thephoton approaching the loop was always much more

computationally intensive than the path of the photonleaving the loop.

One thing that was not done in this analysis wasto transform the source and observer’s coordinatesinto an orthonormal basis in order to compute thedeflection angles and redshift, justified on the basisthat the source and observer are sufficiently far fromthe string that the use of the Minkowski metric is agood approximation. For most phenomena, this wouldonly give rise to an O(h) correction to O(h) effects,and as the distance from the observer to the stringgrows, the effects can only shrink. However, the effectson redshift may be more pronounced, because of thedependence of whether the photon was emitted duringa peak or a trough of the gravitational radiation.

B. Generic Lensing

Our first example of cosmic string microlensingdemonstrated all of the expected effects predicted an-alytically from analysing the infinite straight stringsystem. However, it represented a carefully chosen sys-tem where we expected microlensing to occur, usinggeodesics that grazed the side of the loop appropri-ately, and was also performed at a very strong stringtension. For our second analysis of string microlensing,we decided to choose an arbitrary direction (distinctfrom the previous direction), and place the source andobserver on opposite sides of the loop in that direction.The observer was placed along the z-axis at (0, 0, 5),and the source at (0, 0,−5). Euclidean geodesics con-necting the two thus run directly through the center ofthe string loop. We also decreased the string tensionby an order of magnitude to 10−4. This configurationdemonstrated all of the effects we previously identified,and also contained some unexpected surprises.

The retarded image of the string is shown in Figures13. We see that in each half of the period, we canexpect two microlensing events to occur, as the stringloop crosses the Euclidean geodesic twice in each im-age, once diagonally to the right, and once diagonallyto the left. Note that the cusp on the bottom left,visible in red in the top plot, is visible for a numberof images. Conversely, the cusp on the right, barelyvisible at all in the top right corner of the bottom plot,is only visible for a single image. The reason behindthis is that cusp in the top image is moving away fromthe observer close to the speed of light, and so theretarded image sees it at a number of instants in time.The cusp in the bottom image, however, is movingtowards the observer close to the speed of light, andso the window in which the retarded image sees thecusp is significantly shortened.

Our heuristic picture is confirmed in the plot ofdistance of closest approach as a function of arrival

14

T 8 2 T 8 3 T 8 4 T 8 5 T 8 6 T 8 7 T 8 T

0.05

0.10

0.15

0.20

r

0.260 0.265 0.270 0.275t0

0.0075

0.015

0.0225

0.03

r

0.535 0.540 0.545 0.550t0

0.0075

0.015

r

0.820 0.825 0.830 0.835t0

0.0075

0.015

r

1.095 1.100 1.105 1.110 1.115t0

0.0075

0.015

0.0225

0.03

r

Figure 12. Plot of the distance of closest approach between a geodesic and the string loop as a function of arrival time.The red shaded background is indicative of flux from the source across the period, in arbitrary units. The lower plotshows a zoomed in version of the four microlensing events. Recall that T = L/4 = π/2 in our units. Again, the fourindividual curves are colour-coded to identify which curve is which on different plots, and the colour also changes toreflect the existence of multiple images at that time.

15

Figure 13. Plots of the retarded image of a loop throughoutits period, shown in two halves (10 steps each). The reddot is the position of the source, and the loop is shownfrom the perspective of the observer. In the top image, theloop segment that crosses the line between the observerand the source is closer to the source than the observer;the reverse is true for the bottom image.

time, shown in Figure 12. We see that indeed, fourmicrolensing events occur. Because the string ten-sion is an order of magnitude lower this time, thetheoretical prediction at which microlensing eventsoccur is substantially lower, and extracting data inthe microlensing regime was computationally inten-sive, as the points needed to be an order of magnitudecloser to the string than in the previous case. We seethat the theoretical estimate of when microlensingshould occur is again correct to within a factor of afew, but once again, provides an underestimate at thebeginning of the lensing event, and an overestimate atthe end. This effect is likely related to the velocity ofthe nearest string segment and the angle of approachof the geodesic.

We see that there are four jagged peaks in the curveswhich are again explained by different segments ofthe string becoming closer. Looking at the zoomed-inplots of the microlensing events, note that the plotsoccur in pairs: the top left and bottom right plots arealmost identical, while the top right and bottom leftplots are also almost identical in timing, distance, andslopes for all curves. The duration of the microlensingevents were, in order, approximately 0.7%, 0.65%,0.65% and 0.7% of the total period. These are ap-proximately twice as long as the predicted minimumduration for a microlensing event, which can again be

very reasonably explained through appropriate anglesand velocity-dependent factors. Although we presentonly approximate figures, reflecting uncertainty inthe precise timing, we note that the duration of thefirst and final events were numerically identical tofour significant figures, as were the second and thirdevents, further suggesting some symmetry betweenthe events.

We conjecture that this symmetry lies in the Burdenloop, which oscillates in a symmetric fashion. Lookingat Figure 13, we see that the first two microlensingevents occur when the geodesics pass near segmentsof the string closer to the source, while the other twooccur for segments of the string nearer the observer.However, because of the loop’s symmetry, the seg-ments of string at the front and the back are movingat the same velocity, which leads to the similaritiesin the microlensing events. The symmetry cannot beexact, however, as the geodesic’s behavior before andafter the close encounter with the string is different inthe different cases (in one case, it’s entering the mid-dle of the string, and in the other, it’s leaving). Theduration of a microlensing event is primarily deter-mined through the details of the close encounter withthe string, while the complicated dynamics outsidethis regime have little impact upon this result.

The linear approximation at the point of closest ap-proach was once again investigated for each geodesic.In this case, the typical size of the metric perturba-tion was h ∼ 10−3, spiking up to h ∼ 0.014 for thevery closest geodesics, again showing that the entiresimulation was safely in the linear regime. Comparedto our previous simulation, the decreased string ten-sion is slightly offset by the nearer distance of closestapproach. The analytic estimate from Eq. (36) withα ∼ 0.2 and rmin ∼ 0.075 yields h ∼ 0.004, which iswell within the expected regime.

The next plots, Figure 14, show the angular positionof the source on the sky, as seen by the observer.Again, the rotation is chosen arbitrarily. As expected,there are four separate curves in the sky. Contraryto the previous example, where the outside imagemoved a substantial angle through the sky, here, theoutside images are the red and blue curves, which havemuch smaller footprints on the plot. The images thatarise from threading the loop move a more significantangle through the sky. The similarities in these twocurves are ascribed to the previously noted symmetry.Again, when microlensing solutions exist, the anglebetween them is roughly 4πGµ, in agreement withthe theoretical prediction.

There are a number of interesting features to thisplot. When microlensing solutions exist, the curvestraced in the sky change direction abruptly. Fur-thermore, they move in the direction of where theirpartner solution lies. Looking at the previous plots in

16

-0.0015 -0.0010 -0.0005 0.0000Dx

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

Dy

- 0.00125 - 0.0012Dx0.0016

0.0018

0.002

Dy

0.000075 0.00022Dx0.

0.0011

0.0022

Dy

- 0.000535 - 0.00047Dx0.0026

0.00285

0.0031

Dy

- 0.00005 0.0001Dx0.0005

0.0015

0.0025

Dy

Figure 14. Plot of the angle (in radians) at which the geodesics arrive at the observer plotted in an x− y plane, wherethe −z direction points directly at the source. The bottom four plots show zoomed in versions of the individual curves.

17

T 8 2 T 8 3 T 8 4 T 8 5 T 8 6 T 8 7 T 8 T0.00820

0.00825

0.00830

0.00835

0.00840

0.00845

0.00850

0.00855

Dt

T 8 2 T 8 3 T 8 4 T 8 5 T 8 6 T 8 7 T 8 T0

0.00001

0.00002

0.00003

0.00004

Dx

Figure 15. Above: plot of the time delay of geodesics as afunction of arrival time, compared to the Euclidean traveltime.Below: plot of the increased Euclidean distance throughgeodesic bending as a function of arrival time.

Figure 12, we see that the microlensing solutions onlyoccupy a small portion of the time along the curves,but the angular distance travelled in that time can besignificant, especially in the outside curves. A subtlefeature, the small blip in the blue curve in the topleft plot, turns out be quite interesting and we willreturn to this in a moment.

We next look at the time delay and Euclidean dis-tance increase, as shown in Figure 15. We continueto see the general trends previously identified. Thedistance increase cannot account for the time delay, sothe bulk of it must come from the Shapiro time delay.The time delays between different curves again crosssomewhere during the microlensing events. The Eu-clidean distance increase is greatest for geodesics thatdo not pass through the loop, showing that they aremore bent. A subtle feature of note in these plots isin the time delay of the purple curve near t ∼ 5T/16,which contains an elbow. Again, we return to thisfeature later.

We now turn to the redshift of the geodesics inFigure 16. The general trend described by the Kaiser-Stebbins effect continues to hold here: photons areblueshifted when the nearest string loop segment is

T 8 2 T 8 3 T 8 4 T 8 5 T 8 6 T 8 7 T 8 T

-0.003

-0.002

-0.001

0.000

0.001

0.002

0.003

Df f

Observer

0.998

1.002

1.004

Df f

Figure 16. Above: plot of the redshift (fobs − fsrc)/fsrcfor geodesics as a function of arrival time.Below: plot of redshift in a geodesic as a function ofposition. The x-axis ranges from the source to the observer,while the y-axis shows the redshift. Three curves areplotted, at times corresponding to the beginning (top,blue), middle (middle, purple) and end (bottom, gold) ofthe brown curve (near 5T/8). These curves are all fromgeodesics that pass inside the loop.

moving away from the geodesic, and redshifted whenit is moving towards the geodesic. We also see someinteresting new features. In the top graph, we see aspike in the purple graph around the same positionas the previously noted elbow and a blip in the bluecurve, around the position it showed on the angulardeflection plot, suggesting that whatever caused thosefeatures previously also impacts the redshift.

The other interesting feature to note is in the lowerplot. Previously, there were troughs and crests to aphoton propagating against the gravitational radia-tion into the string loop. Here, the redshift curveslook much more like the absorption of a high-finessecavity. We conjecture that this arises because of thedirection in which the geodesics approach. Recallthat the string loop is mostly in the x-z plane, andthe geodesics move mostly in the z direction. Thismeans that both the near and the far side of the loopcan generate metric peturbations at the fundamentalfrequency, and constructive interference gives rise tothe large spikes in the redshift. These spikes are even

18

Observer

0.998

0.999

1.001

1.002

1.003

1.004

Df f

Figure 17. Plot of the redshift along a geodesic at threetimes: just before, during, and just after the spike in thepurple curve.

Figure 18. Retarded image of the string loop from theperspective of the observer at the time of the feature inthe blue curve. The image is colour-coded by the velocityof the string, with red being close to the speed of light,and blue being much slower (around 0.1c). The formationof the cusp on the right of the curve is evident, which ismoving roughly towards the observer.

stronger than the potential well in the middle of theloop. We will loosely refer to these perturbations asradiation even though they occur in the near zone ofthe system. On the far side of the loop, the photonsonce again ride the gravitational radiation as it fallsoff as 1/r.

Let us now turn to the feature in the purple curvethat was clearly evident in the redshift plot and alsovisible in the time delay plot. The reason for thisspike can be understood in terms of the crests of thegravitational radiation. Once per period, the sourceis going to emit radiation starting on a crest; as thephoton leaves the crest, it will be redshifted downthe crest. In Figure 17, we see the redshift curvesbefore, during and after the crest passes the source,which confirm this as the source of the feature. Thisbehavior was not identified in the previous lensingsimulation, as the crests of the gravitational radiationwere not nearly so pronounced.

The final feature to explain is the blip in the angular

0.7 Observer

0.9990

0.9995

1.0000

1.0005

1.0010Df f

Figure 19. Plot of the redshift along a geodesic at threetimes: at the time of the feature in the blue curve (gold),and at two later times. This plot only shows the last 40%of the distance from the source to the observer, in orderto highlight the unexpected features of the tail end of thegeodesic.

deflection of the blue curve, and the accompanyingblip in the redshift plots. This feature is serendipitous:it turns out that the position of the observer is withina cone of cusp radiation, as one of the cusps fromthe loop radiates in the +z direction. The fact thatthe cusp radiation is responsible is corroborated bythe retarded image of the string at the time of thatblip, as shown in Figure 18: a small red dot to theright is moving at the speed of light; it happens to bemoving roughly in the direction of the observer. Wecan also look at the redshift along the geodesic thatarrives at the time of that blip; it shows an obviousupswing at the very end that is not expected from the1/r decay of gravitational radiation as photons leavethe string loop, see Figure 19. Even more telling isthe redshift along geodesics some time later, whichshow features suggesting that they too have beenaffected by a burst of radiation passing through. Wefind that a null geodesic connects these events to thespacetime position of the cusp. The blue curve inFigure 16 displays a corresponding blip associatedwith the same phenomenon. Although the cone ofcusp radiation is typically thought of as being quitenarrow, the opening angles required here are ∼ 0.15radians to affect the observer, and up to ∼ 0.3 radiansto create the feature in the purple curve in Figure19. A similar cone of cusp radiation will affect thesource, but this signal has likely been drowned out bythe dynamics as photons propagate through the innerpart of the string loop.

VII. CONCLUSION

In this paper, we presented numerical solutions formicrolensing from cosmic string loops. Two specific

19

configurations were analyzed in detail; one was care-fully constructed to yield microlensing solutions anddemonstrate the robustness of our method, while thesecond was chosen more arbitrarily and investigatedat smaller string tension. Both configurations showedclear evidence of the existence of microlensing solu-tions.

A number of theoretical predictions from the infinitestraight string were tested in the loop configurations.First and foremost, the existence of multiple geodesicsbetween the source and observer was verified. The the-oretical prediction of the distance of closest approachat which multiple solutions occur was validated, andthe deficit angle prediction for the angle of deflectionbetween multiple solutions was also shown to holdtrue. The Kaiser-Stebbins effect on photon redshiftwas also demonstrated. Each of these effects wereshown to be qualitatively true, sometimes with O(1)corrections from the string dynamics.

We identified a number of features of cosmic stringmicrolensing that are absent for the infinite straightstring. The most significant of these was the con-tribution to the time delay arising from the Shapiroeffect. Another was the magnitude of the deflectionof a geodesic from the string’s mass, as opposed tothe deficit angle. It was found that this deflectionangle is of the same order as the deficit angle. Theeffect of the cone of cusp radiation on photons was ob-served, which caused a redshift and angular deflection.This was despite not selecting geodesics that passedparticularly close to a cusp. Finally, a redshift effectbased on the gravitational wave phase at the sourceemission was observed. This was particularly evidentin the second configuration we analyzed, where anunexpected resonance gave rise to strongly peakedpulses of gravitational radiation.

A number of intriguing behaviors were also found,including the crossing of curves in the time delayplots, and the angular deflection of geodesics duringmicrolensing events pointing towards each other. Itwas also noted that the condition under which mi-crolensing begins and ends in terms of the requireddistance of closest approach has an O(1) correctionthat relates to the velocity of the appropriate segmentof string. In the configurations investigated here, wefound microlensing with impact parameters furtherfrom the string at the start and closer to it at theend than anticipated. Further investigations will benecessary to understand whether these observationshold more generally than the two situations analyzedhere.

The basic goal of this program of work is to providereliable information for the calculation of microlens-ing rates, as well as to provide accurate descriptions

of microlensing events for observers. Although thenumerical examples presented involved unphysicallylarge string tensions (Gµ of 10−3 and 10−4 respec-tively) we identified how the phenomena scale withstring tension so the results may be applied to physicalstring tensions of interest.

There is much further work to be done. Our firsttask will be to generalize our code to handle kinksand straight segments of string. While these featureswill likely have interesting microlensing signatures intheir own right, physical strings are also highly likelyto possess kinks. Our second task will be to evaluatethe magnification profile during microlensing events.While conventional wisdom suggests that flux doublesduring a microlensing event, this is unlikely to be thefull story, and it will be interesting to see just how themagnification changes during a microlensing event.

Once the tools to investigate these effects are inhand, there are two distinct lines of inquiry. Thefirst involves investigating individual configurationsin detail, with the aim of understanding the identi-fied unexplained behaviors and identifying interestingphysical effects in cosmic string microlensing. Thiswould be particularly useful in order to understandthe behavior of the magnification of images duringmicrolensing events. This would also involve investi-gating effects from different string configurations, aswell as the effect of small scale string structure.

The second path involves calculating the microlens-ing signatures for a large number of configurations ofstrings and source/observer positions. The aim wouldbe to identify how the magnification profile, durationand frequency of microlensing events scale with loopsize, string tension, observer and source distance, andstring and source velocity. An accurate understandingof the appropriate scaling relations would allow formore precise rate predictions to be made, which wouldin turn lead to more accurate bounds on cosmic stringtensions, and possibly even the observation of cosmicstrings in our galaxy.

Ever-more capable optical surveys are beingplanned and carried out, and we are hopeful thatmicrolensing will give us a window into high energyprocesses from the earliest period of the universe’shistory.

ACKNOWLEDGMENTS

We thank Eanna Flanagan, Leo Stein and HenryTye for useful conversations. We gratefully acknowl-edge the support of the John Templeton Foundation(Univ. of Chicago 37426-Cornell FP050136-B).

20

[1] G. Hinshaw et al. (WMAP), Astrophys.J.Suppl. 208,19 (2013), 1212.5226.

[2] P. Ade et al. (Planck Collaboration) (2013),1303.5082.

[3] E. J. Copeland, R. C. Myers, and J. Polchinski, JHEP0406, 013 (2004), hep-th/0312067.

[4] D. F. Chernoff and S. H. Tye (2007), 0709.1139.[5] A. Vilenkin and E. P. S. Shellard, Cosmic Strings

and Other Topological Defects (Cambridge UniversityPress, 1994).

[6] T. Vachaspati and A. Vilenkin, Phys. Rev. D 31, 3052(1985).

[7] J. Christiansen, E. Albin, T. Fletcher, J. Gold-man, I. Teng, et al., Phys.Rev. D83, 122004 (2011),1008.0426.

[8] J. Abadie et al. (LIGO Scientific Collaboration,Virgo Collaboration), Phys.Rev. D85, 122001 (2012),1112.5004.

[9] J. Aasi, J. Abadie, B. Abbott, R. Abbott, T. Abbott,et al. (2013), 1310.2384.

[10] R. van Haasteren, Y. Levin, G. Janssen, K. Lazaridis,M. Kramer, et al., Mon.Not.Roy.Astron.Soc. 414,3117 (2011), 1103.0576.

[11] C. Dvorkin, M. Wyman, and W. Hu, Phys.Rev. D84,123519 (2011), 1109.4947.

[12] P. Ade et al. (Planck Collaboration) (2013),1303.5085.

[13] E. J. Copeland, L. Pogosian, and T. Vachaspati,Class.Quant.Grav. 28, 204009 (2011), 1105.0207.

[14] S. Olmez, V. Mandic, and X. Siemens, Phys.Rev.D81, 104028 (2010), 1004.0890.

[15] D. F. Chernoff (2009), 0908.4077.

[16] D. F. Chernoff and S. H. Tye (to appear).[17] A. A. de Laix, Phys.Rev. D56, 6193 (1997), astro-

ph/9705223.[18] A. de Laix, L. M. Krauss, and T. Vachaspati,

Phys.Rev.Lett. 79, 1968 (1997), astro-ph/9702033.[19] J.-P. Uzan and F. Bernardeau, Phys.Rev. D63,

023004 (2001), astro-ph/0004105.[20] B. Shlaer and M. Wyman, Phys.Rev. D72, 123504

(2005), hep-th/0509177.[21] A. A. de Laix and T. Vachaspati, Phys.Rev. D54,

4780 (1996), astro-ph/9605171.[22] D. Huterer and T. Vachaspati, Phys.Rev. D68,

041301 (2003), astro-ph/0305006.[23] K. Kuijken, X. Siemens, and T. Vachaspati,

Mon.Not.Roy.Astron.Soc. (2007), 0707.2971.[24] D. F. Chernoff and R. Graziani (to appear).[25] S. Refsdal, MNRAS 128, 295 (1964).[26] B. Paczynski, Astrophys.J. 371, L63 (1991).[27] A. Vilenkin, Astrophys.J. 282, L51 (1984).[28] N. Kaiser and A. Stebbins, Nature 310, 391 (1984).[29] T. Vachaspati, Nucl.Phys. B277, 593 (1986).[30] A. Vilenkin, Nature (London) 322, 613 (1986).[31] A. Stebbins, Astrophys.J. 327, 584 (1988).[32] A. Buonanno and T. Damour, Phys.Rev. D60, 023517

(1999), gr-qc/9801105.[33] S. A. Hughes, C. R. Keeton, P. Walker, K. T. Walsh,

S. L. Shapiro, et al., Phys.Rev. D49, 4004 (1994).[34] B. Gough, GNU Scientific Library Reference Manual

(Network Theory Ltd., 2009), 3rd ed.[35] C. Burden, Phys.Lett. B164, 277 (1985).[36] I. I. Shapiro, Phys. Rev. Lett. 13, 789 (1964).