Exact Wave Propagation in a Spacetime with a Cosmic String

0.72 MB·English

by  Teruaki Suyama

KUNS-1997
Exact Wave Propagation in a Spacetime with a Cosmic String
Teruaki Suyama,1 Takahiro Tanaka,1 and Ryuichi Takahashi2
1Department of Physics, Kyoto University, Kyoto 606-8502, Japan
2Division of Theoretical Astrophysics, National Astronomical Observatory of Japan, Mitaka, Tokyo 181-8588,Japan
(Dated: today)
WepresentexactsolutionsofthemasslessKlein-Gordonequationinaspacetimeinwhichaninfi-
nitestraightcosmicstringresides. Thefirstsolutionrepresentsaplanewaveenteringperpendicular
to the string direction. We also present and analyze a solution with a static point-like source. In
the short wavelength limit these solutions approach the results obtained by using the geometrical
optics approximation: magnification occurs if the observer lies in front of the string within a strip
of angular width 8πGµ, where µ is the string tension. We find that when the distance from the
observer to the string is less than 10−3(Gµ)−2λ ∼ 150Mpc(λ/AU)(Gµ/10−8)−2, where λ is the
wave length, the magnification is significantly reduced compared with the estimate based on the
6
geometrical optics due to the diffraction effect. For gravitational waves from neutron star(NS)-NS
0
0 mergers the several lensing eventsper year may be detected byDECIGO/BBO.
2
PACSnumbers: 98.80.Cq,97.60.Lf
n
a
J
I. INTRODUCTION For some time, cosmic string was a candidate for

4 the seed of structure formation of our universe, but
2
this possibility was ruled out by the measurements of
Typical wavelength of gravitational waves from astro-
the spectrum of cosmic microwave background (CMB)
2 physical compact objects such as BH(black hole)-BH bi-
anisotropies[30, 31]. The current upper bound on the
v nariesisinsomecasesverylongsothatwaveopticsmust
9 be used instead of geometrical optics when we discuss dimensionless string tension Gµ is around 10−7 10−6,
∼
8 whichcomesfromtheobservationsofCMB[32,33,34,35]
gravitational lensing. More precisely, if the wavelength
0 and/or the pulsar timing [36, 37, 38, 39]. Although cos-
becomes comparable or longer than the Schwarzschild
2 micstringcannotoccupydominantfractionoftheenergy
radius of the lens object, the diffraction effect becomes
1
density of the universe, its non-negligible population is
5 important and as a result the magnification factor ap-
still allowed observationally[40, 41]. In fact, Sazhin et
0 proaches unity [1, 2, 3, 4, 5]. Mainly due to the
/ possibility that the wave effects could be observed by al.[42, 43] reported that CSL-1, which is a double image
h of elliptical galaxies with angular separation 1.9 arcsec,
future gravitational wave observations, several authors
p
could be the first case of the gravitational lensing by a
- [6, 7, 8, 9, 10, 11, 12, 13, 14, 15] have studied wave ef-
o fects in gravitationallensing in recent years. cosmic string with Gµ 4 10−7.
≈ ×
tr In most of the works which studied gravitationallens- We study in detail wave effects in the gravitational
s ing phenomenon in the framework of wave optics, iso- lensing byaninfinite straightcosmic string. InRef. [12],
a
: lated and normal astronomical objects such as galaxies wavepropagationaroundacosmicstringwasstudiedbut
v are concerned as lens objects. Recently Yamamoto and they put the waveform around the string by hand. 1
i
X Tsunoda[12] studied waveeffects in gravitationallensing Their prescription is correct only in the limit of geomet-
r by an infinite straight cosmic string. The metric around ricaloptics,whichbreaksdownwhenthewavelengthbe-
a a cosmic string is completely different from that around comeslongerthanacertaincharacteristiclength. Inthis
a usual massive object. paper, we present exact solutions of the (scalar) wave
Cosmic strings generically arise as solitons in a equation in a spacetime with a cosmic string. We an-
grand unified theory and could be produced in the alytically show that our solutions reduce to the results
early universe as a result of symmetry breaking phase of the geometrical optics in the short wavelength limit.
transition[16, 17]. If symmetry breaking occurred after We derive a simple analytic formula of the leading or-
inflation, the strings might surviveuntil the presentuni-
verse. Recently, cosmic strings attract a renewed inter-
est partly because a variant of their formation mecha-
nism was proposed in the context of the brane inflation 1 After submitting this paper, we have noticed a paper [44] in
which the solutions of the wave equations around the cosmic
scenario[18, 19, 20, 21, 22, 23, 24]. In this scenario infla-
string are given, though the apparent expressions are different
tion is driven by the attractive force between parallelD-
from those given in this paper. In [44] the author estimated
branesandparallelantiD-branesinahigherdimensional theamplitudeofthediffractedwavetobesuppressedbyO(Gµ)
spacetime. When those brane-anti-brane pairs collide comparedwiththatcorrespondingtothegeometricaloptics. We
and annihilate at the end of inflation, lower-dimensional showthattheimportanceofthediffractioneffectsaredetermined
by the combination of three parameters, Gµ, the distance from
D-branes, which behave like monopoles, cosmic strings
the string to the observer and the wavelength and that the rel-
or domain walls from the view point of four-dimensional ative amplitude of the diffracted wave can be O(1) for realistic
observers,are formed generically [25, 26, 27, 28, 29]. astrophysical situations.
2
dercorrectionstothegeometricalopticsduetothefinite Friedmannuniverseissuppressedbythesquareofthera-
wavelength effects and also an expression for the long tiobetweenthewavelengthandtheHubblelength,which
wavelength limit. Interference caused by the lensing re- can be neglected in any situations of our interest.
mains due to the diffraction effects even when only a Ourgoalofthissectionistoconstructasolutionofthe
single image can be seen in the geometrical optics. This waveequationwhichcorrespondstoaplanewaveinjected
fact increases the lensing probability by cosmic strings. perpendicularly to and scattered by the cosmic string.
This paper is organized as follows. In section II, we This situation occurs if the distance between the source
construct a solution of the wave equation on a back- and the string is infinitely large. In order to construct
ground spacetime with an infinite straight cosmic string such a solution, we introduce a monochromatic source
in the case that a source of the wave is located infinitely uniformly extended in the z-direction and localized in
far. An extension to the case in which a point source is r θ plane,
−
locatedatafinitedistanceisgiveninAppendixB.Insec-
B
tion III, we study properties of the solution obtained in S = δ(r ro)δ(θ π)e−iωt, (2)
sec. IIindetail. InsectionIV,wefocusoncompactbina- (1 ∆) − −
−
ries as the sourcesof gravitationalwavesand discuss the
where ω is the frequency and we have introduced B, a
possible effects due to finiteness of the lifetime and the
constantindependent of∆,to adjustthe overallnormal-
frequencyevolutionofthebinariesonthedetectionofthe
ization when we later take the limit r . The factor
gravitationalwaves which pass near a cosmic string. We (1 ∆)−1 appearsbecauseθ-coordinaote→us∞edinthemet-
also give a rough estimate for the event rate of the lens- −
ric (1) differs from the usual angle
ingofgravitationalwavesfromNS-NSmergersassuming
DECIGO/BBO. Section V is devoted to summary.
ϕ (1 ∆)θ. (3)
≡ −
Here we consider a uniformly extended source instead of
II. A SOLUTION OF THE WAVE EQUATION apointsourcesincetheformeriseasiertohandle. When
AROUND AN INFINITE STRAIGHT COSMIC the limit r is taken, the answers are identical in
o
→ ∞
STRING these two cases. The case with a point-like source at a
finite distance is more complicated. This case is treated
A solution of Einstein equations around an infinite in Appendix B.
straight cosmic string to first order in Gµ is given by Now the wave equation that we are to solve is
[45]
∂2 1 ∂ 1 ∂2
+ + +ω2 φ(r,θ)
d2s= dt2+dr2+(1 ∆)2r2dθ2+dz2, (1) ∂r2 r∂r (1 ∆)2r2∂θ2
− − (cid:18) B − (cid:19)
= δ(r r )δ(θ π). (4)
o
where (r,z,θ) is a cylindrical coordinate(0 θ < 2π) 1 ∆ − −
≤ −
and 2π∆ 8πGµ is the deficit angle around the cosmic
≈ Since φ(r, θ) satisfies the same equation (4) as φ(r,θ)
string. Spatial part of the above metric describes the −
does, φ(r,θ) is even in θ. Thus, it can be expanded as
Euclidean space with a wedge of angular size 2π∆ re-
moved. Due to the deficit angle around a string, double
∞
images of the source are observed with an angular sepa- φ(r,θ)= f (r)cosmθ. (5)
m
ration<2π∆ when a sourceis locatedbehind the string
m=0
X
in the ∼limit of geometrical optics. In general for a wave
with a finite wavelength, some interference pattern ap- From Eqs. (4) and (5), the equations for fm(r) are
pears. An exact solution of Einstein equations around a
d2 1 d ν2
finitethicknessstringhasbeenalreadyobtained[46],but + +ω2 m f (r)
we use the metric (1) as a background since the string dr2 rdr − r2 m
thickness is negligibly small compared with the Einstein (cid:18) ( 1)m B (cid:19)
=ǫ − δ(r r ), (6)
m o
radius, πD∆, where D is the distance from the ob- 1 ∆ 2π −
≈ −
server to the string.
Throughoutthepaper,weconsiderwavesofamassless where ǫo 1,ǫm 2(m 1) and νm (1 ∆)−1m.
≡ ≡ ≥ ≡ −
scalar field instead of gravitational waves for simplicity, The solution of Eq. (6) except for r = ro is a linear
combinationofBesselfunction and Hankelfunction. We
but the wave equations are essentially the same in these
two cases. An extension to the cosmological setup is impose that the wave φ is regular at r = 0 and pure
out-going at infinity. Further, imposing that the wave is
straightforwardlydone by adding an overallscale factor.
In that case the time coordinate t is to be understood continuous at r =ro, fm(r) becomes
as the conformal time. The wave equation remains un-
changed if we consider a conformally coupled field, but fm(r)=Nm Hν(1m)(ωro)Jνm(ωr)Θ(ro−r)
it is modified for the other cases due to curvature scat- (cid:16)
tering. The correctiondue to curvature scattering of the +Jνm(ωro)Hν(1m)(ωr)Θ(r−ro) , (7)
(cid:17)
3
where Θ(x) is the Heaviside step function. Substitut- where the contour of the integral C is such as shown in
ing Eq. (7) into Eq. (6), the normalization factor N is Fig. 1, Eq. (10) can be written as
m
determined as
J (ξ) 1 1
B ǫ ( 1)m φ(ξ,θ)= 0 + dt eξsinht
Nm = 1 ∆ m2−πω −1−∆ 1−∆2iπ ZC
×−Jνm(ωro)Hν(1m)′(ωro)−Hν(1m)(ωro)Jν′m(ωro) −1 ×m∞=0e−1m−t∆+π2mi−2i(m1−π∆∆)(eimθ+e−imθ).(13)
Brh0ǫm( 1)m i X
= − , (8) When t is in the segment of the integration contour C
4i(1 ∆)
− along the imaginary axis, the summation over m does
where denotesadifferentiationwithrespecttotheargu- not converge because the absolute value of each term in
′
ment. From Eqs. (7) and (8) with the aid of the asymp- the summation is all unity. In order to make the se-
toticformulaeoftheBesselandHankelfunctions,φ(r,θ) ries to converge, we need to think that the integration
for r can be written as contour C is not exactly on the imaginary axis but t al-
o
→∞
ways has a positive real part. For bookkeeping purpose,
φ(r,θ)= −iB ro eiωro−iπ4 we multiply each term in the sum by a factor e−ǫm (ǫ
2√2(1 ∆) πω is an infinitesimally small positive real number). Then
− r Eq. (13) becomes
× ∞ ǫmime−2i(m1−π∆∆)Jνm(ωr)cosmθ. (9) J (ξ)
0
m=0 φ(ξ,θ)= +ψ(ξ,θ)+ψ(ξ, θ), (14)
X −1 ∆ −
Wedeterminetheoverallnormalizationofthesourceam- −
plitude B,independently ofGµ,sothat Eq.(9)becomes where ψ(ξ,θ) is defined by
aplanewaveeiωrcosθ whenGµ=0. Thisconditionleads
1 1 eξsinht
to B =−2q2rπoωe−iωro−iπ/4. Then, finally φ becomes ψ(ξ,θ):= 1−∆2iπ ZCdt 1−e−t1−−t∆∗ , (15)
φ(r,θ)= 1 1∆ ∞ ǫmime−2i(m1−∆∆π)Jνm(ωr)cosmθ. (10) with
− m=0 π α(θ)
X t := ǫ+i i , (16)
∗ − 2 − √ξ
III. LIMITING BEHAVIORS OF THE and α(θ):=(π∆ (1 ∆)θ)√ξ.
SOLUTION − −
Now we find that all we need to evaluate is ψ(ξ,θ) in
orderto obtainanapproximateformula for φ(ξ,θ). This
A. Approximate waveform in the wave zone integralwill notbe expressedby simple knownfunctions
ingeneral,buttheintegrationcanbeperformedbyusing
The solution (10) describes the waveformpropagating the method of steepest descent in the limit ξ 1.
≫
arounda cosmic string. But it is not easy to understand TheintegrandofEq.(15)hastwosaddlepointslocated
thebehaviorofthesolutionbecauseitisgivenbyaseries. at t = t iπ/2 and t = t iπ/2 in the vicinity of
+
In fact, it takes much time to perform the summation in the integra≈tion contour C.−W≈e−should also notice that
Eq.(10)numericallyforarealisticvalueoftensionofthe the integrandhasapole att=t , whichis alsoinfinites-
string,say,Gµ<10 6becauseofslowconvergenceofthe imally close to the contour of th∗e integral C. This pole
−
series. Inpartic∼ularitisnotmanifestwhethertheampli- is located near the saddle point at t = t as far as ∆
+
fication of the solution in the short wavelength limit co- andθ aresmall. Hencethe treatmentofthesaddlepoint
incideswiththeonewhichisobtainedbythegeometrical at t = t is much more delicate than that of the saddle
+
optics approximation. Therefore it will be quite useful if point at t = t . We only discuss the saddle point at
onecanderiveasimpleranalyticexpression. Herewere- t=t , then th−e case at t=t is a trivial extension.
+
duce the formula by assuming that the distance between When (t) > 0, (t) < iπ− or (t) < 0, (t) > iπ,
ℜ ℑ 2 ℜ ℑ 2
the string andthe observeris muchlargerthanthe wave whichcorrespondstoshadedregionsinFig.1,eξsinht di-
length, vergesinthelimitξ . Ifα(θ)>0,the poleatt=t
is in the bottom-left→un∞shaded region. In this case w∗e
ξ ωr 1, (11)
cannot deform the contour to the direction of the steep-
≡ ≫
est descent at t=t without crossing the pole at t=t .
which is valid in almost all interesting cases. +
The deformed contour which is convenient to apply th∗e
UsinganintegralrepresentationoftheBesselfunction,
method of the steepest descent is such that is shown as
C˜ in Fig. 1. When we deform the integration contour
1 fromC to C˜, therearisesanadditionalcontributioncor-
Jν(ξ)= 2iπ dt eξsinht−νt, (12) responding to the residue at t = t when α(θ) > 0. On
ZC ∗
4
the other hand, if α(θ) < 0, the pole is in the top-left σ(θ)eiξ−2iα2(θ)Erfc |α(θ)|e−iπ/4
shaded region. In this case, we can deform the contour − √π √2
(cid:18) (cid:19)
of the integral to the direction of the steepest descent +(θ θ). (21)
→−
without crossing the pole t . Hence no additional term
arises. ∗ For illustrative purpose, we compared the estimate
From these observations, we find that it is necessary given in Eq. (17) with the exact solution Eq. (10) in
toevaluatethe integral(15)separatelydependingonthe Fig.3. They agreequite well atξ 1. The deficit angle
≫
signature of α(θ). Though the calculation itself can be andthe observer’sdirectionarechosentobe ∆=0.0025
done straightforwardly, it is somewhat complicated be- and θ =0, respectively.
cause the saddle point and the pole are close to each
Im
other. When the pole islocatedinside the regionaround
the saddle point that contributes dominantly to the in-
tegral, a simple Gaussian integral does not give a good
approximation. Detaileddiscussionsaboutthispointare
givenin Appendix A.Here we onlyquote the finalresult
which keeps terms up to O(1/√ξ),
α(θ)
ψ(ξ,θ) exp iξcos Θ(α(θ)) Re
≈ √ξ
(cid:18) (cid:19)
σ(θ) iα(θ) i
exp iξ+ α˜2(θ)
− √π (1 ∆)√ξ − 2
(cid:18) − (cid:19)
α˜(θ)
Erfc σ(θ) e−iπ/4
× √2
(cid:18) (cid:19)
1 1 e iξ+iπ/4
−
+ , (17)
√2πξ1−∆1−e1−i∆(π−α(θ)/√ξ)
where
FIG. 1: Black, dotted and dashed lines are contours of the
α˜(θ) := i(1−∆) ξ 1−exp i(1 α(∆θ))√ξ ,(18) ionfteeξgsrianlhCt.,C˜andCH,respectively. ±iπ2 arethesaddlepoints
(cid:20) (cid:18) − (cid:19)(cid:21)
p
σ(θ) := sign(α(θ)), (19)
and
+
Erfc(x):= ∞dte−t2. (20)
Zx
We are mostly interested in the cases with ∆,θ 1. (cid:35)
≪ (cid:50)
Then, we have α(θ)/√ξ 1, and thereforeα˜(θ) reduces
(cid:152)
≪
to α(θ). The second term in Eq. (17) is the contribution observer
(cid:51) (cid:49)
fromtheintegralaroundthesaddlepointatt=t along
+
thecontourC˜. Thistermisnotmanifestlysuppressedby
1/√ξ. As far as α(θ) is fixed, this term does not vanish
inthelimitξ . Ofcourse,ifwefix∆andθfirst,and
(cid:36)
→∞
takethelimitξ ,theargumentoftheerrorfunction
→∞
goes to + and the function itself vanishes. However,
∞
α(θ) vanishes at θ =π∆/(1 ∆). Hence even for a very
−
large value of ξ there is always a region of θ in which
this second term cannot be neglected. However, for θ
FIG. 2: Configuration of the source, the cosmic string and
in such a region, α(θ) cannot be very large. Therefore,
the observer. A and B are the positions of a source. O and
we can safely drop the second term in the exponent. On
P are the positions of the cosmic string and the observer,
the other hand, the last term in Eq. (17), which is the respectively. In this figure, the wedge AOB is removed and
contribution from the saddle point at t = t , is always thusA and B must be identified.
suppressed by 1/√ξ. Hence, this term does n−ot give any
significant contribution for ξ 1. The first term in
≫
Eq. (14) can be dropped in the same manner for ξ 1.
≫
Keepingonlythetermswhichpossiblyremaininthelimit
B. Geometrical optics limit
ξ , we finally obtain
→∞
α(θ) Geometrical optics limit corresponds to the limit ξ
φ(ξ,θ) exp iξcos Θ(α(θ)) →
≈ √ξ with∆andθfixed. Inthislimitα(θ)alsogoesto+ ,
(cid:18) (cid:19) ∞ ∞
5
FIG. 3: Comparison between the exact solution Eq. (10) and the approximate one Eq. (17). π∆ is 0.0025. Black line and
dotted one correspond to the exact solution and the approximate one, respectively. We see that except for small ξ the dotted
line overlaps the black one. In theright panel, therelative error is about 10−3.
andhence the errorfunction inEq.(21) vanishes. Hence where ~x represents a two-dimensional vector on the lens
the waveform in the geometrical optics limit, which we planeandT(~x)representsthesummationoftimeofflight
denote as φ , becomes ofthelightrayfromthesourcetothepoint~xonthelens
go
plane and that from the point ~x to the observer. ~x
φ (ξ,θ) = eiξcos(π∆+ϕ)Θ(π∆+ϕ) j
go is a stationary point of T(~x), and n = 0,1/2,1 when
j
+eiξcos(π∆−ϕ)Θ(π∆ ϕ), (22) ~x is a minimum, saddle and maximum point of T(~x),
− j
where ϕ is defined by Eq. (3). respectively. The amplitude ratio u(~x)1/2 is written as
| |
Since φ and hence φ are even in θ, it is sufficient to
go
considerthe casewithθ >0. InFig.2,the configuration u(~x)=1/det[δab ∂a∂bψ(~x)], (26)
−
of the source, the lens and the observer is drawn in the
where ψ(~x) in Eq. (26) is the deflection potential [47]
coordinates in which the deficit angle 2π∆ is manifest,
i.e., the wedge AOB is removedfrom the spacetime. Both which is the integralof the gravitationalpotential of the
points A and B indicate the location of the source. The lens along the trajectorybetween the source andthe ob-
lines OA and OB are to be identified. The angle made server. Eq.(25)representsthatthewaveformisobtained
by these two lines is the deficit angle. The locations of by taking the sum of the amplitude ratio u(~xj)1/2 of
| |
the string and the observer are represented by O and P, each images with the phase factor eiωT(~xj)−iπnj. If the
respectively. In our current setup the distance between lens is the straight string, the spacetime is locally flat
O and A (= r ) is taken to be infinite. When ϕ > π∆, everywhere except for right on the string. This means
o
only the source A can be seen from the observer. This that the deflection potential ψ(~x) is zero and hence the
corresponds to the fact that only the first term remains amplitude ratio is unity for all images [47] and the tra-
for ϕ>π∆ in Eq. (22). For ϕ>π∆, we have jectory where the time of flight T(~x) takes the extremal
value is a geodesic in the conical space, and T(~x) of any
φ (ξ,θ)=eiξcos(ϕ+π∆). (23)
go geodesictakesminimum,whichmeansn =0. Thereare
j
This is a plane wave whose traveling direction is ϕ = two geodesics if the observer is in the shaded region in
π∆, which is the direction of −A→P in Fig. 2 in the limit Fig. 2. The time of flight along the trajectory AP is
−
roF=or|−A→|Oϕ||→<π∞∆., φgo is TA =rol→im∞|−A→P|≈ro+rcos(π∆+ϕ), (27)
φ (ξ,θ)=eiξcos(ϕ π∆)+eiξcos(ϕ+π∆). (24)
go − where r −O→P. The time of flight along the trajectory
≡ | |
This is the superpositionof two plane waveswhose trav- BP is obtained by just replacing ϕ with ϕ. Hence, sub-
−
eling directions are different by the deficit angle 2π∆. stituting (27) into (25), we find that the waveform in
Hence amplificationofthe imagesandinterferenceoccur the geometricaloptics is the same as Eq. (24) except for
for ϕ <π∆ as expected. an overall phase eiroξ. This factor has been already ab-
| |
As we shall explain below, Eq. (22) coincides with the sorbedin the choiceofthe normalizationfactorB in our
one derivedunder the geometricaloptics. In geometrical formula (10).
optics, wave form is given by [11] We define the amplification factor
φ = u(~x )1/2exp[iωT(~x ) iπn ], (25) φ(ξ,θ)
go | j | j − j F(ξ,θ)= , (28)
Xj φUL(ξ,θ)
6
whereφ istheunlensedwaveform. UsingEq.(24),the T(~x), which we call the image points, contribute to the
UL
amplification factor of φ for ϕ <π∆ is given by waveform. The first length scale is r = α( θ)/√ξ r
go s
| | | ± | ×
whichisdefinedastheseparationbetweenanimagepoint
Fgo(ξ,θ) 2e−i2ξ(π∆)2cos(π∆ξϕ), (29) andthestringonthelensplane. Inthispictureweexpect
≈ thatpathswhosepathlengthislongerorshorterthanthe
where we have assumed ϕ and ∆ are small and dropped valueatanimagepointbyaboutonewavelengthwillnot
terms higher than quadratic order. It might be more give a significant contribution because of the phase can-
suggestive to rewrite the above formula into cellation. Namely, only the paths which pass within a
certainradiusfromanimagepointneedto be takeninto
F (ξ,θ) 2cos(π∆ωy), (30) account. Then such a radius will be given by r =√λr,
| go |≈ F
which we call Fresnel radius. Namely, a wave with a fi-
where y =rsinϕ. The distance from a node to the next nite wavelength can be recognized as an extended beam
of when the observer is moved in y-direction is λ/π∆, whose transverse size is given by r . The ratio of these
F
where λ is a wavelength. This oscillation is seen in the two scales gives α( θ):
right panel of Fig. 3. ±
√2πr
s
α( θ) = .
| ± | r
F
C. Quasi-geometrical optics approximation
When r r , i.e., α( θ) 1, the beam width is
s F
≫ ± ≫
In the previous subsection, we have derived the wave- smallerthantheseparation. Inthiscasethebeamimage
form in the limit ξ, α( θ) which corresponds to is not shadowed by the string, and therefore the geo-
the geometrical opti|cs ±app|ro→xim∞ation. Here we expand metrical optics becomes a good approximation. When
the waveform (21) to the lowest order in 1/α( θ). This rs <rF, i.e.,
± ∼
includes the leading order corrections to the geometrical
α( θ)<1, (33)
opticsapproximationduetothefinitewavelengtheffects.
±
∼
For the same reason as we explained in the previous
we cannot see the whole image of the beam, truncated
subsection, we assume that ∆ and ϕ are small. Using
at the location of the string. Then the diffraction ef-
the asymptotic formula for the error function Eq. (A6),
fect becomes important. The ratio of the beam image
theleadingordercorrectionduetothe finitewavelength,
eclipsedby the string determines the phase shift andthe
which we denote as δφ , is obtained as
qgo amplificationof the wavecoming from eachimage. If we
substitute ϕ 0 as a typical value, we obtain a rough
eiξ+iπ/4 1 1 | | ≈
δφ (ξ,θ) = + criterion that the diffraction effect becomes important
qgo
− √2π (cid:18)α(θ) α(−θ)(cid:19) when
eiξ+iπ/4 2π∆
= − √2πξ (π∆)2 ϕ2, (31) λ>2π(π∆)2r, (34)
− ∼
As is expected, the correction blows up for ϕ π∆, or ξ <(π∆)−2 in terms of ξ.
| | ≈
where α(θ) orα( θ) vanishes,irrespectivelyofthe value Th∼esamelogicappliesforausualcompactlensobject.
−
ofξ. Insuchcases,wehavetoevaluatetheerrorfunction In this case the Fresnel radius does not change but the
directly, going back to Eq. (21). typical separation of the image from the lens is given
The expression on the first line in Eq.(31) manifestly by the Einstein radius r √4GMr, where M is the
E
≈
depends only on α( θ) aside from the common phase mass of the lens. Then the ratio between r and r is
factor eiξ. This featu±re remains true even if we consider given by r /r = GM/λ, which leads tEo the uFsual
E F
a small value of α( θ). This can be seen by rewriting criterion that the diffraction effect becomes important
Eq. (21) as ± when λ>GM[1, 2,p3, 4, 5].
From∼the above formula (31), we can read that the
eiξ−2iα2(θ) α(θ) leading order corrections scales like λ/r. This de-
φ(ξ,θ) Erfc − +(θ θ).(32) ∝
≈ √π √2i →− pendencies onλ andr differ fromthe cases that the lens
(cid:18) (cid:19) p
is composed of a normal localized object, in which the
The commonphaseeiξ doesnotaffectthe absolutemag- leading order correction due to the finite wavelength is
nitude of the wave. Except for this unimportant overall (λ/M) [15].
O
phase,the waveformiscompletelydeterminedbyα( θ). The condition for the diffraction effect to be impor-
±
Thegeometricalmeaningoftheseparametersα( θ)is tant (33) can be also derived directly from Eq. (31). In
±
the ratio of two length scales defined on the lens plane. order that the current expansion is a good approxima-
To explain this, let us take the picture that a wave is tion, φ must be smaller than φ . This requires that
qgo go
composed of a superposition of waves which go through 1/α( θ) 1, which is identical to (33).
± ≫
various points on the lens plane. In the geometrical op- We plot the absolute value of the amplification fac-
tics limit the paths passing through stationary points of tor under the quasi-geometrical optics approximation as
7
But the deviation of the amplification from unity out-
side the wedge ϕ > π∆ is rather small except for the
special case ξ(π∆)2 1: for ξ(π∆)2 1 the magnifica-
≈ ≪
tion is inefficient and for ξ(π∆)2 1 the magnification
≫
itself does not occur. Hence the increase of the event
rates of lensing by cosmic strings compared with the es-
timateunderthegeometricalopticsapproximationcould
be important only when the relation ξ(π∆)2 1 is sat-
isfied. If we take D = 1028cm and ω = 10 ≈3Hz which
−
is in the frequency band of LISA(Laser Interferometer
Space Antenna)[48], we findthat the typicalvalue ofGµ
is 2 10 9.
−
≈ ×
FIG. 4: The absolute value of the amplification factor as a So far, we have considered the stringy source rather
functionofξ forθ=0. Blacklineanddashedonecorrespond than a point source. Extension to a point source can be
to Eq. (21) and the quasi-geometrical optics approximation, doneinasimilarmannertothecaseofthestringysource
respectively. The string tension is chosen to be Gµ=10−2.
and is treated in Appendix B. The result is
1 ωrr
dashedline inFig.4. We findthat the quasi-geometrical φ(r,θ,z) eiωD o,θ , (36)
≈−4πD F D
optics approximation is a good approximation for ξ > (cid:16) (cid:17)
∆−2. For ξ <∆−2, the quasi-geometricaloptics approx∼- where D = (r+r )2+z2 is the distance between
o
imationgive∼salargeramplificationfactorthantheexact the source and the observer. , which is defined by
one. Eq. (B14), is rpelated to ψ as F
Inthequasi-geometricalopticsapproximation,wefind
from Eqs. (24) and (31) the absolute value of the ampli-
(x,θ)= e iξψ(ξ,θ) +(θ θ). (37)
fication factor for ϕ=0 is F − ξ x →−
(cid:12) →
(cid:2) (cid:3)(cid:12)
1/2 Hence φ for the point sourc(cid:12)e is similar to that for the
2 ξ π
F(ξ,0) 2 1 cos (π∆)2+ . stringy source. In particular, assuming that ∆,ϕ 1,
| |≈ " −sπξ(π∆2 (cid:18)2 4(cid:19)# and keeping terms which could remain for ωr,ωro ≪ 1,
≫
(35) we have
Fromthisexpression,wefindthatthepositionofthefirst
pwehaikchofcathnebaemaplslioficvaetriiofinedfafcrtoomrlFieisga.t4ξ. ≈Fo4r.ξ25<×∆(π∆2)t−h2e, F ωDrro,θ ≈e−2iωrDro(π∆−ϕ)2
− (cid:16) 1 (cid:17) ϕ π∆ ωrr
presentapproximationisnotvalid,butwekno∼wthatthe Erfc − o +(θ θ). (38)
amplificationfactorshouldconvergetounity inthe limit ×√π (cid:18) √2i r D (cid:19) →−
ξ 0, where r is much larger than r .
F s
→
We show in Fig. 5 the absolute value of the amplifica-
tion factor as a function of ϕ for four cases of ξ around
∆ 2. Top left, top right, bottom left and bottom right D. Simpler derivation of Eq. (32).
−
panels correspond to ξ(π∆)2 = 0.5,1,2 and 4, respec-
tively. BlackcurvesareplotsforEq.(21)andthe dotted We havederivedanapproximatewaveform(32)which
ones are plots for the quasi-geometrical optics approxi- is valid in the wave zone from the exact solution of the
mation. Asisexpected,theerrorofthequasi-geometrical waveequation Eq. (10). Here we show that Eq. (32) can
optics approximation becomes very large near ϕ = π∆, be obtained by a more intuitive and simpler method. In
where α(θ) vanishes. As the value of ξ increases,the an- the path integral formalism [11], the wave form is given
gularregioninwhichthequasi-geometricalopticsbreaks bythe sumofthe amplitude exp(iωT(s))forallpossible
down is reduced. paths which connect the source and the observer. Here
Interestingly, the absolute value of the amplification T(s) is the time of flight along the path s. If the cosmic
factor deviates from unity even for ϕ > π∆ which is string resides between the source and the observer, the
not observed in the geometrical optics l∼imit. This is a wave form will be given by the sum of two terms one
consequence of diffraction of waves, the amplitude of os- of which is obtained by the path integral over the paths
cillation of the interference pattern becomes smaller as whichpassthroughtheuppersideofthestring(y >0)in
θ becomes larger, which is a typical diffraction pattern Fig.2,andtheotherthroughthelowersideofit(y <0).
formed when a wave passes through a single slit. The The waveform coming from the former contribution will
broadeningofthe interferencepatterndue tothe diffrac- be given by
tion effect means that the observers even in the region
|mϕi|c>stπri∆ngc.an detect signatures of the presence of a cos- A ∞ dzQ ∞dyQeiω(|→−AQ|+|→−QP|), (39)
Z−∞ Z0
8
FIG. 5: Black line and dotted one correspond to Eq. (21) and the quasi-geometrical optics approximation, respectively. The
string tension is chosen to be Gµ=10−3.
whereQ=(0,yQ,zQ)isapointonthelensplanespecified with very long wavelengths do not feel the local struc-
byx=0. Onecandeterminethenormalizationconstant ture of string. However, uniform amplification of waves
Abyalittlemoredetailedanalysis,butwedonotpursue shouldoccur asa resultof totalenergyflux conservation
it further here. By integrating Eq. (39), we recover the because the area of the asymptotic region at a constant
first term in Eq. (32). distance from the source is reduced due to the deficit
angle. In this sense such modes feel the existence of a
string.
E. Long wavelength limit
For completeness, we consider the case in which the IV. CONNECTIONS TO OBSERVATIONS
wavelength is longer than the distance from the string
ξ < 1. In this limit, the first few terms in Eq. (10) A. Compact binary as a source
do∼minate, and we find
Inthissection,weconsidercompactbinariesassources
φ(ξ,θ) 1 +ie−∆log2−π∆/2iξ1+∆cosθ. (40) of gravitational waves. Gravitational waves from com-
≈ 1 ∆ Γ(1+∆) pact binaries are clean in the sense that the waves are
−
almost monochromatic: the time scale for the frequency
In particular, for ξ 0 Eq. (40) becomes (1 ∆)−1 to change is much longer than the orbital period of the
→ −
which is larger than unity. This differs from the cases of binaryexceptforthephasejustbeforeplunge. Hencein-
gravitationallensing by a normalcompact object, where terference between two waves coming from both sides of
the amplification becomes unity in the long wavelength the cosmic string could be observed by future detectors.
limit. Thereasonwhytheamplificationdiffersfromunity Sinceeachcompactbinaryhasafinitelifetime,lensing
even in the long wavelength limit is that the space has eventscanbeclassifiedroughlyintotwocases. Ifthedif-
a deficit angle and hence the structure at the spatial in- ferencebetweenthe times offlightalongtwo geodesicsis
finity is different from the usual Euclidean space. Waves largerthanthelifetimeofthebinary,wewillobservetwo
9
independent waves separately at different times. On the whereweassumedthatS(t)isreal. SubstitutingEq.(36)
otherhand, ifthe time delayis shorterthanthe lifetime, to the above expression, we have
what we observe is the superposition of two waves.
Theremaininglifetime ofthebinaryTlife whenthepe- φ(t,~x) 1 ∞dω e iω(t D) ωrro,θ S
riod of the gravitational waves measured by an observer ≈−4πD − − F D ω
is PGW is estimated as Z0 (cid:16) +(cid:17)c.c.(46)
1 (1+η)1/3 P 5/3
T 9.2 10 4 GW P , Eq.(46)isageneralformulawhichappliestoanytime
life ≈ × − (1+z )5/3 η GM GW dependent source. Here we consider the special case in
S (cid:18) (cid:19)
(41) which S(t) takes the form
where η is the mass ratioof the binary(η 1),M is the
massofthemoremassivestarinthebinary≤andz isthe t
source redshift. S S(t)=cos dt′ Ω(t′) , (47)
(cid:18)Z0 (cid:19)
The time delay T is
delay
with
rr
o
T 2 ϕπ∆. (42)
delay
≈ D Ω(t)=ω +ω˙ t, (48)
o o
Takingthetypicalvaluesofparametersasrr /D =1Gpc
o where ω˙ /ω 2 1 and ω > 0 are assumed. This rep-
andϕ=π∆, the conditionTlife Tdelay givesthe upper o o ≪ o
≫ resents a quasi-monochromaticsourcewith its frequency
bound on the mass M,
slowly changing. Then S is
ω
M ≪8×103(1+η3η/5)1/5(cid:18)1π0∆−5(cid:19)−6/5(cid:18)1P0G3sWec(cid:19)8/5M⊙. Sω = √21πω˙ e−i(ω+2ωω˙oo)2+iπ/4+ei(ω−2ωω˙oo)2−iπ/4 . (49)
(43) o (cid:18) (cid:19)
The time scale for the orbital frequency of the binary
Substituting Eqs. (38) and (49) into Eq. (46), and using
to change is the same order as T . Hence the condi-
life the method of the steepest descent, we have
tion T T implies that the frequencies of two
life delay
≫
waves are almost the same. The left and right panels
1 ϕ π∆ Ω(T(ϕ))rr
in Fig. 6 which correspond to different frequencies of φ(t,~x) Erfc − o
gravitationalwaves show the region where the condition ≈ 4π√πD √2i r D !
Eq. (43) is satisfied for three different values of string e−iT(ϕ)(ωo+21ω˙oT(ϕ))+c.c.
parameter ∆. The shaded area represent the parameter × +(ϕ ϕ), (50)
regionbeyondthe detector’ssensitivities. Inthe left and →−
right panels we assumed, respectively, that the thresh- with
old value for detection in strain amplitude for LISA and
rr
DECIGO(DECihertz Interferometer Gravitational wave T(ϕ)=t D+ o(π∆ ϕ)2. (51)
− 2D −
Observatory)[49]/BBO(Big Bang Observer)[50], which
are given by 10−20Hz−1/2 and 10−23Hz−1/2. We find Thisrepresentsasuperpositionoftwowavescomingfrom
that both cases Tlife Tdelay and Tlife Tdelay can oc- both sides of the string whose arrival times differ by
≫ ≪
cur both for LISA and BBO/DECIGO. T(ϕ) T( ϕ) = 2rroπ∆ϕ. In the preceding subsec-
| − − | D | |
tions,westudythewaveformsobservedintwocaseswith
T T and T T .
life delay life delay
B. Waveform ≫ ≪
We can easily extend our waveform (10) to the case 1. T ≫T
life delay
that the frequency of the source changes in time. Let us
write the source as 1 S(t)δ(r r )δ(θ π)δ(z). The
1 ∆ − o − As we have explained in the preceding subsection,
Fourier transformatio−n of S(t) is defined by
what we observe is a superposition of two waves in this
case. Becausetherelativephasedifferenceofthesewaves
S(t)= ∞ dω e iωtS . (44) slowlyincreasesordecreasesintimeduetothefrequency
− ω
Z−∞ change of the binary source and the optical path differ-
ence between two geodesics, we will observe the beat if
Denotingthesolutionφ(t,~x)foramonochromaticsource
the amplitude of the integrated relative phase difference
obtainedintheprevioussectionsbyφ (~x),φcanbewrit-
ω over observation time is larger than (1).
ten as O
Theconditionthatthebeatisobservedcanbederived
φ(t,~x)= ∞dω e−iωtSωφω(~x)+c.c., (45) aTs fo,lltohwesn. fIrfowmeEdeqn.o(t5e0)thtehteotinatleogbrsaetrevdatrieolnatpiveeriopdhabsye
Z0 obs
10
FIG.6: Plots ofregions whereEq.(43)issatisfied forthreedifferentvaluesof thestringparameter. Left andright panelsare
for 10−3Hz and 0.1Hz which are the frequency bands LISA and DECIGO have best sensitivities. Shaded regions are plotted
undertheassumptions that thesignals satisfy SN >10, theredshift of thesource is 1 and 3 year observations.
difference is 2π∆ϕDω˙ T , where both r and r are as-
o obs o
sumed to be O(D). Hence we can observe the beat if
1
T > . (52)
obs
2π∆ϕDω˙
∼ o
Because T is roughly the same as the time scale for
life
the frequency of the binary to change,i.e. T ω /ω˙ ,
life o o
∼
Eq. (52) can be written as
T
T > life . (53)
obs
2π∆ϕDω
∼ o
If T is fixed, e.g. T 3yr for LISA, Eq. (53) is FIG. 7: Plot of the region where Eq. (54) is satisfied. The
obs obs ∼ frequencyofthegravitationalwavesisassumedtobe10−3Hz.
written as an lower bound on M. For T = 3yr and
obs
P =103sec, Eq. (53) becomes
GW
We show in fig. 8 the amplification of the wave corre-
M >2.6 (1+η)1/5 π∆ −6/5 PGW 13/5M . sponding to the first term in Eq. (50 ) as a function of
∼ × η3/5 (cid:18)10−5(cid:19) (cid:18)103sec(cid:19) ⊙ ϕ ∆π normalizedby1/ ωrro/D,whichisnothingbut
(54) −
α(θ) in the case discussed in Sec.III. We find that the
WeshowinFig.7theregionwhereEq.(54)issatisfiedfor − p
amplificationapproacheszeromoreslowlyforϕ π∆>0
LISA with T = 3yr. We find that if Gµ < 2.8 10 8 −
whichisabouobtsoneorderofmagnitudebelow∼thec×urre−nt andoscillatesaroundunityforϕ−π∆<0andtheangu-
larsizeinwhichnon-trivialoscillationsduetothediffrac-
upper bound, LISA will detect the beat of gravitational
tioneffectcanbeobservedisgivenby1/ ωrr /D. Since
o
wavesforallobservablerangesin(µ,M)spaceaslongas T (rr /D)ϕπ∆<T ω 1 in the present case,
Tlife ≫Tdely 2. wdeelhayav≈e (rroo/D)(π∆)2 >li(frer≪o/D−)π∆ϕp 1. Therefore
this angular size of oscil∼lation is much sm≫aller than π∆.
Hence itwillbe verydifficult to detecta lensing eventin
2. T ≪T
life delay which this diffraction effect is relevant.
IfT T ,weobservethe waveformofeitherthe
delay life
≫
first term or the second one in Eq. (50) at a given time. C. Estimation of the event rate
In this section, we estimate the detection rate of the
gravitationallensingcausedbycosmicstringsforplanned
2 Sincethelensingprobabilityisnotexpectedtobehigh,weneed gravitationalwavedetectorssuchasLISA,DECIGOand
a large number of events to detect a lensing event. In such a
BBO.
situation, what gravitational wave detectors can detect is a su-
It is well known that string network obeys the scal-
perposition of various waves. Hence, signal will almost always
havebeatevenifweignorethelensingeffect. ing solution where the appearance of the string network