Detection of cosmic superstrings by geodesic test particle motion

1.1 MB·English

by  Betti Hartmann

Detection of cosmic superstrings by geodesic test particle motion
Betti Hartmann (a),∗ Claus La¨mmerzahl (b),(c),† and Parinya Sirimachan (a)‡
(a) School of Engineering and Science, Jacobs University Bremen, 28759 Bremen, Germany
(b) ZARM, Universita¨t Bremen, Am Fallturm, 28359 Bremen, Germany
(c) Institut fu¨r Physik, Universita¨t Oldenburg, 26111 Oldenburg, Germany
(Dated: January 28, 2011)
(p,q)-strings are bound states of p F-strings and q D-strings and are predicted to form at the
end of brane inflation. As such these cosmic superstrings should be detectable in the universe. In
this paper we argue that they can be detected by the way that massive and massless test particles
moveinthespace-timeofthesecosmicsuperstrings,inparticularwestudysolutionstothegeodesic
equationinthespace-timeoffieldtheoretical(p,q)-strings. Thegeodesicscanbeclassifiedaccording
tothetest particle’s energy,angular momentumand momentum in thedirection ofthestring axis.
1 We discuss how the change of the magnetic fluxes, the ratio between the symmetry breaking scale
1 and the Planck mass, the Higgs to gauge boson mass ratios and the binding between the F- and
0 D-strings, respectively, influence the motion of the test particles. While massless test particles can
2 only move on escape orbits, a new feature as compared to the infinitely thin string limit is the
existence of bound orbits for massive test particles. In particular, we observe that - in contrast to
n
the space-time of a single Abelian-Higgs string - bound orbits for massive test particles in (p,q)-
a
string space-times are possible if the Higgs boson mass is larger than the gauge boson mass. We
J
also compute the effect of the binding between the p- and the q-string on observables such as the
7
light deflection and the perihelion shift. While light deflection can also be caused by other matter
2
distributions,thepossibilityofanegativeperihelionshiftseemstobeafeatureoffinitewidthcosmic

strings that could lead to the unmistakable identification of such objects. In Melvin space-times,
]
h which are asymptotically non-conical, massive test particles have to move on bound orbits, while
t massless test particles can only escape to infinityif their angular momentum vanishes.
-
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e PACSnumbers: 11.27.+d,98.80.Cq,04.40.Nr
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∗ [email protected]
† [email protected]
‡ [email protected]
2
I. INTRODUCTION
Cosmic strings are topological defects that are predicted to have formed via the Kibble mechanism [1] during one
of the phase transitions in the early universe and in the field theoretical description [2] can be considered to be an
example of a topological soliton. Due to the fact that these objects can be extremely heavy (up to 1012 kg/m) they
werebelievedtobe apossiblesourceofthe densityperturbationsthatledtostructureformationandthe anisotropies
in the cosmic microwave background (CMB) [3]. However, the detailed measurement of the CMB power spectrum
as obtained by COBE, BOOMERanG and WMAP showed that cosmic strings cannot be the main source for these
anisotropies.
In recent years cosmic strings gained renewed interest due to the possible connection to the fundamental entities
of String Theory [4]. Brane inflation is a popular inflationary model that can be embedded into String Theory and
predicts the formation of cosmic string networks at the end of inflation [5]. E.g. in the framework of type IIB
String Theory the inflaton field corresponds to the distance between two Dirichlet branes with 3 spatial dimensions
(D3-branes) and inflation ends when these two branes collide and annihilate. The production of strings (and lower
dimensional branes) then results from the collision of these two branes. Each of the original D3-branes has a U(1)
gauge symmetry that gets broken when the branes annihilate. If the gauge combination is Higgsed, magnetic flux
tubes of this gauge field carrying Ramond-Ramond (R-R) charge are D-branes with one spatial dimension, so-called
D-strings. When the gauge combination is confined the field is condensated into electric flux tubes carrying Neveu
Schwarz-NeveuSchwarz (NS-NS) charges and these objects are fundamental strings (F-strings) [6]. D-strings and F-
strings are so-calledcosmic superstrings [4] which seem to be a generic prediction of supersymmetric hybrid inflation
[7] and grand unified based inflationary models [8]. D- and F-strings, however, have different properties than the
usual (solitonic)cosmic strings. The probability of intercommutationof solitonic strings is equal to one but less than
one in the case of cosmic superstring. Therefore, solitonic strings do not merge, while cosmic superstrings tend to
form bound states. When p F-strings and q D-strings interact, they can merge and form bound states, so-called
(p,q)-strings [9] whose properties have been investigated [10]. Even though the origin of (p,q)-strings is type IIB
string theory, their properties can be investigated in the framework of field theoretical models [11–14]. The influence
of gravity on field theoretical (p,q)-strings has been studied in [15].
Since there are good reasons for cosmic superstrings to be a consequence of String Theory it is very exciting to
search for observational consequences of their existence. There has been considerable effort in numerically modeling
cosmicstringnetworkstoobtainCMB powerandpolarizationspectra [16]. Comparisonwithobservationshasshown
that cosmic strings might well contribute considerablyto the energy density of the universe. There is another way to
detect cosmic strings in the universe, namely through the motion of test bodies in such string space-times. The test
particlemotionindifferentspace-timescontainingcosmicstringshasbeeninvestigatedin[17–21],while the complete
set of orbits of test particles in the space-time of black hole piercedby an infinitely thin cosmic string has been given
for a Schwarzschild black hole in [22] and for a Kerr black hole in [23].
In this paper we follow the latter approach and use the field theoretical model discussed in [15] to describe (p,q)-
strings by two coupled Abelian-Higgs models in curved space-time. For vanishing coupling between the two sectors,
the modelcorrespondsto the Abelian-Higgsmodelcoupledminimally to gravity. This modelhassolutionsdescribing
strings with finite core width that have been investigated in [24, 25]. Geodesics in this space-time have been studied
recently[26]andcanonlybegivennumerically. Herewewouldliketoextendthisinvestigationtothefieldtheoretical
description of cosmic superstrings.
Our paper is organized as follows: in Section II, we discuss the field theoretical model that possesses (p,q)-string
solutionsandwealsoworkoutthegeodesicequation. InSectionIII wediscussournumericalresults,inparticularwe
give examples of orbits and demonstrate how the ratio between the symmetry breaking scale and the Planck mass,
the ratios between Higgs and gauge boson masses, the magnetic fluxes and the binding between the F- and D-string
influence our results. We conclude in Section IV.
II. THE MODEL
A. The space-time of a (p,q)-string
The field theoretical model to describe gravitating (p,q)-strings reads [15]
1
S = d4x√ g R+ , (1)
m
Z − (cid:18)16πG L (cid:19)
3
where R is the Ricci scalar and G is Newton’s constant. The matter Lagrangian is given by [11]
m
L
1 1
=D φ(Dµφ)∗ F Fµν +D ξ(Dµξ)∗ H Hµν u(φ,ξ) (2)
m µ µν µ µν
L − 4 − 4 −
withthecovariantderivativesD φ= φ-ie A φ,D ξ= ξ-ie B ξofthetwocomplexscalarfields(Higgsfields)
µ µ 1 µ µ µ 2 µ
∇ ∇
φandξ andthe fieldstrengthtensorsF = A A =∂ A ∂ A , H = B B =∂ B ∂ B of
µν µ ν ν µ µ ν ν µ µν µ ν ν µ µ ν ν µ
∇ −∇ − ∇ −∇ −
twoU(1)gaugepotentialA ,B withcouplingconstantse ande . denotesthegravitationalcovariantderivative.
µ ν 1 2 µ
∇
Finally, the potential V(φ,ξ) reads:
λ λ
u(φ,ξ)= 1(φφ∗ η2)2+ 2(ξξ∗ η2)2 λ (φφ∗ η2)(ξξ∗ η2) , (3)
4 − 1 4 − 2 − 3 − 1 − 2
where λ and λ are the self-couplings of the two scalar fields, while λ >0 is the coupling between the two sectors.
1 2 3
η and η are the vacuum expectation values of the scalar fields.
1 2
In order for both U(1) symmetries to spontaneously break which then leads to the formation of strings we have to
require that the (absolute) minimum of the potential (3) is at non-vanishing values of φ and ξ . This leads to the
| | | |
requirement [11]
λ λ >4λ2 . (4)
1 2 3
The most general static cylindrically symmetric line element invariant under boosts along the z-direction is
ds2 =N(ρ)2dt2 dρ2 L(ρ)2dϕ2 N(ρ)2dz2 . (5)
− − −
For the matter and gauge fields, we apply the Ansatz [2]
1
φ(ρ,ϕ)=η h(ρ)einϕ , A dxµ = (n P(ρ))dϕ (6)
1 µ
e −
1
1
ξ(ρ,ϕ)=η f(ρ)eimϕ , B dxµ = (m R(ρ))dϕ , (7)
2 µ
e −
2
where n and m are integers indexing the vorticity of the two Higgs fields around the z-axis and correspond to the
degree of the map from S1 S1, where the homotopy group is π (S1) = Z. In our field theoretical model of
1
→
(p,q)-strings the p corresponds to the winding n and the q to the winding m.
We can then do the following rescaling
ρ L
ρ , L (8)
→ e η → e η
1 1 1 1
such that the total Lagrangianonly depends on the following dimensionless coupling constants
e η λ
γ =8πGη2 , g = 2 , q = 2 , β = i , (9)
1 e η i e2
1 1 1
where i = 1,2,3. γ is proportional to the ratio between the Planck mass M = G−1/2 and the symmetry breaking
Pl
scale η . Moreover, √β is proportional to the ratio between the Higgs mass M = √λ η and the corresponding
1 1 H,1 1 1
gauge boson mass M =√2e η , while √β /g is proportional to the ratio between the Higgs mass M =√λ η
W,1 1 1 2 H,2 2 2
and the corresponding gauge boson mass M = √2e η . Each of the strings possesses a scalar core with width
W,2 2 2
ρ M−1 and a gauge field core with width ρ M−1, i = 1,2. Note that with the rescaling (8) the width of
H,i ∼ H,i W,i ∼ W,i
the gauge field cores is ρ 1/√2, ρ 1/(gq√2) while the widths of the scalar cores is given by ρ =1/√β ,
W,1 W,2 H,i i
∼ ∼
i=1,2.
The variation of the action (1) with respect to the matter fields leads to the following equations [15]
(N2Lh′)′ P2h 1∂u
= + , (10)
N2L L2 2∂h
(N2Lf′)′ R2f 1∂u
= + , (11)
N2L L2 2∂f
L N2P′ ′
=2h2P , (12)
N2 (cid:18) L (cid:19)
L N2R′ ′
=2g2f2R , (13)
N2 (cid:18) L (cid:19)
4
where the prime denotes the derivative with respect to ρ and the potential u reads
β β
u(h,f)= 1(h2 1)2+ 2(f2 q2)2 β (h2 1)(f2 q2) . (14)
3
4 − 4 − − − −
The variation of (1) with respect to the metric leads to the Einstein equations
1
R = γ T g T , (15)
µν µν µν
− (cid:18) − 2 (cid:19)
where T is the trace of the energy momentum tensor. Using our Ansatz these read [15]
(LNN′)′ (P′)2 (R′)2
=γ + u , (16)
N2L (cid:20) 2L2 2g2L2 − (cid:21)
(N2L′)′ 2h2P2 2R2f2 (P′)2 (R′)2
= γ + + + +u . (17)
N2L − (cid:20) L2 L2 2L2 2g2L2 (cid:21)
In addition there is a constraint equation that is not independent. This reads
N′L′ (N′)2 (P′)2 (R′)2 h2P2 R2f2
2 + =γ (h′)2+(f′)2+ + u . (18)
NL N2 (cid:20) 2L2 2g2L2 − L2 − L2 − (cid:21)
Thesetofdifferentialequationscanbesolvedonlynumericallysubjecttoanappropriatesetofboundaryconditions.
The requirement of regularity at ρ=0 leads to the following conditions
h(0)=0, f(0)=0, P(0)=n, R(0)=m (19)
for the matter fields and
N(0)=1, N′(0)=0, L(0)=0, L′(0)=1 (20)
for the metric fields, while the requirement of finiteness of the energy per unit length leads to
h( )=1, f( )=q , P( )=0 , R( )=0 . (21)
∞ ∞ ∞ ∞
The inertial energy per unit length E(n,m) of the (p,q)-string is given by
in
E(n,m) = √ g T0dρdϕ (22)
in Z − 3 0
∞ (P′)2 (R′)2 h2P2 R2f2
=2π NL (h′)2+(f′)2+ + + + +u dρ , (23)
Z (cid:18) 2L2 2g2L2 L2 L2 (cid:19)
0
where g is the determinant of the 2+1-dimensional space-time given by (t,ρ,ϕ). Note that there is also another
3
notionofenergyinthis space-time,namely thatofthe Tolmanenergy[24,25]. Thisdefines the gravitationallyactive
mass.
In the Bogomolnyi-Prasad-Sommerfield (BPS) limit [27] given by β = β β = 2, β = 0 and with the choice
1 2 3
q =g =1 we havethat Tρ =Tϕ =0 suchthat it followsfrom (16) that N(ρ) ≡1. The remaining BPSequations are
ρ ϕ ≡
Ph Rf
h′ = , f′ = , (24)
L L
P′ R′
=h2 1 , =f2 1 , (25)
L − L −
for the matter fields and
L′′ 2h2P2 2R2f2
= γ + +(h2 1)2+(f2 1)2 (26)
L − (cid:20) L2 L2 − − (cid:21)
for the non-trivial metric function. The solutions fulfill an energy bound such that
E(n,m) =2π(n+m) . (27)
in
Note that in this limit the widths of the scalar cores ρ become equal to the widths of the respective gauge field
H,i
cores ρ , i=1,2.
W,i
5
The binding energy per unit length of a (p,q)-string E(n,m) can be defined as
b
E(n,m) =E(n,m) nE(n,0) mE(0,m) . (28)
b in − in − in
Finally the (p,q)-string possesses magnetic fields in z-direction B~ =B ~e and B~ =B ~e with
1 1 z 2 2 z
P′ R′
B = , B = , (29)
1 2
− L − L
where B and B are given in units of M2 . The magnetic fluxes then read
1 2 W,1
Φ =2πn , Φ =2πm (30)
M,1 M,2
and are obviously quantized. Hence, changing the winding numbers n and m changes the magnetic fluxes along the
(p,q)-string.
B. The geodesic equation
The Lagrangian describing geodesic motion of a test particle in the static cylindrically symmetric space-time
g
L
(5) reads
dxµdxν dt 2 dρ 2 dϕ 2 dz 2
=g =ε=N2 L2 N2 , (31)
g µν
L dτ dτ (cid:18)dτ(cid:19) −(cid:18)dτ(cid:19) − (cid:18)dτ(cid:19) − (cid:18)dτ(cid:19)
where ε=0,1for masslessor massivetestparticles,respectively andτ is anaffine parameterthatcorrespondsto the
proper time for massive test particles moving on time-like geodesics. The space-time has three Killing vectors ∂ , ∂
∂t ∂ϕ
and ∂ which lead to the following constants of motion: the energy E, the angular momentum L along the string
∂z z
axis (z-axis) and the momentum p
z
dt dϕ dz
N2 =:E , L2 =:L , N2 =:p . (32)
z z
dτ dτ dτ
Using the rescaling (8) the constants of motion must be rescaled according to E E/(e η ), p p /(e η ), L
1 1 z z 1 1 z
L /(e η )2. We then find from (31) → → →
z 1 1
dt 2 dρ 2 dϕ 2 dz 2 E2 p2 dρ 2 L2
ε=N2 L2 N2 = − z z . (33)
(cid:18)dτ(cid:19) −(cid:18)dτ(cid:19) − (cid:18)dτ(cid:19) − (cid:18)dτ(cid:19) N2 −(cid:18)dτ(cid:19) − L2
Using the constants of motion we find from (31)
1 dρ 2 E2 ε 1 1 p2 L2
= − E2 1 + z + z . (34)
2(cid:18)dτ(cid:19) 2 − 2(cid:20) (cid:18) − N2(cid:19) N2 L2(cid:21)
The left hand side of (34) is always positive and E2 ε is a constant of motion. Following [28] we can then rewrite
−
this equation as
2
1 dρ
= V (ρ) , (35)
eff
2(cid:18)dτ(cid:19) E −
where
1 1 p2 L2
V (ρ)= E2 1 + z + z (36)
eff 2(cid:20) (cid:18) − N2(cid:19) N2 L2(cid:21)
is the effective potential and = (E2 ε)/2. Note that with this definition the effective potential becomes explicitly
E −
energy-dependent.
6
In the following, we would like to find t(ρ), ϕ(ρ) and z(ρ). For this, we rewrite the geodesic equation in the form
L dρ
z
dϕ= , (37)
±L(ρ)2 E2−p2z L2z ε 1/2
(cid:16)N(ρ)2 − L(ρ)2 − (cid:17)
p dρ
z
dz = , (38)
±N(ρ)2 E2−p2z L2z ε 1/2
(cid:16)N(ρ)2 − L(ρ)2 − (cid:17)
Edρ
dt= . (39)
±N(ρ)2 E2−p2z L2z ε 1/2
(cid:16)N(ρ)2 − L(ρ)2 − (cid:17)
The solutionforeachcomponentcanthen be calculatedasa functionofρ by using numericalintegrationmethods.
III. NUMERICAL RESULTS
We have solved the set of differential equations (10) - (17) numerically using the ODE solver COLSYS that uses a
Newton-Raphsonadaptivegridmethod[29]. The relativeerrorofthe solutionsisonthe orderof10−13 -10−10. Each
component of the geodesic equation can then be integratednumerically by using the integrating function quad,i.e. a
recursive adaptive Simpson quadrature in MATLAB with an absolute error tolerance 10−8. However the numerical
profiles of the metric functions N(ρ) and L(ρ) must first be interpolated. This was done using a piecewise cubic
Hermiteinterpolatingpolynomial,i.e. withpchipinMATLAB.Withthisprocedureitispossibletoobtainasmooth
curve for the effective potential.
In the following we will distinguish between bound orbits, escape orbits and terminating orbits. Note that when
we talk about bound, escape and terminating orbits we are referring to the motion in the x–y–plane. The particles
can, of course, move along the full z–axis from to + for p =0.
z
−∞ ∞ 6
Bound orbits are orbits on which test particles move from a minimal value of ρ, ρ >0 to a maximal value of ρ,
min
ρ < and back again. These orbits have hence two turning points with (dρ/dτ)2 =0. On escape orbits, on the
max
∞
other hand, particles come from ρ= , reach a minimal value of ρ, ρ >0 and move back to ρ= , which means
min
that escape orbits have only one turn∞ing point with (dρ/dτ)2 = 0. Looking at (35) it is obvious tha∞t turning points
arelocatedatthoseρatwhich V (ρ)=0. Finally,terminatingorbitsareorbitsthatendatthestringaxisρ=0.
eff
E−
For all our calculations we have chosen q =1 and g =1.
A. Generalities
Solutions to the model (1) have been extensively studied previously. The Table I summarizes the particular cases.
Solution β1, β2 β3 γ Studiedin
Abelian-Higgs string in flat space-time β1 =β2 6=0 β3 =0 γ =0 [2]
Abelian-Higgs string in curved space-time β1 =β2 6=0 β3 =0 γ 6=0 [24], [25]
(p,q)-stringin flat space-time β1 =β2 =2 β3 6=0 γ =0 [11]
(p,q)-stringin curvedspace-time β1 =β2 =2 β3 6=0 γ 6=0 [15]
TABLE I.Known string solutions of the model (1).
It has been observedin [24] that there aretwo types of solutions if one couples the Abelian-Higgs model minimally
to gravity: string solutions and Melvin solutions which exist for the same values of the parameters in the model.
These differ by their asymptotic behaviour of the metric functions at infinity.
1. String solutions
The string solution behaves like
N(ρ )=c , L(ρ )=c ρ+c , c >0 , (40)
1 2 3 2
→∞ →∞
7
where c , c and c are constants depending on n, m, g, γ and β , i = 1,2,3. For β = 0 it has been found [24, 25]
1 2 3 i 3
that c >1 for β =β β <2, c <1 for β =β β >2 and c =1 in the BPS limit β =β β =2.
1 1 2 1 1 2 1 1 2
≡ ≡ ≡
A solution with the asymptotics (40) describes a conical space-time with deficit angle δ given by
δ =2π(1 c ) (41)
2
−
In linear order the deficit angle δ is given by the product of the coupling γ and the inertial energy per unit length
E(n+m) with δ γE(n+m). As such the constant c = 1 for γ = 0 (or E(n+m) = 0) and c decreases for either γ
in ∼ in 2 in 2
or E(n+m) increasing. If the coupling γ or the energy per unit length is too large then c < 0 and the deficit angle
in 2
δ > 2π. In this case the solution would have a singularity at a finite, parameter-dependent value of ρ = ρ with
0
L(ρ = ρ ) = 0, while N(ρ ) stays finite. These solutions are the so-called super-massive string solutions [30] (or
0 0
invertedstringsolutions). Wewillnotconsiderthesekindofsolutionsinthispaperandwillalwaysassumethedeficit
angle to be smaller than 2π.
The “force” exerted on a test particle corresponds to the right hand side of
d2ρ E2 p2 L2
= − z N′+ z L′ . (42)
dτ2 −(cid:18) N3 (cid:19) (cid:18)L3(cid:19)
Note that for string solutions the effective potential tends asymptotically to a constant with V (ρ )
eff
E2(c21−1)+p2z and hence there is no force exerted on test particles far from the string. While the force→asso∞ciat→ed
2c21
to the angular momentum L is always repulsive, the total force close to the string can either be attractive or
z
repulsive. Since E2 p2 0 and N >0 this depends on the sign of N′ (see more details below).
− z ≥
For ρ 1 the string solutions behave like
≪
N(ρ 1) 1+O(ρ2) , L(ρ 1) ρ . (43)
≪ ∼ ≪ ∼
Hence there is an infinite potential barrier at ρ =0 for test particles with non-vanishing angular momentum L , i.e.
z
these test particles can never reach the string axis at ρ = 0 since their is no force to counterbalance the repulsive
centrifugal force. On the other hand, for L = 0 the effective potential tends to a constant V (ρ 0) p2/2.
Hence particles with E2 ε<p2 can reach tzhe string axis. Since E2 >p2 these terminating orbietffs are→only→posszible
− z z
for massive test particles with ε=1.
Infinitely thin cosmic strings The infinitely thin limit corresponds to the case where both the width of the scalar
core as well as that of the gauge field core tend to zero. The string is hence a 1-dimensional object that can e.g.
be described by the Nambu-Goto action. In this case the metric function N(ρ) 1 (or some other constant that
≡
can be absorbed into the definition of t) and L(ρ) c ρ for ρ > 0. In this case, the only component in the force
2
≡
(42) exerted on a particle is the repulsive angular momentum contribution. Hence, bound orbits are not possible
in this space-time. This can also easily be understood when noting that the space-time of an infinitely thin cosmic
string is locally flat [3] and geodesics are just straight lines. The fact that bound orbits are possible in a finite width
cosmicstringspace-timeisrelatedtothefactthatclosetothestringaxistheconicalspace-timeissmoothedonscales
comparabletothe widthofthestring. Theexistenceofboundorbitsin“pure”cosmicstringspace-times[31]ishence
a new feature when considering cosmic strings with finite width.
2. Melvin solutions
The Melvin solutions exist for the same parameter values as the string solutions, but have a different asymptotic
behaviour:
N(ρ ) a ρ2/3 , L(ρ ) a ρ−1/3 , (44)
1 2
→∞ → →∞ →
whereagaina anda areparameterdependentpositiveconstants. Thisspace-timeisnotasymptoticallyflatandthe
1 2
proper length of a curve with t=const., ρ=const., z =const and ϕ=0 2π is s=2πa ρ−1/3. This tends to zero
2
→
for ρ . For the Melvin space-time with the asymptotic behaviour (44) the effective potential tends to infinity
asymp→tot∞ically with V (ρ ) ρ2/3 for L = 0. Hence there is an infinite potential barrier at infinity for test
eff z
→ ∞ ∝ 6
particles with non-vanishing angular momentum, i.e. these particles can never reach infinity. This is related to the
fact that the total force (42) on a test particle is always attractive at large ρ in Melvin space-times. For L =0, the
z
effective potential tends to E2/2 for ρ . Hence, the asymptotic value of the effective potential is always larger
→ ∞
than (for massive test particles) or equal to (for massless test particles) . Massive test particles moving on radial
E
geodesics can thus not reach infinity, while massless test particles have a turning point at infinity.
For ρ 1 the Melvin solutions behave like the string solutions (43).
≪
8
B. Geodesic motion in (p,q)-string space-times: string solutions
We will mainly discuss the geodesic motion in space-times with the asymptotic behaviour (40) since we believe
this to be the physically relevant case. However, since the Melvin solution is a solution to the Abelian-Higgs model
coupled minimally to gravity,we will also comment on this below.
1. The effective potential
The case β = 0, β = β β has been discussed for n = m = 1 in [26]. It was found that bound orbits are only
3 1 2
≡
possible for β < 2 and for massive particles. In fact, in order to have bound orbits we need (at least) two turning
pointsofthemotion,i.etwointersectionpointsbetweenV and . Notethatfor finiteandlargerthantheminimal
eff
E E
valueoftheeffectivepotentialwewillalwayshaveoneintersectionpointforL =0duetotheinfinitepotentialbarrier
z
6
at small ρ such that escape orbits always exist. However, bound orbits are only possible if in addition the effective
potential has local minima and maxima with dVeff =0. At these local extrema we should then have
dρ
E2 p2 N(ρ)3 L′(ρ)
− z = . (45)
L2 N′(ρ) L(ρ)3
z
Since E2 p2 > 0, N(ρ) > 0, L(ρ) > 0, L′(ρ) > 0 this equation has only solutions for N′(ρ) > 0. For β = 0 it
− z 3
has been observed[26] that the metric function N(ρ) is either monotonically decreasing(for β >2) or monotonically
increasing (for β < 2), while N(ρ) 1 in the BPS limit β = 2. Hence the sign of N′(ρ) doesn’t change and in
≡
particular,boundorbitsareonlypossibleforβ <2. Inthiscasetheenergy-momentumpartoftheforce(42)becomes
attractiveforβ <2,i.e. ifthewidthofthescalarcoreislargerthanthe widthofthegaugefieldcoreandcanbalance
the repulsive partassociatedto the angular momentum. Onthe other hand for β =2 (β >2) the width of the scalar
core is equal (smaller) than the width of the gauge field core. We observe that this leads to a vanishing (repulsive)
energy-momentum part in the force (42) and only escape orbits are possible.
This is different when β > 0. We will first discuss the case n = m = 1. The behaviour of the metric function
3
N(ρ) of a (1,1)-string for γ = 0.3 and different choices of β , i = 1,2,3 is shown in Fig. 1. In all cases the blue
i
dotted-dashedline correspondstoβ =0andforcases(a),(b)and(c)the greensolidlinecorrespondstoβ β(max)
3 3 ≈ 3
with β(max) √β β /2 the maximally allowed value for a given choice of β and β . For (a) β = 1, β = 2, (b)
3 ≡ 1 2 1 2 1 2
β =β =2and(c)β =β =1the increaseofβ leadsto anincreaseofthe asymptoticvalue ofN(ρ) forallchoices
1 2 1 2 3
of the β , i = 1,2,3. Hence, the increased binding between the p- and the q-string pronounces the effect already
i
observed in the β =0 limit. Note that while bound orbits are not possible in the BPS limit β =β =2 for β =0
3 1 2 3
bound orbits do exist for β > 0 and β = β = 2 (which, of course, no longer corresponds to a BPS limit). For (d)
3 1 2
β = 1.5,β = 6, (e) β = 2,β = 4.5 and (f) β = 2.25,β = 4 the metric function N(ρ) can have a local minimum
1 2 1 2 1 2
if β < β(cr)(β ,β ), i.e. if the binding between the strings is not too large. This is new as compared to the β = 0
3 3 1 2 3
limit. We find that β(cr)(1.5,6) 0.7, β(cr)(2,4.5) 0.66, β(cr)(2.25,4) 0.51.
3 ≈ 3 ≈ 3 ≈
Obviously, for (a) β < 2, β = 2, (b) β = β = 2 and (c) β < 2, β < 2 the metric function N(ρ) increases
1 2 1 2 1 2
monotonically,whilefortheothercasesN(ρ)canfirstdecreasefromN(ρ=0)=1,havealocalminimumatρ=ρ
min
with N(ρ ) < 1 and then increase again to N(ρ 1) > 1. This has important consequences for the shape of the
min
≫
effectivepotentialasdiscussedbelowandcanbeunderstoodasfollows: considerthe(1,1)-stringtobeasuperposition
of a (1,0)-string and a (0,1)-string. Now, for β =0, these two strings do not interact. In this case, we know that for
3
β >2, i=1,2the metric function N(ρ) wouldmonotonically decrease,while for β <2, j =1,2 the metric function
i j
N(ρ) would monotonically increase. Superposing a string with β > 2 and one with β < 2 leads than to a metric
i j
functionN(ρ)thatfirstdecreasesandthanincreaseagain. Notethattheoppositeisnotpossiblesincethe scalarcore
of a string with β >2 is smaller than that of a string with β <2.
i j
Thisbehaviourofthe metricfunctionN(ρ)leadstothe observationthatthe effectivepotentialcanhaveanegative
minimum which for p = 0 is located exactly at ρ . In the case β = 0, the effective potential can have a local
z min 3
minimum for β <2, but this will always be positive valued since N′(ρ)>0 means N(ρ)>1. This is shown in Fig.2
for a particle with E = 0.01, L = 0.03 and p = 0. For cases (a), (b) and (c) the effective potential is positive for
z z
all values of ρ for non-vanishing L or p , while it can become negative for the other cases. In fact, the potential
z z
becomes positive everywhere for β β(cr). This will have influence on the existence of bound orbits as discussed
3 ≈ 3
below. In particular if the potential has a negative valued minimum as is e.g. the case for β = 1.5, β = 6 and
1 2
β =0.7, particles with <0, i.e. E2 <1 can move on bound orbits.
3
E
We have also investigated how the metric function N(ρ) changes when changing the winding numbers n, m and
hence the magnetic fluxes along the string. Our results are shown in Fig.3 for γ =0.3.
9
β = 1.00, β = 2.00, γ = 0.30 β = 2.00, β = 2.00, γ = 0.30 β = 1.00, β = 1.00, γ = 0.30
1 2 1 2 1 2
1.1 1.1 1.1
β3 = 0.00 β3 = 0.00 β3 = 0.00
1.08 β3 = 0.35 1.08 β3 = 0.50 1.08 β3 = 0.25
β3 = 0.70 β3 = 0.95 β3 = 0.49
1.06 1.06 1.06
N
N N
1.04 1.04 1.04
1.02 1.02 1.02
1 1 1
0.98 0.98 0.98
0 5 10 15 0 5 10 15 0 5 10 15
ρ ρ ρ
(a) β1=1,β2=2 (b) β1=β2=2 (c) β1=1,β2=1
β = 1.50, β = 6.00, γ = 0.30 β = 2.00, β = 4.50, γ = 0.30 β = 2.25, β = 4.00, γ = 0.30
1 2 1 2 1 2
1.1 1.1 1.1
11..0068 00..99078.990855 1.5 2 2.5 3 3.5 ββββ3333 ==== 0001....03720005 11..0068 001...9909909901682 0 0.2 0.4 0.6 0.8 1 ββββ3333 ==== 0000....02680265 11..0068 001...9909901891 0 0.5 1 1.5 2 2.5 3 3.5 4 ββββ3333 ==== 0000....04560510
N
N1.04 N1.04 1.04
1.02 1.02 1.02
1 1 1
0.98 0.98 0.98
0 5 10 15 0 5 10 15 0 5 10 15
ρ ρ ρ
(d) β1=1.5,β2=6 (e) β1=2,β2=4.5 (f) β1=2.25,β2=4
FIG. 1. The metric function N(ρ) of a (1,1)-string is shown for γ =0.3 and different choices of β1, β2 and β3.
Weobservethattheincreaseinthetotalmagneticfluxalongthestringincreasestheasymptoticvalueofthemetric
function N(ρ) if N′(ρ) > 0. The qualitative features do not change. If a minimum of the metric function exists for
n=m=1 it exists for all choices of n and m (see Fig.3(d)) and if N′(ρ)>0 for n=m=1 this will be the same for
other choices of n and m.
Note that the profiles of the metric functions L(ρ) for all cases are similar to those for β = 0. The deviation of
3
L′(ρ 1) = c from one determines the deficit angle of the space-time and depends on the inertial mass per unit
2
≫
length. Thisisshownforγ =0.3anddifferentchoicesofn,mandβ inFig.4. Forsolutionswithβ =β the slopeof
i 1 2
L(ρ) at infinity decreases with increasing sum n+m p+q. This is natural since an increase in the windings leads
≡
to an increase in the mass per unit length and hence to an increase of the deficit angle. Moreover, for a given sum
p+q the solutions with p=q have the biggest slope of L(ρ) at infinity. This is related to the fact that the solutions
with equal winding are bound strongest (see also the results in [15]).
2. Classification of solutions
The geodesics can be classified according to the test particles energy E, angular momentum L and momentum in
z
the direction of the string axis p . Intersection points of with the effective potential, i.e. points where = V
z eff
E E
correspond to turning points of the motion. The maximum and minimum of the effective potential determine the
largest,respectively smallestpossible value of for bound orbits. The effective potential is determined by the metric
E
functions N and L as well as the constants of motion. Choosing β , i = 1,2,3, γ and n, m we find the numerical
i
profiles of N and L. For a given L2 (and ε = 0 or ε = 1) there is an E2 such that the value of is equal to the
maximal value of the effective potenztial V (ρ) and one E2 such that is equal to the minimal vaElue of V (ρ). In
eff eff
E
the former case, the corresponding orbit is an unstable circular orbit, while in the latter the orbit is a stable circular
orbit.
Defining µ := E2 and ν := L2 we can then plot the domain of existence of bound orbits in the µ-ν-plane. Our
z
results for massive particles with p = 0 are given in Fig.s 5, 6 for n = m = 1. Fig.5 corresponds to the case of a
z
(p,q)-string space-time with monotonically increasing N(ρ) and Fig. 6 to the case of a (p,q)-string space-time which
10
β = 1.00, β = 2.00, γ = 0.30 β = 2.00, β = 2.00, γ = 0.30 β = 1.00, β = 1.00, γ = 0.30
1 2 1 2 1 2
0.14 0.14 0.14
0.01.21 βββ333 === 000...037050 0.01.21 βββ333 === 000...059005 0.01.21 βββ333 === 000...024059
0.08 0.08 0.08
V V V
eff eff eff
0.06 0.06 0.06
0.04 0.04 0.04
0.02 0.02 0.02
0 0 0
−0.02 −0.02 −0.02
0 5 10 15 0 5 10 15 0 5 10 15
ρ ρ ρ
(a) β1=2,β2=2 (b) β1=β2=2 (c) β1=2,β2=1
β = 1.50, β = 6.00, γ = 0.30 β = 2.00, β = 4.50, γ = 0.30 β = 2.25, β = 4.00, γ = 0.30
1 2 1 2 1 2
000...0011.8241 −−−000..0.0000..10000550151 0 1 2 3 ββββ3333 ==== 0001....03720005 000...0011.8241 00000....0000.00000012468 0 0.5 1 1.5 2 2.5 3 ββββ3333 ==== 0000....02680265 000...0011.8241 −−−01234321 0x 10−30.5 1 1.5 2 2.5 3 ββββ3333 ==== 0000....04560510
V V V
eff0.06 eff0.06 eff0.06
0.04 0.04 0.04
0.02 0.02 0.02
0 0 0
−0.02 −0.02 −0.02
0 5 10 15 0 5 10 15 0 5 10 15
ρ ρ ρ
(d) β1=1.5,β2=6 (e) β1=2,β2=4.5 (f) β1=2.25,β2=4
FIG. 2. The effective potential Veff(ρ) in the space-time of a (1,1)-string is shown for γ = 0.3 and different choices of β1, β2
and β3. HereE =0.01, Lz =0.03 and pz =0.
hasN′(ρ)=0atsomenon-vanishing,finitevalueofρ. Thebluedashedandsolidblacklinefrom(µ ,0)to(µ ,ν )and
1 3 3
(µ ,0)to (µ ,ν ), respectively,representthe choiceof(E, L , p )forstableandunstablecircularorbits,respectively,
2 3 3 z z
and bound orbits exist in the colored domain between the two bounding curves. (µ ,ν ) corresponds to the largest
3 3
possible values of µ and ν for bound orbits. M1 denotes the domain in the µ-ν-plane in which is smaller than the
E
minimumofthe effective potentialandhencethere arenosolutionsto the geodesicequation. M4 denotesthe domain
in which is larger than the maximum of the effective potential and only escape orbits are possible. In M2 and M3
E
on the other hand bound orbits are possible. In M2 is smaller than the asymptotic value of the effective potential,
E
but largerthan the minimum of V and only bound orbits are possible. In M3 is largerthan the asymptotic value
eff
E
of the effective potential but smaller than the maximum of V . Hence, in M3 there are bound orbits, but escape
eff
orbits are also possible.
For β = β = 2 and β > 0 we find that µ = 1 for all values of β , while µ as well as (µ ,ν ) increase with
1 2 3 1 3 2 3 3
increasing β . While for β =β =2, β =0 no bound orbits exist at all [26], bound orbits are possible for β =0.1
3 1 2 3 3
and the domain of existence of bound orbits in the µ-ν-plane is extending for increasing β (compare the plots for
3
β =0.1 and β =0.75). The existence of bound orbits in the limit where M =M , i=1,2 is new as compared
3 3 H,i W,i
to the β =0 case.
3
This,however,isnottheonlydifferenceascomparedtothespace-timeofanAbelian-Higgsstring. Asstatedabove
we find that it is possible to have negative valued minima of the effective potential in (p,q)-string space-times. This
leadstotheobservationthatmassivetestparticleswithµ<1cannowmoveonboundorbits. Thisisanewfeatureas
comparedto the β =0 case,wherewe hadtorequirethat µ>1. This means thattestparticles withless energycan
3
moveonboundorbitsin(p,q)-stringspace-timesascomparedtotheβ =0case,whichcorrespondstothespace-time
3
of two non-interacting Abelian-Higgs strings. This is clearly seen in Fig. 6 for γ = 0.35, β = 10, β = 3.6 and two
1 2
different values of β . While for M > M , i = 1,2 and in the β = 0 limit no bound orbits exist [26] they exist
3 H,i W,i 3
in a small domain of the µ-ν-plane for sufficiently large β . The extension of the domain in the µ-ν-plane for which
3
bound orbits exist increases with increasing β , i.e. the values of µ , µ and (µ ,ν ) increase.
3 1 2 3 3
The change of the µ-ν-plot of a (p,q)-string with γ = 0.2, β = 8, β = 0.5, β = 0.99 resulting from the change
1 2 3