Local cosmic strings in Brans-Dicke theory with a cosmological constant

Ozgur Delice*Department of Physics, Bogazici University, 34342 Bebek, Istanbul, Turkey

(Received 9 August 2006; published 25 September 2006)

It is known that Vilenkin’s phenomenological equation of state for static straight cosmic strings isinconsistent with Brans-Dicke theory. We will prove that, in the presence of a cosmological constant, thisequation of state is consistent with Brans-Dicke theory. The general solution of the full nonlinear fieldequations, representing the interior of a cosmic string with a cosmological constant, is also presented.

DOI: 10.1103/PhysRevD.74.067703 PACS numbers: 11.27.+d, 04.20.Jb, 04.50.+h

I. INTRODUCTION

Cosmic strings are linear topological defects which mayhave been produced in the early universe during phasetransitions in general unified theories [1,2]. They werebelieved to have important implications in cosmologyand astrophysics such as their role in primordial densityfluctuations in the early universe where galaxies etc. form.Although accumulation of observational data ruled out thispossibility and they cannot contribute more than �10%[3,4] to the density fluctuations, the interest in cosmicstrings was renewed [5] due to several reasons, such astheir relations with fundamental strings in string theory [6–8].

An infinitely long, straight, local gauge string is charac-terized by (i) a special form for the energy-momentumtensor of the string T0

0 � Tzz � 0 and others vanishing;(ii) a Lorentz boost invariance along the symmetry (z) axis,due to Vilenkin [9]. The investigation of an Abelian-Higgstype theory for a Nielsen-Olesen type U�1� vortex [10,11]strengthens the above prescription for a straight string.Exact interior thick string solutions satisfying the aboveenergy-momentum tensor have been found by Gott [12],Hiscock [13], and Linet [14].

The Brans-Dicke (BD) theory [15] is a natural general-ization of Einstein’s general relativity (GR) where thegravitational coupling constant is replaced by a scalar field.This theory has some desirable properties; for instance, itobeys the weak equivalence principle and it is also com-patible with the solar system experiments for certain rangesof its parameter [16]. It has been applied to several issuesranging from inflation schemes in cosmology to quantumgravity. The inclusion of scalar fields in GR is also sug-gested by different theories, for example, as a dilaton fieldwhich naturally arises in the low energy limit of stringtheories or in the Kaluza-Klein theories [17].

In the framework of scalar-tensor theories, cosmicstrings are investigated in detail [18–27]. Sen et al. [23]and later Arazi et al. [24] demonstrated that the BD fieldequations are inconsistent with a local string having theabove energy-momentum tensor. Moreover, Gregory and

Santos [25] studied the cosmic string as an isolated self-gravitating Abelian-Higgs vortex in dilaton gravity, andthey showed that the above energy-momentum tensor re-sults in a contradiction in the field equations in general, andone must take into account the other stresses as nonvanish-ing. However, their investigation also shows that if thereexists a mass term in the theory, then the contradiction isremoved and Vilenkin’s prescription might be appropriate.The inconsistency is also removed if we have a moregeneral scalar-tensor theory [26], or the string is nonstatic[27]. In this report, I will show that, in the presence of acosmological constant, Vilenkin’s prescription is consis-tent with BD theory. I also present some exact solutions ofthe BD field equations satisfying the above energy-momentum tensor and representing an infinitely long,static straight, local gauge cosmic string.

II. FIELD EQUATIONS

The Brans-Dicke theory is described, in the presence ofa cosmological constant, by the action

S �Zd4x

��������gp

����R� 2�� �

!�g��@��@��

�

� Sm��; g�; (1)

here R is the Ricci scalar, � is the cosmological constant, gis the determinant of the spacetime metric g��, � is theBrans-Dicke scalar field, ! is the Brans-Dicke parameter,and Sm denotes the action of matter fields �. We use unitsin which c � @ � 1 and mostly plus signature. The action(1) yields the field equations

G�� ��g�� �T����!

�2

��;��;� �

1

2g���

;��;�

�

�1

���;�;� � g�����; (2)

�2!� 3��� � �2��� T��: (3)

Here, T�� is the energy-momentum tensor of the matterfields. Note that, in this frame (1), the energy-conservationequation*Electronic address: odelice@boun.edu.tr

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r�T�� � 0 (4)

holds.Let us consider a general cylindrically symmetric, static

metric

ds2 � e2�K�U��dr2 � dt2� � e2Udz2 � e�2UW2d�2; (5)

where t, r, z, ’ denote the time, the radial, the axial, andthe angular cylindrical coordinates with the ranges �1<t, z <1, 0 r <1, 0 � 2�, and the metric func-tionsK,U,W and the BD scalar field� are the functions ofr only. For the string, we consider Vilenkin’s prescriptionand take the only nonvanishing terms of the energy-momentum tensor of the form

T00 � Tzz � ���r�: (6)

Then, the field equations are

�W00

W� K0

W0

W�U02 �

!2

��0

�

�2� �K0 �U0�

�0

���00

�

�W0

W�0

��

���

��

�e2�K�U�;

(7)

K0W0

W�U02 �

!2

��0

�

�2� �K0 �U0�

�0

��W0

W�0

�

��e2�K�U�; (8)

W00

W� 2U00 � 2U0

W0

W� K00 �U02

� �!2

��0

�

�2��00

��W0

W�0

��U0

�0

�

�

���

��

�e2�K�U�; (9)

K00 �U02 � �!2

��0

�

�2�U0

�0

���00

���e2�K�U�;

(10)

�00 ��0W0

W� �

2

2!� 3���� ��e2�K�U�: (11)

The energy-conservation equation (4) yields K0� � 0.Thus, in order to have a string (� � 0), we must have

K0 � 0; (12)

and hereafter we take eK � 1. Using this and Eqs. (8) and(10), we find

�00 ��0W0

W� �2��e�2U: (13)

Considering this equation with (11) and taking into ac-count (12), we have

� � 2�!� 1���: (14)

This expression says that, for � � 0, we must have either� � 0, i.e. no string at all, or !! 1, if we have � � 0.Thus, if we insist on � � 0, the solution reduces to thecorresponding solution in GR. This fact was noticed by[23] and they concluded that BD theory is inconsistent withlocal strings, since for this case the only solution is thesame as the GR one. However, for � � 0, ! � �1 wehave � � 0, and for BD theory with a cosmological con-stant, Vilenkin’s prescription for local strings is consistent.

III. SOLUTIONS

Having proved that we can have a nontrivial string forBD theory with a cosmological constant, let me introduceexact solutions representing the interior regions of thestring. Note that for local gauge strings we must have boostinvariance along t, z directions and this requires U � 0.Then, the field equations reduce to the following set:

�00 ��0W0

W� �2��; (15)

!2

��0

�

�2�W0

W�0

���; (16)

��W�00

�W� �2��2�!�; (17)

together with (14). Now, we limit our analysis to !>�2.(For !<�2, the results are the same as �! ��.) Withthis limitation, we have two different sets of solutionsdepending on the sign of �.

A. � > 0

For this case Eq. (17) can be integrated to give

W� � A sin��r� � B cos��r�; �> 0; (18)

where

� ������������������������2��2�!�

p> 0: (19)

We choose B � 0 due to cylindrical symmetry. By consid-ering Eqs. (15) and (16) we find

W�r� � Asin�!�1�=�!�2���r�tan�=�!�2����=2�r�; (20)

��r� � sin1=�!�2���r�tan��=�!�2����=2�r�; (21)

where � � 1. The metric has the form

ds2 � �dt2 � dr2 � dz2 �W2d�2: (22)

Regularity conditions on the axis for cylindrical symmetryrequire W�0� � 0 and W0�0� � 1. The first one is auto-matically satisfied and the second one is satisfied if

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A � 2�!=�2!�4�=����������������������!� 2��

p: (23)

Hence the solution is regular and free of a conical singu-larity on the axis. For � � 1 it is also free of any singularityon the axis since scalars constructed from curvature ele-ments are regular. For example, the Ricci scalar is

R �2��!� �!� 1�2�1� cos��r���

�!� 2� cos���=2�r�(24)

and is regular on the axis. Other scalars have similarexpressions. Thus, we see that the interior string solutionis smooth and regular near the axis. However, the solutionis singular at r � rc � �=�2��, hence the radius of thestring should be smaller than rc.

The mass per unit length of the string having radius r0

can be calculated using

m �Z 2�

0d�

Z r0

0��r�W�r�dr: (25)

This yields

m � 2�2!=�!�2��!� 1�

�!� 2�sin2���=2�r0�: (26)

The solution with � � �1 must be excluded since it issingular on the axis.

B. � < 0

For this case, from (17) and cylindrical symmetry wemust have

W� � A sinh�r� (27)

with

���������������������������j2��!� 2�j

q; (28)

which yields, with the metric of the form (22),

W�r� � Asinh�!�1�=�!�2��r�tanh�=�!�2���=2�r�; (29)

��r� � sinh1=�!�2��r�tanh��=�!�2���=2�r�; (30)

where � � 1. The analyses of the regularity conditionsand curvature scalars show that the solution with � � 1 issmooth, regular, and free of any singularity. Unlike �> 0,there is no upper limit on the value of the radius of thestring. The solution with � � �1 is singular on the axisand must be excluded. The mass per unit length of thissolution (� � 1) with radius r0 is

m � 2�2!=�!�2��!� 1�

�!� 2�sinh2��=2�r0�: (31)

For all of these solutions � and m are positive.

IV. CONCLUSIONS

In this report we have proved that, unlike the case � � 0[23,24], in the presence of a cosmological constant, �, thephenomenological equation of state for local cosmicstrings T0

0 � Tzz � �> 0 is consistent with Brans-Dicke theory (for ! � �1;�2). This result is in accor-dance with the analysis of Gregory and Santos [25]. Wehave also presented exact solutions for full Brans-Dickefield equations with a � term for this configuration anddiscussed some of their properties. Further properties ofthese solutions with possible exterior solutions, analoguesof GR results [28,29], are under investigation [30] and willbe presented elsewhere.

There is an interesting duality between GR and BDtheories for cosmic strings. In GR, Vilenkin’s ansatz is agood choice; however, when there is a cosmological con-stant, one must take into account the other stresses asnonvanishing, otherwise there is no static solution [29].In BD theory, however, as we have shown, the situation isreversed, and if there is no cosmological term, then there isno solution, but if there is a cosmological constant, then wecan have strings with the above properties in BD theory.

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