Vacuum polarization by a cosmic string in de Sitter spacetime
In this paper we investigate the vacuum polarization effect associated with a quantum massive scalar field in a higher dimensional de Sitter spacetime in the presence of a cosmic string. Because this investigation has been developed in a pure de Sitter space, here we are mainly interested on the effects induced by the presence of the string. So this analysis is developed by expressing the complete Wightman function as the sum of two terms: The first one corresponds to the bulk where the cosmic string is absent and the second one is induced by the presence of the string. By using the Abel-Plana summation formula, we show that for points away from the string the latter is finite at the coincidence limit and it is used to evaluate the vacuum averages of the square of the field and the energy-momentum tensor induced by the cosmic string. Simple asymptotic formulae are obtained for these expectation values for points near the string and at large distances from it. It is shown that, depending on the curvature radius of de Sitter spacetime, two regimes are realized with monotonic and oscillatory behavior of the vacuum expectation values at large distances.
PACS numbers: 03.70.+k, 98.80.Cq, 11.27.+d
De Sitter (dS) space is the curved spacetime which has been most studied in quantum field theory during the past two decades. The main reason resides in the fact that it is maximally symmetric and several physical problems can be exactly solvable on this background; 111De Sitter space enjoys the same degree of symmetry as the Minkowski one  moreover the importance of these theoretical analysis increased by the appearance of the inflationary cosmology scenario . In great number of inflationary models, approximated dS spacetime is employed to solve relevant problems in standard cosmology. During an inflationary epoch, quantum fluctuations in the inflaton field introduce inhomogeneities and may affect the transition toward the true vacuum. These fluctuations play important role in the generation of cosmic structures from inflation. More recently astronomical observations of high redshift supernovae, galaxy clusters and cosmic microwave background  indicate that at the present epoch, the Universe is accelerating and can be well approximated by a world with a positive cosmological constant.
Cosmic strings are linear topologically stable gravitational defects which appear in the framework of grand unified theories. These objects could be produced in very early Universe as a consequence of spontaneous breakdown of gauge symmetry [4, 5]. Although recent observations data on the cosmic microwave background have ruled out cosmic strings as the primary source for primordial density perturbation, they are still candidate for the generation of a number of interesting physical effects such as gamma ray burst , gravitational waves  and high energy cosmic rays . Recently, cosmic strings have attracted renewed interest partly because a variant of their formation mechanism is proposed in the framework of brane inflation -.
Many of high energy theories of fundamental physics are formulated in higher dimensional spacetimes. Although topological defects have been first analyzed in four-dimensional spacetime [4, 5], they have been considered in the context of braneworld as well. In this scenario the defects live in a -dimensions submanifold embedded in a -dimensional Universe. The cosmic string case corresponds to two additional extra dimensions. In this context the gravitational effect of global strings has been considered in [12, 13] as responsible for the compactification from six to four spacetime dimensions, naturally producing the observed hierarchy between electroweak and gravitational forces. The analysis of quantum effects for various spin fields on the background of dS spacetime, have been discussed by several authors (see, for instance,  and - and references therein). Also for the cosmic string spacetime these analysis have considered in - and -, for scalar and fermionic fields respectively.
In this paper, we shall investigate the one-loop quantum effects arising from vacuum fluctuations associated with massive scalar field on the background of higher dimensional de Sitter spacetime considering the presence of a cosmic string in it (for cosmic strings in background of dS spacetime see -). Consequently these effects will take into account the presence of the curvature of the spacetime and the non-trivial topology associated with the two-dimensional conical sub-space. The results obtained here can be used, in particular, for the investigation of the effects of the quantum fluctuations induced by the string in the inflationary phase. Though the cosmic strings produced in phase transitions before or during early stages of inflation would have been drastically diluted by the expansion, the formation of defects during inflation can be triggered by a coupling of the symmetry breaking field to the inflaton field or to the curvature of the background spacetime (see, for example, ). The cosmic string can also be continuously created during inflation by quantum-mechanical tunnelling . Another class of models to which the results of the present paper are applicable corresponds to string-driven inflation where the cosmological expansion is driven entirely by the string energy . The problem under consideration is also of separate interest as an example with gravitational and topological polarizations of the vacuum, where all calculations can be performed in a closed form.
The paper is organized as follows. In section 2 we present the background associated with the geometry under consideration and the solution of the Klein-Gordon equation admitting an arbitrary curvature coupling. Also we present the complete Wightman function and consider the special case where the planar angle deficit in the cosmic string subspace is an integer fraction of . In sections 3 and 4 we calculate the vacuum expectation values of the field squared and the energy-momentum tensor induced by the cosmic string. Finally the main results of this paper are summarized in section 5. Appendix contains some technical details of the obtainment of the Wightaman function in a more workable form.
2 Wightman function
The main objective of this section is to calculate the positive frequency Wightman function associated with a massive scalar field in a higher-dimensional de Sitter spacetime taking into account the presence of a cosmic string. This quantity is important to the calculation of scalar vacuum averages. In order to do that we first obtain the complete set of eigenfunctions for the Klein-Grodon equation admitting an arbitrary curvature coupling. The geometry associated with the corresponding background spacetime is given by the following line element:
where and define the coordinates on the conical geometry, , and the parameter is related with the cosmological constant and Ricci scalar by the formulae
being the dimension of the spacetime given by . The parameter is bigger than unity and codifies the presence of the cosmic string. 222It can be shown that by defining the coordinates , they represent a hyperboloid embedded in a -dimensional conical spacetime. For further analysis, in addition to the synchronous time coordinate we shall use the conformal time defined according to
In terms of this coordinate the above line element takes the form
The line element (1) can also be written in static coordinates. For simplicity we consider the corresponding coordinate transformation in the case with . This transformation is given by the relations
and the line element takes the form
with . Introducing a new angular coordinate , from (6) we obtain the static line element of the de Sitter spacetime with deficit angle previously discussed in . In this paper it is shown that to leading order in the gravitational coupling the effect of the vortex on de Sitter spacetime is to create a deficit angle in the metric (6).
The field equation that will be considered is
where is an arbitrary curvature coupling constant. The complete set of solutions of this equation in the coordinate system defined by (4) is:
where , ,
In (8), and represent the Hankel and Bessel functions respectively, and is the set of quantum numbers, being . The coefficient can be found by the orthonormalization condition
where the integral is evaluated over the spatial hypersurface , and represents the Kronecker-delta for discrete indices and Dirac-delta function for continuous ones. This leads to the result
for the normalization coefficient.
We shall employ the mode-sum formula to calculate the positive frequency Wightman function:
with , . In appendix A we show that the expression for the Wightman function can be transformed to the form
In (14) the prime on the sign of summation means that the term should be halved.
The analysis of the vacuum polarization in the higher dimensional dS spacetime, with toroidal compactification of extra dimensions, have been developed in [30, 31, 32]. Here we are mainly interested in quantum effects induced by the presence of the cosmic string. In order to investigate these effects we introduce below the subtracted Wightman function
As the presence of the string does not change the curvature for the background manifold for , the structure of the divergences in the coincidence limit is the same for both terms on the right hand side. Hence, for these points the function is finite in the coincidence limit. Regarding to the value of the parameter , two different approaches to obtain will be provided in the follows.
2.1 Special case with an integer
Now the integral over is evaluated with the help of formula
where is the hypergeometric function.
Introducing the notation
the Wightman function is presented by a sum of images of the dS Wightman function,
In (19) we have introduced the notations
The term of (19) is divergent at the coincidence limit and coincides with the Wightman function in dS spacetime. Finally, the part in the Wightman function induced by the cosmic string, , is given by
which is finite at the coincidence limit.
Note that by making use of the relation
we can express the function in terms of the associated Legendre function of the first kind. In the case of massless field and conformal coupling,
one has and for the function (20) we find .
2.2 General case
For the general case of the subtracted Wightman function can be expressed by
A more convenient expression for this function can be provided by using the Abel-Plana formula (see, for instance, ) for the summation over :
Now we can see that in the evaluation of the difference the terms coming from the first integral on the right of the Abel-Plana formula cancel out and one obtains
with the notation
As a result the subtracted Wightman function takes the form
For a conformally coupled massless scalar field we have :
In this case the problem under consideration is conformally related to the problem with a cosmic string in the Minkowski spacetime and the corresponding subtracted Wightman functions are connected by the standard conformal transformation.
3 The computation of
The formal expression for the vacuum expectation value (VEV) of the field squared is given by evaluating the Wightman function at the coincidence limit:
However this procedure provides a divergent result and to obtain a finite and well defined value some renormalization procedure is necessary. The important point here is that the presence of the cosmic string does not introduce additional curvature. Thus, the divergences are contained in the part corresponding to the pure dS spacetime and the string induced part is finite for points outside the string. As we have already extracted the first part, the renormalization procedure is reduced to the renormalization of the dS part in the absence of the string which is already done in literature [15, 16, 17]. So according to the fact that
we can conclude that
Due to the maximal symmetry of the dS spacetime the VEV does not depend on the spacetime point. In the special case of being an integer number the resulting formulae are essentially simplified and in the following we present the corresponding calculations separately.
3.1 Special case
For integer we use (22). The VEV is promptly obtained:
Note that the VEV (34) is a function of the ratio which is the proper distance from the string, , measured in the units of the dS curvature radius . In the discussion below, for the evaluation of the VEV of the energy-momentum tensor we will need also the covariant d’Alembertian for the string induced part in the VEV of the field squared:
where the prime means the derivative with respect to . The second derivative in this expression can be excluded by using the equation for the function which is easily obtained from the hypergeometric equation:
For the first derivative of this function one has
In figure 1 we have plotted the string induced part in the VEV of the field squared versus the ratio (proper distance from the string measured in units of ) for minimally coupled scalar field () with (left panel) and (right panel) in 4-dimensional spacetime. The numbers near the curves correspond to the values of the parameter . For the left panel the parameter is real and for the right one this parameter is imaginary. In the latter case the oscillatory behavior of the VEV is seen at large distances from the string. A simple formula for the asymptotic behavior of the VEV of the field squared at large distances from the string will be given below for the general case of the parameter .
For a conformally coupled massless scalar field form (34) one finds
For even values of , the summation on the right-hand side of the above expression can be obtained in a closed form by use of the formulae
for the sum
In particular, for a 4-dimensional spacetime, , we need . Consequently,
For a six-dimensional spacetime, is a long expression obtained by the recurrence relation above, however we find . In this case we have
3.2 General case
For the general case of the parameter we have an integral representation for the VEV of the field squared by using (29):
where and in the discussion below we use the notation
As before, we see that the contribution induced by the cosmic string in the VEV of the field squared depends on the ratio . For the covariant d’Alembertian of the string induced part in the VEV of the field squared we find
Note that for the derivatives of the function we have the relations
For a conformally coupled massless scalar field, , the formula (44) reduces to
The general formula for the string induced part in the VEV of the field squared is simplified in asymptotic regions of small and large distances. For small values of the ratio (points near the cosmic string) the argument of the Mac-Donald function in (44) is large and to the leading order one has . With this approximation the integral over is evaluated by using the formula 
and one finds
In this formula and in the discussion below we use the notation
Comparing with (49), we conclude that at small distance, the leading order term of the string induced part in the VEV of the field squared coincides with the corresponding result for a massless conformally coupled field.
At large distances from the string, , we replace the Mac-Donald function in by the corresponding asymptotic expression for small values of the argument. After that the integral over is evaluated by formula (50) and for real values of the parameter one finds
For a conformally coupled massless scalar field this result coincides with the exact formula (49). For imaginary values of , in the similar way, we have the following asymptotic estimate
where and the phase are defined by the relation
Hence, in this case, at large distances from the string, the string induced part decays as and the damping of the corresponding VEV has an oscillatory nature.
4 Vacuum expectation value of the energy-momentum tensor
In this section we shall analyze the contribution to the VEV of the scalar energy-momentum tensor induced by the cosmic string. As in the previous section we may write
where is the corresponding VEV in dS spacetime when the string is absent. The latter does not depend on the spacetime point and is well-investigated in literature [15, 16, 17]. It corresponds to a gravitational source of the cosmological constant type and, in combination with the initial cosmological constant , given by (2), leads to the effective cosmological constant , where is the Newton gravitational constant. The string induced contribution is obtained by use of the formula
is the Ricci tensor for the background spacetime.
4.1 Special case
First we consider the special case of integer . By using the expressions for the Wightman function and the VEV of the field squared, after long calculations the diagonal components of the energy-momentum tensor are presented in the form (no summation over )
where we have defined
In (59) we have introduced the notations
For the non-zero off-diagonal component one has
We recall that for a conformally coupled massless scalar field one has and, as it can be easily seen from (62), the off-diagonal component vanishes. The explicit expression for the part directly follows from formulae (34) and (36):
Now it can be checked that the components of the energy-momentum tensor satisfy the trace relation
In particular, for a conformally coupled massless scalar this tensor is traceless and the trace anomalies are contained in the purely dS part only. Note that from the covariant conservation equation we have also the following relations between the components of the energy-momentum tensor:
These relations are fullfilled separately for pure and string induced parts in the VEV of the energy-momentum tensor.
In figure 2 the string induced part in the VEV of the energy density is depicted as a function of in the case of a minimally coupled scalar field with (left panel) and (right panel) in 4-dimensional dS spacetime. For the right panel the parameter is imaginary and the decay of the VEV is oscillatory. In the next subsection this feature will be seen from the corresponding asymptotic formula.
4.2 General case
For an arbitrary value of , we need the formulae (29) and (44). For this case the calculations of the components of the energy-momentum tensor are much longer. The final results for the diagonal components are given below (no summation over ):
where is defined by the relation (60) and the following notations are introduced
In addition to diagonal components there is also non-zero off-diagonal component
It can be explicitly checked that the string induced parts in the components of the VEV of the energy-momentum tensor satisfy the trace relation (64).
For a conformally coupled massless scalar field one has and the off-diagonal component (68) vanishes. For the diagonal components in this case one has (no summation over )
where the function is defined by relation (52).
The asymptotic behavior of the string induced part in the VEV of the energy-momentum tensor at small and large distances is investigated in the way similar to that used above for the case of the field squared. For points near the string, , to the leading order we have
For the off-diagonal component the corresponding formula takes the form