# Electromagnetic vacuum stresses and energy fluxes induced
by a cosmic string in de Sitter spacetime

A. A. Saharian1, V. F. Manukyan1, V. Kh. Kotanjyan1,2, A. A. Grigoryan1

1Department of Physics, Yerevan State University,
1 Alex Manoogian Street, 0025 Yerevan, Armenia
2Institute of Applied Problems in Physics NAS RA,
25 Hr. Nersessian Street, 0014 Yerevan, Armenia Corresponding author. E-mail
address<EMAIL_ADDRESS>
###### Abstract

For the electromagnetic field in $(D+1)$-dimensional locally de Sitter (dS)
spacetime, we analyze the effects of a generalized cosmic string type defect
on the vacuum expectation value of the energy-momentum tensor. For the Bunch-
Davies vacuum state, the topological contributions are explicitly extracted in
both the diagonal and off-diagonal components. The latter describes the
presence of radially directed energy flux in the vacuum state. It vanishes for
$D=3$ because of the conformal invariance of the electromagnetic field and is
directed towards the cosmic string for $D\geq 4$. The topological
contributions in the vacuum stresses are anisotropic and, unlike to the
geometry of a cosmic string in the Minkowski spacetime, for $D>3$ the stresses
along the directions parallel to the string core differ from the energy
density. Depending on the planar angle deficit and the distance from the
cosmic string, the corresponding expectation values can be either positive or
negative. Near the cosmic string the effect of the gravitational field on the
diagonal components of the topological part is week and the leading terms in
the respective expansions coincide with the expectation values for a cosmic
string in background of Minkowski spacetime. The spacetime curvature
essentially modifies the behavior of the topological terms at proper distances
from the cosmic string larger than the dS curvature radius. In that region,
the topological contributions in the diagonal components of the energy-
momentum tensor decay in inverse proportion to the fourth power of the proper
distance and the energy flux density behaves as inverse-fifth power for all
values of the spatial dimension $D$. The exception is the energy density in
the special case $D=4$. For a cosmic string in the Minkowski bulk the energy
flux is absent and the diagonal components are proportional to the $(D+1)$th
power of the inverse distance.

Keywords: Cosmic string, de Sitter spacetime, electromagnetic vacuum

##  1 Introduction

The canonical quantization of fields is based on the expansion of the field
operator in terms of a complete set of mode functions being the solutions of
the classical field equation. That expansion defines the annihilation and
creation operators and then the space of the Fock states is constructed. The
vacuum is defined as a state of a quantum field that is nullified by the
action of the annihilation operator. The mode functions employed in the
quantization procedure reflect both the local and global properties of
background spacetime and the same holds for the vacuum state. In particular,
among the interesting directions in the investigations of the quantum vacuum
is the dependence of its properties on the spatial topology. Here we study the
polarization of the electromagnetic vacuum on background of
$(D+1)$-dimensional de Sitter (dS) spacetime in the presence of a topological
defect that is a generalization of a straight cosmic string in
(3+1)-dimensional spacetime. The cosmic strings are formed in the early
universe as a result of symmetry breaking phase transitions (for reviews see
[1, 2, 3, 4]). The interior energy density of those defects is determined by
the energy scale at which the phase transition takes place and the observation
of cosmic strings at recent epoch of the universe expansion may provide an
important window to high-energy physics. The gravitational field around of
those kinds of topological defects is a source of a number of physical effects
including the gravitational lensing, the generation of gravitational waves and
gamma ray bursts. The cosmic strings also give rise to non-Gaussian
distorsions in the fluctuations of the cosmic microwave background [5].

Our consideration of dS spacetime as a background geometry is motivated by
several reasons. First of all, its high degree of symmetry allows to obtain
closed analytic expressions for the cosmic string induced local
characteristics of the vacuum state. This will shed light on combined effects
of local geometry and topology in more complicated backgrounds. Another reason
is related to the profound role of dS spacetime in cosmology. The most
inflationary models employ the dS exponential expansion in the early universe
to provide natural solutions to a number of problems in the Standard Big-Bang
cosmology (see [6, 7, 8]). From the other side, the observational data on the
temperature anisotropies of cosmic microwave background, high redshift
supernovae and galaxy clusters [9]-[15] indicate that the recent expansion of
the universe is dominated by a source of the cosmological constant type. The
relative contribution of the latter to the total energy density will increase
during the expansion and the corresponding geometry asymptotes dS spacetime in
the future.

The quantum field-theoretical effects in fixed dS background were investigated
in a large number of papers. Different coordinate systems have been
considered, including global, planar, static and hyperbolic coordinates. In
particular, motivated by an important role of vacuum fluctuations during the
inflationary phase in the formation of large scale structures in the post
inflationary era, the dynamics of vacuum fluctuations of scalar fields have
attracted great deal of attention. The properties of those fluctuations are
encoded in the observed spectrum of temperature anisotropies for cosmic
microwave background radiation and this opens a possibility to test the
effects of gravity on quantum matter. A similar amplification of the
electromagnetic field quantum fluctuations by the dS expansion can serve as a
mechanism for the generation of extragalactic magnetic fields. However, for
this mechanism to work, it is required to break the conformal invariance of
the electromagnetic field in (3+1)-dimensional spacetime. That can be done by
an additional coupling of the electromagnetic field with other degrees of
freedom, for example, by adding a factor in the Maxwell kinetic term that
depends on other fields. Another possibility for breaking the conformal
symmetry is to consider models with extra spatial dimensions (for the
generation of large scale magnetic fields in those types of models see, for
example, [16, 17, 18]). This is one of the reasons for our discussion of
background spacetime with general number of dimensions. Another reason is that
the background geometry we consider can also be generated by fundamental
strings in superstring theories. A mechanism for the formation of those types
of objects within the framework of brane inflation models has been discussed
in [5, 19, 20, 21]. In models with fundamental strings as cosmic strings,
depending upon the number of compact dimensions, the spatial dimension of the
effective theory may vary from 3 to 9.

The polarization of vacuum by a cosmic string has been widely considered in
the literature for both neutral and charged quantum fields. That was mainly
done for an idealized model where the influence of cosmic string on the
spacetime geometry outside the core is reduced to the generation of planar
angle deficit determined by the energy density inside the core. Exact analytic
results for the local characteristics of the vacuum are obtained for highly
symmetric background geometries, including Minkowski, dS and anti-de Sitter
(AdS) spacetimes. Cosmic string type topological defects in dS and AdS
spacetimes were considered in [22]-[27]. The influence of the cosmological
expansion on the properties of scalar vacuum around a straight cosmic string
in flat Friedmann-Robertson-Walker type models has been discussed in [28] for
special values of the planar angle deficit $2\pi-\phi_{0}$ corresponding to
integer values of the parameter $q=2\pi/\phi_{0}$. For those values the Green
function in a conical space is expressed as an image sum over the Green
functions in the absence of cosmic string. The vacuum polarization around a
straight cosmic string in dS spacetime for general values of the planar angle
deficit in the cases of massive scalar and fermionic fields has been
investigated in [29, 30]. The correlators for the electromagnetic field and
the vacuum expectation values of the electric and magnetic fields squared were
discussed in [31, 32]. Additional topological effects on the properties of the
scalar and fermionic vacua induced by compactification of a cosmic string
along its axis in locally dS bulk were studied in [33, 34, 35]. The
polarization of the scalar and fermionic vacua around a cosmic string in
background of AdS spacetime has been investigated in [36, 37]. The combined
effects of a cosmic string, compact dimensions and branes on the local
characteristics of the vacuum in locally AdS spacetime were discussed in
[38]-[43]. The finite temperature effects on the expectation values of the
charge and current densities for a charged scalar field in the presence of a
cosmic string and compact dimension on locally AdS bulk were considered
recently in [44].

The organization of the present paper is as follows. The problem setup and the
mode functions for the electromagnetic field are presented in the following
section. Section 3 considers the vacuum expectation values (VEVs) for the
diagonal components of the energy-momentum tensor. The contributions induced
by the cosmic string are separated explicitly and their asymptotics are
investigated. Special cases are discussed and the results of numerical
analysis are presented. In addition to the diagonal components, the vacuum
energy-momentum tensor has a nonzero off-diagonal component that describes
energy flux along the radial direction. This is a cosmic string induced effect
and is studied in Section 4. The main results of the paper are summarized in
Section 5.

##  2 Setup and electromagnetic mode functions

As a background geometry we consider $(D+1)$-dimensional spacetime having the
line element

$ds^{2}=dt^{2}-e^{2t/\alpha}[dr^{2}+r^{2}d\phi^{2}+\left(d\mathbf{z}\right)^{2}],$
(2.1)

where $\mathbf{z=}\left(z^{3},...,z^{D}\right)$ with $-\infty<z^{i}<+\infty$,
$-\infty<t<+\infty$, $0\leqslant r<\infty$, and
$0\leqslant\phi\leqslant\phi_{0}$. In the special case $\phi_{0}=2\pi$ this
line element describes $(D+1)$-dimensional dS spacetime sourced by the
cosmological constant $\Lambda=D(D-1)/(2\alpha^{2})$ and foliated by flat
spaces with cylindrical coordinates $(r,\phi,\mathbf{z})$. For $\phi_{0}<2\pi$
and at the points $r>0$ the local geometrical characteristics coincide with
those for dS spacetime. In particular, for the Ricci scalar one has
$R=D(D+1)/\alpha^{2}$. The geometry described by (2.1) is a generalization of
an idealized cosmic string, being a linear topological defect in $D=3$, for
spatial dimensions $D>3$. The defect core, given by $r=0$, presents a
$(D-2)$-dimensional spatial hypersurface. Introducing the new time coordinate
$\tau$ in accordance with $\tau=-\alpha e^{-t/\alpha}$, $-\infty<\tau<0$, the
line element is written in the form conformally flat for $r>0$:

$ds^{2}=\frac{\alpha^{2}}{\tau^{2}}[d\tau^{2}-dr^{2}-r^{2}d\phi^{2}-\left(d\mathbf{z}\right)^{2}].$
(2.2)

Similar to the case of the Minkowski bulk, for the model with $D=3$ the
deficit angle $2\pi-\phi_{0}$ in dS spacetime is induced by the vortex
solution of the Einstein-Abelian-Higgs equations in the presence of a
cosmological constant [22].

In quantum field theory the notion of the vacuum state, denoted as
$|0\rangle$, has a global nature and its properties are sensitive to both
local and global characteristics of the background spacetime. In the geometry
under consideration, the nontrivial spatial topology induced by the planar
angle deficit gives rise to additional vacuum polarization for quantum fields.
Here we are interested in the topological effects on the VEV of the energy-
momentum tensor $T_{i}^{l}$ for the electromagnetic field with the vector
potential $A_{i}(x)$. Our approach for evaluation of the expectation value
$\langle 0|T_{i}^{l}|0\rangle\equiv\langle T_{i}^{l}\rangle$ will be based on
the direct summation over the complete set of electromagnetic modes for the
background geometry (2.2). We will denote by
$\\{A_{(\beta)i}(x),A_{(\beta)i}^{\ast}(x)\\}$ the corresponding set for the
vector potential, where $\beta$ is the collective notation for the quantum
numbers specifying the solutions of Maxwell’s equations in $(D+1)$-dimensional
spacetime. With
$F_{(\beta)lm}=\partial_{l}A_{(\beta)m}-\partial_{m}A_{(\beta)l}$ being the
electromagnetic field tensor for the modes, the mode-sum for the vacuum
energy-momentum tensor is expressed as

$\langle
T_{i}^{l}\rangle=\frac{1}{4\pi}\underset{\beta}{\sum}\left[\frac{1}{4}\delta_{i}^{l}F_{(\beta)}^{mn}F_{(\beta)mn}^{\ast}-F_{(\beta)i}^{\cdot
m}F_{(\beta)\cdot m}^{l\ast}\right],$ (2.3)

where $F_{(\beta)}^{il}=g^{im}g^{ln}F_{(\beta)mn}$. In what follows it is
convenient to work in the coordinates $(\tau,r,\phi,\mathbf{z})$ with the
metric tensor
$g_{il}=\left(\alpha/\tau\right)^{2}\mathrm{diag}(1,-1,-r^{2},-1,\ldots,-1)$.
We will fix the vector potential by the gauge conditions $A_{0}=0$ and
$\sum_{i=1}^{D}\partial_{i}(\sqrt{|g|}A^{i})=0$ with
$\sqrt{|g|}=\left(\alpha/\eta\right)^{D+1}r$ and $\eta=|\tau|$.

The electromagnetic cylindrical modes have been discussed in [45] for
$(D+1)$-dimensional dS spacetime and were generalized for the presence of a
cosmic string in [31]. One has $D-1$ polarization states and we will
distinguish them by the index $\sigma=1,\ldots,D-1$. In addition to this
quantum number, the electromagnetic modes are specified by the wave vector
$\mathbf{k}=(k_{3},\ldots,k_{D})$ in the subspace covered by the coordinates
$\mathbf{z}$, by the radial quantum number $\gamma$, $0\leq\gamma<\infty$, and
by the azimuthal quantum number $m$, $m=0,\pm 1,\pm 2,\ldots$. Consequently,
the set $\beta$ is specified as $(\beta)=(\gamma,m,\mathbf{k},\sigma)$.
Introducing the notation $q=2\pi/\phi_{0}$, the vector potentials for separate
polarizations are expressed as [31]

$\displaystyle A_{(\beta)l}(x)$ $\displaystyle=$
$\displaystyle\eta^{\frac{D}{2}-1}Z_{\frac{D}{2}-1}(\omega\eta)\left\\{\begin{array}[]{c}\left(0,iqm/r,-r\partial_{r},0,\ldots,0\right)\\\
\left(0,\omega\epsilon_{\sigma
n}+i\omega^{-1}\mathbf{k}\cdot{\boldsymbol{\epsilon}}_{\sigma}\partial_{n}\right)\end{array}\right.$
(2.9) $\displaystyle\times J_{q|m|}(\gamma
r)e^{iqm\phi+i\mathbf{k}\cdot\mathbf{z}},\;\begin{array}[]{l}\sigma=1\\\
\sigma=2,\ldots,D-1\end{array},$

where $\omega=\sqrt{\gamma^{2}+|\mathbf{k}|^{2}}$, $n=1,\ldots,D$,
$J_{\nu}(y)$ is the Bessel function,
$\mathbf{k}\cdot\mathbf{z}=\sum_{i=3}^{D}k_{i}z^{i}$, and
$\mathbf{k}\cdot{\boldsymbol{\epsilon}}_{\sigma}=\sum_{i=3}^{D}k_{i}\epsilon_{\sigma
i}$, with $\epsilon_{\sigma 1}=\epsilon_{\sigma 2}=0$. In (2.9), the function
$Z_{\nu}(y)$ is the linear combination of the Hankel functions
$H_{\nu}^{(1)}(y)$ and $H_{\nu}^{(2)}(y)$,
$Z_{\nu}(y)=\sum_{i=1,2}c_{\beta}^{(i)}H_{\nu}^{(i)}(y)$. The polarization
vectors ${\boldsymbol{\epsilon}}_{\sigma}$ obey the orthonormalization
condition

$\sum_{l,n=3}^{D}\left(\delta_{nl}-\frac{k_{l}k_{n}}{\omega^{2}}\right)\epsilon_{\sigma
l}\epsilon_{\sigma^{\prime}n}=\frac{\gamma^{2}}{\omega^{2}}\delta_{\sigma\sigma^{\prime}}.$
(2.10)

The orthonormalization condition for the mode functions is written as

$\int
d^{D}x\sqrt{|g|}[A_{(\beta^{\prime})l}^{\ast}\nabla^{0}A_{(\beta)}^{l}-(\nabla^{0}A_{(\beta^{\prime})l}^{\ast})A_{(\beta)}^{l}]=4i\pi\delta(\gamma-\gamma^{\prime})\delta(\mathbf{k}-\mathbf{k}^{\prime})\delta_{mm^{\prime}}\delta_{\sigma\sigma^{\prime}},$
(2.11)

where $\nabla^{i}=g^{ik}\nabla_{k}$ and $\nabla_{k}$ is the covariant
derivative operator. This condition determines the one of the coefficients
$c_{\beta}^{(i)}$ in the linear combination of the Hankel functions in the
definition of the function $Z_{\nu}(y)$. Similar to that for pure dS bulk, the
second coefficient is fixed by the choice of the vacuum state. Here we will
assume that the electromagnetic field is prepared in the state that is the
analog of the Bunch-Davies vacuum for a scalar field in dS spacetime. For that
state $c_{\beta}^{(2)}=0$ and in the expressions (2.9) for the mode functions

$Z_{\frac{D}{2}-1}(\omega\eta)=c_{\beta}^{(1)}H_{\frac{D}{2}-1}^{(1)}(\omega\eta).$
(2.12)

With this choice, the normalization coefficient is found from (2.11):

$|c_{\beta}^{(1)}|^{2}=\frac{q}{4(2\pi\alpha)^{D-3}\gamma}.$ (2.13)

Note that this coefficient is the same for all the polarizations.

For $D=3$ one has
$\eta^{D/2-1}H_{D/2-1}^{(1)}(\omega\eta)=-ie^{-i\omega\tau}\sqrt{2/\pi\omega}$
and the mode functions (2.9) coincide with those for a cosmic string in the
Minkowski bulk. Of course, this is a consequence of the conformal invariance
of the electromagnetic field in 4-dimensional spacetime. For $D>3$ and in the
limit $\alpha\rightarrow\infty$ for a fixed value of the synchronous time $t$,
from (2.9) the mode functions are obtained for a cosmic string in
$(D+1)$-dimensional Minkowski bulk.

##  3 Diagonal components of the energy-momentum tensor

Having the complete set of the electromagnetic modes we can evaluate the VEV
of the energy-momentum tensor by making use of the formula (2.3). The
expression in the right-hand side of (2.3) is divergent and a regularization
is required. For example, the point splitting procedure can be employed with
the combination of the two-point functions from [31] (for vector field
correlators in dS spacetime see [46]-[53] and references therein). Another
possibility is to introduce a cutoff function, for example, in the form
$e^{-b\omega^{2}}$ with the cutoff parameter $b>0$. The important point to be
mentioned is the following. For $r>0$ the local geometry in the presence of
the cosmic string is the same as that for the pure dS spacetime and, hence,
for those points the divergences in the VEV of the energy-momentum tensor are
the same as well. From here it follows that if one extracts explicitly the
part in the regularized VEV corresponding to the pure dS geometry, then the
renormalization is required for that part only. The remaining topological
contribution is finite in the limit when the regularization is removed. Below
we will extract the pure dS part and the specific regularization scheme is not
essential in that procedure. In the evaluation of the diagonal components we
will regularize the expectation values by introducing the cutoff function
$e^{-b\omega^{2}}$.

###  3.1 General expressions

Substituting the modes (2.9) in (2.3) with the cutoff function
$e^{-b\omega^{2}}$ and by using the relation [31]

$\sum_{\sigma=2}^{D-1}\epsilon_{\sigma n}\epsilon_{\sigma
l}=\frac{\gamma^{2}\delta_{nl}+k_{n}k_{l}}{\omega^{2}},$ (3.1)

with $l,n=3,...,D$, for summation over the polarization states, the
regularized diagonal components of the vacuum energy-momentum tensor are
expressed as (no summation over $i$)

$\displaystyle\langle T_{i}^{i}\rangle_{\mathrm{reg}}$ $\displaystyle=$
$\displaystyle\frac{2^{2-D}q\eta^{D+2}}{\pi^{D/2+1}\Gamma(D/2-1)\alpha^{D+1}}\sideset{}{{}^{\prime}}{\sum}_{m=0}^{\infty}\int_{0}^{\infty}dk\,k^{D-3}\int_{0}^{\infty}d\gamma\gamma
e^{-b\omega^{2}}$ (3.2)
$\displaystyle\times\sum_{l=1,2}t_{l}^{(i)}[k,\gamma,J_{qm}(\gamma
r)]K_{\frac{D}{2}-l}(\omega\eta e^{-i\pi/2})K_{\frac{D}{2}-l}(\omega\eta
e^{i\pi/2}),$

where $k=|\mathbf{k}|$, $K_{\nu}(x)$ is the Macdonald function. The prime on
the sign of summation means that the term $m=0$ is taken with coefficient 1/2.
The Macdonald function is introduced instead of the Hankel function in the
expression for the electromagnetic modes. In (3.2), the function containing
the radial dependence is defined as

$t_{l}^{(i)}\left[k,\gamma,f\left(y\right)\right]=(a_{l}^{(i)}k^{2}+b_{l}^{(i)}\gamma^{2})\left[f^{\prime
2}(y)+\delta_{i}\frac{q^{2}m^{2}}{y^{2}}f^{2}(y)\right]+\left[\left(D-3\right)c_{l}^{(i)}k^{2}+d_{l}^{(i)}\gamma^{2}\right]f^{2}(y),$
(3.3)

where

$\delta_{i}=\left\\{\begin{array}[]{ll}1,&i=0,3,\ldots,D\\\
-1,&i=1,2\end{array}\right..$ (3.4)

For the coefficients in the definition of the function
$t_{l}^{(i)}\left[k,\gamma,f\left(y\right)\right]$ one has

$\displaystyle a_{l}^{(0)}$ $\displaystyle=$
$\displaystyle\left(2,2\right),\;b_{l}^{(0)}=\left(D-2,1\right),\;c_{l}^{(0)}=\left(1,1\right),\;d_{l}^{(0)}=\left(1,D-2\right),$
$\displaystyle a_{l}^{(1)}$ $\displaystyle=$
$\displaystyle\left(0,0\right),\;b_{l}^{(1)}=\left(2-D,-1\right),\
c_{l}^{(1)}=\left(1,-1\right),\;d_{l}^{(1)}=\left(-1,2-D\right),$
$\displaystyle a_{l}^{(2)}$ $\displaystyle=$
$\displaystyle\left(0,0\right),\;b_{l}^{(2)}=\left(D-2,1\right),\;c_{l}^{(2)}=\left(1,-1\right),\;d_{l}^{(2)}=\left(-1,2-D\right),$
(3.5)

and

$\displaystyle a_{l}^{(p)}$ $\displaystyle=$
$\displaystyle\left(2\frac{D-4}{D-2},-2\right),\;b_{l}^{(p)}=\left(D-4,-1\right),$
$\displaystyle c_{l}^{(p)}$ $\displaystyle=$
$\displaystyle\left(\frac{D-6}{D-2},\frac{4-D}{D-2}\right),\;d_{l}^{(p)}=\left(1,4-D\right),$
(3.6)

for $p=3,\ldots,D$. Here, the first and second terms correspond to $l=1$ and
$l=2$, respectively.

For the further transformation of (3.2) we use the integral representation
[54]

$K_{\nu}(\omega\eta e^{-i\pi/2})K_{\nu}(\omega\eta
e^{i\pi/2})=\frac{1}{2}\int_{0}^{\infty}\frac{dx}{x}\exp\left(-\omega^{2}\frac{x}{4}+\eta^{2}\frac{2}{x}\right)K_{\nu}\left(2\eta^{2}/x\right),$
(3.7)

for the product of the Macdonald functions. Substituting in (3.2), after
integration over $k$ we get (no summation over $i$)

$\displaystyle\langle T_{i}^{i}\rangle_{\mathrm{reg}}$ $\displaystyle=$
$\displaystyle\frac{2^{-D}q\eta^{D+2}}{\pi^{D/2+1}\alpha^{D+1}}\sideset{}{{}^{\prime}}{\sum}_{m=0}^{\infty}\int_{0}^{\infty}\frac{dx}{x}u^{D/2}e^{2\eta^{2}/x}\int_{0}^{\infty}d\gamma\gamma
e^{-\gamma^{2}/u}$ (3.8)
$\displaystyle\times\sum_{l=1,2}t_{l}^{(i)}(\gamma^{2}/u,J_{qm}(\gamma
r))K_{\frac{D}{2}-l}(2\eta^{2}/x),$

where $u=4/(x+4b)$ and

$\displaystyle t_{l}^{(i)}(v,f\left(y\right))$ $\displaystyle=$
$\displaystyle\left[\left(\frac{D}{2}-1\right)a_{l}^{(i)}+b_{l}^{(i)}v\right]\left[f^{\prime
2}(y)+\delta_{i}\frac{q^{2}m^{2}}{y^{2}}f^{2}(y)\right]$ (3.9)
$\displaystyle+\left[\left(D-3\right)\left(\frac{D}{2}-1\right)c_{l}^{(i)}+d_{l}^{(i)}v\right]f^{2}(y).$

The integrals over $\gamma$ are evaluated on the base of the formula [55]

$\int_{0}^{\infty}d\gamma\ \gamma e^{-\gamma^{2}/u}J_{qm}^{2}(\gamma
r)=\frac{u}{2}e^{-r^{2}u/2}I_{qm}(r^{2}u/2),$ (3.10)

with $I_{\nu}(x)$ being the modified Bessel function, and by using the
formulas presented in [31]. This gives (no summation over $i$)

$\displaystyle\langle T_{i}^{i}\rangle_{\mathrm{reg}}$ $\displaystyle=$
$\displaystyle\frac{(\eta/r)^{D+2}q}{2^{D/2}\pi^{D/2+1}\alpha^{D+1}}\sideset{}{{}^{\prime}}{\sum}_{m=0}^{\infty}\int_{0}^{\infty}\frac{dx}{x}w^{D/2+1}e^{2\eta^{2}/x}$
(3.11)
$\displaystyle\times\sum_{l=1,2}K_{\frac{D}{2}-l}(2\eta^{2}/x)\hat{W}_{l}^{(i)}\left(e^{-w}I_{qm}(w)\right),$

where $w=2r^{2}/(x+4b)$ and the operators $\hat{W}_{l}^{(i)}$ are defined by
the expression

$\hat{W}_{l}^{(i)}=b_{l}^{(i)}w\partial_{w}^{2}+\left[(D/2-1)a_{l}^{(i)}+b_{l}^{(i)}(1+w)+d_{l}^{(i)}w\right]\partial_{w}+e_{l}^{(i)},$
(3.12)

for the components with $i=0,3,...,D,$ and

by

$\hat{W}_{l}^{(i)}=(d_{l}^{(i)}-b_{l}^{(i)})w\partial_{w}+e_{l}^{(i)},$ (3.13)

for $i=1,2$. Here the notations

$e_{l}^{(0)}=\frac{D-1}{2}\left(D,D\right),\;e_{l}^{(i)}=\frac{D-1}{2}\left(D-4,2-D\right),$
(3.14)

are introduced with $i=1,2,\ldots,D$.

Next, in order to separate the topological contribution, we use the formula
[56]

$q\sideset{}{{}^{\prime}}{\sum}_{m=0}^{\infty}e^{-w}I_{qm}\left(w\right)=\sideset{}{{}^{\prime}}{\sum}_{j=0}^{[q/2]}e^{-2ws_{j}^{2}}-\frac{q}{\pi}\sin(q\pi)\int_{0}^{\infty}dz\frac{e^{-2w\cosh^{2}z}}{\cosh(2qz)-\cos(q\pi)},$
(3.15)

where $s_{j}=\sin(j\pi/q)$ and $[q/2]$ is the integer part of $q/2$. The prime
on the summation sign in the right-hand side means that the terms $j=0$ and
$j=q/2$ (for even values of $q$) should be taken with additional coefficient
$1/2$. In the case $q=1$ the $j=0$ term remains only. This shows that the part
in the VEV corresponding to the $j=0$ term in (3.15) corresponds to the
regularized VEV of the energy-momentum tensor in dS spacetime in the absence
of the cosmic string. It is presented as (no summation over $i$)

$\langle
T_{i}^{i}\rangle_{\mathrm{reg}}^{\mathrm{(dS)}}=\frac{\alpha^{-D-1}}{\left(2\pi\right)^{D/2+1}}\int_{0}^{\infty}dy\frac{y^{D/2}e^{y}}{(1+2by/\eta^{2})^{D/2+1}}\sum_{l=1,2}K_{\frac{D}{2}-l}(y)e_{l}^{(i)}.$
(3.16)

The renormalized VEV of the energy-momentum tensor in dS spacetime, $\langle
T_{i}^{l}\rangle_{\mathrm{ren}}^{\mathrm{(dS)}}$, has been widely investigated
in the literature and our main concern here will be the cosmic string induced
contribution. From the maximal symmetry of the dS spacetime and of the Bunch-
Davies vacuum state the structure $\langle
T_{i}^{l}\rangle_{\mathrm{ren}}^{\mathrm{(dS)}}=\mathrm{const}\cdot\delta_{i}^{l}$
is expected for the geometry in the absence of the cosmic string.

For the topological part induced by the cosmic string one has (no summation
over $i$)

$\langle T_{i}^{i}\rangle_{\mathrm{t}}=\underset{b\rightarrow
0}{\lim}\left[\langle T_{i}^{i}\rangle_{\mathrm{reg}}-\langle
T_{i}^{i}\rangle_{\mathrm{reg}}^{\mathrm{(dS)}}\right].$ (3.17)

For $r>0$ the limit in the right-hand side is finite. From (3.11), by using
(3.15), we find the following representation (no summation over $i$)

$\langle
T_{i}^{i}\rangle_{\mathrm{t}}=\frac{2\alpha^{-D-1}}{(2\pi)^{D/2+1}}\left[\sideset{}{{}^{\prime}}{\sum}_{j=1}^{[q/2]}t^{(i)}\left(r/\eta,s_{j}\right)-\frac{q}{\pi}\sin(q\pi)\int_{0}^{\infty}dz\frac{t^{(i)}\left(r/\eta,\cosh
z\right)}{\cosh(2qz)-\cos(q\pi)}\right].$ (3.18)

In (3.18) we have used the notation

$t^{(i)}(x,y)=\int_{0}^{\infty}duu^{\frac{D}{2}}e^{u-2ux^{2}y^{2}}\sum_{l=1,2}K_{\frac{D}{2}-l}(u)f_{l}^{(i)}(x,y,u),$
(3.19)

with the functions

$f_{l}^{(i)}(x,y,u)=\left[4b_{l}^{(i)}ux^{2}y^{2}-2(b_{l}^{(i)}+d_{l}^{(i)})ux^{2}-((D-2)a_{l}^{(i)}+2b_{l}^{(i)})\right]y^{2}+e_{l}^{(i)},$
(3.20)

for $i=0,3,...,D$, and

$f_{l}^{(i)}(x,y,u)=2(b_{l}^{(i)}-d_{l}^{(i)})ux^{2}y^{2}+e_{l}^{(i)},$ (3.21)

for $i=1,2$. The topological part of the VEV depends on $\eta$ and $r$ in the
form of the ratio $r/\eta$. This property is a consequence of the maximal
symmetry of dS spacetime. By considering that $\alpha r/\eta$ is the proper
distance from the string, we see that $r/\eta$ is the proper distance,
measured in units of the dS curvature scale $\alpha$.

Note that the function (3.19) can also be written in the form

$\displaystyle t^{(i)}(x,y)$ $\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}du\,u^{\frac{D}{2}}e^{u-2ux^{2}y^{2}}\sum_{l=1,2}K_{\frac{D}{2}-l}(u)$
(3.22)
$\displaystyle\times\left[2a_{li}^{(4)}ux^{2}y^{4}+a_{li}^{(3)}ux^{2}y^{2}+a_{li}^{(2)}y^{2}+\frac{D-1}{4}a_{li}^{(1)}\right],$

with $i=0,1,2,3$. The coefficients are given by the expressions

$\displaystyle a_{li}^{(1)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}D&D-4&D-4&D-4\\\
D&2-D&2-D&2-D\end{array}\right),$ (3.25) $\displaystyle a_{li}^{(2)}$
$\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}2\left(2-D\right)&0&0&2(4-D)\\\
1-D&0&0&D-1\end{array}\right),$ (3.28)

and

$\displaystyle a_{li}^{(3)}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}1-D&3-D&D-1&3-D\\\
1-D&D-3&D-1&D-3\end{array}\right),$ (3.31) $\displaystyle a_{li}^{(4)}$
$\displaystyle=$ $\displaystyle\left(\begin{array}[]{cccc}D-2&0&0&D-4\\\
1&0&0&-1\end{array}\right).$ (3.34)

For odd values of $D$ the integral in (3.22) is expressed in terms of
elementary functions. By analogy with an ideal fluid, the diagonal components
$\langle T_{i}^{i}\rangle_{\mathrm{t}}$ of the energy-momentum tensor can be
interpreted as topological contributions to the vacuum pressures along
respective directions, $P_{\mathrm{t}i}=-\langle
T_{i}^{i}\rangle_{\mathrm{t}}$ with $i=1,2,\ldots,D$. These pressures are
anisotropic.

It can be checked that the diagonal components obey the trace relation

$\langle T_{i}^{i}\rangle_{\mathrm{t}}=(3-D)\langle L\rangle_{\mathrm{t}},$
(3.35)

where the topological part in the VEV of the Lagrangian density
$L=-F_{il}F^{il}/(16\pi)$ is given by

$\langle
L\rangle_{\mathrm{t}}=\frac{2\alpha^{-D-1}}{(2\pi)^{D/2+1}}\left[\sum_{l=1}^{[q/2]}g_{L}(r/\eta,s_{l})-\frac{q}{\pi}\sin(q\pi)\int_{0}^{\infty}dy\frac{g_{L}(r/\eta,\cosh
y)}{\cosh(2qy)-\cos(q\pi)}\right],$ (3.36)

with the function

$\displaystyle g_{L}(x,y)$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}du\,u^{\frac{D}{2}}e^{u-2x^{2}y^{2}u}\left\\{K_{\frac{D}{2}-2}(u)\left[2ux^{2}y^{2}\left(2y^{2}-D+1\right)+\left(D-1\right)\left(D/2-2y^{2}\right)\right]\right.$
(3.37)
$\displaystyle-\left.K_{\frac{D}{2}-1}(u)\left[2ux^{2}y^{2}\left(2(D-2)y^{2}-D+1\right)+(D-1)D/2-4(D-2)y^{2}\right]\right\\}.$

The electric and magnetic contributions to the VEV of the Lagrangian density
have been discussed in [31]. By using the corresponding expressions for the
VEVs of the squared electric and magnetic fields, we can check that the
formula (3.36) is obtained. In the special case $D=3$ the electromagnetic
field is conformally invariant and the topological part is traceless. The
trace relation $T_{i}^{i}=(3-D)L$ in classical electrodynamics directly
follows from the definition of the energy-momentum tensor. In quantum field
theory anomalies may arise in the corresponding relation between the VEVs. The
well-known example is the trace anomaly for the energy-momentum tensor of a
conformally coupled massless scalar field. In the problem at hand we have
local dS geometry for $r>0$ and the trace anomaly is contained in the pure dS
part of the VEV. The trace relation for the topological parts in the VEVs is
the same as that in classical electrodynamics.

###  3.2 Special cases and asymptotics

In the special cases $D=3$ and $D=5$ from (3.19) one gets

$\displaystyle t^{(i)}(x,y)$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2\pi}}{4(xy)^{4}}A^{(i)},\ D=3,$ $\displaystyle
t^{(i)}(x,y)$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2\pi}}{4(xy)^{6}}(B^{(i)}y^{2}+C^{(i)}x^{2}y^{2}+D^{(i)}),\
D=5,$ (3.38)

where

$\displaystyle A^{(0)}$ $\displaystyle=$ $\displaystyle A^{(1)}=A^{(3)}=1,\
A^{(2)}=-3,$ $\displaystyle B^{(0)}$ $\displaystyle=$ $\displaystyle
B^{(i)}=2,\ B^{(1)}=B^{(2)}=0,$ $\displaystyle C^{(0)}$ $\displaystyle=$
$\displaystyle-C^{(1)}=1,\ C^{(2)}=5,$ $\displaystyle D^{(0)}$
$\displaystyle=$ $\displaystyle D^{(1)}=-2,\ D^{(2)}=10,$ (3.39)

and $C^{(i)}=-1$,$\;D^{(i)}=-2$, for $i=3,4,5$. The corresponding topological
parts in the energy-momentum tensor are expressed in terms of the functions

$c_{n}(q)=\sum_{j=1}^{[q/2]}s_{j}^{-n}-\frac{q}{\pi}\sin(q\pi)\int_{0}^{\infty}dz\frac{\cosh^{-n}z}{\cosh(2qz)-\cos(q\pi)}.$
(3.40)

For even values of $n$, the functions $c_{n}(q)$ can be found by using the
recurrence scheme described in [57]. In particular, one has
$c_{2}(q)=(q^{2}-1)/6$ and

$\displaystyle c_{4}(q)$ $\displaystyle=$
$\displaystyle\frac{q^{2}-1}{90}(q^{2}+11),$ $\displaystyle c_{6}(q)$
$\displaystyle=$ $\displaystyle\frac{q^{2}-1}{1890}(2q^{4}+23q^{2}+191).$
(3.41)

By using these results we find (no summation over $i$)

$\langle T_{i}^{i}\rangle_{\mathrm{t}}=-\frac{A^{(i)}c_{4}(q)}{8\pi^{2}(\alpha
r/\eta)^{4}},$ (3.42)

for $D=3$ and

$\langle
T_{i}^{i}\rangle_{\mathrm{t}}=\frac{\left[B^{(i)}+(r/\eta)^{2}C^{(i)}\right]c_{4}(q)+D^{(i)}c_{6}(q)}{16\pi^{3}(\alpha
r/\eta)^{6}},$ (3.43)

for $D=5$. In the case $D=3$ the off-diagonal components of the energy-
momentum tensor vanish (see below) and from (3.42) one gets (no summation over
$i$) $\langle T_{i}^{i}\rangle_{\mathrm{t}}=(\eta/\alpha)^{4}\langle
T_{i}^{i}\rangle_{\mathrm{t}}^{\mathrm{(M)}}$, where $\langle
T_{i}^{i}\rangle_{\mathrm{t}}^{\mathrm{(M)}}$ is the corresponding VEV in
Minkowski bulk [58, 59]. This is a direct consequence of the conformal
invariance of the electromagnetic field in $D=3$ spatial dimensions and of the
conformal flatness of the background geometry. For $D=3$ the energy density is
negative everywhere, whereas for $D=5$ it can be either negative or positive,
depending on the values of the parameters (see the graphs below).

In a similar way, for $D=7$ we get (no summation over $i$)

$\langle T_{i}^{i}\rangle_{\mathrm{t}}=\frac{3\pi^{-4}}{2^{6}\left(\alpha
r/\eta\right)^{8}}\sum_{l=1}^{3}E_{i+1,l}(r/\eta)c_{2+2l}(q),$ (3.44)

with the $4\times 3$ matrix

$E_{k,l}(x)=\left(\begin{array}[]{ccc}x^{2}(8+9x^{2})&4(2+x^{2})&-3\\\
x^{4}&-4x^{2}&-3\\\ 21x^{4}&28x^{2}&21\\\
x^{2}(8+x^{2})&4(2-x^{2})&-3\end{array}\right),$ (3.45)

where $k=1,2,3,4$ and $l=1,2,3$. The functions $c_{4}(q)$ and $c_{6}(q)$ are
given by (3.41) and the function $c_{8}(q)$ is obtained by using the
recurrence scheme from [57]:

$c_{8}(q)=\frac{q^{2}-1}{28350}\left(3q^{6}+43q^{4}+337q^{2}+2497\right).$
(3.46)

As it will be seen below, the functions $c_{n}(q)$ also appear in the
asymptotic expressions for the components of the energy-momentum tensor. The
function $c_{n}(q)$, $n\geq 1$, is positive for $q>1$ and monotonically
increasing in that region. Additionally, one has $c_{n}(1)=0$ and
$c_{n}(2)=1/2$. We also have the properties $c_{n+1}(q)<c_{n}(q)$ for $1<q<2$
and $c_{n}(q)<c_{n+1}(q)$ for $q>2$. The asymptotic behavior of the function
$c_{n}(q)$ for large values of $q$ can be found from (3.40). For $q\gg 1$ the
dominant contribution in (3.40) comes from the terms in the sum with $j\ll q$
and we get $c_{n}(q)\approx\left(q/\pi\right)^{n}\zeta(n)$, where $\zeta(n)$
is the Riemann zeta function. Note that $\zeta(4)=\pi^{4}/90$,
$\zeta(6)=\pi^{6}/945$, $\zeta(8)=\pi^{8}/9450$, and this estimate is in
agreement with the exact formulas (3.41) and (3.46).

The Minkowskian limit corresponds to $\alpha\rightarrow\infty$ for a fixed
value of the time coordinate $t$. In this case one has $\eta\approx\alpha-t$
and $\eta$ is large. Hence, we need the asymptotic of the function
$t^{(i)}(x,y)$ for small values of $x$. In this limit the dominant
contribution to the integral in (3.19) comes from large values of $u$ and
using the asymptotic expression $K_{\nu}(u)\approx\sqrt{\pi/(2u)}e^{-u}$ for
the Macdonald function for large argument, to the leading order we get the VEV
$\langle T_{i}^{k}\rangle_{\mathrm{t}}^{\mathrm{(M)}}$ for a cosmic string on
the Minkowski bulk. The VEV is diagonal and the expression for the components
with $i=0,3,\ldots,D$ has the form (no summation over $i$) [32]

$\langle
T_{i}^{i}\rangle_{\mathrm{t}}^{\mathrm{(M)}}(r)=\frac{\Gamma(\left(D+1\right)/2)}{\left(4\pi\right)^{(D+1)/2}r^{D+1}}\left[\left(D-3\right)^{2}c_{D-1}(q)-\left(D-1\right)c_{D+1}(q)\right].$
(3.47)

The corresponding energy density can be either positive or negative, depending
on the parameters $q$ and $D$. The radial and azimuthal components are given
by the expressions

$\langle T_{1}^{1}\rangle_{\mathrm{t}}^{\mathrm{(M)}}(r)=-\frac{1}{D}\langle
T_{2}^{2}\rangle_{\mathrm{t}}^{\mathrm{(M)}}(r)=-\frac{\Gamma(\left(D+1\right)/2)}{\left(4\pi\right)^{(D+1)/2}r^{D+1}}\left(D-1\right)c_{D+1}(q).$
(3.48)

In the special case $D=3$ from (3.47) and (3.48) we get the result obtained in
[58, 59]. For the Minkowski bulk, the stresses $\langle
T_{l}^{l}\rangle_{\mathrm{t}}^{\mathrm{(M)}}(r)$ along the directions
$l=3,\ldots,D$ are equal to the energy density $\langle
T_{0}^{0}\rangle_{\mathrm{t}}^{\mathrm{(M)}}(r)$. The corresponding equation
of state, $P_{\mathrm{t}l}^{\mathrm{(M)}}=-\langle
T_{l}^{l}\rangle_{\mathrm{t}}^{\mathrm{(M)}}=-\langle
T_{0}^{0}\rangle_{\mathrm{t}}^{\mathrm{(M)}}$, is of the cosmological constant
type, though the energy density and pressures depend on the radial coordinate.
For dS background geometry with spatial dimensions $D>3$ the stresses along
the directions parallel to the core differ from the energy density.The energy
density corresponding to (3.47) is negative for $D=3,4$. In spatial dimensions
$D\geq 5$ the energy density becomes zero for special values $q_{0}$ of the
parameter $q$. In those dimensions one has $\langle
T_{0}^{0}\rangle_{\mathrm{t}}^{\mathrm{(M)}}>0$ for $1<q<q_{0}$ and $\langle
T_{0}^{0}\rangle_{\mathrm{t}}^{\mathrm{(M)}}>0$ for $q>q_{0}$. The critical
value $q_{0}$ increases with increasing $D$. For example, we have $q_{0}=2$
for $D=5$ and $q_{0}\approx 3.83,4.86$ for $D=6,7$, respectively. For large
values of $q$, by using the corresponding asymptotic for the function
$c_{n}(q)$, we get (no summation over $i$)

$\langle
T_{i}^{i}\rangle_{\mathrm{t}}^{\mathrm{(M)}}(r)\approx\frac{\left(1-D\right)\zeta(D+1)}{\pi^{3(D+1)/2}(2r)^{D+1}}\Gamma\left(\frac{D+1}{2}\right)q^{D+1},$
(3.49)

for $i=0,1,3,\ldots,D$.

Now let us consider the asymptotic behavior of the diagonal components for the
energy-momentum tensor at large and small distances from the cosmic string. At
small proper distances compared to the curvature radius of the background
spacetime one has $r/\eta\ll 1$ and we need the asymptotic of the function
$t^{(i)}(x,y)$ for $x\ll 1$. For those $x$ the dominant contribution to the
integral in (3.19) comes from large values of $u$. By using the corresponding
asymptotic for the Macdonald function we get

$t^{(i)}(x,y)\approx\frac{\sqrt{\pi}\Gamma\left((D+1)/2\right)}{2^{D/2+1}\left(xy\right)^{D+1}}t^{(i)}(y),$
(3.50)

where $t^{(i)}(y)=\left(D-3\right)^{2}y^{2}-D+1$ for $i=0,3,\ldots,D$, and
$t^{(1)}(y)=-t^{(2)}(y)/D=1-D$. Substituting those asymptotics in (3.18) we
see that, to the leading order (no summation over $i$),

$\langle T_{i}^{i}\rangle_{\mathrm{t}}\approx\langle
T_{i}^{i}\rangle_{\mathrm{t}}^{\mathrm{(M)}}(\alpha r/\eta),\;r/\eta\ll 1.$
(3.51)

Hence, the leading terms in the asymptotic expansions of the diagonal
components near the string coincide with the corresponding expressions for the
Minkowski bulk with the distance from the string replaced by the proper
distance in the dS bulk. In particular, the energy density near the cosmic
string is negative in spatial dimensions $D=3,4$. As it has been discussed
above, for $D\geq 5$ for critical values $q=q_{0}$ the energy density $\langle
T_{0}^{0}\rangle_{\mathrm{t}}^{\mathrm{(M)}}$ becomes zero. Near those points
and for dS bulk, in the expansion over $\alpha r/\eta$ the next to the leading
terms should be kept. For large values of $q$ the energy density near the
cosmic string is negative with the asymptotic obtained from (3.49) making the
replacement $r\rightarrow\alpha r/\eta$. Near the cosmic string the radial
stress is negative and the azimuthal stress is positive, $\langle
T_{1}^{1}\rangle_{\mathrm{t}}<0$ and $\langle
T_{2}^{2}\rangle_{\mathrm{t}}>0$.

At large distances from the cosmic string one has $r/\eta\gg 1$ and in (3.18)
the asymptotic for the function $t^{(i)}(x,y)$ is required for large values of
the first argument. For $D=3$ we have a conformal relation with the VEV in the
Minkowski bulk and $\langle T_{i}^{i}\rangle_{\mathrm{t}}\propto
1/(r/\eta)^{4}$ for all values of the ratio $r/\eta$. In order to estimate the
integral for $D\geq 4$, we note that in the limit under consideration the
dominant contribution to the integral in (3.19) comes from the region near the
lower limit and we use the asymptotic expression for the Macdonald function
for small argument. In the leading order, this gives

$t^{(i)}(x,y)\approx 2^{D/2-1}\frac{\Gamma(D/2-1)}{16(xy)^{4}}t_{0}^{(i)},$
(3.52)

with the notations

$\displaystyle t_{0}^{(0)}$ $\displaystyle=$
$\displaystyle(D-1)(D-4),\;t_{0}^{(2)}=D(D-1),$ $\displaystyle t_{0}^{(i)}$
$\displaystyle=$ $\displaystyle(D-3)(D-6)-2,\;i=1,3,...,D.$ (3.53)

The exception is the case $D=4$ for the function $t^{(0)}(x,y)$:

$t^{(0)}(x,y)\approx-\frac{3}{2}\frac{\ln(xy)}{(xy)^{6}},\;D=4.$ (3.54)

With the asymptotic (3.52), from (3.18) we find the leading behavior at large
distances (no summation over $i$)

$\langle
T_{i}^{i}\rangle_{\mathrm{t}}\approx\frac{\Gamma(D/2-1)t_{0}^{(i)}c_{4}(q)}{32\pi^{D/2+1}\alpha^{D+1}(r/\eta)^{4}},\;D\geq
4,$ (3.55)

except $i=0$ for $D=4$. For the energy density in the special case $D=4$ one
gets

$\langle
T_{0}^{0}\rangle_{\mathrm{t}}\approx-\frac{3c_{6}(q)\ln(r/\eta)}{8\pi^{3}\alpha^{4}(r/\eta)^{6}},\;D=4,$
(3.56)

and the decay is stronger. At large distance from the cosmic string the
topological contribution to the energy density is negative for $D=3,4$ and
positive for $D>4$. The stresses $\langle T_{i}^{i}\rangle_{\mathrm{t}}$, with
$i=1,3,...,D$, are negative for $3\leq D\leq 6$ and positive for $D>6$. The
stress $\langle T_{2}^{2}\rangle_{\mathrm{t}}$ is positive for $D\geq 3$. It
is of interest to note that the topological contributions in the diagonal
components decay at large distances as the inverse fourth power of the proper
distance from the cosmic string in all spatial dimensions $D\geq 3$. The
exception is the energy density in 4-dimensional space with the leading term
(3.56). This behavior is in contrast to the geometry of a cosmic string in the
Minkowski bulk where the VEV decays like $1/r^{D+1}$.

Finally, we turn to the asymptotic for large values of the parameter $q$ and
fixed $r/\alpha$, $q\gg 1,r/\alpha$. The dominant contribution to the
topological parts (3.18) come from the terms in the sum over $j$ with
$j\pi/q\ll 1$. For those $j$ one has $s_{j}\approx$ $j\pi/q\ll 1$ and we need
the asymptotic of the function $t^{(i)}(x,y)$ for $y\ll 1$ and $xy\ll 1$. In
the region under consideration the integral in (3.22) is dominated by the
contribution from large values of $u$. By using the corresponding asymptotic
for the function $K_{D/2-l}(u)$, to the leading order we get

$t^{(i)}(x,y)\approx\frac{\sqrt{\pi}\left(1-D\right)}{2^{D/2+1}\left(xy\right)^{D+1}}\Gamma\left(\frac{D+1}{2}\right)\left[A^{(i)}-\delta_{2}^{i}(D-3)\right],$
(3.57)

for $i=0,1,2,3$. Substituting this, with $y=s_{j}\approx$ $j\pi/q$, in (3.18),
the sum over $j$ is approximated by the Riemann zeta function $\zeta(D+1)$ and
for the leading term one finds (no summation over $i$)

$\langle
T_{i}^{i}\rangle_{\mathrm{t}}\approx\frac{\left(1-D\right)\zeta(D+1)}{\pi^{3(D+1)/2}\left(2\alpha
r/\eta\right)^{D+1}}\Gamma\left(\frac{D+1}{2}\right)\left[A^{(i)}-\delta_{2}^{i}(D-3)\right]q^{D+1},$
(3.58)

for $q\gg 1$. We can show that in the special cases $D=3,5,7$ this general
result is in agreement with (3.42), (3.43), and (3.44). For $D=3$ the estimate
(3.58) directly follows from (3.42) with combination of the approximation
$c_{n}(q)\approx\left(q/\pi\right)^{n}\zeta(n)$ for large $q$. For $D=5$ and
$D=7$ we note that the dominant contributions to the corresponding expressions
come from the terms containing $c_{6}(q)$ and $c_{8}(q)$, respectively, and
the same approximation for those functions gives the result (3.58). The
leading term (3.58) is traceless and the stresses $\langle
T_{i}^{i}\rangle_{\mathrm{t}}$ with $i=1,3,\ldots,D$, coincide with the energy
density $\langle T_{0}^{0}\rangle_{\mathrm{t}}$. The latter is negative for
all values of $D$. In the Minkowskian limit the estimate agrees with (3.49).

In Figure 1 we have displayed the dependence of the diagonal components of the
topological contributions in the VEV of the energy-momentum tensor, $\langle
T_{i}^{i}\rangle_{\mathrm{t}}$ (in units of $1/\alpha^{D+1}$), on the ratio
$r/\eta$ (proper distance from the cosmic string in units of the dS curvature
radius $\alpha$). The numbers near the curves correspond to the value of the
index $i$ and the left and right panels are plotted for $D=5$ and $D=6$,
respectively. For the planar angle deficit we have taken the value
corresponding to $q=1.5$. The dashed curves on both panels present the energy
flux (see below). For both cases $D=5,6$ the energy density is positive. For
$D=3,4$ the corresponding energy density is negative. In general, as it has
been demonstrated by the asymptotic analysis, depending on the values of $D$
and $q$, the energy density can be either positive or negative. For the values
of the parameters corresponding to Figure 1 the radial and azimuthal stresses
are monotonic functions of $r/\eta$, whereas the stresses $\langle
T_{i}^{i}\rangle_{\mathrm{t}}$, $i=1,3,\ldots,D$, are positive near the cosmic
string and negative al large distances.

|
---|---
Figure 1: The radial dependence of the topological contributions in the vacuum
energy-momentum tensor for spatial dimensions $D=5$ (left panel) and $D=6$
(right panel). The numbers near the curves correspond to the value of the
index $i$ of the diagonal components $\langle T_{i}^{i}\rangle_{\mathrm{t}}$
and the graphs are plotted for $q=1.5$. The dashed curves present the energy
flux $\langle T_{0}^{1}\rangle_{\mathrm{t}}$.

To show the dependence on the planar angle deficit, in Figures 2 and 3 we have
presented the topological contributions in the diagonal components of the
vacuum energy-momentum tensor, $10^{2}\alpha^{D+1}\langle
T_{i}^{i}\rangle_{\mathrm{t}}$, as functions of the ratio $r/\eta$ and of the
parameter $q$ for the spatial dimension $D=5$. The left and right panels of
Figure 2 correspond to the energy density ($i=0$) and the radial stress
($i=1$), respectively. The left and right panels in Figure 3 present the
azimuthal stress ($i=2$) and the axial stress ($i=3$). Depending on the values
of $q$ and $r/\eta$, the energy density and the axial stress corresponding to
the topological contributions can be either positive or negative. The radial
azimuthal stresses are monotonic functions of both variables. These features
are in agreement with the asymptotic analysis described above.

|
---|---
Figure 2: The topological contributions in the vacuum energy density, $10^{2}\alpha^{6}\langle T_{0}^{0}\rangle_{\mathrm{t}}$ (left panel), and in the radial stress, $10^{2}\alpha^{6}\langle T_{1}^{1}\rangle_{\mathrm{t}}$ (right panel), versus the proper distance from the cosmic string (in units of the curvature radius $\alpha$) and the parameter $q$ determining the planar angle deficit. The graphs are plotted for the model with $D=5$. |
---|---
Figure 3: The same as in Figure 2 for the azimuthal stress
$10^{2}\alpha^{6}\langle T_{2}^{2}\rangle_{\mathrm{t}}$ (left panel), and the
axial stress $10^{2}\alpha^{6}\langle T_{3}^{3}\rangle_{\mathrm{t}}$ (right
panel).

##  4 Energy flux

In the previous section we have investigated the diagonal components of the
energy-momentum tensor. The problem under consideration is not homogeneous
with respect to the coordinates $r$, $\eta$ and, in addition to the diagonal
components, one has a nonzero off-diagonal component $\langle
T_{0}^{1}\rangle$ that corresponds to the energy flux along the radial
direction. The respective mode sum is obtained from (2.3) with the mode
functions (2.9). Again, by using (3.1) for the summation over polarizations,
one gets

$\langle
T_{0}^{1}\rangle=\frac{2i(D-3)q\eta^{D+2}}{(2\pi)^{D}\alpha^{D+1}}\partial_{r}\sideset{}{{}^{\prime}}{\sum}_{m=0}^{\infty}\int
d\mathbf{k}\int_{0}^{\infty}d\gamma\,\omega\gamma K_{\frac{D}{2}-2}(\omega\eta
e^{-\pi i/2})K_{\frac{D}{2}-1}(\omega\eta e^{\pi i/2})J_{qm}^{2}(\gamma r).$
(4.1)

The radial derivative in the right-hand side excludes the contribution of the
part corresponding to the dS geometry in the absence of the cosmic string.
That part is zero as a consequence of the problem symmetry. The nonzero energy
flux is a purely topological effect induced by the string. It is finite for
$r>0$ and the cutoff function can be removed from the beginning.

For the further transformation of the energy flux we use the integral
representation

$\displaystyle K_{\nu-1}(e^{-i\pi/2}\omega\eta)K_{\nu}(e^{i\pi/2}\omega\eta)$
$\displaystyle=$
$\displaystyle-\frac{i}{2\eta\omega}\int_{-\infty}^{+\infty}dy\,e^{-2\nu
y}\int_{0}^{\infty}\frac{dx}{x}\left(\nu-\frac{1}{x}\right)$ (4.2)
$\displaystyle\times\exp\left(-x\eta^{2}\omega^{2}\sinh^{2}y-\frac{1}{x}\right).$

The latter is obtained from the representation [54]

$K_{\nu}(e^{-i\pi/2}\omega\eta)K_{\nu}(e^{i\pi/2}\omega\eta^{\prime})=\frac{1}{2}\int_{-\infty}^{+\infty}dy\,e^{-2\nu
y}\int_{0}^{\infty}\frac{du}{u}e^{-u/2-\omega^{2}\beta/(2u)},$ (4.3)

with the notation $\beta=2\eta\eta^{\prime}\cosh(2y)-\eta^{2}-\eta^{\prime
2}$, taking the derivative with respect to the argument of the second
Macdonald function in (4.3) and then passing to the limit
$\eta^{\prime}\rightarrow\eta$. Substituting (4.2) (with $\nu=D/2-1$) into the
expression (4.1), the integral over $\gamma$ is evaluated by using (3.10).
After some transformations we find

$\displaystyle\langle T_{0}^{1}\rangle_{\mathrm{t}}$ $\displaystyle=$
$\displaystyle\frac{(3-D)q\eta}{2\left(2\pi\right)^{D/2+1}\alpha^{D+1}}\int_{0}^{\infty}du\,u^{\frac{D}{2}-1}e^{u}K_{\frac{D}{2}-1}(u)$
(4.4)
$\displaystyle\times\partial_{r}\left(2+r\partial_{r}\right)e^{-u(r/\eta)^{2}}\sideset{}{{}^{\prime}}{\sum}_{m=0}^{\infty}I_{qm}(u(r/\eta)^{2}).$

Next, by using the formula (3.15), one gets

$\langle
T_{0}^{1}\rangle_{\mathrm{t}}=\frac{8(D-3)r}{\left(2\pi\right)^{D/2+1}\alpha^{D+1}\eta}\left[\sum_{j=1}^{[q/2]}t^{(01)}(r/\eta,s_{j})-\frac{q}{\pi}\sin(q\pi)\int_{0}^{\infty}dy\frac{t^{(01)}(r/\eta,\cosh
y)}{\cosh(2qy)-\cos(q\pi)}\right],$ (4.5)

where the notation

$t^{(01)}(x,y)=y^{2}\int_{0}^{\infty}du\,u^{\frac{D}{2}}\left(1-uy^{2}x^{2}\right)K_{\frac{D}{2}-1}(u)e^{u-2x^{2}y^{2}u},$
(4.6)

has been used. Similar to the case of the diagonal components, the energy flux
density $\langle T_{0}^{1}\rangle_{\mathrm{t}}$ depends on the radial and time
coordinates through the ratio $r/\eta$ that presents to the proper distance
from the cosmic string measured in units of the dS curvature radius $\alpha$.
An equivalent expression for the function $t^{(01)}(x,y)$ is given by

$t^{(01)}(x,y)=\frac{y^{2}}{2}\int_{0}^{\infty}du\,u^{\frac{D}{2}+1}e^{u-2x^{2}y^{2}u}\left[K_{\frac{D}{2}-2}(u)-K_{\frac{D}{2}-1}(u)\right].$
(4.7)

This shows that this function is negative. For $\langle
T_{0}^{1}\rangle_{\mathrm{t}}>0$ the energy flux is directed from the cosmic
string and for $\langle T_{0}^{1}\rangle_{\mathrm{t}}<0$ it is directed
towards the string.

As we could expect from the conformal relation with the problem of cosmic
string in the Minkowski bulk, the energy flux vanishes for $D=3$. For other
odd values of the spatial dimension $D$ the integral in (4.6) is expressed in
terms of the elementary functions. In particular, we get

$\langle
T_{0}^{1}\rangle_{\mathrm{t}}=-\frac{c_{4}(q)}{8\pi^{3}\alpha^{6}\left(r/\eta\right)^{5}},$
(4.8)

for $D=5$ and

$\langle
T_{0}^{1}\rangle_{\mathrm{t}}=-\frac{3}{8}\frac{(r/\eta)^{2}c_{4}(q)+c_{6}(q)}{\pi^{4}\alpha^{8}(r/\eta)^{7}},$
(4.9)

for $D=7$. In both these cases $\langle T_{0}^{1}\rangle_{\mathrm{t}}<0$.

For large values of the curvature radius, corresponding to the Minkowskian
limit, the dominant contribution in the integral representation (4.6) for the
function $t^{(01)}(r/\eta,y)$ comes from the integration range with large
values of $u$. To the leading order we get

$\langle
T_{0}^{1}\rangle_{\mathrm{t}}\approx-\frac{\Gamma\left((D+1)/2\right)}{\left(4\pi\right)^{(D+1)/2}\alpha
r^{D}}(D-3)^{2}c_{D-1}(q).$ (4.10)

As we could expect, in the Minkowskian limit ($\alpha\rightarrow\infty$) the
energy flux vanishes. The asymptotic of the energy flux near the cosmic
string, corresponding to $r/\eta\ll 1$, is found in a similar way with the
result

$\langle
T_{0}^{1}\rangle_{\mathrm{t}}\approx-\frac{(D-3)^{2}\Gamma\left((D+1)/2\right)}{\left(4\pi\right)^{(D+1)/2}\alpha^{D+1}\left(r/\eta\right)^{D}}c_{D-1}(q),\;r/\eta\ll
1.$ (4.11)

In order to find the behavior of the energy flux at large distances from the
string, $r/\eta\gg 1$, it is more convenient to use the representation (4.7)
for the function $t^{(01)}(x,y)$. In the limit under consideration one has
$x\gg 1$ and the main contribution to the integral in (4.7) gives the region
near the lower limit. By using the asymptotics for the Macdonald function for
small argument, we can see that, to the leading order,

$\langle
T_{0}^{1}\rangle_{\mathrm{t}}\approx-\frac{(D-3)\Gamma(D/2-1)c_{4}(q)}{8\pi^{D/2+1}\alpha^{D+1}\left(r/\eta\right)^{5}}.$
(4.12)

In the special cases $D=5,7$ this result agrees with (4.8) and (4.9).

The asymptotic behavior of the energy flux for large values of $q$ is studied
in the way similar to that we have used for the diagonal components. From
(4.6), for $xy\gg 1$ we find

$t^{(01)}(x,y)\approx\frac{\sqrt{\pi}\left(3-D\right)}{2^{D/2+3}x^{D+1}y^{D-1}}\Gamma\left(\frac{D+1}{2}\right).$
(4.13)

In combination with (4.5) this gives the following leading term in the
expansion over $1/q$:

$\langle
T_{0}^{1}\rangle_{\mathrm{t}}\approx-\frac{(D-3)^{2}\zeta\left(D-1\right)q^{D-1}}{2^{D+1}\pi^{3D/2-1/2}\alpha^{D+1}\left(r/\eta\right)^{D}}\Gamma\left(\frac{D+1}{2}\right).$
(4.14)

By taking into account that $c_{n}(q)\approx\left(q/\pi\right)^{n}\zeta(n)$
for $q\gg 1$, in the special cases $D=5,7$ the general estimate (4.14) agrees
with (4.8) and (4.9).

Figure 4 presents the energy flux versus the radial distance (left panel) and
the parameter $q$ (right panel). The graphs on the left and right panels are
plotted for $q=1.5$ and $r/\eta=1$, respectively. The numbers near the curves
are the corresponding values of the spatial dimension. For all the cases
$\langle T_{0}^{1}\rangle_{\mathrm{t}}<0$ and the energy flux is directed
towards the cosmic string.

|
---|---
Figure 4: The energy flux as a function of the distance from the cosmic string
(left panel) and as a function of the parameter $q$ (right panel). The left
panel is plotted for $q=1.5$ and for the right panel we have taken $r/\eta=1$.
The numbers near the curves present the values of the spatial dimension $D$.

As an additional check for the formulas given above we can show that the
components of the topological contribution to the vacuum energy-momentum
tensor obey the covariant conservation equation $\nabla_{i}\langle
T_{l}^{i}\rangle_{\mathrm{t}}=0$. In the problem at hand two relations are
obtained between the components corresponding to the equations with $l=0$ and
$l=1$. By taking into account that all the components depend on $r$ and $\eta$
in the form of the ratio $w=r/\eta$, they are written as

$\displaystyle\left(w\partial_{w}+D+1\right)\langle
T_{0}^{0}\rangle_{\mathrm{t}}$ $\displaystyle=$
$\displaystyle-\left(\partial_{w}+\frac{1}{w}\right)\langle
T_{0}^{1}\rangle_{\mathrm{t}}+\langle T_{i}^{i}\rangle_{\mathrm{t}},$
$\displaystyle\left(w\partial_{w}+D+1\right)\langle
T_{0}^{1}\rangle_{\mathrm{t}}$ $\displaystyle=$
$\displaystyle\left(\partial_{w}+\frac{1}{w}\right)\langle
T_{1}^{1}\rangle_{\mathrm{t}}-\frac{1}{w}\langle T_{2}^{2}\rangle_{t}.$ (4.15)

With these relations, the components $\langle T_{i}^{i}\rangle_{t}$,
$i=2,3,\ldots,D$, are expressed in terms of $\langle T_{0}^{0}\rangle_{t}$,
$\langle T_{0}^{1}\rangle_{t}$, and $\langle T_{1}^{1}\rangle_{t}$. We recall
that the components of the energy-momentum tensor were considered in the
coordinate system $(\tau,r,\phi,\mathbf{z})$ with the conformal time $\tau$.
For the components in the coordinates $(t,r,\phi,\mathbf{z})$, with $t$ being
the synchronous time, one gets (no summation over $i$): $\langle
T_{\mathrm{(s)}i}^{i}\rangle_{\mathrm{t}}=\langle
T_{i}^{i}\rangle_{\mathrm{t}}$ and $\langle
T_{\mathrm{(s)}0}^{1}\rangle_{\mathrm{t}}=e^{-t/\alpha}\langle
T_{0}^{1}\rangle_{\mathrm{t}}$. The topological contribution in the vacuum
energy contained in the spatial volume $V$ is expressed as
$E_{V}^{\mathrm{(t)}}=\int_{V}d^{D}x\,e^{Dt/\alpha}\langle
T_{\mathrm{(s)}0}^{0}\rangle_{\mathrm{t}}$. Taking the region $\\{r_{1}\leq
r\leq r_{2},\,0\leq\phi\leq\phi_{0},\,0\leq z^{i}\leq L_{i}\\}$,
$i=3,\ldots,D$, as the volume $V$, from the first equation (4.15) for the
corresponding energy $E_{V}^{\mathrm{(t)}}=E_{r_{1}\leq r\leq
r_{2}}^{\mathrm{(t)}}$ we get

$\partial_{t}E_{r_{1}\leq r\leq r_{2}}^{\mathrm{(t)}}=\phi_{0}L_{3}\cdots
L_{D}e^{Dt/\alpha}\left[-r\langle
T_{\mathrm{(s)}0}^{1}\rangle_{\mathrm{t}}|_{r_{1}}^{r_{2}}+\frac{1}{\alpha}\int_{r_{1}}^{r_{2}}dr\,r\sum_{i=1}^{D}T_{i}^{i}\right].$
(4.16)

By taking into account that $\phi_{0}r_{l}L_{3}\cdots L_{D}e^{(D-1)t/\alpha}$
is the proper surface area of the spatial hypersurface
$\\{r=r_{l},\,0\leq\phi\leq\phi_{0},\,0\leq z^{i}\leq L_{i}\\}$, from (4.16)
we see that $\langle T_{0}^{1}\rangle_{\mathrm{t}}$ is the energy flux per
unit proper surface area. The second term in the square brackets of (4.16)
corresponds to the work done by the vacuum stresses. We recall that for large
$r$ the energy density $\langle T_{0}^{0}\rangle_{t}$ behaves as $1/r^{4}$
(with exception for the case $D=4$, see (3.56)) and, hence, the energy
$E_{r_{1}\leq r\leq r_{2}}^{\mathrm{(t)}}$ is finite in the limit
$r_{2}\rightarrow\infty$.

In the discussion above we have considered an idealized defect with zero
thickness core. With this idealization, the topological contributions in the
diagonal components diverge on the core as $1/r^{D+1}$ and the off-diagonal
component behaves like $1/r^{D}$. Similar to the case of the Minkowskian bulk,
discussed in [60], we can consider a simple model of finite core of radius $a$
with the interior geometry described by the nonsingular metric tensor
$g_{il}=\left(\alpha/\tau\right)^{2}\mathrm{diag}(1,-1,-u^{2}(r),-1,\ldots,-1)$
for $r<a$. The geometry of the core is specified by the regular function
$u(r)$. The exterior metric, $r>a$, as before, is given by (2.2). Matching of
the interior and exterior geometries at $r=a$ is done by the Israel condition.
This condition relates the derivative of the function $u(r)$ at $r=a$ with the
surface energy-momentum tensor localized on the core boundary (see [60] for
the Minkowski bulk). For the electromagnetic modes in the region $r>a$,
instead of the Bessel function $J_{q|m|}(\gamma r)$, now the linear
combination $J_{q|m|}(\gamma r)+b_{\beta}Y_{q|m|}(\gamma r)$, with the Neumann
function $Y_{q|m|}(\gamma r)$, will appear. The coefficient $b_{\beta}$ is
obtained from the matching conditions between the interior and exterior modes
at $r=a$. In general, it will depend on the polarization and codifies the
information about the interior geometry. The simplest model would be the core
with perfectly reflecting boundary $r=a$ that induces the boundary condition
$\left.n^{i}\,{}^{\ast}F_{ii_{1}\cdots i_{D-1}}\right|_{r=a}=0$, where $n^{i}$
is the normal vector of boundary and ${}^{\ast}F_{ii_{1}\cdots i_{D-1}}$ is
the dual tensor for $F_{ik}$. For $D=3$ this corresponds to the boundary
condition on the surface of a perfect conductor. The effects of a conducting
cylindrical shell on the local characteristics of the electromagnetic vacuum
in the geometry of a cosmic string in (3+1)-dimensional Minkowski bulk have
been studied in [61]. The influence of a cylindrical boundary in
$(D+1)$-dimensional dS spacetime, described by the line element (2.2) with
$\phi_{0}=2\pi$ (the cosmic string is absent) on the VEVs of the sguared
electric and magnetic fields and on the vacuum energy-momentum tensor is
discussed in [45].

In the problem at hand, the effect of a perfectly reflecting finite core of
the cosmic string on the vacuum energy-momentum tensor in the region $r>a$ can
be investigated in the way similar to that used in [45] for the dS bulk
without a cosmic string. The coefficient $b_{\beta}$ in the exterior mode
functions is determined by the boundary condition at $r=a$. Convenient
expressions for the finite core contributions in the VEVs are obtained by
rotating the respective integrals over $\gamma$ in the complex plane (for the
corresponding procedure see [45]). In this way the integrands are expressed in
terms of the modified Bessel functions and the corresponding integrals
exponentially converge in the upper limit for $r>a$. As a consequence, the
vacuum energy-momentum tensor is decomposed as $\langle
T_{i}^{l}\rangle=\langle T_{i}^{l}\rangle_{\mathrm{dS}}+\langle
T_{i}^{l}\rangle_{\mathrm{t}}+\langle T_{i}^{l}\rangle_{\mathrm{fc}}$, where
$\langle T_{i}^{l}\rangle_{\mathrm{dS}}$ is the VEV in dS spacetime, $\langle
T_{i}^{l}\rangle_{\mathrm{t}}$ is the VEV considered in the discussion above,
and $\langle T_{i}^{l}\rangle_{\mathrm{fc}}$ is the part induced by the finite
core. When the angular deficit is absent, $\phi_{0}=2\pi$, at large distances
from the core, $r/a,r/\eta\gg 1$, the boundary induced contribution $\langle
T_{i}^{l}\rangle_{\mathrm{fc}}$ in spatial dimensions $D>4$ decays as [45]
$1/[\left(r/\eta\right)^{D+2}\ln(r/a)]$ for the diagonal components and like
$1/[\left(r/\eta\right)^{D+1}\ln(r/a)]$ for the component $\langle
T_{0}^{1}\rangle_{\mathrm{fc}}$. For $D=4$ an additional factor $\ln(r/\eta)$
appears for the diagonal components. We expect a similar behavior in the
presence of a cosmic string (the corresponding investigation will be presented
elsewhere) and the part $\langle T_{i}^{l}\rangle_{\mathrm{t}}$ will be
dominant compared to the contribution $\langle
T_{i}^{l}\rangle_{\mathrm{fc}}$.

As it follows from the discussion above, the effects of cosmic string on the
electromagnetic vacuum in spatial dimensions $D>3$ have qualitatively new
features which are partially related to the absence of conformal invariance.
These effects may have interesting implications in string theory motivated
models where the role of cosmic strings is played by fundamental strings [62].
In those models the spatial dimension of the effective theory depends on the
specific compactification scheme and may take values in the range $3\leq D\leq
9$. Another class of string theory inspired models with cosmic string type
structures were discussed recently within the framework of brane inflationary
models (see [21] for a review). The breakdown of conformal invariance for the
electromagnetic field is required in inflationary models of the generation of
large-scale magnetic fields from quantum fluctuations in the dS expansion
stage (for the discussion of various mechanisms see, e.g., [17, 63, 64]). A
possible mechanism, widely considered in the literature, is based on non-
minimal interactions of the electromagnetic field with other fields. Another
possibility, related to the dynamical evolution of extra spatial dimensions
before a radiation-dominated epoch in models with $D>3$, has been discussed in
[16]. If the preceding stage corresponds to the inflationary phase with dS
expansion then the modifications of the electromagnetic vacuum fluctuations
discussed above will be codified in large-scale perturbations of the magnetic
field around the cosmic string in the post-inflationary epoch. An interesting
signature for the presence of extra dimensions would be the energy flux in the
radial direction (for features of inflationary models with cosmic strings see
[19]). Several mechanisms have been discussed in the literature for the
formation of cosmic strings during inflation [1]-[4]. They include the direct
interaction between the inflaton and the field responsible for symmetry
breaking, an additional coupling to the background curvature and quantum-
mechanical tunnelling [65, 66, 67]. Among the interesting directions for the
further research could be the back-reaction of the vacuum polarization we have
discussed here on the spacetime geometry.

##  5 Conclusion

The expectation value of the energy-momentum tensor is among the central
objects in quantum field theory on curved backgrounds. In addition to being an
important local characteristic of a given state for quantum fields, it appears
as a source of the gravitational field in semiclassical Einstein equations and
determines the back reaction effects of quantum matter on the spacetime
geometry. We have discussed the combined effects of the background geometry
and topology on the VEV of the energy-momentum tensor for the electromagnetic
field in $(D+1)$-dimensional locally dS spacetime. The nontrivial topology is
generated by the presence of a topological defect that is a generalization of
a cosmic string in 4-dimensional spacetime. The corresponding geometry is
described by the line element (2.2) where the information on the defect is
codified in the angle deficit $2\pi-\phi_{0}$.

For evaluation of the VEV of the energy-momentum tensor we have employed the
mode-sum formula (2.3), with the mode functions for the vector potential given
by (2.9). The regularization of the mode-sum is done by introducing the cutoff
function $e^{-b\omega^{2}}$. The application of the formula (3.15) allowed us
to extract from the expectation values the contributions corresponding to the
pure dS bulk when the cosmic string is absent. For spacetime points outside
the defect core, $r>0$, the presence of the cosmic string does not alter the
local geometry and, hence, does not induce new divergences in the vacuum
energy-momentum tensor compared to those for dS spacetime. Consequently, the
topological contributions in the VEVs are finite for $r>0$ and the
regularization in the corresponding expressions can be directly removed
passing to the limit $b\rightarrow 0$ for the parameter in the cutoff
function. The diagonal components of the topological contribution to the VEV
of the energy-momentum tensor are given by (3.18), where the function
$t^{(i)}(x,y)$ is defined as (3.19), or equivalently, as (3.22). An additional
nonzero off-diagonal component of the vacuum energy-momentum tensor,
describing energy flux along the radial direction, is given by the expression
(4.5) with two equivalent expressions, (4.6) and (4.7), for the function
$t^{(01)}(x,y)$ in the integrand. Because of the maximal symmetry of dS
spacetime, the VEVs depend on the time and radial coordinates in terms of the
proper radial distance form the cosmic string core, given by the combination
$\alpha r/\eta=re^{t/\alpha}$. For the topological contributions we have
explicitly checked the trace relation (3.35) and the covariant conservation
equation. In the problem at hand the latter is reduced to the set of equations
(4.15).

To clarify the dependence of the topological terms on the parameters of the
problem special cases and asymptotic regions have been considered. In the
limit $\alpha\rightarrow\infty$ with fixed $t$, from the expressions for the
diagonal components of the energy-momentum tensor in dS spacetime the cosmic
string induced vacuum energy-momentum tensor is obtained in the Minkowski
bulk. In this special case the stresses along the directions parallel to the
core of the defect are equal to the energy density and they are given by
(3.47), whereas the radial and azimuthal components are expressed as (3.48).
All the components are monotonic functions of the radial coordinate with power
law decay $1/r^{D+1}$. The off-diagonal component vanishes in the Minkowskian
limit. The leading term in the corresponding expansion over $1/\alpha$ is
expressed as (4.10). The electromagnetic field is conformally invariant in
4-dimensional dS spacetime and the corresponding topological contributions are
obtained from the expressions in the Minkowski bulk with the radial coordinate
replaced by the proper distance from the string $\alpha r/\eta$. For odd
values of the spatial dimension $D$ the expressions for the topological terms
in dS bulk are further simplified by evaluating the integrals in (3.22) and
(4.7). The corresponding VEVs are expressed in terms of the functions (3.40)
with even values of the index $n$. Those functions are polynomials of degree
$n$ (see (3.41), (3.46)).

For points near the cosmic string, corresponding to small proper distances
compared to the curvature radius of dS spacetime, the influence of the
gravitational field on the diagonal components of the topological terms is
weak. The leading terms in the expansion over $r/\eta$ coincide with the
corresponding results in the Minkowski bulk where the radial distance is
replace by the proper distance $\alpha r/\eta$. The energy flux along the
radial direction is an effect induced by the gravity and the asymptotic of the
corresponding component of the energy-momentum tensor near the cosmic string
is given by (4.11). The effects of gravity on the topological contributions in
the VEV of the energy-momentum tensor are essential at proper distances from
the cosmic string larger than the dS curvature radius. For $r/\eta\gg 1$ and
$D\geq 4$ the corresponding asymptotic for the diagonal components is given by
(3.55). The exception is the energy density for $D=4$ with the asymptotic
behavior (3.56). For the energy flux density the decay at large distances is
stronger, like $1/\left(r/\eta\right)^{5}$. Note that in contrast to the
problem of cosmic string in the Minkowski bulk, for dS background geometry the
degree of power law decay of the topological contributions at large distances
does not depend on the spatial dimension.

## Acknowledgments

A.A.S. was supported by the grant No. 21AG-1C047 of the Science Committee of
the Ministry of Education, Science, Culture and Sport RA. V.Kh.K. was
supported by the grants No. 22AA-1C002 and No. 21AG-1C069 of the Science
Committee of the Ministry of Education, Science, Culture and Sport RA.

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