Formation and Evolution of Wakes in the Spacetime Generated by a Cosmic String in f(R) Theory of Gravity ()
1. Introduction
Extended Theories of Gravity have recently received increasingly attention in issues such as dark matter [1] [2] [3] [4] and dark energy [5] [6] [7]. In particular, the
theories of gravity have been suggested as a possible alternative to explain the late time cosmic speed-up experienced by our universe [8] [9]. Such theories avoid the Ostrogradski’s instability that can otherwise prove to be problematic for general higher derivatives theories [10] [11]. Besides, these theories could be probed by the recent LIGO detections [12] and by gravitational lensing observations [13].
On the other hand, it is well known that several types of topological defects may have been created by the vacuum phase transitions in the early universe [14] [15]. In particular, cosmic strings have been extensively studied in many kinds of alternative gravity theories, notably as scalar-tensor and
theories where many aspects and applications were developed [16] [17] [18] [19].
In this paper, our main purpose is to study the formation and evolution of wakes in cylindrically symmetric solutions in the framework of
theories in vacuum. In particular, we aim to explore the propagation of particles and light in a
cosmic string. We compare our results with the wakes formed by cosmic string solutions obtained in General Relativity and the Scalar-Tensor Theories of Gravity.
This paper is structured as follows. In Section 2, we briefly review the
theories of gravity (for comprehensive reviews in
gravity see [20] ). In Section 3, for the sake of completeness, we review the obtention of the cylindrically symmetric solutions in vacuum. We give special attention to the cosmic string solution. Is it important to stress that, in this section, we follow the reference [21] ? In Section 4, the formation and evolution of wakes in the background of a
are studied and these are our original work. Finally, in the conclusion we summarize our main results and discuss some perspectives.
2.
Gravity in the Metric Formalism—A Brief Review
In this section, we will mainly follow the references [20].
The action associated with the modified theories of gravity coupled with matter fields is given by:
(1)
where
is an analytical function of the the Ricci scalar, R,
and
corresponds to the action associated with the matter fields. By using the metric formalism, the field equations become:
(2)
in which
is the geometric energy-momentum tensor, namely
(3)
with
.
The standard minimally coupled energy-momentum tensor,
, derived from the matter action, is related to
by
(4)
Thus the field equations can be written as
(5)
Taking the trace of the above equation we get
(6)
which express a further scalar degree of freedom that arises in the modified theory. Through this equation it is possible to express
in terms of its derivatives and the trace of the matter energy-momentum tensor, as follows
(7)
Substituting the above expression into (5) we obtain
(8)
From this expression we can see that the combination below
(9)
with fixed indices, is independent of the corresponding index. So, the following relation
(10)
holds for all
and
.
3. Vacuum Cylindrically Symmetric Solutions in
Gravity: A Brief Review
In this section we review how to derive the field equations associated with cosmic string system in the context of modified theories of gravity following Azadi et al. [21].
The
field equations in vacuum are given by
(11)
Taking the trace of Equation (11), we have
(12)
Now, since we are interested in obtaining static solutions with cylindrical symmetry in vacuum, we will work with a general metric in Weyl coordinates
given by
(13)
where
and
are functions of r only.
The non-zero components of the Ricci tensor are:
(14)
where prime (') indicates derivative with respect to r. Therefore, the scalar curvature is
(15)
Replacing
of Equation (12) in Equation (11), finally we get:
(16)
Defining
(which is the equivalent of
in the absence of matter) as
(17)
we can easily see that
(18)
which is a scalar quantity. This means that
for any
, which implies that we can replace Equation (11) by
,
and
. After some straightforward calculation, we finally get, respectively
(19)
(20)
(21)
Therefore, any group of functions
and
which satisfies the equations above is a solution of the modified gravity equations in vacuum.
Since these equations are highly nonlinear, we will consider the particular case where
. This is justified by the fact that we are interested in cosmic string solution and the external metric of a cosmic string is locally flat.
Field Equations Solutions for the Special Case R=0
Let us start deriving Equation (11) with respect to r,
(22)
In the particular case where R= constant, Equation (22) implies that
. As a consequence, Equations (17-19) become
(23)
(24)
(25)
In order to solve this non-linear equations system, let us first sum up Equations (22) and (23). In doing this, we get
(26)
Defining
in Equation (26), we get
(27)
which can be rewritten as
(28)
Integrating Equation (28), we obtain
(29)
where
is a constant to be determined later.
Now, subtracting Equation (23) from Equation (22), we get
(30)
and we find
(31)
where
is a constant to be determined later.
Replacing Equations (29) and (31) in Equation (21), we get a differential equation for the function
, which is
(32)
In order to solve our equations, we will make the hypothesis that
is a linear function of r. This is justified by cosmic string solutions in either General Relativity or Scalar-tensor gravities. Therefore,
. Hence, we have
(33)
(34)
(35)
In order to satisfy Equation (32), the constants
and
must obey the following relations
(36)
We can easily see that the metric functions (33-35) satisfy Equation (32) in the particular case where
.
Redefining the quantities
and
and making
without any loss of generality, we can write down the metric in Weyl coordinates as
(37)
Defining the quantities
and
and defining new coordinates such as
(38)
(39)
In doing this, the metric reduces to a very simple form1
(40)
Applying the complex transformation
and
, we get a well known metric [22]
(41)
which, apart from the sigh
is pretty much the same as the Levi-Civita static cylindrically symmetric solution in General Relativity [23].
When
, the spacetime (41) becomes2
(42)
It is very easy to see that this spacetime is locally flat but not globally Euclidean. This spacetime is conical, as we can see in Figure 1, with a deficit angle equal to
(43)
as long as
which imposes some constraints on the constants
and
.
4. Formation and Evolution of Wakes in the Spacetime of a
Cosmic String
In this section we will study the formation and evolution of some structures when a cosmic string moves through a region containing baryonic matter.
4.1. The Formation of Wakes
Let us suppose that the cosmic string moves with a constant velocity
in the x-axis. Since
is constant in (42), the string does not exert any gravitational force on test particles. However, test particles do suffer a perturbation when passing through a cosmic string. To see that, let us consider that one is in a comoving frame in which the string is at rest and, by a Lorentz transformation, the baryonic matter is approaching the string with the velocity
. From Figure 2, we can see how the velocity of the test particles are perturbed and we can calculate this perturbation. Of course, it depends on the deficit angle and, hence, on the parameters of the
theory
(44)
where
, considering
.
In this way, particles which move in regions where
may collide with particles which move in regions where
after passing through the string and form stable structures called “wakes”, see Figure 3.
4.2. The Evolution of Wakes: The Zel’dovich Approximation
Let us now make a quantitative description of the accretion problem using the Zel’dovich approximation [24], which consists in considering the Newtonian
Figure 2. (a) A cosmic string moving with constant velocity vc. (b) Particles moving with constant velocity vc in the comoving string frame.
Figure 3. Particles collision with impact parameters equal to R and R'.
accretion problem in an expanding Universe by means of the method of linear approximation.
Suppose that the wake is formed at
, where
is the time in which matter starts to dominate over radiation. The physical trajectory of a particle can be written as
(45)
where
is the unperturbed comoving position and
is the comoving displacement developed as a consequence of the gravitational attraction induced by the wake on the particle.
The equation of motion in the Newtonian limit is given by
(46)
where
is the Newtonian potential which obeys the Poisson equation3
(47)
The matter density
is determined in terms of the background density
as [24]
(48)
which gives
(49)
where
and
are the radial components of these quantities.
Replacing Equation (49) in the Poisson Equation (47), we obtain
(50)
If we replace the relation
in the equation above, we get the linearized equation for
(51)
Since we are working in the matter era,
. Hence, Equation (51) becomes
(52)
The equation above is the well-known Euler equation. Applying appropriate conditions such as
and
the solution is
(53)
The comoving coordinate
can be calculated using the fact that
, which means that, eventually, the particle stops expanding with the Hubble flow and starts to collapse onto the wake. Therefore, we get
(54)
Hence, we are now able to compute the wake’s thickness
and surface density
[25]
(55)
(56)
Finally, we obtain
(57)
(58)
where
, which means that the wake’s thickness and density depend on the parameters of the
theory.
5. Final Remarks
In this work we considered cylindrically symmetric solutions in the framework of
theory of gravity. These solutions were obtained in vacuum regime and in the particular case where
. A cosmic string solution was of special interest and we studied the formation and evolution of wakes in this spacetime. Comparing our results with those obtained previously in the literature [25] [26], both in General Relativity and in Scalar-Tensor theories, respectively, we can see that they resemble the GR wakes instead of the scalar-tensor ones, as we would expect since
is constant and there is no gravitational force exerted by the
cosmic string in the same way as the GR cosmic string. However, for a precise comparison and further numerical evaluation, we must consider the internal cosmic string matter configuration because we need to determine the metric constants
. In particular, they must obey the GUT cosmic string order of magnitude for all parameters.
As we expected, all wake’s physical quantities depended on the parameters of the particular theory of gravity under consideration. But, again, in order to make a quantitative evaluation, we need to consider not a vacuum solution but a full energy-momentum tensor for the internal cosmic string configuration in the same way as [16] [27]. This is under consideration now.
As a future perspective of this work, we plan to study the
cosmic string as a generator of the rotational curves in galaxies [1]. This work will come as a forthcoming paper.
Acknowledgments
P. O. Mesquita and M. E. X. Guimarães would like to thank PIBIC/CNPq (Conselho Nacional de Desenvolvimento Cientfico e Tecnológico) for a support during the preparation of this work.
NOTES
1The calculations are long but straightforward.
2We will suppress the symbol from now on because it will not cause any confusion.
3In order to avoid confusion, we have changed the radial coordinate for r instead of
which represents now the matter density.