Alignment of Quasar Polarizations on Large Scales Explained by Warped Cosmic Strings

The recently discovered alignment of quasar polarizations on very large scales could possibly explained by considering cosmic strings on a warped five dimensional spacetime. Compact objects, such as cosmic strings, could have tremendous mass in the bulk, while their warped manifestations in the brane can be consistent with general relativity in 4D. The self-gravitating cosmic string induces gravitational wavelike disturbances which could have effects felt on the brane, i.e., the massive effective 4D modes (Kaluza-Klein modes) of the perturbative 5D graviton. This effect is amplified by the time dependent part of the warp factor. Due to this warp factor, disturbances don't fade away during the expansion of the universe. From a non-linear perturbation analysis it is found that the effective Einstein 4D equations on an axially symmetric spacetime, contain a"back-reaction"term on the righthand side caused by the projected 5D Weyl tensor and can act as a dark energy term. The propagation equations to first order for the metric components and scalar-gauge fields contain $\varphi$-dependent terms, so the approximate wave solutions are no longer axially symmetric. The disturbances, amplified by the warp factor, can possess extremal values for fixed polar angles. This could explain the two preferred polarization vectors mod $(\varphi, 90^o)$.


Introduction
Physicists speculate that extra spatial dimensions could exist in addition to our ordinary 4-dimensional faded away, mainly because of the inconsistencies with the power spectrum of the CMB. Moreover, they will produce a very special pattern of lensing effect, not found yet by observations. New interest in cosmic strings arises when it was realized that cosmic strings could be produced within the framework of string theory inspired cosmological models. Investigations on cosmic strings in warped brane world models show consistency with the observational bounds [4,5]. The warp factor makes these strings consistent with the predicted mass per unit length on the brane, while brane fluctuations can be formed dynamically due to the modified energy-momentum tensor components of the scalar-gauge field. This effect is triggered by the time-dependent warp factor. The recently discovered "spooky" alignment of quasar polarization over a very large scale [6] good be well understood by the features of the cosmic strings in brane world models and could be the first evidence of the existence of these strings.
In section 2 we will outline the warped 5-dimensional model. In section 3 we apply the multiple-scale approximation in order to find an wave-like solution to first order of the Einstein and matter field equations.

The warped 5D model
Let us consider the warped five-dimensional Friedmann-Lemaître-Robertson-Walker (FLRW) model in cylindrical polar coordinates The function W is the warp factor and y the extra (bulk) dimension. Here ψ and γ are functions of (t, r), while W is a function of (t, r, y). Our 4-dimensional brane is located at y = 0. All standard model fields resides on the brane, while gravity can propagate into the bulk. We consider a scalar-gauge field on the brane in the form [7] with η the vacuum expectation value of the scalar field and ǫ the coupling constant. As potential we take the well-known "mexican hat" potential V (Φ) = 1 8 β(Φ 2 −η 2 ) 2 . From the Einstein equations on the 5-dimensional spacetime one obtains a solution for the warp factor W[5] with d i and α some constants and Λ 5 the bulk cosmological constant. The first term in Eq.(3) is just the warp factor of the Randall-Sundrum model. The second term modifies the effective 4D Einstein equations. The Einstein field equations induced on the brane can be derived using the Gauss-Codazzi equations and the Israel-Darmois junction conditions. The modified Einstein equations become [8] 4 with 4 G µν the Einstein tensor calculated on the brane metric 4 g µν = 5 g µν − n µ n ν and n µ the unit vector normal to the brane. In Eq.(4) the effective cosmological constant Λ ef f = 1 2 (Λ 5 + κ 2 4 Λ 4 ) = 1 2 (Λ 5 + 1 6 κ 4 5 Λ 2 4 ) and Λ 4 is the vacuum energy in the brane (brane tension). The latter equality sign is a consequence of the relation between the 4-and 5-dimensional Planck mass in the braneworld approach, κ 4 5 = 6 Λ4 . If in addition the brane tension is related to the 5-dimensional coupling constant and the cosmological constant by 1 6 Λ 2 4 κ 4 5 = −Λ 5 , then Λ ef f = 0 and we are dealing with the RS-fine tuning condition [2]. The first correction term S µν in Eq.(4) is the quadratic term in the energy-momentum tensor arising from the extrinsic curvature terms in the projected Einstein tensor The second correction term E µν in Eq.(4)is given by and is a part of the 5D Weyl tensor and carries information of the gravitational field outside the brane and is constrained by the motion of the matter on the brane, i.e., the Codazzi equation. The scalar-gauge field equation becomes [7] with D µ Φ ≡ 4 ∇ µ Φ + iǫA µ Φ, 4 ∇ µ the covariant derivative with respect to 4 g µν , ǫ the gauge coupling constant and the star represents the complex conjugated. F µν is the Maxwell tensor. From Eq.(4) together with the matter field equations Eq. (7), one obtains a set of partial differential equations, which can be solved numerically [5]. Because gravity can propagate in the bulk, the cosmic string can build up a huge mass per unit length ( or angle deficit) Gµ ∼ 1 by the warp factor and can induce massive KK-modes felt on the brane, while the manifestation in the brane will be warped down to GUT scale, consistent with observations. Disturbances in the spatial components of the stress-energy tensor cause cylindrical symmetric waves, amplified due to the presence of the bulk space and warp factor. They could survive the natural damping due to the expansion of the universe. These disturbances could have a profound influence on the expansion of the universe. There could even be a "self-acceleration" without the need of an effective brane cosmological constant [9].
Besides the numerical solutions of the field equations, one should like to find an approximate wave solution where one can recognize the nonlinear features. In order to keep track of of the different orders of approximation, we will apply a multiple-scale analysis in the next section.

Nonlinear wave approximation
A linear approximation of wave-like solutions of the Einstein equations is not adequate in the case of high energy or strong curvature. There is a powerful approximation method to study non-linear gravitational waves without any averaging scheme. The method is called a "two-timing" or "multiple-scale" method, because one considers the relevant fields V i in point x on a manifold M dependent on different scales (x, ξ, χ, ...) [10,11,12]: Here ω represents a dimensionless parameter, which will be large (the "frequency", ω >> 1). So 1 ω is a small expansion parameter. Further, ξ = ωΘ(x), χ = ωΠ(x), ... and Θ, Π, ... scalar (phase) functions on M. The small parameter 1 ω can be the ratio of the characteristic wavelength of the perturbation to the characteristic dimension of the background. On warped spacetimes it could also be the ratio of the extra dimension y to the background dimension or even both. One is interested in an approximate solution of the metric and matter fields. If one substitute the series Eq.(8) into the field equations, one obtains a formal series where now n runs from −m to ∞, with m a constant. One says that Eq. (8) is an approximate wavelike solution of order n of the field equations if F (n) i (x, ξ, χ, ...) = 0 for all n. The method is very useful when one encounters non-uniformity in a regular perturbation expansion, i.e., the appearance of secular terms. In general relativity, this will occur when high-frequency gravitational waves interact with the background metric or the curvature is strong due to the presence of compact objects. On our 5D spacetime, we expand withḡ µν the background metric andΦ,Ā µ the background scalar and gauge fields. Let us consider, for the time being, only rapid variation in the direction of l µ transversal to the sub-manifold Θ = constant (One could also consider independent rapid variation transversal to the sub-manifold Π = constant). We can now define with l µ ≡ ∂Θ ∂x µ . We expand the several relevant tensors, for example, with where the colon represents the covariant derivative with respect to the 4ḡ µν . These expressions can also be calculated on 5 g µν . We substitute the expansions into the effective brane Einstein equations Eq.(4) and subsequently put equal zero the various powers of ω. We then obtain a system of partial differential equations for the fieldsḡ µν , h µν , k µν and the scalar gauge fieldsΦ, Ψ, Ξ,Ā µ , B µ ≡ [B 0 , B 1 , 0, B, 0] and C µ . The perturbations can be ϕ-dependent. The ω (−1) Einstein equation becomes and the ω (0) equation The contribution from the bulk space, E (−1) µν , must be calculated with the 5D Riemann tensor If we consider l µ l µ = 0, i.e., the eikonal equation, then one obtains from Eq. (15) which in other contexts is used as gauge conditions. It turns out that the contribution from the E One also needs the Ricci tensor 4 R (0) µν in Eq.(16), which is given by ( for σ = τ ) From the Einstein equations Eq.(16), one can deduce a set of partial differential equations (PDE's) when one imposes additional gauge conditions. As a simplified model, we take h 22 = −h 11 , h 34 = h 35 = h 45 = h 14 = h 15 = 0 (leaving 4 independent h µν terms), we have 7 unknown functions for the background and first order perturbations:W 1 ,ψ,γ,ḣ 13 ,ḣ 11 ,ḣ 44 andḣ 55 . One can also integrate the equation Eq.(16) with respect to ξ. If we suppose that the perturbations are periodic in ξ, we then obtain the Einstein equations with back-reaction terms: where we took Λ ef f = 0 for the RS fine-tuning and τ de period of the high-frequency components. One can say that the term − E ∂ ϕ ḣ 11 + e 2γ r 2ḣ 44 −W 2 1 e 2γ−2ψḣ 55 +k 24 −k 14 = −2κ 2 4XP e 2γ−2ψW 2 1 sin ϕΨ, We notice that in our simplified case of radiative coordinates Θ(x µ ) = t+ r, the equations for the background metric separates from the perturbations. So this example is very suitable to investigate the perturbation equations. For the first order gauge field perturbation B µ we used the condition l µ B µ = 0, which is a consequence, as we will see, of the gauge field equations. So B µ can be parameterized as The propagation equation forḣ 55 yieldsḣ 55 = F 1 (t + r)F 2 (ϕ, y, ξ), which is expected, because the brane part ofḣ 55 must be separable from the bulk part. We omitted for the time being, the κ 4 5 contribution. It is manifest that to zero order there is an interaction between the high-frequency perturbations from the bulk, the matter fields on the brane and the evolution ofḣ ij , also found in the numerical solution [5]. We observe again that the bulk contributionḣ 55 is amplified byW 2 1 . It is a reflection of the massive KK modes felt on the brane. The contribution ofḣ 55 in Eq.(29) disappears when ∂ tψ − ∂ rψ + 1 2r = 0. In the static case this results in a solutionψ = a log r + b (a=1/2 in our case). This solution is of less physical significance because a distant test particle in this field will be repelled from the cylinder for 0 < a < 1 [14]. The equations for the matter fields can be obtained in a similar way. The equation for the backgroundΦ becomesD αD αΦ − The equation forĀ µ is the same as in the unperturbed situation. For the first order perturbations we obtain ( for l α C α = 0) For these matter field equations one needs the condition l αĀ α = 0, otherwise the real and imaginary parts oḟ Ψ interact as the propagation progresses. From Eq.(26), Eq.(27) and Eq.(33) we observe on the right hand side ϕ-dependent terms, amplified by W 2 1 . So the approximate wave solution is no longer axially symmetric, also found by [10]. After integration with respect to ϕ, we obtain from Eq.(27) ( for k 14 = k 24 ) This means that the (r, r) first order disturbanceḣ 22 (ḣ 22 = −ḣ 11 )) could have its maximum for fixed angle ϕ amplified by the warp factor W 2 1 . If we choose for example Ψ = cos ϕΨ(t, r, ξ), then the last term in Eq.(34) becomes κ 2 4XP e 2γ−2ψW 2 1 cos 2ϕΨ, which has two extremal values on [0, π] mod ( 1 2 π). The energy-momentum tensor component 4 T This angle-dependency could be an explanation of the recently found spooky alignment of the rotation axes of quasars over large distances in two perpendicular directions.
The next step is to investigate the higher order equations in ω , which will provide the propagation equations of k µν and back-reaction terms in the background field equations Eq.(22), (23) and Eq.(24). In this way, one can construct an approximate solution of the Einstein and scalar-gauge field equations and one can keep track of the different orders of perturbations.

Conclusions
A nonlinear approximation of the field equations of the coupled Einstein-scalar-gauge field equations on a warped 5D spacetime is investigated. To zeroth order in the expansion parameter it is found that the evolution of the perturbations on the brane is triggered by the electric part of the 5D Weyl tensor and carries information of the gravitational field outside the brane. The warpfactor in the nominator in front of the bulk contributions will cause a huge disturbance on the brane and could act as dark energy. It turns out that the first order disturbances are no longer axially symmetric. This means that wave-like disturbances in the energy-momentum tensor components can have preferred ϕ directions perpendicular to each other. This could be an explanation of the alignment of the preferred directions of the quasar polarization axes.