On the distribution of polarization directions
in quasar samples
D. Sáez and J.A. Morales
Departamento de Astronomı́a y Astrofı́sica, Universidad de Valencia, Burjassot,
Valencia, Spain. diego.saez@uv.es and antonio.morales@uv.es
Summary. Recent observations of the polarization angles of some quasar samples
strongly suggest that these angles are not randomly distributed on Gpc scales. Linear
vector perturbations (vortical velocity fields) of the concordance universe produce
rotations of the quasar polarization directions, which depend on both the quasar
redshifts and the line of sight. The scales of these linear perturbation are assumed
to be so large that they are compatible with the observed properties of the cosmic
microwave background. Appropriate vector perturbations and quasar distributions
are used to estimate the largest rotations produced by linear vector modes. These
rotations reach values of a few degrees. Although these values seem to be small to
explain the correlations of the observed polarization directions, larger scales and
nonlinear evolution could lead to greater rotations.
1 Introduction
Various decades ago [1], it was claimed that the distribution of QSO polarization directions has a certain dipolar form. Recently, by using both modern
statistical techniques and recent observations (see e.g. [2] and [3]), correlations
and alignments in the polarization directions of some samples of quasars have
been pointed out. A globally rotating universe has been proposed to explain
these unexpected observations.
Against a global rotation, we propose the existence of standard vector perturbations of the concordance model; namely, very large scale vortical velocity
fields. As it is proved below, perturbations of this type produce a certain gravitational rotation of the polarization directions depending on both the emission
redshift and the line of sight and, consequently, these rotations may generate
significant correlations in an initially uncorrelated distribution (at emission
time) of polarization position angles.
The spatial scales of our vector modes are assumed to be greater (smaller)
than Lmin = 104 M pc (Lmax = 5 × 104 M pc); having a power spectrum
proportional to k −3 ; hence, scales close to Lmax have more power than those
close to Lmin . Only the first multipoles of the Cosmic Microwave Background
2
D. Sáez and J.A. Morales
(CMB) might become slightly affected by the existence of vector perturbations with these small scales and, as it is well known, the observed values of
these multipoles ([4]) are too small to be explained in the framework of the
concordance model; therefore, our modes are compatible with current CMB
observations.
As it is well known, the concordance model is a flat Friedmann-RobertsonWalker (FRW) universe with cosmological constant and cold dark matter,
which simultaneously explains many recent observations (see e.g. [5]). In this
model, the reduced Hubble constant is h = 10−2 H0 ' 0.71, where H0 is the
Hubble constant in units of Km s−1 M pc−1 , and the density parameters of
vacuum energy and matter (baryonic plus dark) are ΩΛ ' 0.73 and Ωm ' 0.27
(ΩΛ + Ωm = 1), respectively.
In this paper, it is assumed that scalar, vector and tensor modes are all
linear and, then, it is well known that their evolutions are independent; by this
reason, we can hereafter assume that only vector perturbations are present in
the universe.
Let us close this section with some remarks on notation: Greek (Latin)
indices run from 0 to 3 (1 to 3) and units are defined in such a way that
c = 8πG = 1, where c is the speed of light and G is the gravitational constant.
Symbol a stands for the scale factor, whose present value is assumed to be
a0 = 1 (flat universe), η is the conformal time, ηµν is the Minkowski metric,
and n is the unit vector in the observation direction. Whatever quantity A
may be, Ae stands for its values at QSO emission time.
2 Rotations of the QSO polarization angle produced by
vector modes
The line element of the flat FRW background and that of the perturbed (real)
universe can be written as follows:
ds2 = a2 (η) ηµν dxµ dxν = a2 (η) [−dη 2 + dr2 + r2 (dθ2 + sin2 θdφ2 )]
(1)
and
ds2 = a2 (η) (ηµν + hµν )dxµ dxν ,
(2)
respectively, where the small first order quantities hµν defines the perturbation. Our calculations are done in the conformal Newtonian gauge [6], in
which, one can write:
h00 = hij = 0,
h0i = (h1 , h2 , h3 ) ≡ h
(3)
In this gauge, vectors {er , eθ , eφ } are strictly orthogonal (condition hij =
0) and, then, the polarization vector P of the radiation received along the
direction er can be written in the form:
P = cos ψ eθ + sin ψ eφ ,
(4)
On the distribution of polarization directions in quasar samples
3
which indicates that the polarization angle, ψ, is that formed by P and eθ .
After defining the polarization angle, a formula for its total variation, δψ,
from emission to observation, can be easily obtained taking into account the
parallel propagation of vector P along radial null geodesics. The final formula
for this variation reads as follows:
Z
1 re
δψ = −
(∇ × h) · n dr ;
(5)
2 0
note that ∇ is the covariant derivative with respect the background flat 3dimensional metric; thus, δψ is obtained by integrating the curl of the vector
perturbation along the line of sight.
3 Calculating δψ rotations in Fourier space
For pure vector modes, the divergence of the vector field h vanishes and, then,
this field can be expanded in terms of the so-called vector harmonics, which
are solutions of the Helmholtz vector equation (see e.g., [6] and [7]). In the
flat case, the form of these harmonics is as follows:
Q± (r, k) = ± (κ) exp(ik · r) ,
(6)
±
k2 k3
σ
1√
√1
± k1kk3 − i k2 , ±
2 = σ 2 ± k + i k1 , 3 = ∓ k 2 ,
1√
where ±
1 = σ1 2
1
p
and σ1 = k12 + k22 ; therefore, in flat universes, expansions in terms of vector
harmonics are also plane wave expansions.
R
The vector field h can be written in the form h(η, r) = − f (η, r, k)d3 k,
where function f is the following linear combination:
f (η, r, k) = B + (η, k) Q+ (r, k) + B − (η, k) Q− (r, k) .
(7)
The expansion of h involve the coefficients B ± (η, k).
There are two additional physical fields with vector parts: the peculiar velocity, v, and the traceless tensor, Eij , describing anisotropic stresses. The coefficients involved in the expansions of these fields are v ± (η, k) and Π ± (η, k),
respectively.
All the above expansions can be introduced into Einstein equations [6]
to get the relations satisfied by functions B ± (η, k), v ± (η, k), and Π ± (η, k).
The resulting equations can be easily solved by assuming that we work: (i) in
the matter dominated era (quasar emission), (ii) in the absence of anisotropic
stresses (Π ± = 0) and, (iii) by using the variables vc± = v ± − B ± (which are
gauge invariant) and B ± . The solution has the form:
±
vc± (η, k) = vc0
(k)/a(η) ,
±
where vc0
are the present values of functions vc± and
(8)
4
D. Sáez and J.A. Morales
±
B ± (η, k) = 6H02 Ωm vc0
(k)/k 2 a2 (η) ;
(9)
hence, functions vc± ∝ a−1 and B ± ∝ a−2 describe decaying vector modes.
For Π ± 6= 0, either constant or growing modes can appear. Although this condition requires the existence of new fields at the scales of interest, our ideas
about the structure of the universe at these enormous scales are pure speculations and, consequently, the existence of unknown fields is fully admissible.
Finally, from the above expansions in vector harmonics and Eq. (5), the
rotation δψ can be calculated using the following formula:
Z re
dr
δψ = 3H02 Ωm
[n · F (r)] ,
(10)
2 (r)
a
0
where
F (r) =
Z
+ +
− −
vc0
(κ) − vc0
(κ)
exp(ik · r) d3 k .
k
(11)
The integral in Eq. (10) can be performed (along radial null geodesics of the
background) by using the vector field F (r), whose components can be seen
as Fourier transforms according to Eq. (11).
Furthermore, the expansions of vectors h and v c read as follows:
Z ±
vc0 (k) ±
2
−2
(κ) exp(ik · r) d3 k ,
(12)
h(η, r) = −6H0 Ωm a (η)
k2
Z
±
−1
v c (η, r) = a (η) vc0
(k) ± (κ) exp(ik · r) d3 k
(13)
and, consequently, the components of vector fields h and v c are also Fourier
±
transforms depending on quantities vc0
(k). This fact suggests a method to
±
superimpose distributions of vector modes: we first assign vc0
(k) quantities to
the nodes of the Fourier k-box and, then, we use the Fast Fourier Transform
(FFT) to get F , h and v c in position space. The box size and the resolution
of the Fourier transforms must be fixed. In position space, this size must be
greater than the maximum spatial scale (Lmax = 5 × 104 M pc) involved in
the simulation, whereas the cell size must be smaller than the minimum scale
(Lmin = 104 M pc). These large spatial scales require very big boxes but not
great spatial resolutions, e.g., a good choice for the above values of Lmax
and Lmin is the following: a box size Lbox = 2 × 105 M pc and a cell size
∆ ' 390 M pc (512 cells per box edge).
After fixing the box and the cell sizes and before performing the Fourier
±
transforms, quantities vc0
(k) are assigned according to the following criteria:
±
1. The four components of the two complex numbers vc0
(k) are statistically
independent among them.
2. Each of these numbers has a vanishing mean and a power spectrum of the
form A/k n .
±
± ∗
3. The relation vc0
(k) = −[vc0
] (−k) is satisfied to ensure that all the Fourier
transforms give real numbers.
On the distribution of polarization directions in quasar samples
5
In this way, modes with scales between Lmin and Lmax are statistically superimposed in our simulations. For A = 4.5 × 10−4 and n = 3, the resulting
metric perturbation field h(r) appears to be smaller than ∼ 0.2 –at any node–
for z = 2.6 and, moreover, the values of v c (r) are of the order of 10−3 at the
same redshift; hence, this superimposition of decaying vector modes evolves in
the linear regime all along the redshift interval (0, 2.6). Finally, the integral in
Eq. (10) is performed –along background null geodesics– for all the directions
pointing towards the 3072 pixels of a HEALPIx (Hierarchical Equal Area Isolatitude Pixelisation of the Sphere) map, see [8]. For all these directions, the
integration is calculated from z = 2.6 to z = 0.
Fig. 1. Full sky representation of δψ for constant z = 2.6
Table 1. Multipoles of three realizations. First row corresponds to the map displayed
in Fig. 1
realization
C1
C2
C3
1
2
3
7.82
12.51
16.00
0.19
0.24
0.46
0.048
0.072
0.029
C4
0.0002
0.0002
0.0016
Results are presented in Fig. 1, where |δψ| takes on maximum values close
to 3. No small spots appear in these maps, which suggests that only the first
multipoles are significant in this sky realization. The same occurs for other
realizations of linear vector modes. The multipoles corresponding to three
of these realizations are presented in Table 1, where we see that the dipole
dominates. These multipoles of δψ, namely, the C` quantities for 1 ≤ ` ≤ 4,
6
D. Sáez and J.A. Morales
have been calculated by using the code ANAFAST of the HEALPIx package
(as it is done in the case of temperature maps of the CMB).
4 Discussion and perspectives
Vector perturbations are decaying modes, except in the presence of unknown
large scale fields producing anisotropic stresses. The possible existence of this
kind of fields (justifying condition Π ± 6= 0) deserves attention. Such fields
could generate nonlinear vector modes at low redshifts, nevertheless, the nonlinear evolution of these modes [6] is being studied yet. By this practical reason, only linear vector modes evolving in the absence of anisotropic stresses
(Π ± = 0) are considered along this paper. In such a case, the angle δψ corresponding to quasars with z < 2.6 only can reach values of a few degrees.
These values seem to be too small to explain the correlations observed in
quasar polarization orientations, see e.g., [1], [2] and [3]. Larger values would
require large scale nonlinear vector perturbations at low redshifts.
Many years ago, Skrotskii [9] used Maxwell equations to prove that the
polarization vector rotates as the radiation crosses a perturbed Minkowski
space-time describing a slowly rotating body. We have considered the same
effect in another kind of cosmological space-time involving vortical motions.
The quasi dipolar form of the δψ angular distributions appearing in our simulations (see Fig. 1 and Table 1) strongly remember Birch observations [1].
Acknowledgments This work has been supported by the Spanish Ministerio
de Educación y Ciencia, MEC-FEDER projects AYA2003-08739-C02-02 and
FIS2006-06062.
References
1.
2.
3.
4.
5.
6.
7.
8.
P. Birch: Nature, 298, 451 (1982)
D. Hutsemékers: A&A, 332, 410 (1998)
D. Hutsemékers et al: A&A, 441, 915 (2005)
G. Hiwshaw et al: astro-ph/0603451 (2006)
D. N. Spergel et al: astro-ph/0603449 (2006)
J.M. Bardeen: Phys. Rev., 22D, 1882 (1980)
W. Hu, & M. White: Phys. Rev., 56D, 596 (1997)
K.M. Górski, E. Hivon, & B.D. Wandelt: In Proceedings of the MPA/ESO
Conference on Evolution of Large Scale Structure, ed by A.J. Banday, R.K.
Sheth, L. Da Costa (1999) [astro-ph/9812350]
9. G. V. Skrotskii: Dokl. Akad. Nauk. USSR 114, 73 (1957). Englihs translation:
Sov. Phys.–Dokl. 2, 226 (1957).