Eur. Phys. J. C (2012) 72:1928DOI 10.1140/epjc/s10052-012-1928-y

Regular Article - Theoretical Physics

A complete 3D numerical study of the effectsof pseudoscalar–photon mixing on quasar polarizations

Nishant Agarwal1, Pavan K. Aluri2, Pankaj Jain2,a, Udit Khanna2,3, Prabhakar Tiwari21McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA2Department of Physics, Indian Institute of Technology, Kanpur 208016, India3Present address: Harish-Chandra Research Institute, Allahabad 211019, India

Received: 19 October 2011 / Revised: 29 December 2011 / Published online: 20 March 2012© Springer-Verlag / Società Italiana di Fisica 2012

Abstract We present the results of three-dimensional simu-lations of quasar polarizations in the presence of pseudosca-lar–photon mixing in the intergalactic medium. The inter-galactic magnetic field is assumed to be uncorrelated inwave vector space but correlated in real space. Such a fieldmay be obtained if its origin is primordial. Furthermorewe assume that the quasars, located at cosmological dis-tances, have negligible initial polarization. In the presenceof pseudoscalar–photon mixing we show, through a directcomparison with observations, that this may explain the ob-served large scale alignments in quasar polarizations withinthe framework of big bang cosmology. We find that the sim-ulation results give a reasonably good fit to the observeddata.

1 Introduction

In a recent paper [1] it was shown that pseudoscalar–photonmixing in the presence of correlated intergalactic magneticfields might provide an explanation for the observed largescale alignment of quasar polarizations at visible wave-lengths [2–4]. The alignment is seen over cosmologicallylarge distances of order Gpc [2–8]. In [4], the authors ar-gue that it is very unlikely that the effect originates dueto interstellar extinction. The effect is puzzling within theframework of big bang cosmology since we do not expectastrophysical objects to be correlated over such large dis-tances. There is also direct evidence that other properties ofthese objects, such as polarization position angles at radiofrequencies or position angles of parsec-scale jets, do notshow any large scale alignment [9]. Hence the effect doesnot seem to be intrinsic to these sources and might arise dueto propagation. Furthermore the propagation effect must not

a e-mail: pkjain@iitk.ac.in

affect radio polarizations since these do not show any align-ment. Pseudoscalar–photon mixing is a good candidate toexplain the alignment of quasar optical polarizations since itis negligible at radio frequencies. Large scale correlations inthe intergalactic magnetic field could lead to such an align-ment over large distances [1].

In [1], it was assumed that the intergalactic magneticfield may be obtained by a simple cosmological evolutionof the primordial magnetic field [10–14] within the frame-work of big bang cosmology. It was assumed that visible ra-diation from distant quasars is unpolarized at the source. Itacquires polarization due to its mixing with hypothetical lowmass pseudoscalars in background magnetic fields [15–26].Such pseudoscalars arise in many extensions of the stan-dard model of particle physics [27–35]. It was shown thatthe linear polarization angle of electromagnetic radiationfrom quasars separated over large distances becomes aligneddue to correlations in the background magnetic field [1].In [1] the authors restricted their attention to quasars thatare aligned in one dimension, along the line of sight. Inthe current paper we present the results of a general three-dimensional numerical analysis of this problem, and showby a direct comparison with observations that such an effectcan explain the observed large scale alignments in quasarpolarizations.

The mixing of photons with pseudoscalars has manyinteresting astrophysical and cosmological implications[5, 36–53]. It affects both the intensity and polarizationof radiation. Furthermore it also changes its spectral char-acteristics [54–58]. Extensive laboratory and astrophysicalsearches of these particles have led to stringent limits ontheir masses and couplings [38, 47, 59–79]. Some astrophys-ical observations also appear to indicate a positive signatureof pseudoscalar–photon mixing [53, 80].

The intergalactic magnetic field is not known verywell. In most treatments of pseudoscalar–photon mixing

Page 2 of 10 Eur. Phys. J. C (2012) 72:1928

through intergalactic space it is assumed that the backgroundmedium may be split into a large number of uncorrelated do-mains. The domain size is typically taken to be of order Mpcwith the magnetic field in each domain of order nG [43, 44,46]. The magnetic field in different domains points in ran-dom directions and is uncorrelated. Here we assume that thebackground magnetic field is of primordial origin [10–14].We also consider discretized real space consisting of a largenumber of cells or domains of equal size. Each domain isassumed to be cubical of side rG. The j th component of themagnetic field, Bj (r), in real space is given by the discreteFourier transform,

Bj (r) = 1

V

∑bj (k)eik.r , (1)

where V is the volume in real space and bj (k) is the j thcomponent of the magnetic field in wave vector space. Themagnetic field is assumed to be uniform in each domain. Wemay write the two point correlations of bj (k) as

⟨bi(k)b∗

j (q)⟩ = δk,qPij (k)M(k) = δk,qσ 2

ij (k), (2)

where σ 2ij (k) = Pij (k)M(k), k = |k| and Pij (k) = (δij −

kikj

k2 ) is the projection operator. The function M(k) shows apower law behavior,

M(k) = AknB , (3)

where nB is the power spectral index. The power law de-pendence of this correlation function would lead to correla-tions in the magnetic field in real space over large distances.This is in contrast to the assumption made in most treatmentsof pseudoscalar–photon mixing through intergalactic space[43, 44, 46]. The constant A in (3) is fixed by demandingthat∑

i

⟨Bi(r)Bi(r)

⟩ = B20 , (4)

where we assume the value of about 1 nG [12, 81, 82] for B0.The value of the spectral index nB has been obtained

by making a best fit to the matter and CMB power spec-trum [81]. This fit gives, nB = −2.37. A fully scale invari-ant power spectrum would correspond to the value nB = −3.We expect that the intergalactic medium may be turbulent onshort distance scales. In this case we may expect the mag-netic field to obey a Kolmogorov power spectrum for dis-tance scales much smaller than the size of the system. A sim-ilar model is also applicable to the galactic medium, whereboth the plasma density and the magnetic field are knownto show a Kolmogorov power spectrum [83, 84]. Hence itmight be more appropriate to assume that nB has some de-pendence on k. Here we shall simply treat nB as a fixedparameter, independent of k. The spectrum may also have

a low k cutoff, kmin, which will imply a cutoff on corre-lations in real space for distances larger than rmax = k−1

min.Here we shall assume that rmax is larger than the size of oursystem, which we take to be of the order of a few Gpc. Hencewe may simply set it equal to infinity. The large scale cor-relations induced among quasar polarizations crucially de-pend on this parameter. If this parameter is much smallerthan 1 Gpc, then the mechanism proposed in [1] may notwork. Alternatively we may argue that the observation oflarge scale correlations in quasar polarizations might be anindication that rmax ≥ 1 Gpc. We emphasize that a magneticfield with such large distance correlations may be generatedwithin the framework of the big bang model. The perturba-tions generated at the time of inflation have wavelengths ex-tending up to the horizon in the current era. The Cosmic Mi-crowave Background (CMB) anisotropies show significantpower in the quadrupole, which implies correlations overthe entire sky. The matter spectrum, however, gets signifi-cantly modified due to evolution after decoupling and corre-lations at such large distances get suppressed. However, themagnetic field need not evolve in the same manner as thenon-relativistic matter.

In our analysis we first ignore cosmological evolutionand assume a flat space-time. We are interested in radiationfrom quasars located at redshifts of order unity. In this caseour neglect of cosmological evolution would lead to an er-ror of order unity, which is comparable to other errors, suchas in the modeling of the intergalactic magnetic field. So farmost of the literature on pseudoscalar–photon mixing in abackground magnetic field has ignored cosmological evo-lution. We next also take cosmological evolution into ac-count.

A useful statistic to test for alignment of quasar polariza-tions is defined as follows [2, 6]. We first define a measureof dispersion, dk , in the neighborhood of the kth quasar,

dk = 1

nv

nv∑

i=1

cos[2(ψi + �i→k) − 2ψk

]. (5)

Here ψi are the polarization angles of the nearest neighborsof the quasar at position k. We include a total of nv nearestneighbors. The nearest neighbors also include the polariza-tion angle of the kth quasar. The concept of nearest neigh-bors requires a measure of distance. In a curved space wedo not have a unique definition of distance. Here we usecomoving distance as well as angular diameter distance forthis purpose. When we include cosmological evolution, weavoid this problem by locating our simulated sources at theprecise positions of the observed sources. The polarizationsat two different angular positions are compared by makinga parallel transport from their location to the position of thekth quasar along the great circle joining these points on thecelestial sphere. This induces the factor �i→k , included in

Eur. Phys. J. C (2012) 72:1928 Page 3 of 10

the definition of dk . The parallel transport is required sincewe are comparing polarizations located at different points onthe surface of the celestial sphere. If this parallel transportfactor is not included then the resulting statistic is not in-variant under coordinate transformations [6]. We next max-imize dk as a function of ψk . The resulting value of ψk

is interpreted as the mean polarization angle at the posi-tion k and the corresponding maxima of the function in (5)gives an estimate of dk . The statistic may now be defined as[2, 6],

SD = 1

ns

ns∑

k=1

dk|max, (6)

where ns is the total number of sources in the data. A largevalue of SD indicates a strong alignment between polariza-tion vectors. In our analysis we shall compute this statis-tic theoretically and compare the corresponding values ob-tained by observations [2–4, 6] in order get an estimate ofthe model parameters.

A potential problem with our hypothesis that the align-ment is caused by pseudoscalar–photon mixing is discussedin Ref. [85]. It is observed that in most cases these quasarsshow negligible circular polarization [4, 86]. So far this hasbeen seen only in a small sample. Pseudoscalar–photon mix-ing predicts a larger circular polarization and hence may notconsistently explain the data. The authors argue that if weassume the incident light to be natural white light, then dueto decoherence the circular polarization is predicted to bezero, consistent with observations. However, circular polar-ization is observationally found to be very small even witha broadband filter. In this case theory does indicate a sig-nificant circular polarization which is not in agreement withthe data. Here we address this issue partially. We point outthat there exist many other effects which may cause decoher-ence and hence reduce the predicted circular polarization.For example, in most treatments the medium is assumed tobe uniform. Alternatively one assumes a large number of do-mains with different properties but each individual domainis assumed to be uniform. In reality the medium, at any rea-sonable length scale, shows fluctuations both in space andtime. Since the background magnetic field is not uniform,the pseudoscalar particle produced would not have the sameenergy and momentum as that of the incident photon. Thisacts as another source of decoherence and will suppress thecircular polarization. We show this explicitly by consider-ing a model space varying magnetic field. We find that ifthe magnetic field shows rapid variations in space, then thecircular polarization is greatly reduced in comparison to thelinear polarization. Furthermore, we argue that a time vary-ing medium may also lead to suppression of circular polar-ization.

2 Pseudoscalar–photon mixing in an expandinguniverse

In this section we give the basic equations of pseudoscalar–photon mixing in an expanding universe [87–89]. We as-sume the standard Friedmann–Robertson–Walker (FRW)metric, with the curvature parameter k = 0, using conformaltime. The metric may be written as

gμν =

⎛

⎜⎜⎝

a2 0 0 00 −a2 0 00 0 −a2 00 0 0 −a2

⎞

⎟⎟⎠ = a2ημν, (7)

where a is the scale parameter and ημν the Minkowski met-ric.

2.1 Mixing with pseudoscalars

The action of the electromagnetic field Fμν coupled to apseudoscalar field φ may be written as

S =∫

d4x√−g

[−1

4FμνF

μν − 1

4gφφFμνF

μν

+ 1

2

(ω2

pa−3)AμAμ + 1

2gμνφ,μφ,ν − 1

2m2

φφ2]. (8)

Here gφ is the coupling of φ with the electromagnetic field.We have also written an effective photon mass term. Thiscorresponds to the effective mass ω2

pa−3 it acquires due tothe medium, where ωp is the plasma frequency. In (8) wehave ignored any self couplings of the scalar field. The equa-tion of motion for the scalar field may be written as

−∂2χ

∂η2+ ∇2χ = −gφ

a

(a2E

) · (a2B) + m2

φa2χ, (9)

where we have replaced φ by χa

. In (9), Bi + Bi =12εijka

2Fjk is the magnetic field, Ei = a2F 0i the usual elec-tric field, B the background magnetic field, and B the mag-netic field of the wave. The Maxwell equations in curvedspace-time may be written as

∇.(a2E

) = −gφ

a∇χ.

(a2B

) +(

ω2p

a

)A0, (10)

∇ × (a2E

) + ∂(a2B)

∂η= 0, (11)

with

∂(a2B)

∂η= 0, (12)

Page 4 of 10 Eur. Phys. J. C (2012) 72:1928

∇ × (a2B) − ∂(a2E)

∂η= gφ

a

(∇χ × a2E)

+ gφ

a

(a2B

)∂χ

∂η+

(ω2

p

a

)A, (13)

∇.(a2B

) = 0, (14)

where A represents the vector potential. In the above equa-tions we have assumed a uniform background magnetic fieldand approximated B + B ≈ B as |B| � |B|. Here we alsoignore the term containing derivatives of the scale factor,a, since the time scale for cosmological evolution is muchlarger in comparison to other times scales in the problem.Now we take the curl of Faraday’s law, and use the aboveequations to get the wave equation for E,

−∂2(a2E)

∂η2+∇2(a2E

) = ω2p

a

(a2E

)+ gφ

a

(a2B

)∂2χ

∂η2. (15)

The time dependence of the resulting solution may be ex-pressed as

(a2E

) ∝ exp(−iωη) = exp

[−iω

∫dt

a(t)

]. (16)

We also note that ω(t)a(t) = ω0, where ω0 is the frequencyat the current time.

2.2 Mixing solution

We choose the comoving coordinate system with the z-axisas the direction of propagation. In a particular domain themagnetic field is assumed to be uniform. Only the compo-nent of E parallel to the background magnetic field mixes

with χ . We also define A = (a2E)ω

and replace (a2E) byωA. We write the field equations of A‖ and χ as

(ω2 + ∂2

z

)(A‖χ

)− M

(A‖χ

)= 0, (17)

where

M =⎛

⎝ω2

p

a− gφ

a(a2 B⊥)ω

− gφ

a(a2 B⊥)ω m2

φa2

⎞

⎠ . (18)

Here, B⊥ is the transverse component of B and ω is the fre-quency of radiation at the propagation domain. Due to ex-pansion the observed energy today is redshifted. Hence wereplace ω → ω

a. Finally, the mixing matrix may be written

as

M =⎛

⎝ω2

p

a− gφ

a2 (a2 B⊥)ω

− gφ

a2 (a2 B⊥)ω m2φa2

⎞

⎠ . (19)

We solve the mixed field equations in a manner similar tothat in Refs. [23, 24]. We diagonalize the mixing matrix M

by an orthogonal transformation, OMOT = MD , where

O =(

cos θ − sin θ

sin θ cos θ

), (20)

where the angle θ can be expressed as

tan 2θ = 2gφωa−2(a2 B)

(ω2

p

a− m2

φa2)

. (21)

We assume that the mass of the pseudoscalar (mφ) is negli-gible compared to the plasma frequency (ωp).

3 Simulations

We first generate the magnetic field numerically in wavevector space. This is relatively straightforward since it is un-correlated for different wave vectors. The projection opera-tor implies that the component of the magnetic field parallelto k is zero. The two orthogonal components are uncorre-lated. It is simplest to use polar coordinates (k, θ,φ) in wavevector space. Equation (2) implies that for any wave vectork, bk = 0 and bθ and bφ are uncorrelated. We may, therefore,generate these by assuming the Gaussian distribution,

f(bθ (k), bφ(k)

) = N exp

[−

(b2θ (k) + b2

φ(k)

2M(k)

)], (22)

where N is the normalization factor. This represents an un-correlated Gaussian distribution for the two components ofthe magnetic field in Fourier space, corresponding to thewave vector k. Once we have these two components we canobtain the three Cartesian components of the magnetic fieldin wave vector space. Next we use (1) to obtain the threeCartesian components of the magnetic field in real space.The Gaussian random variates are generated by using theNumerical Recipes [90] code gasdev. The discrete Fouriertransform is obtained by using the Recipes code fourn.

The optical polarizations are propagated, taking pseudo-scalar–photon mixing into account by the procedure de-scribed in Ref. [79]. We need to propagate over a large num-ber of magnetic domains from the quasar to the observer.We choose a fixed “external” coordinate system. The trans-verse component of the magnetic field in each domain isaligned at some angle to the fixed coordinate axes. In orderto propagate through each domain we first rotate the coordi-nates so that the transverse magnetic field aligns along oneof the coordinate axes. We then use standard expressionsfor pseudoscalar–photon mixing [79] in order to evaluatethe correlation functions of the electromagnetic and pseu-doscalar fields after propagation through the domain. We

Eur. Phys. J. C (2012) 72:1928 Page 5 of 10

then rotate back to the fixed coordinate system. This pro-cedure is repeated for propagation through each domain tillthe wave reaches the observer.

4 Results

In this section we present the results of our numerical anal-ysis. We first consider the case of flat space-time, ignoringcosmological evolution. Next we give results for the case ofexpanding universe. We set the following parameter values:plasma density ne = 10−8 cm−3, frequency ν = 106 GHz,coupling gφ = 6.4 × 10−11 GeV−1 and mass mφ = 0. Weshall use the spectral index nB = −2.37 but will also ex-plore a range of values of this parameter. We perform mostof our simulations on a grid size of 1024 × 1024 × 1024,with the observer located at the center of the grid. The gridsize is determined by the computing power available to us.

4.1 Flat space-time

Here we present our results ignoring cosmological evolu-tion. We assume 200 quasars distributed randomly in space.The resulting statistic SD as a function of the distanceamong sources is shown in Fig. 1. Results are presentedfor four different values of the domain size rG = 4, 8, 16,32 Mpc. The number of points on the grid are suitably scaledso as to keep the total volume fixed. In Fig. 1 the statis-tic is computed by including all the sources that lie withina particular distance from a source. This fixes the numberof nearest neighbors to be included for any distance. Thereal data have a total of 355 sources. The relationship be-tween the number of nearest neighbors and the distance forthe real data is shown in Fig. 2. We find that for a distance of25 Mpc, the number of nearest neighbors is approximately 1.It becomes close to 2 only at a distance of about 200 Mpc.

Fig. 1 The statistic SD as a function of the distance between sourcesfor different choices of the domain size rG (in Mpc) and the number ofpoints on the grid NG. Here we have set nB = −2.37

In Fig. 3 we show the fluctuations in the statistic SD for agiven choice of parameters nB = −2.37, rG = 8 Mpc andthe number of points on the grid, NG = 1024. Results areshown for four different seeds used to initialize the simula-tions. We see significant fluctuations and hence to comparewith data it is better to take the mean over several differ-ent computations. Finally in Fig. 4 we compare the simu-lation results with observations. The simulation results areshown for a mean of five different initializing seeds choos-ing nB = −2.37 and −2.98. Here we have set rG = 8 Mpcand the number of points on the grid equal to 1024. We alsoshow results for the case where the background magneticfield is taken to be completely random. We find that the re-sults for correlated magnetic fields show reasonable agree-ment with data. However, those with a random magneticfield are systematically below observations.We also find a rel-atively mild dependence on the exponent nB . We concludethat pseudoscalar–photon mixing in a background magneticfield, described by (1) and (2), gives reasonable agreement

Fig. 2 The number of nearest neighbors as a function of distance. Theupper curve (dashed) corresponds to angular diameter distance assum-ing a vacuum dominated universe. The lower curve (solid) is obtainedassuming comoving distance in a matter dominated universe

Fig. 3 The fluctuations in the statistic SD for different choices of therandom seed. Here we have set nB = −2.37, rG = 8 Mpc and the num-ber of points on the grid NG = 1024

Page 6 of 10 Eur. Phys. J. C (2012) 72:1928

Fig. 4 Comparison of the simulation results with observed data. Thedotted curve with large gaps shows the statistic obtained using obser-vations. The remaining curves are obtained by simulations using pa-rameters nB = −2.37 (dotted with small gaps), nB = −2.98 (solid),and a completely random magnetic field (dashed). Here we have setrG = 8 Mpc

with observations. The small deviations from the data canbe easily adjusted by fine tuning the parameter values.

4.2 Expanding universe

In this section we present our results for the case of an ex-panding universe. We consider 355 sources, to exactly matchthe number of sources observed in the real data. The posi-tions of these sources are also taken to be the same as inthe data. The medium parameters, i.e. the magnetic fieldstrength, the plasma density are taken to be the same as inthe earlier non-expanding universe. Here these values are as-sumed to be their comoving values. The observed frequencyω of the electromagnetic wave is assumed to be 2 eV. Thecomoving domain size is taken to be 15 Mpc. The Hubbleconstant is set equal to 2.133h × 10−42 GeV, with h = 0.7.Furthermore, we assume a matter dominated universe. Aswe shall see our results for the expanding universe are com-parable to those obtained for flat space-time and hence suchdetails do not make a very large difference to our final re-sults.

We test our model against the data of the 355 sources ob-served by Hutsemékers et al. [2–4]. On placing the sourcesin our simulations at the same positions as in the data, weperform the propagation and calculate the Stokes param-eters. We calculate the coordinate invariant statistics SD ,given in (6), for the observed polarizations and for the po-larizations obtained from our simulations. The distributionof the linear polarization both for observed data and simula-tions is shown in Fig. 5. For the simulations we show resultsfor nB = −2.37 and −2.95.

In Fig. 6, we show the statistic SD for the data as wellas the simulations. The simulation results are shown for amean of five computations. We see that for both nB = −2.37

Fig. 5 The polarization histograms for the observed 355 sources (solidblack curve) and for our simulations in an expanding universe. Thered dashed curve and the green dash-dotted curve show results fornB = −2.95 and nB = −2.37, respectively (Color figure online)

Fig. 6 The coordinate invariant statistic SD as a function of thenumber of nearest neighbors for the 355 sources. The simulated re-sults are shown by green triangles (nB = −2.37) and red squares(nB = −2.95), while the black crosses represent the observed data(Color figure online)

and −2.95 the simulations show reasonable agreement withdata. As mentioned above here we have set the domain sizerG = 15 Mpc. It might be more reasonable to choose asmaller domain size. We find that as we reduce the domainsize to rG = 12 Mpc, our statistic, SD , becomes significantlylarger than data. Hence we find even larger correlations inthe optical polarizations. It may be possible to play with pa-rameters to make our simulation results agree with obser-vations in this case also. For example, this may be accom-plished by decreasing the background magnetic field and/orthe pseudoscalar–photon coupling constant gφ . However wepostpone a detailed fit to future work since it requires exten-sive numerical simulations.

We may get some idea about the dependence of the statis-tic, SD , on the domain size by performing simulations ona smaller grid. For this purpose we choose a grid size of256×256×256. We again take all the 355 sources but place

Eur. Phys. J. C (2012) 72:1928 Page 7 of 10

Fig. 7 The coordinate invariant statistic SD as a function of the num-ber of nearest neighbors for 355 sources. The sources are located atthe same angular positions as real sources but their radial distance isreduced so that we can use a smaller grid of size 256 × 256 × 256.Results are shown for four different domain sizes rG = 10 Mpc (redsquares), 12 Mpc (green triangles), 14 Mpc (blue inverted triangles)and 16 Mpc (purple circles), after averaging over 40 different simula-tions. Here we have set nB = −2.37 (Color figure online)

them at a smaller radial distance with their polarizations andangular positions taken to be same as that of real data. Theresulting statistic, SD , as a function of the number of near-est neighbors is shown in Fig. 7 for four different domainsizes. The results are shown after averaging over 40 differ-ent simulations in order to reduce the effect of fluctuations.We find that SD shows an oscillatory behavior as a functionof the domain size. It increases as we increase the domainsize from 10 Mpc to 14 Mpc. However with a further in-crease from 14 Mpc to 16 Mpc it starts to decrease. A similartrend is seen for a larger grid of 512 × 512 × 512. This sug-gests that for the range of parameters we consider, SD showsan oscillatory dependence on the domain size. Furthermorethe variation is significant but not very large. These resultssuggest that for the distance scale corresponding to the realdata, the results may also show oscillations. Hence we ex-pect to find a suitable fit to data even for smaller domainsizes with other parameters similar to what were chosen forthe fit shown in Fig. 6.

5 Mixing in a more general background

As mentioned in the introduction a potential problem withour hypothesis is that the observed circular polarization inquasars is found to be negligible [4, 86]. However our modelpredicts a large circular polarization. We argued in the intro-duction that our model of the background medium, i.e. themagnetic field and the plasma density, may be unrealistic.The medium is expected to show fluctuations in time andspace at any length scale relevant to propagation over cos-mological distances. Here we have assumed that the mag-

netic field and the plasma density are time independent. Fur-thermore we have assumed that the magnetic field is uniformin a particular domain. In this section we determine howthese assumptions affect our predictions for circular polar-ization.

5.1 Space dependent magnetic field

We first examine how small scale spatial fluctuations in themagnetic field affect final results. We will continue to as-sume a uniform plasma density throughout this section. Asdiscussed above, the magnetic field is likely to have fluctua-tions even on the scale of a single domain. Here we assumea simple model of space varying magnetic field, given by

B = B0(cos2 αx + sin2 αy

)(23)

where

α = 2π

λ× z (24)

and λ is the wavelength for the B field fluctuations. The rateat which the magnetic field changes with position z can bevaried by changing the wavelength. Furthermore we pointout that the field has been chosen such that its mean is notzero. The A and χ field equations with this background canbe written as

(ω + i∂z)

⎛

⎝A1

A2

χ

⎞

⎠ − M

⎛

⎝A1

A2

χ

⎞

⎠ = 0, (25)

where

M =

⎛

⎜⎜⎝

ω2p

2ω0 − gφ

2 B0 cos2 α

0ω2

p

2ω− gφ

2 B0 sin2 α

− gφ

2 B0 cos2 α − gφ

2 B0 sin2 αm2

φ

2ω

⎞

⎟⎟⎠ .

(26)

Here, we have approximated (ω2 + ∂2z ) ≈ 2ω(ω + i∂z) [20]

and we have taken A as (A1 x + A2 y). These equations aresame as those given in Sect. 2.2 with the scale factor a = 1.

Here we do not assume that the background magneticfield is changing sufficiently slowly so that we can choosebasis vectors such that one of them points parallel to thetransverse background magnetic field at any point z alongthe path of the wave. This approximation would be valid ifλ is very large but would break down for small λ. Insteadwe directly solve these equations numerically in terms ofthe density matrix, which is defined as

ρ(z) =⎛

⎝〈A∗

1(z)A1(z)〉 〈A∗1(z)A2(z)〉 〈A∗

1(z)χ(z)〉〈A∗

2(z)A1(z)〉 〈A∗2(z)A2(z)〉 〈A∗

2(z)χ(z)〉〈χ∗(z)A1(z)〉 〈χ∗(z)A2(z)〉 〈χ∗(z)χ(z)〉

⎞

⎠ .

Page 8 of 10 Eur. Phys. J. C (2012) 72:1928

Fig. 8 The variation of linear (solid curve) and circular polarization(dashed curve) with respect to the wavelength λ of the backgroundmagnetic field

(27)

The Stokes parameters can be written as

I (z) = ⟨A∗

1(z)A1(z)⟩ + ⟨

A∗2(z)A2(z)

⟩, (28)

Q(z) = ⟨A∗

1(z)A1(z)⟩ − ⟨

A∗2(z)A2(z)

⟩, (29)

U(z) = ⟨A∗

1(z)A2(z)⟩ + ⟨

A∗2(z)A1(z)

⟩, (30)

V (z) = i(⟨

A∗1(z)A2(z)

⟩ − ⟨A∗

2(z)A1(z)⟩)

. (31)

We assume that the wave is initially unpolarized andpropagate in this medium over a distance of 10 Mpc. Theresulting circular polarization |V |/I and the linear polariza-tion

√Q2 + U2/I is shown in Fig. 8 as a function of the

wavelength λ of the background magnetic field. The figureclearly shows that for small λ, which corresponds to rapidfluctuations, the circular polarization is almost negligibleas compared to the linear polarization. The value of linearpolarization for small λ is comparable to that at large λ,where it starts to show large fluctuations. The circular po-larization becomes comparable to linear polarization if thebackground varies slowly. Hence we clearly see that circu-lar polarization is much smaller than linear polarization ifthe background magnetic field shows spatial fluctuations onsufficiently small length scales.

Here we have presented results only for a single domain.We do not attempt the more ambitious task of integratingover all the domains. That would be much more compu-tationally intensive in comparison to the calculation per-formed in this paper since it would involve solving a sys-tem of differential equations in each domain. Furthermore itwould depend on the precise model we use for the magneticfield in each domain. In any case our analysis in this sectionclearly shows that there exists a very reasonable model ofmagnetic field which would lead to highly suppressed circu-lar polarization.

5.2 Time dependent background magnetic field

Another important issue that we consider is the possible timedependence of the background magnetic field. If the back-ground magnetic field is time dependent, the pseudoscalarproduced due to mixing with photons will not have the samefrequency as that of the incident photon. Hence its corre-lation with photons may be significantly reduced. The cir-cular polarization arises primarily due to reconversion ofpseudoscalars back into photons. This generates a relativephase among the different components of the electromag-netic wave and hence leads to circular polarization. We seethis explicitly by considering the evolution of the densitymatrix in a particular domain, given in Ref. [79], reproducedhere for convenience,

⟨A‖(z)A∗‖(z)

⟩ = 1

2

⟨A‖(0)A∗‖(0)

⟩[1 + cos2 2θ

+ sin2 2θ cos[z(�φ − �A)

]]

+ 1

2

⟨φ(0)φ∗(0)

⟩[sin2 2θ

− sin2 2θ cos[z(�φ − �A)

]]

+{

1

2

⟨φ(0)A∗‖(0)

⟩[sin 2θ cos 2θ

− sin 2θ cos 2θ cos[z(�φ − �A)

]

− i sin 2θ sin[z(�φ − �A)

]]

+ c.c.

}(32)

⟨A⊥(z)A∗⊥(z)

⟩ = ⟨A⊥(0)A∗⊥(0)

⟩(33)

⟨A⊥(z)A∗‖(z)

⟩ = ⟨A⊥(0)A∗‖(0)

⟩[cos2 θeiFz + sin2 θeiGz

]

+ ⟨A⊥(0)φ∗(0)

⟩[sin θ cos θ

(eiFz − eiGz

)].

(34)

Here the vector A = E/ω, ω is the frequency of the electro-magnetic wave, φ is the pseudoscalar field and A‖ and A⊥respectively represent the components of A, parallel and per-pendicular to the transverse component of the backgroundmagnetic field. These equations give the density matrix el-ements after propagating through one domain, assuming aspace and time independent magnetic field and plasma den-sity. The circular polarization is governed by the correla-tion 〈A⊥(z)A∗‖(z)〉. Let us assume that initially this cor-relation is zero. Then after propagating through one do-main, it becomes non-zero only if initially the correlation〈A⊥(0)φ∗(0)〉 is non-zero. We have assumed that at thesource all the cross correlations are zero, i.e. the beam isunpolarized and the pseudoscalar field is zero. Hence cir-cular polarization can be produced only if, after propagat-ing through some domains, the correlation 〈A⊥φ∗〉 becomes

Eur. Phys. J. C (2012) 72:1928 Page 9 of 10

non-zero. However here we argue that if the backgroundmagnetic field is time-dependent, the frequency of the pseu-doscalar field produced due to mixing would be differentfrom that of the electromagnetic field. Hence its correlationwould be significantly reduced.

The pseudoscalar field equation may be written as

∂2φ

∂t2− ∇2φ + m2

φφ = gφE · B, (35)

where B represents the background magnetic field. Let usassume that the background magnetic field also undergoesfluctuations in time. Hence we may make a Fourier decom-position of this field. Here we assume that it contains onlyone significant component with frequency Ω , i.e. |B| ∼cos(Ωt). The frequency of the electric field is ω. It is clearfrom (35) that the frequency of the pseudoscalar producedhas to be ω′ = ω ± Ω .

Let us assume that the incident beam has a narrow spec-tral distribution, specified by the function A(ω), which de-notes one of the components of the electromagnetic wave.We require correlations such as 〈A(ω′)φ∗(ω′)〉, where ω′ =ω ± Ω . Since φ(ω′) ∝ A(ω)e−iω′t , we have

⟨A

(ω′)φ∗(ω′)⟩ ∝ ⟨

A(ω′)A∗(ω)

⟩. (36)

It is clear that if ω′ is sufficiently different from ω, this cor-relation would be vanishingly small. We expect this corre-lation to decrease rapidly with increase in the difference be-tween the two frequencies. We point out that the backgroundmedium is expected to have fluctuations on all time scales.It is only after averaging out short time fluctuations that weexpect it to show a somewhat smooth behavior. Hence itmay not be reasonable to assume that Ω is close to zero.We point out that, although these temporal fluctuations maysignificantly affect the circular polarization, the mixing phe-nomenon would still generate significant linear polarization,as long as the mean value of the transverse component ofthe magnetic field is significantly different from zero. Thisis because the electromagnetic wave component parallel tothe transverse magnetic field will decay into pseudoscalarsand the perpendicular component will not. Hence we findthat circular polarization may be significantly reduced com-pared to linear polarization if the background magnetic fieldshows sufficiently rapid fluctuations in time.

6 Conclusions

We have shown by direct simulations, both in a static andexpanding universe, that the hypothesis of pseudoscalar–photon mixing is able to explain the observed alignment ofoptical polarizations from distant quasars for standard pa-rameter values of the intergalactic magnetic field and plasma

density. The value of pseudoscalar–photon coupling is takento be close to its observational limit. The simulation resultsshow reasonable agreement with data both with the distri-bution of linear polarization as well as the coordinate in-variant statistic. Given the uncertainties in the medium pa-rameters we have not performed the exercise of finding thebest fit to the data. In any case before attempting a de-tailed fit it is necessary to perform further calculations toconclusively establish that pseudoscalar–photon mixing isindeed responsible for the alignment. One potential prob-lem with this explanation, as discussed in the introduction,is the relatively large circular polarization predicted by thisphenomenon [85]. However, there are many sources of de-coherence which need to be properly taken into accountbefore making a firm conclusion on the strength of circu-lar polarization. We have explicitly shown that if the mag-netic field shows fluctuations over the distance scale of in-dividual domains, the circular polarization predicted by thisphenomenon is significantly reduced in comparison to lin-ear polarization. We find this phenomenon even for a rela-tively large wavelength of fluctuations, of order 1 Mpc, ofthe background magnetic field. Furthermore the agreementwe find with data is potentially encouraging and hence thisproposal may be taken seriously. It is possible that even ifthis precise mechanism is found not to explain the data con-sistently, a suitable generalization might work. We postponesuch issues for future research.

Acknowledgements We thank Archana Kamal for useful discus-sions. We thank Subhayan Mandal for sharing with us his unpublishedwork on axion-photon mixing in an expanding universe.

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