Dependence of Residual Rotation Measure (RRM) on Intervening MgII Absorbers at Cosmic Distances
aa r X i v : . [ a s t r o - ph . C O ] J u l Mon. Not. R. Astron. Soc. , 1–6 (2013) Printed 19 May 2018 (MN L A TEX style file v2.2)
Dependence of Residual Rotation Measure (RRM) onIntervening Mg ii Absorbers at Cosmic Distances
Ravi Joshi ⋆ , Hum Chand ⋆ Aryabhatta Research Institute of observational sciencES (ARIES), Manora Peak, Nainital − Accepted —. Received —; in original form —
ABSTRACT
We investigate the dependence of residual rotation measure (RRM) on interveningabsorption systems at cosmic distances by using a large sample of 539 SDSS quasars inconjunction with the available rotation measure catalog at around 21cm wavelength.We found an excess extragalactic contribution in standard deviation of observed RRM( σ rrm ) of about 8.11 ± .
83 rad m − in our sample with intervening Mg ii absorber ascompare to the sample without Mg ii absorber. Our results suggest that interveningabsorbers could contribute to the enhancement of RRM at around 21cm wavelength,as was found earlier for RM measurements at around 6cm wavelength. Key words: galaxies: distances and redshifts, magnetic fields – polarization, quasars:absorption lines – quasars: objects: general – intergalactic medium – techniques: spec-troscopic
Magnetic field plays a key role in the structural and dynam-ical evolution of the Universe (e.g., Mestel & Paris 1984;Rees 1987), but there are no methods for its direct measure-ment. Faraday Rotation (FR) is one of the powerful probesto measure the strength of magnetic field over the cosmictime scale (e.g, Bernet et al. 2008, 2012; Hammond et al.2012; You et al. 2003; Welter et al. 1984; Kronberg et al.2008; Kronberg & Perry 1982; Kronberg et al. 1977;Kronberg & Simard-Normandin 1976). This Rotation Mea-sure (RM) is defined as the change in observed polarizationangle (∆ χ ) per unit change in observed wavelength square(∆ λ ). For a linearly polarized radio source at cosmologicalredshift ( z s ) it is given by (Bernet et al. 2012): RM ( z s ) = ∆ χ ∆ λ = 8 . × Z z s n e ( z ) B k ( z )(1 + z ) dldz dz, (1)where RM is in units of rad m − , the free electronnumber density, n e , is in cm − , the magnetic field compo-nent along the line-of-sight, B || , is in Gauss, and the co-moving path increment per unit redshift, dl/dz , is in par-sec. However, the observed RM has contributions from twocomponents, namely extragalactic radio source and ionized ⋆ E-mail: [email protected](RJ); [email protected](HC) medium of our Galaxy. As a result of this, it is not straightforward to quantify the FR contribution coming from the ex-tragalactic radio source. This extragalactic component alsoincludes the contributions from intervening galaxies and/ortheir halos, protogalaxies, intergalactic clouds, an intraclus-ter gas consisting of widespread coexpanding diffuse inter-galactic medium and intrinsic to the quasar. Therefore, theextragalactic RM can only be studied in terms of the resid-ual rotation measure (RRM), after removing the Galacticcomponent from the observed RM.Earlier studies of RM on its redshift evolu-tion has shown that the RM dispersion of quasarsincreases at high redshift (Kronberg et al. 2008;Kronberg & Simard-Normandin 1976; Rees & Reinhardt1972), in contrast to the (1 + z ) − dilution effect onRM (e.g., see Eq. 1). This led to the conclusion thatthe magnetic field strength as traced by the RM of highredshift galaxies is at least comparable to the currentepoch (Kronberg et al. 2008). The possible origin of thishigh magnetic field at cosmic distances remain ambiguous,however, such high fields could be either intrinsic to thequasars (e.g., arising in its immediate environment) or dueto the intervening environments along the lines-of-sightbetween the polarized source and the observer. Bernet et al.(2008) have probed the latter possibility from the analysisof high resolution optical spectra of 76 quasars. Theyhave shown that the quasars with strong Mg ii absorptionline systems are unambiguously associated with larger c (cid:13) J oshi & Chand
RM, inferred from their RM observation at around 6cmwavelength. In other words, the major contribution to theextragalactic component of observed RM comes from theintervening galaxies. However, in contrast to the case withMg ii absorption systems having rest frame equivalent width( EW r ) greater than 0.3˚A, Bernet et al. (2010) have shownthat the weaker systems do not contribute significantly tothe observed RM of the background quasars. The abovediscrepancy is attributed to the higher impact parametersof weak systems compared to strong ones.Recently, Hammond et al. (2012) made a catalog ofRRM available for 3651 radio sources that were observed ataround 21cm. They reported an observed standard deviationin RRM of 23.2 rad m − from a mixed sample of quasarsand galaxies having a mixture of sightlines with and withoutMg ii absorber. Further, subtracting the possible errors con-tributing to this measured standard deviation, such as: (i)the measurement errors of individual RM of 11 rad m − asgiven in the catalog by Taylor et al. (2009); (ii) error associ-ated with the galactic rotation measure (GRM) calculationsof 6 rad m − (Oppermann et al. 2012); and (iii) 12 −
17 radm − from the RM fluctuations on smaller angular scalesthan are being sampled by above GRM (Stil et al. 2011).The remaining contribution from extragalactic componentis found to be typically around 10 −
15 rad m − , similar toSchnitzeler (2010). Importantly, with this extensive study,they could not reproduce any significant redshift evolutionof RRM as seen by other studies (e.g., Kronberg et al. 2008;Welter et al. 1984).To explain the above discrepancy, Bernet et al. (2012)have proposed a model consisting of partially inhomoge-neous rotation measure screen, which causes wavelength de-pendent depolarization. As a result the depolarization intheir model toward longer wavelength such as close to 21cmused in Hammond et al. (2012) data set will be larger thanthe shorter wavelength close to 6cm used in Bernet et al.(2008), which has been attributed to the above discrepancy.This was supported by their result that the RM distribu-tion with and without Mg ii absorber do differ from data setbased on 6cm, unlike no such difference seen on their 21cmdata set. However, it should be noted that this data set hav-ing RM measurements at both the above wavelengths (i.e.,21cm and 6cm) consist of only 54 radio source sightlines. Inview of the important consequences of the above results, itis very important to carry out the analysis by using largersample size to detect and quantify any effect of interveningabsorbers on the extra-galactic component of RRM. Thisforms the main motivation of our present work, by carry-ing out the analyses for subsample of sightlines with andwithout Mg ii absorber in the parent sample of 567 SDSSquasars and by using the RRM-redshift catalog providedby Hammond et al. (2012).This paper is organized as follows. Section 2 describethe selection of the samples of RRM data set, while Section3 gives methodology used in the analysis. In Section 4, wepresent the results of our analysis, followed by the discussionand conclusions in Section 5. Our sample is collected from catalog produced byHammond et al. (2012), which consists of 3651 sources athigh galactic latitude, having | b | > ◦ . The catalog isconstructed by assigning redshift for polarized radio sourcecataloged by Taylor et al. (2009) using an optical databaseof, e.g., NED and SIMBAD and optical surveys namelySDSS-DR8 , 6DFGS , 2dfGRS and 2QZ/6QZ (e.g., seetheir online Table 1). We have applied following criteria toselect our sample from the above mentioned catalog.(i) Firstly, we have restricted ourselves only to those ra-dio sources whose optical association are assigned as quasarusing SDSS database due to the advantage of the availabilityof their spectra from SDSS archive. Of the 3651 source in thecatalog of Hammond et al. (2012), this selection filter left uswith 860 polarized radio sources having SDSS spectra. Outof them, we only consider the most common designated ra-dio to optical association namely class ‘A’, which representsan unresolved NVSS radio source (observed at 21cm) thatalign closely with the corresponding optical counterpart (i.e.,the NVSS-optical offset <
15 arcsec) and also have a FIRST counterpart, which is used to make the radio-optical asso-ciation. Further, the class ‘A’ in Hammond et al. (2012) issubdivided into seven subclasses A(i)-A(vii), depending onthe radio morphology in FIRST survey. To minimize the un-certainty of radio and optical association, we have includedonly those subclasses having at the most 3 matches in FIRSTimage within 30 arcsec of the optical position, leading to theexclusion of sources belonging to A(vi), which have morethan 3 such associations. In addition, we have included theclass ‘B’ sources, representing a similar situation to class ‘A’by using only NVSS, either due to the non-detection or theabsence of data set in FIRST survey. This left us with asample of 730 polarized radio sources having SDSS spectra.(ii) Secondly, we have excluded all the quasars havingemission redshift outside the range of 0 . z em . ii doublet, if present. Similarly, upper z em constrainis to ensure that Mg ii emission line does not fall above thehighest SDSS wavelength, which is 9200˚A, so that ambigu-ity of any Mg ii absorber falling above the spectral coverage NED=NASA/IPAC Extragalactic Databasehttp://ned.ipac.caltech.edu SIMBAD=Set of Identifications, Measurements and Bibliogra-phy for Astronomical Data; http://simbad.u-strasbg.fr/simbad/ SDSS=Sloan Digital Sky survey DR8;http://skyserver/sdss3.org/dr8/en/ FIRST=The VLA Faint Images of the Radio Sky at Twentycentimeters survey; http://sundog.stsci.edu/c (cid:13) , 1–6
RM and intervening Mg ii absorbers can be avoided. These filters have reduced the sample to 615quasars.(iii) Finally, to minimize the uncertainty in the radio tooptical association, we have removed those quasars for whichthe optical and radio sightlines are separated by more than 7arcsec (similar to Bernet et al. 2008), resulting in the sampleof 567 quasars, for our analysis. ii absorption systems The identification of the Mg ii absorption doublet in thenormalized continuum spectrum, was carried out using theprocedure discussed in detail by Joshi et al. (2013, under re-view). Briefly, the procedure initially fits a continuum to theSDSS spectroscopic data, employing a first principal compo-nent analysis (PCA) as a guess for the Ly α and C iv emis-sion lines (i.e., from 1000˚A-2000˚A in the rest-frame). Next,a b-spline algorithm was used to fit the underlying residualcontinuum, which results roughly in a power-law with broademission features superposed.The procedure automatically also searches for absorp-tion features in the normalized spectrum redward of Ly α .The search was carried out for absorption features, fittedwith Gaussian profile by taking an initial FWHM of 2.5pixels, with the additional requirement that the minimumseparation between lines of doublet should be about 2 timesthe FWHM. Out of all such cases, the final selection wasmade by accepting only the lines which are above 3 timesthe rms( σ ) noise in the spectrum.The absorption features thus identified for each quasar,were searched for absorption line pairs. For this purposethe procedure first computed redshift of a given absorptionfeature, assuming it to be Mg ii λ ii λ ii λ ii absorption system, was that at least twoof these three lines must be present at the expected loca-tions above a 3 σ threshold. Equivalent widths of all the ac-cepted Mg ii absorption lines were then measured by sum-ming the difference from unity of the flux in the normalizedobserved-frame spectrum, within about 10 pixels wide boxes( ∼ ii lines.As a further check, we also carried out a visual confir-mation of each absorption system identified via the aboveautomated procedure. This step is important since (i) ourautomated search do not carry out the line profile match-ing and hence can result in over counting the Mg ii doubletrather than missing out any genuine system, and (ii) anyvisually noticed uncertainty in the continuum level couldhave significantly distorted the estimate of EW r , renderingthe strong/weak classification of absorption systems unre-liable. In this process of visual scrutiny, we first looked forthe strongest five Fe ii lines corresponding to the candi-date Mg ii doublet. We then made a velocity plot of Mg i ,Mg ii and the five Fe ii lines, so as to match by eye the lineprofile and strength to that expected on the basis of the lineoscillator strengths. Thus, in the spectra of the 567 quasars,we visually inspected all 673 Mg ii absorption systems candi-dates and confirmed 256 of them. Continuum fitting of each confirmed Mg ii absorption system was then further checkedby plotting the fitted continuum to the spectral segment con-taining the Mg ii doublet. In all cases where the continuumfitting over the relevant spectral segment seemed unsatisfac-tory, we refined the local continuum fit and recomputed the EW r (Mg ii ). To test the ‘intervening hypothesis’, we haveremoved those 28 quasars having associated Mg ii absorberswith relative velocity < − in the spectra. Thisleads to a final sample of 539 quasars, which is used in restof our analysis. Among these 539 quasars, 388 are withoutMg ii absorber, while 119 have one and 32 have more thanone Mg ii absorber in their spectra. The quasars with strong Mg ii absorption systems alongtheir line-of-sight are found to have broad RM distributionfrom the analysis of 6cm data set (Bernet et al. 2008). How-ever, in a recent study no such signature is observed withthe 21cm data set by Bernet et al. (2012), though with thenominal sample size consisting of only 54 radio source sight-lines (see their Figure 3). In this work, to test the abovediscrepancy, in Figure 1 ( left panel ), we have shown the cu-mulative distribution of RM for sightlines with Mg ii (withnumber n MgII >
0, dashed-dotted line; n MgII >
1, dashedline) and without Mg ii ( n MgII = 0, thick line) absorberusing our large data set of 539 sightlines having RM mea-surement at around 21cm wavelength. The K − S test rulesout the null hypothesis for the quasar subsets, (i) withoutand with at least one Mg ii absorber and (ii) without andwith at least two Mg ii absorbers, at a confidence level of48% and 66% respectively, which is statistically nonsignifi-cant using our this modest sample size. We also notice herethat the major difference in RM distribution can be seenat around 30 rad m − , which is similar to the average con-tribution of 20 rad m − from the foreground galactic RM(GRM) component (Bernet et al. 2008). To see any such ef-fect of GRM, we have plotted the cumulative distribution ofRRM (i.e., GRM subtracted RM), shown in the right panel of Figure 1. The two distribution without and with at leastone Mg ii absorber are found to be different with the K − Stest giving P null = 0 .
21. Similarly, a P null = 0 .
32, is seen be-tween the quasar subsets having at least two Mg ii absorberand no absorber, implying that the hypothesis of quasarswith and without Mg ii absorber have a similar distributionis ruled out at a confidence level of 79%.To elucidate the effect of intervening absorber on RRM,in Figure 2, we have plotted the histogram of RRM forthe quasars with absorber (shaded region) and without ab-sorber (thick line), after normalizing with total quasar countwithin respective subsets. At first look, it appears that theRRM distribution for the sightlines that passes through theMg ii systems is significantly broader than that for whichabsorption is absent. This reflects in the standard deviationof RRM for the quasars with and without Mg ii absorberbeing 18.93 ± .
05 rad m − and 17.11 ± .
69 rad m − , respec-tively. Here, the error on standard deviation of RRM is com-puted by propagation of errors in individual RRM value byassuming their Gaussian nature, as follows : c (cid:13)000
69 rad m − , respec-tively. Here, the error on standard deviation of RRM is com-puted by propagation of errors in individual RRM value byassuming their Gaussian nature, as follows : c (cid:13)000 , 1–6 J oshi & Chand -50 0 50 100 RM (rad m -2 )0.00.20.40.60.81.0 C u m u l a t i v e F r a c t i on Quasars with no absorberQuasars with atleast one absorberQuasars with atleast two absorber -50 0 50 100 RRM (rad m -2 )0.00.20.40.60.81.0 C u m u l a t i v e F r a c t i on Quasars with no absorberQuasars with atleast one absorberQuasars with atleast two absorber-10 -5 0 5 100.10.20.30.40.50.60.7
Figure 1.
Left panel:
Cumulative distribution of the Rotation Measure (RM), for the quasar with no absorber (thick line), with atleast one absorber (dashed-doted line) and with at least two absorber (dashed line). right panel: same as left, for the Residual RotationMeasure (RRM) measurements, the inset displays the zoom-in on the maximum distance between the distribution functions. -100 -50 0 50 10000.020.040.060.080.1
Figure 2.
Histogram of Residual Rotation Measure (RRM),normalized by the total quasars counts in the subsets of quasarwith ( EW r (2796) > ii absorber (shaded region) andwithout absorber (thick line). δσ = 1 σ (N − vuut N X i=1 ( x i − x ) δx i + N X i=1 x i − x ! δx , (2)Where, σ is the standard deviation of RRM, x i ’s arethe observed RRM values, x is mean RRM value and δx i ’s, δx are their respective errors.To quantify the excess standard deviation ( σ ex ) seen Figure 3.
Distribution of rest frame equivalent width with resid-ual rotation measure (RRM) for the quasars with Mg ii absorber. along the sightlines with intervening absorbers, we have sub-tracted the standard deviation of RRM for the quasar sub-sample with and without Mg ii absorber in the quadraturei.e. σ ex = q σ − σ , (3)The excess in the standard deviation is found to be8.11 ± .
85 rad m − , i.e., at 2 . σ level, where the associatederror is computed with error propagation as c (cid:13) , 1–6 RM and intervening Mg ii absorbers Figure 4.
Histogram of the fractional polarization ( p ) for thequasars with Mg ii absorber (shaded region) and without absorber(thick line). δσ ex = 1 σ ex q σ δσ + σ δσ , (4)This high significance excess at 2 . σ level in standarddeviation is much higher than the above discussed 79% con-fidence level implied by the K − S test for the subsample ofquasar with and without Mg ii absorber. One possibilitycould be that the error bars in RRM measurements may beunderestimated. To quantify it, we have computed the re-duced χ value for our subsample without Mg ii absorber, byassuming the expected value of RRM measurements to be(i) the mean value of RRM measurements, and (ii) as zerovalue. In both the cases, the reduced χ values are found tobe around 2.87. Now, assuming that for true error bars inRRM measurements for subsample without Mg ii absorbershould result this reduced χ to be around unity, suggestsus to scale up all the error bars on our RRM measurementsby square root of 2.87. Repeating the above analysis withthese scaled error bars on RRM, our revised standard devi-ation for quasars with and without Mg ii absorber becomes18.93 ± − and 17.11 ± − respectively.Similarly, the excess standard deviation (using Eq 3) now be-come 8.11 ± − , i.e., at 1.7 σ level. Given that ourK-S test also give the similar confidence level (i.e., 79%) itappear that our earlier result with higher significance of 2 . σ level perhaps may be due to this underestimation of errorbars in RRM measurements. Therefore, henceforth through-out the analysis, we have derived our results using the scalederror bars in RRM measurements.We also note that our result is in good agreement withHammond et al. (2012) and Schnitzeler (2010), where theyhave shown the standard deviation of RRM by its extra-galactic component in the order of 10 −
15 rad m − and ∼ − , respectively. In this 10 −
15 rad m − standarddeviation of RRM by Hammond et al. (2012), they havealso taken into account the possible contribution caused byerrors associated with GRM and RM measurements, by their quadrature subtraction from the observed σ (RRM) value(e.g., see Section 1). Recall that in their study they have useda mixture of sightlines consisting of both with and withoutMg ii absorber. However in our study, as we do have separatesubsample of sightlines with and without Mg ii absorber, itallows us to quantify the contribution in σ ex due to inter-vening absorbers by subtracting the σ (RRM) of later fromthe former in quadrature (e.g see Eq. 3). This being thedifference of two standard deviations in quadrature, makesour result of 8.11 ± .
83 rad m − , free from any error contri-bution to σ (RRM) that are common for both the subsam-ples of with and without Mg ii absorbers, such as associ-ated with the GRM and RM measurements considered inHammond et al. (2012).Now, it is also worth to see the RRM distribution with EW r , as the quasars having absorber with larger EW r areexpected to have larger observed RRM dispersion. In Fig-ure 3, we have plotted the RRM distribution with EW r where the open circle represents the EW r corresponding toan absorber over a sightline, while the star (red) correspondsto the sum of EW r values for the sightlines having morethan one Mg ii absorber. In the absence of obvious trend ofRRM with EW r , we have computed the standard deviationin RRM for the absorber having EW r < EW r > ± .
42 and 19.23 ± .
23 rad m − , respec-tively, with error bars computed using Eq 2. Though thereis a very mild excess in the value of standard deviation ofRRM for larger EW r subsample but it is consistent within1 σ error bars.Hammond et al. (2012) found an anti-correlation be-tween RRM and fractional polarization ( p ). One possiblephysical bias suggested by them was the presence of inter-vening absorbers; which need to be tested by plotting theRRM with ‘ p ’ for a subclass of sample with and withoutMg ii absorber. In this possible scenario, to explain anti-correlation of RRM versus p , one would expect that thefractional polarization observed for a sample with Mg ii ab-sorber (having larger RRM) should be smaller than the sam-ple without Mg ii absorber. To test this hypothesis in Fig-ure 4, we have plotted the histogram of fractional polariza-tion ( p ) for the quasars with (shaded region) and without(thick line) Mg ii absorber. The median (weighted mean)values of p , for sample with and without Mg ii absorber,are found to be 3.50(4.57) and 3.57(5.16), respectively. TheK − S test shows that the null hypothesis is ruled out witha low confidence level of 63.13%. This very mild differencewith present sample size does not allow us to conclude muchon above hypothesis, but a larger sample will be helpful tosay firmly about it.
The excess extragalactic FR due to the presence of a strongintervening absorber in the optical spectrum of quasars isnoticed in many recent studies (Kronberg & Perry 1982;Welter et al. 1984; You et al. 2003; Bernet et al. 2008).Commonly accepted view is that the major contribu-tor could be the intervening galaxies along the sight-line of polarized radio sources. For instance, Bernet et al.(2008) have used a sample of total 76 sightlines withRM observations at around 6cm, and found a statis- c (cid:13)000
The excess extragalactic FR due to the presence of a strongintervening absorber in the optical spectrum of quasars isnoticed in many recent studies (Kronberg & Perry 1982;Welter et al. 1984; You et al. 2003; Bernet et al. 2008).Commonly accepted view is that the major contribu-tor could be the intervening galaxies along the sight-line of polarized radio sources. For instance, Bernet et al.(2008) have used a sample of total 76 sightlines withRM observations at around 6cm, and found a statis- c (cid:13)000 , 1–6 J oshi & Chand tically higher RM for their 7 sources showing morethan one Mg ii absorber. Many other studies havealso shown the increase of RRM with redshift (e.g.,Kronberg et al. 2008; Kronberg & Simard-Normandin 1976;Rees & Reinhardt 1972), albeit by using only a nominalsample size till now. Recently, with a large data set of 3651quasars with RRM observation at 21cm, Hammond et al.(2012) have not found any such signature of RRM evolutionwith redshift. Similarly, the role of intervening absorber inRRM at 21cm was also studied till now by using a smallsample size of only 54 quasars (Bernet et al. 2012). Here,we have addressed these questions by using a larger sam-ple size of the 539 SDSS quasars having RM data set atwavelength at around 21cm, rather than at around 6cm .Our analysis has shown that RRM distribution for thequasars with and without Mg ii absorber do differ at confi-dence level of 79%. This show that there is contribution of in-tervening absorber to enhance the RRM, even at wavelengthof 21cm. Its non-detection in the analysis of Bernet et al.(2012), could be due to their smaller sample size, wherethey have used just a sample of 54 sightlines having RRMmeasurement at 21cm. However, we also note that at 6cmmeasurement of RM, the two distribution with and withoutMg ii absorber do differ with much more confidence levels(e.g, Bernet et al. 2008) than what we have found for oursample of RRM measurement at 21cm. This may be dueto the recent mechanism proposed by Bernet et al. (2012),where the intervening absorbers acts as a RM screens andcauses more depolarization at longer wavelength.We also found that the standard deviation of RRMfor sample with Mg ii absorber do have excess of about8.11 ± .
83 rad m − as compared to the sample withoutMg ii absorber. This is consistent with Hammond et al.(2012) and Schnitzeler (2010) studies, where in their in-dependent analysis they have found the value of extragalac-tic component in RRM standard deviation to be around10 −
15 rad m − and 6 rad m − respectively.Further, we have also computed the standard deviationof RRM for subsets having Mg ii absorber with EW r < EW r > EW r < EW r > ± .
42 and 19.23 ± .
23 rad m − , respectively (seeFigure 3), which is tentatively consistent with above expec-tation.We also notice that the factional polarization ( p ) ofthe quasars with and without Mg ii absorber show nomi-nal difference, with median (weighted mean) values of p , tobe 3.50(4.57) and 3.57(5.16) respectively. This very nomi-nal excess for sightlines without Mg ii absorbers, could bedue to the less depolarization in the absence of interveningabsorbers (e.g., see also Hammond et al. 2012; Bernet et al.2012). However, a larger data set would require to reducethe statistical error bar.At present, with our large data set of 539 sightlines,we could only found a statistical excess of 8.11 ± .
83 radm − in standard deviation of RRM for the sample withMg ii absorber as compared to the sample without Mg ii ab-sorber, by using RRM observation at around 21cm wave-length. This has allowed us to conclude that the interveningMg ii absorber also makes contribution for the excess in standard deviation of RRM observed even at around 21cmwavelength, though perhaps with smaller magnitude thanat 6cm (e.g, Bernet et al. 2008); the theoretical implicationof which needs further explorations. ACKNOWLEDGMENTS
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