Redshift evolution of extragalactic rotation measures
aa r X i v : . [ a s t r o - ph . C O ] M a y Mon. Not. R. Astron. Soc. , 1–9 (?) Printed 18 October 2018 (MN L A TEX style file v2.2)
Redshift evolution of extragalactic rotation measures
J. Xu , and J. L. Han , ⋆ National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Road, Chaoyang District, Beijing 100012, China. School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China. School of Astronomy and Space Science, Nanjing University, Nanjing, China
Accepted 2014 May 19. Received 2014 May 19; in original form 2014 March 19
ABSTRACT
We obtained rotation measures of 2642 quasars by cross-identification of the most updatedquasar catalog and rotation measure catalog. After discounting the foreground Galactic Fara-day rotation of the Milky Way, we get the residual rotation measure (RRM) of these quasars.We carefully discarded the effects from measurement and systematical uncertainties of RRMsas well as large RRMs from outliers, and get marginal evidence for the redshift evolution ofreal dispersion of RRMs which steady increases to 10 rad m − from z = 0 to z ∼ andis saturated around the value at higher redshifts. The ionized clouds in the form of galaxy,galaxy clusters or cosmological filaments could produce the observed RRM evolutions withdifferent dispersion width. However current data sets can not constrain the contributions fromgalaxy halos and cosmic webs. Future RM measurements for a large sample of quasars withhigh precision are desired to disentangle these different contributions. Key words: polarization — intergalactic medium — radio continuum: general — magneticfields
Faraday rotation is a powerful tool to probe the extragalacticmedium. The observed rotation measure of a linearly polarized ra-dio source at redshift z s is determined by the polarization anglerotation ( ψ − ψ ) against the wavelength square ( λ − λ ) RM obs = ψ − ψ λ − λ = 0 . Z z s n e ( z ) B || ( z )(1 + z ) dldz dz. (1)The rotation measure (RM, in the unit of rad m − ) is an integratedquantity of the product of thermal electron density ( n e , in the unitof cm − ) and magnetic fields along the line of sight ( B || , in the unitof µ G) over the path from the source at a redshift z s to us. Here thecomoving path increment per unit redshift, dl/dz , is in parsecs.The observed rotation measure, RM obs , with a uncertainty, σ RM ,is a sum of the rotation measure intrinsic to the source, RM in , therotation measure in intergalactic space, RM IGM , the foregroundGalactic RM,
GRM , from our Milky Way Galaxy, i.e. RM obs = RM in + RM IGM + GRM. (2)It has been found that the RM distribution of radio sources inthe sky are correlated in angular scale of a few degree to a fewtens degree (Simard-Normandin & Kronberg 1980; Oren & Wolfe1995; Han et al. 1997; Stil, Taylor & Sunstrum 2011), which indi-cates the smooth Galactic RM foreground. The extragalactic rota-tion measures is RM in + RM IGM = RM obs − GRM , which ⋆ E-mail: [email protected] is often called as residual rotation measure (RRM), i.e. the resid-ual after the foreground Galactic RM is discounted from the ob-served RM. Because the polarization angle undergoes a randomwalk in the intergalactic space due to intervening magnetoionicmedium, the RRMs from the intergalactic medium should have azero-mean Gaussian distribution. Radio sources at higher redshiftwill pass through more intervening medium, so that variance ofRRMs, V RRM , of a sample of sources is expected to get larger athigher redshifts. Though the measured RM values from a sourcecould be likely wavelength dependent due to unresolved multi-ple components (Farnsworth, Rudnick & Brown 2011; Xu & Han2012; Bernet, Miniati & Lilly 2012), RM values intrinsic to a ra-dio source at redshift z s are reduced by a factor (1 + z s ) due tochange of λ when transformed to the observer’s frame, and for thevariance by a factor (1 + z s ) , the RRMs are therefore often sta-tistically used to probe magnetic fields in the intervening mediumbetween the source and us, such as galaxies, galaxy clusters or cos-mic webs.Previously there have been many efforts to investigate RRMdistributions and their possible evolution with redshift. Withouta good assessment of the foreground Galactic RM in early days,RMs of a small sample of radio sources gave some indicationsfor larger RRM data scatter at higher redshifts, which weretaken as evidence of magnetic field in the intergalactic medium(Nelson 1973; Vallee 1975; Kronberg & Simard-Normandin1976; Kronberg, Reinhardt & Simard-Normandin 1977;Thomson & Nelson 1982). Burman (1974) proposed the steady-state model and found that the variance of RRM approaches a lim-iting value at z s ∼ ; Kronberg, Reinhardt & Simard-Normandin c (cid:13) ? RAS J. Xu and J. L. Han
Figure 1.
Sky view for the foreground Galactic RM derived byOppermann et al. (2012) (the top panel) and by Xu & Han (2014) (the mid-dle panel), and their difference (the bottom panel). See Xu & Han (2014)for details. (1977) suggested σ IGM < rad m − ; Vallee (1975) claimedthe upper limit of intergalactic rotation measure as being 10rad m − . Theoretical models for the random intergalactic mag-netic fields in the Friedmann cosmology (Nelson 1973) and inEinstein-de Sitter cosmology (Burman 1974) and for the uniformfields (Vallee 1975) have been proposed. Thomson & Nelson(1982) summarized the Friedmann model (Nelson 1973) and thesteady-state model (Burman 1974) and also proposed their ownionized cloud model. In Friedmann model, particles conservationis assumed and the field is frozen in the evolving Friedmanncosmology, Thomson & Nelson (1982) showed V RRM increasingwith (1 + z ) ∼ depending on cosmology density Ω M . In thesteady-state model contiguous random cells do not vary with time,which induces the intergalactic V RRM ∝ [1 − (1 + z s ) − ] . Inthe ionized cloud model, the Faraday-active cells with randomfields are in the form of non-evolving discrete gravitationallybound, ionized clouds, so that the final V RRM ∝ [1 − (1 + z s ) − ] .Thomson & Nelson (1982) applied these three models to fitthe increasing RM variance of 134 quasars against redshift, but can not distinguish the models due to large uncertainties.Welter, Perry & Kronberg (1984) made a statistics of the RRMs of112 quasars and found a systematic increase of V RRM with redshifteven up to redshift z above 2. After considering possible contribu-tions to the RRM variance from RMs intrinsic to quasars or fromRMs due to discrete intervening clouds, Welter, Perry & Kronberg(1984) suggest that the observed RRM variance mainly resultsfrom absorption-line associated intervening clouds.Because intervening galaxies are most probable cloudsfor the intergalactic RMs at cosmological distances, ef-forts to search for evidence for the association betweenthe enlarged RRM variance with optical absorption-lines ofquasars therefore have been made for many years, first byKronberg & Perry (1982), later by Welter, Perry & Kronberg(1984); Watson & Perry (1991); Wolfe, Lanzetta & Oren (1992);Oren & Wolfe (1995); Bernet et al. (2008); Kronberg et al.(2008); Bernet, Miniati & Lilly (2010), and most recently byBernet, Miniati & Lilly (2012) and Joshi & Chand (2013). Smalland later larger quasar samples with or without the MgII absorptionlines (e.g. Joshi & Chand 2013), with stronger or weaker MgIIabsorption lines (e.g. Bernet, Miniati & Lilly 2010), with orwithout the Ly α absorption lines (e.g. Oren & Wolfe 1995), arecompared for RRM distributions. In almost all cases, the RRMor absolute values of RMs of quasars with absorption lines showsignificantly different cumulative RM probability distributionfunction or a different variance value from those without absorp-tion lines, and those of higher redshift quasars show a marginallysignificant excess compared to that of lower redshift objects. Mostrecently Joshi & Chand (2013) got the excess of RRM deviation of . ± . rad m − for quasars with MgII absorption-lines.Certainly intervening objects could be large-scale cosmic-web or filaments or super-clusters of galaxies, with a coherencelength much larger than a galaxy, which may result in a pos-sible excess of RRMs (Xu et al. 2006). At least the RRM ex-cess due to galaxy clusters has been statistically detected (e.g.Clarke, Kronberg & B¨ohringer 2001; Govoni et al. 2010). Com-puter simulations for large-scale turbulent magnetic fields to-gether with inhomogeneous density in the cosmic web of tens ofMpc scale have been tried by, e.g., Blasi, Burles & Olinto (1999);Ryu et al. (2008); Akahori & Ryu (2010, 2011), and also comparedwith real RRM data. The RMs from cosmic web probably are verysmall, only about a few rad m − (Akahori & Ryu 2011). The dis-persion of so-caused RRMs is also small, which increases steeplyfor z < and saturates at a value of a few rad m − at z ∼ .Because of the smallness of RM contribution from inter-galactic space, to study the redshift evolution of extragalacticRMs, we have to enlarge the sample size of high redshift ob-jects for RMs and have to reduce the RRM uncertainty. The un-certainty of RRM is limited by not only the observed accuracyfor RMs of radio sources but also the accuracy of estimated fore-ground of the Galactic RMs. The RMs were found to be cor-related over a few tens of degree in the mid-latitude area (e.g.Simard-Normandin & Kronberg 1980; Oren & Wolfe 1995). TheGRM uncertainty in most previous studies is large, around 20rad m − in general, due to a small covering density of availableRMs in the sky. Noticed that RMs have smallest random valuesnear the two Galactic poles (Simard-Normandin & Kronberg 1980;Han, Manchester & Qiao 1999; Mao et al. 2010). To reduce the un-certainty of RRMs, You, Han & Chen (2003) tried to use RMs ofonly 43 carefully selected extragalactic radio sources toward Galac-tic poles, and found only the marginal increase of V RRM with red-shift. c (cid:13) ? RAS, MNRAS , 1–9 edshift evolution of extragalactic rotation measures Table 1. σ RM Ref GRM σ GRM
RRM σ RRM (deg) (deg) (deg) (deg) (rad m − ) (rad m − ) (rad m − ) (rad m − ) (rad m − ) (rad m − )0.0417 30.9331 1.801 110.1507 –30.6630 –37.9 11.0 tss09 –68.8 1.7 30.9 11.10.2542 24.1450 0.300 108.4335 –37.3031 –63.2 14.5 tss09 –57.8 1.9 –5.4 14.60.3867 14.9356 0.399 105.3749 –46.2285 –34.9 3.8 tss09 –17.0 1.2 –17.9 4.00.7050 30.5447 2.300 110.6968 –31.1693 –40.5 13.9 tss09 –68.6 1.7 28.1 14.00.7992 16.4839 1.600 106.5177 –44.8449 –24.3 8.5 tss09 –21.2 1.2 –3.1 8.60.9283 –11.8633 1.300 84.3539 –71.0677 –3.8 13.3 tss09 0.4 1.4 –4.2 13.40.9383 –11.1383 1.569 85.6081 –70.4718 –8.2 9.3 tss09 1.8 1.4 –10.0 9.41.3108 4.4186 1.200 101.7086 –56.5377 13.3 5.6 tss09 –2.7 1.4 16.0 5.81.5942 –0.0733 1.038 99.2808 –60.8590 12.0 3.0 skb81 –5.2 1.6 17.2 3.41.5958 12.5981 0.980 106.1113 –48.7967 –11.2 8.5 tss09 –10.0 1.2 –1.2 8.61.6471 8.8044 1.900 104.5495 –52.4592 –8.0 13.7 tss09 –3.5 1.2 –4.5 13.72.0550 13.6133 1.000 107.1538 –47.9300 0.1 15.9 tss09 –11.9 1.2 12.0 15.92.1925 0.0611 0.505 100.5304 –60.9412 –38.5 18.8 tss09 –5.4 1.5 –33.1 18.92.2071 –0.2778 2.000 100.3199 –61.2648 7.8 8.0 tss09 –5.3 1.5 13.1 8.12.2662 6.4725 0.400 104.4242 –54.8694 –17.1 7.4 tss09 –1.8 1.3 –15.3 7.52.4463 6.0972 2.311 104.5382 –55.2800 10.3 15.4 tss09 –1.7 1.3 12.0 15.52.5758 14.5606 0.901 108.2184 –47.1326 –25.6 10.5 tss09 –14.1 1.1 –11.5 10.62.6196 20.7969 0.600 110.1993 –41.0599 –36.4 16.2 tss09 –36.6 1.6 0.2 16.32.6450 –30.9042 0.999 7.5916 –80.3078 –10.4 9.3 tss09 8.3 0.8 –18.7 9.32.8967 8.3986 1.300 106.3363 –53.1842 3.2 2.0 tss09 –2.9 1.2 6.1 2.33.0346 7.3308 1.800 106.0930 –54.2510 –22.1 14.5 tss09 –2.1 1.2 –20.0 14.63.3363 –15.2297 1.838 84.3517 –75.1678 11.1 13.0 tss09 –0.1 1.2 11.2 13.13.4754 –4.3978 1.075 99.7867 –65.5687 –1.5 4.4 tss09 –0.0 1.4 –1.5 4.63.6575 –30.9886 2.785 5.1071 –81.0824 9.0 2.0 mgh+10 7.7 0.8 1.3 2.23.7604 –18.2142 0.743 77.7952 –77.7642 –2.5 4.9 tss09 3.7 1.0 –6.2 5.04.0000 39.0072 1.721 115.4420 –23.3486 –123.9 4.0 kmg+03 –81.6 2.4 –42.3 4.74.0533 29.7517 1.300 113.8648 –32.4992 –74.3 8.1 tss09 –66.0 1.7 –8.3 8.34.0729 24.9656 1.800 112.9169 –37.2218 –43.1 13.6 tss09 –60.1 1.8 17.0 13.74.1658 25.1747 1.300 113.0655 –37.0299 –77.1 6.9 tss09 –60.9 1.8 –16.2 7.14.2588 32.1558 1.086 114.5124 –30.1529 –42.1 12.5 tss09 –62.4 1.6 20.3 12.6This table is available in its entirety online. A portion is shown here for guidance regarding its form and content. In addition to the previously cataloged RMs(e.g. Simard-Normandin, Kronberg & Button 1981;Broten, MacLeod & Vallee 1988) and published RM data inliterature, Taylor, Stil & Sunstrum (2009) have reprocessed the2-band polarization data of the NRAO VLA Sky Survey (NVSS,Condon et al. 1998), and obtained the two-band RMs for 37,543sources. Though there is a systematical uncertainty of . ± . rad m − (Xu & Han 2014), the NVSS RMs can be used togetherto derive the foreground Galactic RM (Oppermann et al. 2012;Xu & Han 2014), see Fig. 1. Hammond, Robishaw & Gaensler(2012) obtained the RMs of a sample of 4003 extragalactic objectswith known redshifts (including 860 quasars, data not releasedyet) by cross-identification of the NVSS RM catalog sources(Taylor, Stil & Sunstrum 2009) with known optical counterparts(galaxies, AGNs and quasars) in literature, and they concluded thatthe variance for RRMs does not evolve with redshift. Nevertheless,Neronov, Semikoz & Banafsheh (2013) used the same dataset andfound strong evidence for the redshift evolution of absolute valuesof RMs. Further investigation on this controversy is necessary.Recently, Xu & Han (2014) compiled a catalog of reliableRMs for 4553 extragalactic point radio sources, and used aweighted average method to calculate the Galactic RM foregroundbased on the compiled RM data together with the NVSS RMdata. On the other hand, a new version of quasar catalog (Milli- quas) is updated and available on the website , which compiledabout 1,252,004 objects from literature and archival surveys anddatabases. Here we cross-identify the two large datasets, and ob-tained a large sample of RMs for 2642 quasars, which can be usedto study the redshift evolution of extragalatic RMs. We will intro-duce data in the Section 2, and study their distribution in Section 3.We discuss the results and fit the models in Section 4. We obtained the rotation measure data of quasars from the cross-identification of quasars in the newest version of the MillionQuasars (Milliquas) catalog with radio sources in the NVSS RMcatalog (Taylor, Stil & Sunstrum 2009) and the compiled RM cat-alog (Xu & Han 2014). The Million Quasars catalog (version3.8a, Eric Flesch, 2014) is a compilation of all known type Iquasars, AGN, and BL-Lacs in literature. To avoid possible in-fluence on RRMs from different polarization fractions of galax-ies and quasars (Hammond, Robishaw & Gaensler 2012), we takeonly type I quasars in the catalog. We adopt 3 ′′ as the up-per limit of position offset for associations between quasarsand the radio sources with rotation measure data, according toHammond, Robishaw & Gaensler (2012), and finaly get RMs for http://quasars.org/milliquas.htmc (cid:13) ? RAS, MNRAS , 1–9 J. Xu and J. L. Han
Figure 2.
The redshift and RRM distribution for 2642 quasars together with the histogram for RRM uncertainty ( upper left panel ) and 684 quasars with formalRM uncertainty σ RRM − ( upper right panel ). Similar plots for 2202 quasars with only the NVSS RMs of Taylor, Stil & Sunstrum (2009) ( lowerleft panel ) or for 440 quasars with RMs from the compiled RM catalog of Xu & Han (2014) ( lower right panel ). RRM = RM − GRM , and theiruncertainty by σ RRM = p σ + σ , as listed in Table 1.The RRM distributions of 2642 quasars are shown in Fig-ure 2, including the distribution against redshift and amplitude,together with the histograms for RRM amplitude and uncertainty.Most RM data taken from Taylor, Stil & Sunstrum (2009) have a large formal uncertainty and also a previously unknown system-atic uncertainty (Xu & Han 2014; Mao et al. 2010). Because theuncertainty is a very important factor for deriving the redshift evo-lution of the residual rotation measures (see below), the best RRMdata-set for redshift evolution study should be these with a verysmall uncertainty, e.g. σ RRM − . We get a RRM data-set of 684 quasars with such a formal accuracy, without consid-ering the systematic uncertainty, and their RRM distribution isshown in the upper right panel in Figure 2. To clarify the sourcesof RM data, we present the RRM distribution for 2202 quasarswhich have the RM values obtained only from the NVSS RMcatalog (Taylor, Stil & Sunstrum 2009), and also for 440 quasarswhose RM values are obtained from the compiled RM catalog ofXu & Han (2014). The RRM data shown in Figure 2 should be carefully analysed toreveal the possible redshift evolution of the RRM distribution. c (cid:13) ? RAS, MNRAS , 1–9 edshift evolution of extragalactic rotation measures Table 2.
Statistics for real RRM distribution of subsamples in redshift binsSubsamples from the NVSS RM catalog Subsamples from the compiled RM catalogRedshift No. of z median W RRM W RRM0
No. of z median W RRM W RRM0 range quasars (rad m − ) (rad m − ) quasars (rad m − ) (rad m − )2338 quasars of σ RRM
20 rad m − : 2018 NVSS RMs and 320 compiled RMs0.0–0.5 152 0.400 17.7 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± σ RRM
15 rad m − : 1703 NVSS RMs and 312 compiled RMs0.0–0.5 136 0.400 16.9 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± σ RRM
10 rad m − : 1129 NVSS RMs and 296 compiled RMs0.0–0.5 88 0.400 15.4 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± σ RRM − : 406 NVSS RMs and 220 compiled RMs0.0–0.5 40 0.394 13.7 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Figure 3.
The probability distribution function of measured RRM values [solid line for P ( RRM ) ] compared with that of the mock RRM sample with the best W RRM [dotted line for P mock ( RRM ) ] for the subsamples of quasars with the NVSS RMs and σ RRM rad m − in different redshift ranges. The fittingresidues, which is mimic to χ , against various W RRM are plotted in the lower panels, which define the best W RRM and its uncertainty at 68% probability.
Looking at Figure 2, we see that the most of 2642 RRMshave values less than 50 rad m − , with a peak around 0 rad m − .Only a small sample of quasars have | RRM | >
50 rad m − ,which may result from intrinsic RMs of sources or RM contribu-tion from galaxy clusters. The RM dispersion due to foregroundgalaxy clusters is about 100 rad m − (see Govoni et al. 2010;Clarke, Kronberg & B¨ohringer 2001). In this paper we do not in-vestigate the RRMs from galaxy clusters, therefore exclude 91 ob-jects (3.44%) with | RRM | >
50 rad m − , and then 2551 quasars are left in our sample for further analysis. Secondly, most of thesequasars have a redshift z < . Because the sample size for high red-shift quasars is too small to get meaningful RRM statistics, we ex-cluded 62 quasars (2.43%) of z > for further analysis of redshiftevolution. Finally we have RRMs of 2489 quasars with | RRM |
50 rad m − and z < .Noticed in Table 1 that RRMs of these quasars have formaluncertainties σ RRM between 0 and 20 rad m − , which would un-doubtedly broaden the real RRM value distribution and probably c (cid:13) ? RAS, MNRAS , 1–9 J. Xu and J. L. Han
Figure 4.
The same as Fig. 3 but for the subsamples of quasars with the compiled RMs and σ RRM − . Probability distribution function is notsmooth due to small sample size and small RRM uncertainties. Figure 5.
The real dispersion W RRM of RRM distributions as a function of redshift for five subsamples of quasars in five redshift ranges, calculated fordifferent RRM uncertainty thresholds and separately for the NVSS RMs (open circles) and the compiled RMs (filled circles). The median redshift of thesubsample is adopted for each redshift bin. Two dot-dashed lines are the scaled “ALL” and “CLS” model from Akahori & Ryu (2011), the dotted line is theevolving Friedmann model (the EF model by Nelson 1973), and the dashed line is the ionized cloud model (i.e. the IC model by Thomson & Nelson 1982),which are scaled and fitted to the filled circles. bury the possible small excess RRM with redshift. We thereforework on 4 subsamples of these quasars with different RRM uncer-tainty thresholds, σ RRM
20 rad m − , 15 rad m − , 10 rad m − and 5 rad m − . Because the NVSS RMs has an implicit systematicuncertainty of . ± . rad m − (Xu & Han 2014), different fromthat of the compiled RMs which is less than 3 rad m − , we studythe RRM distribution for two samples of quasars separately: onewith RMs taken from the NVSS RM catalog, and the other withRMs from the compiled RM catalog. We divide the quasar samplesinto five subsamples in five redshift bins, z = (0 . , . , (0.5, 1.0),(1.0, 1.5), (1.5, 2.0), and (2.0, 3.0), to check the redshift evolutionof real dispersion of RRM distributions. How to get the real dispersion of RRM distributions, givenvarious uncertainties of RRM values? We here used the bootstrapmethod. It is clear that the probability of a real RRM value followsa Gaussian function centered at the observed RRM value with awidth of the uncertainty value, i.e. p ( RRM ) = 1 √ πσ RRM i e − ( RRM − RRMi )22 σ RRMi , (3)here RRM i is the i th data in the sample, and σ RRM i is its un-certainty. We then sum so-calculated probability distribution func-tion (i.e. the PDF in literature) for N observed RRM values for asubsample of quasars in a redshift range, assuming that there are c (cid:13) ? RAS, MNRAS , 1–9 edshift evolution of extragalactic rotation measures in-significant evolution in such a small redshift range, P ( RRM ) = N X p ( RRM i ) , (4)which contains the contributions from not only real RRM distribu-tion width but also the effect of observed RRM uncertainties.If there is an ideal RRM data set without any measurementuncertainty, the RRM values follow a Gaussian distribution withthe zero mean and a standard deviation of W RRM which is thereal dispersion of RRM data due to medium between sources andus. We generate such a mock sample of RRM data with the sam-ple size 30 times of original RRM data but with a RRM un-certainty randomly taken from the observed RRMs. We sum theRRM probability distribution function for the mock data, as donefor real data. We finally can compare the two probability dis-tribution functions, P ( RRM ) and P mock ( RRM ) , by using the χ test as for two binned data sets (see Sect.14.2 in Press et al.1992). For each of input W RRM , the comparison gives a residual ( P ( RRM ) − P mock ( RRM )) which mimics the χ . For a set ofinput values of W RRM , we obtain the residual curve. Example plotsfor the subsample of quasars with the NVSS RMs and σ RRM
20 rad m − are shown in Figure 3, and for quasars with the com-piled RMs and σ RRM − shown in Figure 4. Obviouslythe best match between P ( RRM ) and P mock ( RRM ) with an in-put W RRM should gives smallest residual, so that we take this best W RRM as the real RRM dispersion. The residual curve, if normal-ized with the uncertainty of the two PDFs which is unknown anddifficult, should give the χ = 1 for the best fit, and ∆ χ = 1 in therange for the doubled residual for the 68% confidence level. There-fore the uncertainty of W RRM is simply taken for the range withless than 2 times of the minimum residual in the residual curve.In the last, note that there is an implicit systematic uncer-tainty of σ sys = 10 . ± . rad m − in the NVSS RMs andthe maximum about σ sys < − in the compiled RMs(Xu & Han 2014), which are inherent in observed RRM values.The above mock calculations have not considered this contribu-tion, and therefore the real dispersion of RRM distribution shouldbe W RRM0 = p W − σ sys . We listed all calculated results of W RRM and W RRM0 for all subsamples of quasars in Table 2. Be-cause almost all W RRM and W RRM0 have a value larger than 10rad m − , the small uncertainty of the systematic uncertainty of lessthan 2 or 3 rad m − does not make remarkable changes on theseresults in Table 2.Figure 5 plots different W RRM0 values as a function ofredshift ( z ) for five subsamples of quasars, calculated forquasar subsamples with different thresholds of RRM uncertain-ties and also separately for quasars with the NVSS RMs and withthe compiled RMs. We noticed that the W RRM0 values obtainedfrom the NVSS RMs and the compiled RMs are roughly con-sistent within error-bars, and that the W RRM0 values obtainedfrom RRMs with different RRM thresholds are also consistentwithin error-bars. In all four cases of different σ RRM thresholds,we can not see any redshift evolution of the W RRM0 of quasarswith only the NVSS RMs, which is consistent with the conclu-sions obtained by Hammond, Robishaw & Gaensler (2012) andBernet, Miniati & Lilly (2012). However, the W RRM0 values sys-tematically increase (from ∼
10 to ∼
15 rad m − ) with the σ RRM thresholds (from 5 to 20 rad m − ), which implies the leakage of σ RRM to W RRM0 even after the simple discounting systematicaluncertainty. There is a clear tendency of the change of W RRM0 forquasars with the compiled RMs, increasing steeply when z < and flattening after z > , best seen from the samples of σ RRM − . This indicates the marginal redshift evolution, which isconsistent with the conclusion given by, e.g. Kronberg et al. (2008)and Joshi & Chand (2013). We therefore understand that the smallamplitude dispersion of RRMs is buried by the large uncertainty ofRRMs, and such real RRM evolution can only be detected throughhigh precision RM measurements of a large sample of quasars infuture. Using the largest sample of quasar RMs and the best determinedforeground Galactic RMs and after carefully excluding the influ-ence of RRM uncertainties and large RRM “outliers”, we obtainedFigure 5 to show the redshift evolution of dispersion of extragalac-tic rotation measures. We now try to compare our results with pre-viously available models mentioned in Section 1.As nowdays, the Λ CDM cosmology is widely accepted. Thenon-evolving steady-state universe is no longer supported by somany modern observations and we will not discuss it. The old co-expanding evolving Friedmann model (Nelson 1973) is ruled outby our RRM data as well (see Figure 5), because the electron den-sity and magnetic field in the model are scaled with redshift via n e = n (1 + z ) and B = B (1 + z ) and the variance of RRMs( ∝ W ) should increase with z . Among the three old models,the ionized cloud (IC) model given by Thomson & Nelson (1982)can really include all possible RM contributions and fit to the data.The ionized clouds along the line of sight can be the gravitationallybounded and ionized objects, which may be associated with proto-galaxies, galactic halos, galaxy clusters or even widely distributedintergalactic medium in cosmic webs. The dashed lines in Figure 5are the fitting to the W RRM0 data by the ionized cloud model. In Λ CDM cosmology, it has the form of V RRM = V Z z s z ) p Ω m (1 + z ) + Ω Λ dz (5)with a fitting parameter V = (0 . n c B || c ) c l c f H ≈ ±
150 rad m − , (6)where n c , B c and l c are the electron density, magnetic field and thecoherence size of a random field size, f is the filling factor, H is the Hubble parameter and c is the light velocity. Current Λ CDMcosmology takes H =70 km s − Mpc − , Ω m =0.3 and Ω Λ =0.7.The RRM variance ( V RRM ∝ W ) in the ionized cloud modelhas a steep increase at low redshift and flattens at high redshift,which fits the W RRM0 data very well (see Figure 5). The simila-tions given by Akahori & Ryu (2011) verified the shape of the RMdispersion curves. We scaled the “ALL” model of Akahori & Ryu(2011) to fit the data, and also scaled their “CLS” model to showthe relatively small amplitude from cosmic webs.For a sample of quasars, the lines of sight for some ofthem pass through galaxy halos indicated by MgII absorp-tion lines which probably have a RRM dispersion of severalrad m − (Joshi & Chand 2013); some quasars behind galaxy clus-ters may have large RRM dispersion of a few tens rad m − (Clarke, Kronberg & B¨ohringer 2001; Govoni et al. 2010); somequasars just through intergalactic medium without such interveningobjects should have a RRM dispersion of 2 ∼ − from thecosmic webs (see the cluster subtracted model of Akahori & Ryu(2011). These different clouds give different V . We noticed, how-ever, that the redshift evolution of RRM dispersions of each kind ofclouds depends only on cosmology (see Eq. 5), not the V . c (cid:13) ? RAS, MNRAS , 1–9 J. Xu and J. L. Han
Figure 6.
The probability distribution function for the absolute values ofRRMs in the upper panel for 146 quasars of z > from the compiledRM catalog (Xu & Han 2014) with RM uncertainty σ RRM − ,which is fitted by the two mock samples with a narrow real RRM dispersion W RRM1 standing for the contributions from galaxy halos and cosmic websand a wide RRM dispersion W RRM2 for the contribution from galaxy clus-ters. The likelihood contours for the best fits by using two dispersions areshown in the lower panel , with the best W RRM values marked as the blackdot.
In principle, we can model the RRM dispersion with a combi-nation of ionized clouds with different fractions, i.e. V = V gala ∗ f gala + V cluster ∗ f cluster + V IGM ∗ f IGM . We checked our quasarsamples in the SDSS suvery area, about 10% to 15% of quasars (fordifferent samples in Table 2) are behind the known galaxy clustersof z . in the largest cluster catalog (Wen, Han & Liu 2012).Quasars behind galaxy clusters have a large scatter in RRM data inFigure 2, mostly probably extended to beyound 50 rad m − , whichgive a wide Gaussian distribution of real RRM dispersions. Thefraction for the cluster contribition is at least f cluster ∼ . − . ,because of unknown clusters at higher redshifts. The fraction forgalaxy halo contribution shown by MgII absorption lines f gala isabout 28% (Joshi & Chand 2013). If we assume the coherence sizeof magnetic fields in these three clouds as 1 kpc, 10 kpc and 1000kpc, the mean electron density as − cm − , − cm − and − cm − , and mean magnetic field as 2 µ G, 1 µ G and 0.02 µ G(e.g. Akahori & Ryu 2011), and the filling factors as 0.00001, 0.001and 0.1 (Thomson & Nelson 1982) for galaxy halos, galaxy clus-ters and intergalactic medium in cosmic webs, we then can estimatethe dispersions of these clouds, which are 7, 11 and 2 rad m − atz=1, respectively. Whatever values for the different ionized clouds,they will have to sum together with various fractions to fit the dis-persions of RRM data.After realizing that the real RRM dispersion of quasars at z > does not change with redshift for each kind of ionizedclouds, we now model the probability distribution function of ab-solute values of RRM data for all 146 quasars with z > fromthe compiled RM catalog, without discarding any objects limitedby redshift and RRM values but with a formal RM uncertainty σ RRM − (see Figure 6). We found that such a prob-ability function can be fitted with two components, one for a small W RRM which stands for the contributions from galactic halos andcosmic webs, and one for a wide W RRM which comes from thegalaxy clusters. Two such muck samples with optimal fractionsare searched for the best match of the probability function. Weget W RRM = 11 . +3 . − . rad m − with a fraction of f =0.65, and W RRM = 51 +135 − rad m − with a fraction of f =0.35 for clus-ters. However, we can not separate the contributions from galaxyhalos and cosmic webs which are tangled together in W RRM .We therefore conclude that the dispersion of RRM data steadyincreases and get the saturation at about 10 rad m − when z > . However, the current RM dataset, even the largest sample ofquasars, are not yet good enough to separate the RM contributionsfrom galaxy halos and cosmic webs due to large RRM uncertain-ties. A larger sample of quasars with better precision of RM mea-surements are desired to make clarifications. ACKNOWLEDGMENTS
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