You are an expert at summarizing long articles. Proceed to summarize the following text: 
there is plenty evidence for magnetic fields on the scale of the earth , the solar system , the interstellar medium , galaxies and clusters of galaxies [ for the extragalactic magnetic field see review by kronberg ( 1994 , k94 ) and references therein ] . we still do nt know however whether any larger scale magnetic fields exist . there are various upper limits on the rms value of the cosmological magnetic field , but the upper limits are still too high to exclude any relevance to magnetic fields observed on smaller scales . cosmological magnetic fields might help to resolve the bothersome question of the origin of the @xmath17 g magnetic field which exists on galactic scales or fields of similar size seen on the scales of galaxy clusters . a primordial ( pre  galaxy formation ) magnetic field amplified by the gravitational collapse , and then by processes like differential rotation and dynamo amplification ( parker 1979 ) , can serve to seed these observed smaller scale magnetic fields . the most stringent upper limits to date for magnetic fields on cosmological scales come from three different sources :    1 .   big bang baryonic nucleosynthesis ( bbn ) . + magnetic fields that existed during the bbn epoch would affect the expansion rate , the reaction rates , the electron density ( and possibly the space - time geometry ) . by taking all these effects into account in the calculation of the element abundances , and then comparing the results with the observed abundances one can set limits on the magnetic field amplitude . the limits for homogeneous magnetic fields on scales @xmath18 ( distances are quoted in comoving coordinates ) and up to the bbn horizon size ( @xmath19 ) are @xmath20 g in terms of today s values ( grasso & rubinstein 1995 , 1996 ; cheng , olinto , schramm , & truran 1996 ) . the relevance of these limits for the maximum value of magnetic field seeds on the subgalactic scale is obvious , but in order for these limits to be relevant to the intergalactic magnetic field , further assumptions about the super - horizon magnetic field power spectrum and the magnetic field generation epoch should be made . 2 .   the cmb radiation . + the presence of magnetic fields during the time of decoupling causes anisotropy , different expansion rates in different directions ( zeldovich & novikov 1975 ) . masden ( 1989 ) provides a useful formula for the connection between the measured temperature fluctuations on a given scale at @xmath21 , the decoupling time , and the limits on the equivalent scale today . for a magnetic field energy density much smaller than the matter energy density , the limit is @xmath22 for typical values of @xmath23 ( the hubble constant in units of @xmath5 km s@xmath6 mpc@xmath6 ) , temperature fluctuations of @xmath24 , @xmath25 and @xmath26 the limit becomes @xmath27 g . recently , barrow , frreira , & silk ( 1997 ) used the cobe 4-year data to constrain the magnetic field on the horizon scale by @xmath28 ng . 3 .   multipole expansion of @xmath0 observations . + passing through a magnetoionic medium , polarized light undergoes rotation of the polarization vector ( faraday rotation , for definition see  [ sec : corr ] ) . the rotation amount is proportional to the dot product of the magnetic field and the light propagation direction , and to the distance the light travels through the medium . if any cosmological magnetic field exists , the farther away the source , the more the polarized light component is rotated . we refer to this as the  monopole " term . this term exists even if the field has no preferred direction on the horizon scale , namely even without breaking the isotropy hypothesis . + if there is a preferred direction to the field on a cosmological scale , a dipole will show up in the @xmath0 measurements across the sky , due to the different sign of the dot product ( and thus the different rotation direction ) . so far attempts have been made only to identify a monopole or a dipole in the magnetic field for scales of @xmath29 ( kronberg 1976 ) and @xmath30 ( valle 1990 ) . the search for a dipole signature in the faraday rotation values for a sample of extra - galactic sources ( qso ) yields limits of @xmath31 ng depending on the assumed cosmology ( kronberg & simard - normandin 1976 , kronberg 1976 ) . valle ( 1990 ) tried to estimate the limits on the dipole and the monopole terms and concluded that an all - prevailing ( up to @xmath32 ) field must be smaller than @xmath33 $ ] ng .    other potential methods of detecting a possible cosmological magnetic field include distortion of the acoustic (  doppler " ) peak in the cmb power spectrum ( adams , danielsson , grasso , & rubinstein 1996 ) and faraday rotation of the cmb radiation polarized components ( loeb & kosowsky 1996 ) . these methods where not designed to probe any magnetic field generated after recombination . their implementation is still pending on the upcoming measurements of the cmb fluctuations and polarization . a method that does probe magnetic fields generated in the post recombination era , relies on cosmic ray ( cr ) detection . the effect of magnetic fields on high energy crs ( @xmath34 ev ) is twofold . it will first alter the energy distribution of crs ( lee , olinto , & sigl 1995 , waxman & miralda - escud 1996 ) and then , if the cr direction is identified and attributed to a known region or source , the magnetic field either smears the directionality ( for field coherent scales much smaller than the distance to the source ) or deflects the cr direction from aligning with the source direction . current limits from this last effect are either very weak or non - existing .    theories for primordial magnetic fields in the framework of structure evolution paradigms have been suggested by a number of authors . magnetic fields on cosmological scales emerge in the framework of these theories in one of three epochs : the inflation era , the plasma era , and the post - recombination era . in the inflation era , magnetic fields form due to quantum mechanical processes ( turner & widrow 1988 ; quashnock , loeb , & spergel 1989 , vachaspati 1991 , ratra 1992a , 1992b , dolgov & silk 1993 ) or possibly magneto - hydrodynamics ( brandenburg , enquist , & olesen 1996 ) . during the plasma era , prior to recombination , magnetic fields form due to vorticity caused by the mass difference between the electron and the proton ( harrison 1973 , but see rees 1987 ) , or where magnetic field fluctuation survival ( no vorticity assumed ) depends on the fluctuation scale ( tajima 1992 ) . other authors have pointed out that even in the post recombination era ( but before galaxy formation ) magnetic fields can be generated by tidal torques ( zweibel 1988 ) , or by cosmic strings wakes ( ostriker & thompson 1987 , thompson 1990 ) . magnetic field generated in the process of structure formation itself , due to falling matter ( pudritz & silk 1989 ) , or star burst regions , are less likely to feed back into the intergalactic space and to affect the cosmological magnetic field . nevertheless , proposals in this spirit were also considered by taking into account wind driven plasma which fills intergalactic space with magnetic fields . whatever the origin of cosmological magnetic fields , each suggested theory , specifies the magnetic field amplitude and power spectrum shape . thus , each of the theories can in principle become refutable if we could measure , or put limits on , the normalized power spectrum of the cosmological magnetic field . this challenge and the realization that with existing or upcoming data , it will be possible to measure the magnetic field power spectrum has led us to develop a method for doing so . we argue that much better estimates ( or upper limits ) for the cosmological magnetic field can be calculated by using the rm correlation matrix . all previous attempts to limit the cosmological magnetic field dipole on very large scales ( @xmath35 ) , suffer from two drawbacks . from a theoretical point of view it is difficult to reconcile the necessary anisotropy such a dipole imposes with other evidence which suggest global isotropy . from a practical point of view , this test is confined to one scale , and does nt allow measurements over a whole range . only by extending the measurement to a whole range , can one estimate the full power spectrum ( ps ) . although contribution from a  random walk " can be useful even if only a monopole is calculated ( see below ) , the information is partial , inferior to the full derivation of the magnetic field by @xmath0 correlations , and sensitive to evolutionary effects . why is the correlation approach more advantageous than other type of statistics and in particular why is it better than the monopole approach ? in the monopole approach we are looking for a correlation between rm values and the source redshift . let s pretend at first that all measurements are exact and there are no noise sources ( we ll notice in  [ sec : calc ] that these two assumptions are far too optimistic ) . consider a scale @xmath36 over which the magnetic field is coherent with an rms value of @xmath10 . a line - of - sight to a source at a distance @xmath37 will cross @xmath38 such regions . there is no correlation in the magnetic field orientation between the regions , and on average each one of them contributes @xmath39 . the overall contribution due to @xmath40 random walk steps sums up to @xmath41 . it is thus clear that the larger @xmath37 , the bigger @xmath42 we expect for an ensemble of sources located at @xmath43 . now let s turn to the correlation approach . two lines - of - sight to two adjacent sources ( separation angle @xmath44 and assume the same redshift for the two ) are observed for correlation between the @xmath0 values of the polarized light emanating from them . if @xmath45 the two light rays undergo _ the same _ rotation in each patch of length @xmath36 ( for small @xmath44 we neglect the difference in the @xmath46 term between the magnetic field and the line - of - sight ) . the ensemble average of the correlation term would then be @xmath47 , with the same proportionality constant as before . in the limit of @xmath48 , the two expressions coincide , but for any value of @xmath49 , the signal from the correlation calculation is amplified by the factor @xmath50 . for example , if @xmath51 @xmath52 and @xmath53 the signal is amplified by more than an order of magnitude . moreover , the correlation provides information about the power spectrum ( different @xmath44 and @xmath37 values ) of the magnetic field that otherwise we would nt be able to obtain . the correlation also makes it easyer to separate the noise from the signal , unlike the monopole approach . we will return to this approximation later on , under more realistic considerations (  [ subsec : ston ] ) , when we attempt to evaluate the number of pairs needed in order to establish a certain signal - to - noise ratio at a given scale ( @xmath44 ) . for now , this simple model demonstrates the advantages of this paper s approach .    in order to relate actual data of rm measurements to the magnetic field , we begin by considering the connection between the two . in section [ sec : corr ] we introduce the magnetic field correlation tensor and its connection to the rm correlation . in section [ sec : calc ] we make use of these definitions and describe the calculation procedure for the @xmath0 correlation from the raw data to the constraints on the magnetic field power spectrum . a key issue is the estimate of noise from non - cosmological contributions to the @xmath0 . the impatient reader can turn immediately to  [ subsec : ston ] to get a rough estimate for the expected signal - to - noise ratio . we demonstrate the procedure in the following section (  4 ) , where we simulate a few realistic examples of cosmological magnetic fields , and exploit them to get rm measurements from which we derive back an estimate for the original power spectrum . the simulations also allow us to perform a realistic error analysis . in section 5 we discuss prospects for applying the procedure to real data and conclude with our results . linearly polarized electro - magnetic radiation of frequency @xmath54 traveling a distance @xmath55 through a non - relativistic plasma medium with the magnetic field @xmath56 , is rotated by the angle @xmath57 . in other words , the polarization angle is changed by ( lang 1978 eqs . 1 - 268 @xmath58 1 - 270 ) @xmath59 where @xmath60 , @xmath61 , and @xmath62 are the electron charge , mass , and number density respectively ( we use units of @xmath63 ) . the rotated polarization vector itself is written @xmath64 a pseudo - vector degenerated in rotation of @xmath65 by @xmath66 . it is related to the stokes parameters via @xmath67 that in turn are expressed by the electric field components perpendicular to the wave propagation direction @xmath68 @xmath69 @xmath70 @xmath71 where time average is explicitly assumed .    in the cosmological context , some of the quantities are functions of time ( or equivalently redshift or distance ) . we assume fully ionized gas ( ionization fraction @xmath72 ) and thus the average number density of free electrons is @xmath73 the baryon number density at a specific location is @xmath74 , with @xmath75 the dimensionless density fluctuation . number densities scale like @xmath76 , where @xmath77 is the scale factor . the frequency scales like @xmath78 , and under the assumption of flux conservation , @xmath79 scales like @xmath80 . the distance unit  @xmath55 " in a robertson - walker metric is given by @xmath81 expressed via @xmath82 the conformal time . since @xmath83 for the redshift @xmath84 , the overall rotation of the polarized radiation of a source at the redshift @xmath84 , the direction @xmath85 as observed from @xmath86 , ( we set @xmath87 for simplicity ) and at frequency @xmath88 , is given by @xmath89 ( 1+z')^3   { \vec b_0}({\hat q}r ' ) \cdot { \hat q } { { \rm d } r ' \over { \sqrt { 1 - kr^{'2 } } } } = { \rm ( { \hat q } , z ) \over \nu_0 ^ 2 } \ , .\ ] ] today s values are all denoted by the subscript  0 " and @xmath90 . the time dependence of @xmath91 is taken into account later on ( eq . [ eq : ups_xi ] ) . the two point correlation function of the rm is defined by @xmath92 see nissen & thielheim ( 1975 ) for a similar definition . throughout the paper , @xmath93 is the notation for an ensemble average . we use the notation @xmath94 and obtain @xmath95(1+z'_1)^3   \int_0^{r_2(z_2 ) } { { \rm d } r'_2 \over { \sqrt { 1 - kr_2^{'2 } } } } \left[1+\delta(\hat{q}_2r_2')\right ] ( 1+z'_2)^3\ ] ] @xmath96 where summation over the spatial components @xmath97 is assumed . the integral ( [ eq : ups_1 ] ) is made out of four terms . two of the terms ( those involve only one power of @xmath98 ) vanish because @xmath99 and we assume vanishing correlation between the density fluctuations and magnetic fluctuations [ @xmath100 due to the vector nature of @xmath101 and the scalar @xmath98 . the two remaining terms are @xmath102 @xmath103 and @xmath104 @xmath105 where @xmath106 is the baryonic matter correlation function that we identify with the matter correlation function . today s value of it is @xmath107 and the missing powers of @xmath108 in eq . ( [ eq : ups_xi ] ) account for the linear evolution of this correlation . the rm correlation function is the sum @xmath109 . the last integral of eqs . ( [ eq : ups_1 ] , [ eq : ups_0 ] , [ eq : ups_xi ] ) is the correlation tensor of the magnetic field defined as @xmath110 . in the cosmological case , we assume isotropy on scales much smaller than the horizon and confine @xmath111 to be a function of @xmath112 only . being a divergence - free field the magnetic field correlation can be written ( monin & yaglom 1975 ) as a combination of parallel ( @xmath113 ) and perpendicular ( @xmath114 ) functions . the parallelism and orthogonality are given with respect to the connecting vector : @xmath115 @xmath116{r_i r_j \over r^2 } + c_{\perp}(r ) \delta^k_{ij}\ , , \ ] ] @xmath117 @xmath118{k_j k_l \over k^2 } + c_{\perp}(k ) \delta^k_{jl}\ , , \label{eq : c_pre_par}\ ] ] where @xmath119 is the kronecker @xmath98-function . in the spectral domain we define the function @xmath120 and express both correlation functions by the function @xmath121 , namely @xmath122 with @xmath123 denoting the spherical bessel function of the @xmath124 order . when people refer to the  magnetic power spectrum " they usually mean @xmath125 which should further be multiplied by @xmath126 in order to get @xmath121 . we shall hereafter comply with this notation and refer to @xmath125 as the three - dimensional  magnetic power spectrum " , @xmath127 . figure  [ fig : b_corr ] show the two functions @xmath128 and @xmath129 in units of @xmath130 for three values of the power index @xmath131 [ @xmath132 . each function is shown with a minimal gaussian smoothing of @xmath133 . naturally , smaller power indices mean larger correlation length .    -1.2truecm 5truecm = 2.7 in -2truecm 0.5truecm = 2.7 in 2truecm = 2.7 in -0truecm    going back to the expression ( [ eq : ups_1 ] ) for the rm correlation we can rewrite it as @xmath134 @xmath135({\hat { q}_1 \cdot \hat r ' ) ( \hat{q}_2 \cdot \hat r ' ) } + c_{\perp}(r'){\hat q}_1 \cdot { \hat q}_2 \right\ } \,.\ ] ] as expected , the rm correlation depends only on the relative distance , @xmath136 , and the angle .    for comoving distances between two sources at @xmath137 and @xmath138 separated by the angle @xmath44 on the sky we use weinberg s ( 1972 , eq . 14.2.7 ) coordinate transformation , implemented by osmer ( 1981 ) for the @xmath139 case ( @xmath140  the deceleration parameter ) and generalized by boyle ( 1986 ) for other @xmath140 values . for the flat universe case ( @xmath141 ) the comoving relative distance reduces to the simple form @xmath142^{1/2}\ , .\ ] ] the distance to a source at the redshift @xmath84 ( no cosmological constant ) is given by ( kolb & turner 1990 , eq . 3.112 ) @xmath143 and can be easily generalized for other cases .    in order to estimate the importance of @xmath144 [ eq . ( [ eq : ups_xi ] ) ] we adopt the apm power spectrum ( baugh & efstathiou 1993 ; tadros & efstathiou 1995 ) with the fitting formula @xmath145 the fitting parameters @xmath146 , @xmath147 , @xmath148 , and gaussian smoothing of @xmath133 . we then derive the real space correlation function by the fourier transform of the smoothed power spectrum . in the range @xmath149 the result is very similar to the derived apm real space correlation function ( no explicit smoothing ) @xmath150 ( baugh 1995 ) . for this choice we get ( regardless of the @xmath127 form , and for @xmath151 ) , @xmath152 . we hence identify @xmath153 hereafter . we relate the magnetic field normalization to the power spectrum by @xmath154 where @xmath155 the factor @xmath156 is due to the definition of @xmath127 [ as @xmath129 ] , and @xmath158 is the gaussian ( top - hat ) window function of radius @xmath159 , in @xmath160 space , @xmath161 \ , ; \quad w_{th } = { 3 \over ( kr)^3 } [ \sin(kr ) - kr\cos(kr ) ] \,.\ ] ] a good benchmark to use for the normalization range is the cmb limit due to its simplicity and the availability of @xmath162 measurements on many scales ( either today or in the near future ) . we therefore work in the range of @xmath163 ng throughout this paper . the observational limits on the magnetic field as deduced from the @xmath0 dipole [ @xmath164 or @xmath165 are thus very different from the limits imposed by cmb fluctuations on @xmath166 scale . we note that by eq . ( [ eq : b_norm ] ) one can not infer limits on the magnetic field magnitude from one scale to another . the limits must involve the power spectrum shape . since no estimate for the latter exists , and since theoretical predictions for it range from power index of @xmath167 to @xmath168 ( ratra 1992a ) , limits on one particular scale hardly limit other scales .    -1.5truecm 5truecm = 2.7 in -3truecm 0.5truecm = 2.7 in 2truecm = 2.7 in -1truecm    figure [ fig : rm_corr ] show the values for the rm correlation function for pairs of sources at the same redshift , and three different spectral indices . the normalization of the magnetic field in each case is @xmath169 ng ( @xmath170 ) . this scale resembles the mean separation between rich clusters . we notice that for close pairs , high redshift sources are preferable in order to get high signal , while a transition typically occurs at a fraction of a degree where it becomes preferable to use lower redshift sources . this transition is due to the uncorrelated magnetic fields experienced by high redshift sources with large separation . only when the two lines - of - sight approach the correlation length , does contribution to the rm correlation becomes substantial . the different , uncorrelated magnetic fields up to the correlation length , play the role of uncorrelated noise , and dominate the correlated signal from the low redshift segment of the line - of - sight .    on the scale @xmath171 , the rm correlation signal still stands on @xmath172 depending on the power spectrum . the raw data consist of @xmath173 extra galactic objects for which polarization measurements are available in @xmath174 wavelengths . broten , macleod , & valle ( 1988 ) , and oren & wolfe ( 1995 ) have emphasized the need for a careful selection of wavelengths for each observed source . for the current application there are additional special requirements that will become clear when we get to the discussion (  [ sec : discuss ] ) . + each of such @xmath174 measurements consists of two types of data : the degree of polarization , and the polarization position angle [ @xmath175 and @xmath176 of eq . ( [ eq : p_vec ] ) ] . two functions are then constructed which give the dependence of these on wavelength . the @xmath0 is computed from the latter by using eq . ( [ eq : p_vec ] ) where @xmath177 . each measured @xmath178 value has an error , @xmath179 , that emerges from the measurement error ( instrumental , ionosphere model , the earth magnetic field model etc . ) and translates to the error in the fit from which the @xmath0 is derived . for typical sources with polarization degree of @xmath180 ( namely @xmath181 of polarized component at a few ghz frequency ) , the measurement error in each datum is of the order of a few degrees ( kato , tabara , inoue & aizu 1987 ; simard - normandin , kronberg , button , 1980 ( skb ) ; oren & wolfe 1995 ) . for realistic @xmath182 , this instrumental error translates to a typical error in the @xmath183 fit of @xmath184 and is dwarfed by other noise terms in the procedure (  [ subsec : gal_mask ] ,  [ subsec : i+f ] ) . we therefore set @xmath185 from now on . the measured rms are the sum of a few contributions , and can not be directly plugged into the expression ( [ eq : ups_2 ] ) to evaluate @xmath186 . + to begin with , there exists the internal @xmath187 for every one of the @xmath173 extragalactic sources . then there is the integrated @xmath188 which is presumably due to the cosmological magnetic field ( and the free electrons ) along the line - of - sight to the source . there may also exist contribution @xmath189 from intervening systems along the line - of - sight ( foreground screen ) either next to the source itself ( the host galaxy of a quasar ) , lyman-@xmath190 systems , galaxies , or clusters of galaxies . before this combined signal gets to the detector it still has to go through the galactic magnetic field , where it is rotated once more by the amount @xmath191 . the final measurement , @xmath178 is thus the linear sum @xmath192 where @xmath193 is the redshift - vector [ actually translated to a distance vector (  2 ) ] for the @xmath124 source . since we are interested in the cosmological contribution to the @xmath0 [ the second term in eq . ( [ eq : lin_sum ] ) ] , the first step should involve  cleaning " the measured signal of all irrelevant extra contributions . we shall attempt to perform this cleaning in a statistical way . there are two alternatives to assess the galactic contribution to the measured @xmath0 . the two ways differ by the population sample used for the assessment . if an independent population at the outskirts of the galaxy exists , for which @xmath0 can be measured , this population can be used to map out the galactic @xmath0 . we shall hereafter use the term `` mask source population '' for the set of objects by which we map the galactic contribution to the rm . an estimate for the thickness of the galactic magnetic layer is @xmath163 kpc ( simard - normandin & kronberg 1980 ) . that means that apart from the galactic center direction , sources of distances that satisfy @xmath194 kpc ( @xmath195 kpc for @xmath196 ) are located beyond the reynolds layer and fully probe the galactic contribution to the rm . we hence consider two possibilities for the mask source population . one natural candidate for this role is the pulsar population , for which @xmath0s are available . in order to make use of the pulsars we need to find a subset of them that reside in the outer part of the milky - way magnetic layer . + a class of pulsars that is especially appropriate for the task of probing the galactic rm contribution is the millisecond pulsars . this population of old pulsars is believed to reside far out of the galactic plane . the number density of millisecond pulsars as implied by a number of surveys at high galactic latitude ( @xmath197 sensitivity at @xmath198 ghz ) ranges between @xmath199 deg@xmath200 , @xmath201 pulsars over the sky ( foster , cadwell , wolszczan , & anderson 1995 ; camilo , nice , & taylor 1996 ) . the alternative mask population for assessment of the galactic @xmath0 contribution is a sub - set of the closest extragalactic sources . the disadvantage of using this population is the need for a large enough source number at low enough redshift , to allow differentiation between the cosmological contribution and the galactic one . exploiting extragalactic sources for the resolution of the galactic contribution affects only slightly the amount of cosmological contribution correlation , as long as the nearby sources are taken within @xmath202 , since the bulk of the sources is at @xmath203 (  [ sec : testing ] ) .    in order to use either one of the galactic mask populations , its number density should exceed a certain minimal number density . we now attempt to assess this minimal number . + the galactic @xmath0 contribution is a two - dimensional field with @xmath0 values . the data on the other hand are made of distinctive sources located at discrete directions . the lines - of - sight to the galactic mask population , do not necessarily coincide with the line - of - sight direction towards the extragalactic sources . the way to circumvent this difficulty is by introducing a _ smoothed _ version of the galactic @xmath0 map , namely the field @xmath204 . the two fields are connected via the smoothing angle @xmath205 , and by a specific choice of a gaussian smoothing for @xmath206 measurements of the galactic @xmath207 located at the directions @xmath208 @xmath209 \over \sum\limits_{i=1}^{n_g } \exp \left[- { ( { \hat z}_i - \hat{z})^2 \over 2 \theta_s^2 } \right ] } \,.\ ] ] when implemented using real data , the smoothing involves weights by measurement errors as well . for the purpose of this paper , we ignore these weights to avoid unnecessary complication . using a smoothed field for the galactic @xmath0 contribution , introduces random noise , @xmath210 , that can be directly evaluated by the very same @xmath206 sample via @xmath211    4truecm    0.5truecm    1.0truecm    4truecm    the noise can be a function of position , it is difficult however to account for a position - dependent noise with low number of @xmath206 . there are a few estimates in the literature for the noise level , @xmath210 , on different smoothing scales . + an upper limit for @xmath210 is given in minter & spangler ( 1996 ) , who calculate the observed `` structure function '' , @xmath212 which is the same as eq . ( [ eq : gal_res ] ) but with an arbitrary lag @xmath44 and the unsmoothed values of the rm . they conclude that @xmath213rad@xmath214m@xmath215 , and drops steeper for @xmath216 . the value of the @xmath212 is an upper limit for @xmath217 as calculated here , due to the smoothing we apply ( and was not applied in the @xmath212 calculation ) . simonetti , cordes , & spangler ( 1984 ) calculate @xmath218 rad@xmath219m@xmath215 , depending on the galactic latitude , for much larger angular scales of @xmath220 ! ( linear scale ) . + simonetti & cordes ( 1986 ) corroborate these results and consider even larger angular scales to obtain typically @xmath221rad@xmath219m@xmath215 on scales less than @xmath222 . + oren & wolfe ( 1995 ) calculate a very similar quantity to @xmath210 ( with a varying top hat window instead of a fixed gaussian ) , and obtain a typical variance of @xmath223 rad@xmath219m@xmath215 on @xmath224 scales . + all of these measurements suggest that for a mask population with number density of about one source per @xmath225deg.@xmath219 , the galactic contribution to the rm ( @xmath226 ) can be resolved with a @xmath227 accuracy of @xmath228 . that means that an isotropic coverage of about @xmath229 mask sources on the sky is enough for achieving this noise level . +  damage control " for ignoring the position dependence can be devised by comparing the  clean " rm field dipole and quadrupole to the galactic direction of the two . these first two multipole should be the most prominent signature of the galactic rm mask . we go back to the suggested mask populations and examine whether they are able to fulfill the requirement of the minimal number density . + the largest pulsar catalog to date [ the electronic version of taylor , manchester , & lyne ( 1993 ) ] consists of @xmath230 pulsars out of which @xmath231 have rm measurements . this catalog was not compiled with a specific emphasis for millisecond pulsars , and therefore only @xmath232 pulsars with rm values lie in galactic latitude @xmath233 . the softer condition @xmath234 ( considering an  rm layer " of @xmath235 kpc ) is fulfilled for @xmath236 pulsars with both rm registered value and estimated distance @xmath37 ( taylor & cordes 1993 ) . + however , these @xmath236 pulsars are still not distributed isotropically across the sky , and tend to concentrate near the galactic plane .    a way to avoid this anisotropy is to compile a sample of millisecond pulsars , that lie far away from the galactic plane . this population has a number density that exceeds the necessary @xmath237 deg.@xmath200 by at least a factor of two and it may be as high as seven times denser than the minimal value . this population can be detected for rm ( fernando camilo , private communication , 1997 ) ms , and five for which @xmath238 ms . the smallest polarized flux at @xmath239ghz for a pulsar with a registered rm is @xmath240mjy ] and ensures the @xmath241 noise level . this kind of sample is ideal for the current paper analysis , as it does not contain any cosmological contribution or internal rm contribution . the main disadvantage of it is that it does not exist yet .    in considering the extragalactic mask population ; for the @xmath242 sources needed to resolve the galactic contribution , it is sufficient to have a sample of @xmath243mjy sensitivity ( at a few ghz ) up to @xmath244 [ see loan , wall , & lahav ( 1997 ) for the number density of radio sources with flux @xmath245 mjy , and dunlop & peacock ( 1990 ) for the combined flat / steep spectrum luminosity function for extrapolation to the @xmath243mjy limit ] .    for this paper we follow oren & wolfe ( 1995 ) and use an extragalactic sample of rm measurements as an upper limit for the noise that appears by the galactic rm removal procedure . the sample we use consists of @xmath246 sources as listed by skb . the skb catalog allows us to realistically mimic the galactic mask , and is an overestimate since we ascribe all the rm of the sources to the galactic contribution . moreover , unlike oren & wolfe we do not exclude  outliers " . this use aims to provide a realistic model of the galactic mask contribution , and allows us to demonstrate the method power . for our purpose it should not be taken literally as the true galactic mask contribution ( even though oren & wolfe did consider it that way ) because no segregation by redshift was applied to choose mask sources from the skb list . we neither attempted to evaluate the internal rm ( by means explained in [ subsec : i+f ] ) in order to minimize the @xmath210 value . + the skb source locations ( projected , and therefore concentrated toward the circumference ) are marked as black dots on figure [ fig : maskmap ] and shown on top of the smoothed galactic @xmath0 field with smoothing scale of @xmath247 . the smooth map is shown in celestial coordinates . figure  [ fig : res_mask ] shows the rm residuals as functions of galactic latitude and longitude , and figure  [ fig : maskres_map ] shows the contour map of the smoothed residual field . both figures ( [ fig : res_mask ] & [ fig : maskres_map ] ) show hardly any longitude dependence of the residuals ( in spite of the solar system off - center position ) , and bigger residuals in the @xmath248 strip about the galactic plane ( the same conclusion of oren & wolfe 1995 ) . in an all - sky coverage of sources , this residual deviation should be of some worry , but as we shall see (  [ subsec : all_sky ] ) , it does nt bias the result for the most likely power spectrum as calculated by the bayesian analysis . if the analysis is confined to high latitudes , then no systematic error due to the galactic mask is expected (  [ sec : testing ] ) . for a few square degrees area , we notice there are typically no significant gradients in the galactic rm field beyond @xmath249 . this allows us to use the smooth value safely when analyzing a small area with extragalactic rm measurements . this value is also consistent with all the abovementioned values of @xmath212 as obtained by more detailed analyses . we would like to get an estimate for the internal @xmath0 contribution and for the foreground screen contribution ( if the latter exists ) . we do not attempt to correct each individual source for the @xmath0 contribution due to the internal and foreground screen terms . we _ do not _ take the measured @xmath0 and subtract an estimate for @xmath250 , we rather attempt to estimate the distribution of @xmath250 , and treat it as another noise term . the difference between the two separate terms in eq . ( [ eq : lin_sum ] ) ( @xmath251 and @xmath189 ) emerges from the fact that while the internal contribution is obtained from the same medium that emits the polarized light , the foreground screen only serves as a filter for an already existing signal . in previous investigations , attempts were made to separate the two contributions ( burn 1966 ; laing 1984 ) , later on it became clear that the signature of the two is rather similar ( tribble 1991 ) . for our purposes we are interested in all previous calculations  left overs " , their signal is our noise and vice versa . for the noise estimate we claim it is legitimate to take the internal contribution and the foreground screen contribution together . we argue that the two can be dealt with simultaneously . to this end we shall use the other piece of information provided by the observations  the polarization degree . tribble ( 1991 ) shows the connection between the observed depolarization ( as a function of wavelength ) and the observed rm for an extended source . using this connection , and provided the polarization measurements are available , a direct estimate for the internal rm can be carried out . recall we do not expect any depolarization due to the cosmological rm since the cosmological coherence we are after is orders of magnitude bigger ( @xmath252 a few @xmath253 ) than the source size . it is interesting to note , that even if we had perfect information about the depolarization due to the source structure and foreground screen contribution and an exact relation between the depolarization degree and the @xmath0 , we could still not subtract this derived @xmath0 from the observed one . this is due to the fact that the polarization degree does nt specify the @xmath0 direction . tribble s statistical connection between the observed depolarization and the observed rm is valid only if the correlation scale of the rm structure function for the source or the foreground screen is much shorter than the telescope beam size . this condition is probably not valid for individual damped lyman@xmath190 systems , and galaxies that serve as foreground screens , and for which long range correlation across the telescope beam may exist . full justification for neglecting damped lyman@xmath190 systems and galaxies exists only if we can avoid lines - of - sight that cross such systems . otherwise the order of magnitude of an intervening galaxy contribution is obtained as follows : for a galaxy seen face - on , where there is a danger of a well ordered magnetic field , the disc thickness , to which the magnetic field is presumably connected is of the order of kpc ( simard - normandin & kronberg 1980 ) . observations show magnetic field magnitude of @xmath254 g , mainly in the disc plane . in order to equate this to a cosmological magnetic field of the order of ng , and coherent scale of @xmath255 , the average baryon number density across the galaxy should be at least fifty times higher than the cosmological @xmath3 . however , we do not expect any correlation between the magnetic field orientation of galaxies along the line - of - sight , or galaxies adjacent ( in angle ) to each other . the worst contribution to the rm , could only come from a random walk of @xmath256 steps where @xmath256 is the number of foreground screens ( intervening systems ) along the line - of - sight . since one can model the probability for an intersecting system along the line - of - sight as a function of the source redshift ( welter , perry , & kronberg 1984 ; lanzetta , wolfe , & turnshek 1995 ) , another noise term can be added . we did not attempt to model this noise term here , because we believe that the best strategy would be to avoid highly contributing ( damped lyman-@xmath190 and lyman limit ) intervening systems . low column density lyman-@xmath190 system contribute very little to the rm signal ( see eq . ( 1.2 ) in welter , perry , & kronberg 1984 ) . an exception may exist , if two sources ( or two separated parts of the same source ) cross the same cluster . in that case the noise of the two sources ( @xmath250 ) may be correlated , but then we can go back to the technique that uses the depolarization , in an attempt to remove the noise correlation .    the best strategy , as stated earlier , would be to avoid intervening systems altogether . there are two methods for doing this . the first method is simply to avoid intervening systems by looking for sources that exhibit no absorption lines in their spectra ( we assume spectrography is carried out anyway , to allow redshift determination ) . the existence of a large fraction of quasar lines - of - sight which do not intersect a dense intervenor is corroborated by mller & jakobsen ( 1990 ) and lanzetta , wolfe , & turnshek ( 1995 ) analyses . the other method is to put a depolarization limit on the observations ( the one imposed by tabara & inoue 1979 ) . such a limit naturally reduces the noise term of @xmath257 and does not affect the correlation signal . as a matter of fact , a sequence of such depolarization limits , may be helpful in constituting the estimate for @xmath257 . there are yet more methods for minimizing the number of sources that manifest internal rm as we shall point out when we consider real observations (  [ sec : discuss ] ) .      in the current stage of research we have very little knowledge about the power spectrum of the cosmological magnetic field . we can hardly even put limits on its integral value , the rms value of the field . there are , however predictions regarding the power spectrum shape and amplitude . in the lack of any observational preference towards any of these predictions ( apart from the limits on its rms value ) , we assume that all models are equally probable . conventional estimates of the @xmath0 correlation , estimates of the sort applied to galaxies are not very useful . it is difficult to get correct error estimate without simulations , that in turn must assume some magnetic field power spectrum , moreover in the traditional correlation calculation procedure , the errors of different bins in @xmath258 space are correlated . on top of that , the quantity we are after is the magnetic field power spectrum , and the inversion from the integrated @xmath0 back to the magnetic field is a non - trivial one . all of the above lead us to the use of bayesian statistics as a tool for finding the best parameters , and their probability , given a model . in the bayesian formalism , the a - posteriori probability density of a model , @xmath259 , given the data , @xmath260 , is @xmath261 as stated earlier , @xmath262 , the model probability density is unknown and therefore assumed uniform for all models . the data probability , in the denominator is the same for all models , and therefore serves as a normalization factor . we are thus left with equivalence between the operation of maximizing the probability of the model , given the data [ @xmath263 and the operation of maximizing the probability of the data , given the model [ @xmath264 . this equivalence allows us to write down the likelihood function for @xmath173 sources with measured @xmath0 . we begin by constructing the @xmath265 symmetric matrix @xmath266 of the expectation values for the @xmath0 correlation ( eq . [ eq : ups_2 ] ) between these sources ( as determined by their position ) . we further assume that the errors in each @xmath0 value are uncorrelated and gaussianly distributed . the variance of the overall error is the sum of the quadratures of the various error sources @xmath267 and the likelihood function becomes @xmath268^{-1/2 }    \exp\left ( -{1\over 2}\sum_{i , j}^{n_{src } } { \rm_i \tilde \upsilon_{ij}^{-1 } \rm_j}\right)\,,\ ] ] where @xmath269 . + the @xmath270 statistic is defined as @xmath271 . the @xmath270 statistics as defined , is a @xmath270 distribution ( of @xmath173 degrees of freedom ) with respect to the _ data points _ , not the model parameters . this @xmath270 is interpreted as a rough estimate for the confidence levels in the _ parameter _ space .      before proceeding to the elaborated description of the various tests we performed with artificial data , we would like to get a rough estimate for the expected rm correlation and the necessary number of sources for a high enough signal - to - noise ratio at various scales . we temporarily abandon the bayesian approach , and focus on @xmath272 . recall that the noise in the cross correlation is not correlated , but may still dominate the signal if not appropriately averaged over many pairs . the signal - to - noise ratio for an individual measurement of a pair at identical redshift , a separation angle @xmath44 , and in the context of a specific model , is given by @xmath273 where in the last equality we assume no dependence between the various noise terms and the model ( cosmology + power spectrum ) . the cosmological parameters we choose to use are as follows : we restrict ourselves to flat cosmology with @xmath274 , we choose @xmath275 ( but the scaling with @xmath4 is straightforward ) . we take the ionization factor to be @xmath276 which according to the gunn - peterson effect ( gunn & peterson 1965 ) is a reasonable choice , and @xmath277 ( tytler , fan , & burles 1996 ) , @xmath278 @xmath279 .    for @xmath280 pairs where each source is a member of one pair only , the signal - to - noise can further be increased by @xmath281 . in figure [ fig : ston ] we plot the necessary number of pairs in order to achieve a signal to noise ratio of @xmath282 . all sources are assumed to be at the same redshift ( @xmath283 , @xmath243 , or @xmath284 ) , we take the noise values @xmath285 ( this is the value for @xmath247 ) . for the noise term due to the internal rm and the ( possible ) foreground screen we use the tabara & inoue ( 1979 ) catalog as a guideline . tabara & inoue listed @xmath286 radio sources in their catalog with @xmath0 @xmath287 . that means a @xmath288 cover , and gives @xmath289 . the most conservative assumption one can make is to attribute all of this @xmath0 to the internal and foreground screen contribution . + the procedure for deriving the internal and foreground screen contribution from the depolarization (   [ subsec : i+f ] ) bears noise itself and is correct only statistically . we model its accuracy as a gaussian with one quarter the width of the observed rm distribution . doing so we end up with @xmath290 . this is a conservative choice , and the true value for @xmath257 is probably much smaller . note for instance the @xmath291 value as obtained by oren & wolfe ( 1995 ) . the measurement errors are neglected (   [ subsec : raw ] ) . two power spectra are considered and they are both normalized to give @xmath292 ng ( the @xmath293 case gives results between the @xmath294 and the @xmath295 power spectra ) . a typical number of pairs for achieving a signal to noise ratio of @xmath195 in the range @xmath296 is between @xmath297 and @xmath298 pairs , depending on the power spectrum . 0.5truecm = 2.7 in 2truecm = 2.7 in    this estimate of @xmath9 is not accurate . to begin with , for @xmath173 sources we expect less than @xmath299 statistically independent pairs . we do not expect all sources to reside at the same redshift , neither do we expect the errors to be identical for all sources . furthermore , the separation between the sources is not fixed , but rather is spread from the arc - minute scale to @xmath300 . due to the need for a realistic error estimate , and a demonstration of an actual implementation of the technique we propose , led us to apply it to mock catalogs , for which the underlying power spectrum is known . as a reliable probe for our method we create artificial catalogs of radio sources for which rm values are measured . we first generate a gaussian random field of the magnetic vector potential , @xmath301 with the power spectrum @xmath302 . the realization takes place on a @xmath303 grid points equally spaced in comoving coordinates . the @xmath127 power spectrum is gaussianly smoothed on the one grid cell scale . the amplitude of the field is arbitrary . we then derive the magnetic field itself by the real space relation @xmath304 translated to @xmath160 space @xmath305 . we use the @xmath160 space symmetries due to the real nature of the @xmath301 field ( no imaginary part ) , and go back to real space to obtain a divergence - free magnetic field with the desired correlation function , and periodic boundary conditions . we check the resultant field by calculating the field divergence about each and every grid point in real space and get @xmath306 .    at this point we select the cosmology into which we embed the simulation . our standard choice is identical to the one described in the signal - to - noise estimate (   [ subsec : ston ] ) . due to the limited dynamical range , we choose the grid scale to represent @xmath307 , and we are therefore confined to a largest simulation wavelength of @xmath308 . we then select the source mask , the source distribution across the sky , and their redshifts . the two - dimensional distribution is taken to be a poisson - like distribution , this choice allows us to have close pairs ( unlike grid selection ) , but to be as conservative as possible in terms of the angular correlation function of the sources . introducing any correlation as observed for radio sources ( kooiman , burne & klypin 1995 ; sicotte 1995 ; cress 1995 ; loan , wall , & lahav 1997 ) can only increase the number of close pairs on small angles , and thus improve the @xmath0 correlation resultant signal ( see fig . [ fig : ston ] ) . we do not take explicitly into account very small angle ( sub arc - minute ) separations that may exist in real data sets for resolved extended sources , whose different parts can be used as more than one background source . we employ two distribution schemes , an all - sky coverage , and a cover of @xmath309 deg.@xmath219 area centered about the north galactic pole . the source redshift distribution is selected according to the @xmath310 of radio sources with flux @xmath311 mjy ( at 4.85 ghz ) as calculated by loan , wall , & lahav ( 1997 ) who used the mean of the theoretical luminosity function models of dunlop & peacock ( 1990 ) and kindly provided their fit . loan ( 1997 ) @xmath310 is peaked around @xmath312 , and has a shape resembling a low redshift truncated gaussian of width ( @xmath313 ) @xmath314 .    in principle , setting flux limit and sky coverage , sets the source number . in practice only a sub - set of all radio sources emit polarized light . a random sample taken from the nvss survey ( condon 1994 ) shows registered polarization angle for @xmath315 of the sources . refraining from sources with intervening systems and large internal rm may further decrease the number . since the source number density in the radio catalogs of this flux limit is a few per square degree , when we prepare an  all - sky " coverage catalog ( see below ) , we are always way below the available number of sources . for smaller angular coverage with higher source density , one has to further decrease the flux limit . for example in the @xmath316 mjy limit we expect @xmath317 sources per square degree ( nvss , condon 1994 ) , and in the @xmath243 mjy limit we expect source density of about @xmath5 per square degree ( the  first " survey ; becker , white , & helfand 1995 ) . the @xmath310 function of loan ( 1997 ) is therefore interpreted as a normalized selection function , with a cutoff at the highest redshift of the catalog . we expect this selection function to provide an underestimate for the number of sources at high redshift , if a lower flux limit is set . since sources at higher redshift typically bear higher signal ( without changing the noise ) , such an imposed selection is a conservative choice in terms of the expected signal - to - noise ratio . we choose , which is peaked in a relatively low redshift , may be useful in mimicking another effect . as the source redshift increases , the probability for intervening systems along the line - of - sight increases as well . if in a sample compilation we try to avoid intervening system , this attempt translates to a steeper fall - off of the selection function . ] + for each selected source , we integrate along the line - of - sight over the dot product between the magnetic field and the line - of - sight direction using the periodic boundary conditions . the integration scheme assigns weights to each integration segment , @xmath160 , according to @xmath318 in order to account for the cosmological evolution . we then add the following noise terms to the resultant @xmath0    1 . internal variation + foreground screens . + we draw this noise term from a gaussian of @xmath319 variance (   [ subsec : ston ] ) . this is the added rm to the cosmological one . however , we also try to imitate the procedure of @xmath257 evaluation by the depolarization . to this end , for a specific value of @xmath320 ( drawn earlier ) , we mimic the recovery of the @xmath257 from the depolarization degree , by listing an @xmath257 value scattered about @xmath320 . the scatter has a gaussian distribution of @xmath321 variance . this is the error in the noise estimate . observer " takes into account the scattered value of the @xmath257 and not the value that was actually added to the integrated cosmological rm . 2 .   galactic mask . + we use the model - estimate of the galactic mask , smoothed to @xmath322 scale ( see   [ subsec : gal_mask ] ) . since the smoothed galactic rm differs from the true @xmath0 to the line - of - sight , we add a @xmath0 drawn from a gaussian distribution with the standard deviation of @xmath323 ( eq . [ eq : gal_res ] ) .    at the end of this process we are left with a mock @xmath0 catalog that consists of source coordinates and redshift , measured @xmath0 value , and the total error in the @xmath0 value ( @xmath257 ) . we proceed by feeding the mock catalog into the maximum likelihood procedure . in the current application , we consider only the right cosmology , the one in which the simulation was embedded . in principle the sensitivity to the cosmology choice can be checked as well . the two variables for the model are the amplitude and the power index . it is really the combination of the two that the likelihood procedure constraints best . + the statistical significance of the @xmath270 distribution ( as defined following eq . ( [ eq : like ] ) ) is obtained in the standard way . the number of degrees of freedom is the source number minus the number of fitting parameters , and the goodness of fit is calculated by taking into account the value of @xmath324 . if the goodness of fit turns out to be very small ( @xmath325 ) , this is a clear indication for convergence to the wrong minimum , and vice versa . for the right solution the minimum value for the @xmath270 per degree of freedom ( the source number ) is typically unity within @xmath326 accuracy . we first explore partial sky coverage mock rm catalogs . figures [ fig : like_part ] show contour maps of @xmath327 with steps of unity . the thicker lines indicate the approximate @xmath328 , and @xmath329 levels of the probability function as ascribed by using two parameter @xmath270 statistics (  [ subsec : bayes ] ) . the power spectrum normalization is our standard @xmath330 ng , depending on the power spectrum . we use @xmath331 sources for the part sky and @xmath309 square degrees coverage . the different source number and @xmath10 values reflect the fact that we have fixed the angular sky coverage . that in turn translates differently for different spectra . a redshift cutoff of @xmath332 has been applied . + figure [ fig : like_part_high ] shows a particular example for the case @xmath295 , where we have increased @xmath333 to the @xmath334 ng level in order to verify the result dependence on @xmath10 .    -1.5truecm 5truecm = 2.7 in -2truecm 0.5truecm = 2.7 in 2truecm = 2.7 in -0truecm    we notice that for all power spectra , the bayesian approach recovers the correct value from the artificial catalog ( marked by a cross sign ) to within the @xmath227 level . the method is more robust with respect to the amplitude determination . the power index is not well bound unless it is very different from the true one . the method ( with the limited number of sources ) tends to recover best the @xmath10 value on a certain scale , which is a combination of the power index and the amplitude . typical values for the ratio between the true @xmath10 and the minimization result for @xmath10 are in the range @xmath335 . partial sky coverage is preferable in terms of the number of small separation pairs , and the indifference to the galactic mask variation as function of direction . a collection of several of these small patches is expected to provide an even better constrain on the magnetic field power spectrum , but the analysis becomes a bit more complicated , as one should also include the large scale correlation between the patches without increasing the noise from the auto correlation to an intolerable level .    given the relatively big errors , we desire off - diagonal terms in the correlation matrix to be as big as possible ( small separation ) . this is meant to avoid heavy weight for diagonal terms which are dominated by noise . the price for choosing small patches with small sky coverage is limited ability to constrain the power index . under special conditions , this ability can be partially recovered by turning to an all sky coverage as we discuss in the next section .    -1.5truecm 5truecm = 2.7 in -1.0truecm      an all sky coverage , with the small correlation scales that are tested here , is impractical without lowering the error level significantly . otherwise the bayesian analysis will all be dominated by the diagonal terms with a large noise added to them and one loses the advantage of the correlation scheme for noise reduction . all sky coverage is however still feasible if we design the sample very carefully . in order to minimize the galactic noise we should observe sources in proximity to the mask sources direction and avoid the necessity for smoothing the galactic mask . instead , the rm values as deduced from the mask source population can simply be subtracted from the observed rm values . the small variation in the galactic magnetic field on small scales ( simonetti , cordes , & spangler 1984 ; simonetti & cordes 1986 ; minter & spangler 1996 ) allows this subtraction to take place . if on top of that one can reduce the noise from internal rm to the level of @xmath336 ( one can always avoid foreground screens ) by multi wavelength observations , then it becomes sensible to exploit an all - sky coverage . figure [ fig : xlike_all_n=0 ] shows an example of a sample taken at the mask source directions with @xmath337 and @xmath338 ng . the rest of the parameters are identical to those of the part sky coverage ( @xmath295 ) of the last section . we notice that the power index is somewhat better constrained in an all sky coverage , especially if the amplitude ( or @xmath10 ) are assumed or known from some different source . the true values are recovered to within the @xmath339 level . moreover such coverage allows us to calculate all prevailing magnetic fields on the sample scale , expressed as @xmath340 with @xmath341 the dirac delta function and @xmath159 the sample depth . notice , however that using the formalism proposed in section [ sec : corr ] , we can not take into account power spectra that are not functions of @xmath342 alone so this test is a bit weaker than the dipole test .    -1.5truecm 5truecm = 2.7 in -1truecm we have presented the formalism for the rm correlation function and demonstrated that by using it we can put limits on the power spectrum of the cosmological magnetic field . these limits are more stringent than limits obtained by any other method that currently exists . limits of @xmath343 level for magnetic field of @xmath198 ng on inter - cluster scales ( @xmath344 ) can be devised with only @xmath345 sources . the increment of the source number reflects linearly in the @xmath10 limits . we performed the statistical analysis following a bayesian approach that seems to be the most adequate for this problem . the correlation method we propose is also less susceptible to systematic effects such as observational bias ( or bias due to evolution ) toward smaller internal rm measures for higher redshift sources . both of these biases tend to lower the expected correlation signal as the source redshifts increase , whereas a cosmological magnetic field tends to raise the correlation signal between such sources .    in order to estimate the statistical significance of limits as derived by the proposed method , we have simulated a few possible cosmological magnetic fields . the simulations allow us to have a fair estimate of the errors involved in the power spectrum derivation . the analysis depends , however on a few assumptions , and most crucially on the ability to eliminate or estimate the foreground screen and internal contribution to the observed rm . this crucial point can be addressed in the process of the sample compilation . reduction of the foreground screen and internal rm contribution is achievable by applying two cutoff criteria to the sample . the first cutoff in minimal polarization degree is meant to avoid sources that exhibit large depolarization degree . these sources are usually interpreted as having a big non - cosmological contribution to their measured rm . the second cutoff is for sources that do not follow a @xmath346 dependence of the rm . following laing s ( 1984 ) recipe , in order to minimize the number of sources with internal rm , the wavelength span should allow detection of deviation from the @xmath346 dependence of the rm for @xmath347 . should this deviation occur , the source is to be eliminated from the catalog as a suspect for substantial internal rm contribution .      the spatial coverage of the sky should be twofold . small patches of a few square degrees enable the small separation pair number needed for noise reduction . small patches , however , can not reveal large correlation scales in an effective way . it is thus desirable to compile a dilute all sky coverage sample , in addition to the high number density patches off the galactic plane . the two sampling strategies cover a large range of potential correlation scales , without the necessity for an all - sky dense sample . the all sky coverage should preferably look for sources in proximity to the mask sources directions in order to minimize the galactic noise term . accurate internal rm for these sources should be evaluated through depolarization measurements in order for them to be useful for the power spectrum evaluation . ultimately , the sample can always be mimicked in terms of the exact source locations in the framework of all and every detected model . exact imitation of the source locations ensures the right correlation terms in the correlation matrix . the limits we have derived here are a combination of limits on the free electron average density , @xmath348 , and the magnetic field . if by any other fashion ( like hii absorption ) an estimate for @xmath349 can be achieved , then the limits on @xmath79 will become more robust . the estimate is also a function of the assumed cosmology , and may be entangled with magnetic field evolution in the post recombination era . we havent modeled such evolution in this paper . this method of rm correlation can further be generalized to the _ smoothed _ rm correlation . the data can be smoothed on a certain angular scale , and then either compared to the expectation value predicted by a model , or inverted numerically to give limits on @xmath79 . the bayesian approach , though ceases to be advantageous in the smoothed case , because even though the noise terms decrease , the smoothing introduces correlation among the errors , and the statistical interpretation becomes more hazardous . a nice feature of the proposed analysis is that it is bound to yield results . these can be either limits on the magnetic field magnitude , or actual detection of its value . either way , these results may help in lifting the curtain over the mystery of the cluster and galactic magnetic fields origin . i would like to thank arthur kosowsky for extensive discussions and helpful assistance , george blumenthal for valuable insight and critical reading of the manuscript , and james bullock for a very carefull examination of the manuscript . + this work was supported in part by the us national science foundation ( phy-9507695 ) . adams , j. , danielsson , u.h . , grasso , d. , & rubinstein , h. 1996 , sissa preprint , astro - ph/9607043 barrow , j.d . , ferreira , p.g . , & silk , j. 1997 , sissa preprint , astro - ph/9701063 baugh , c.m . 1995 , sissa preprint , astro - ph/9512011 baugh , c.m . and efstathiou , g. 1993 , , 265 , 145 becker , r.h . , white , r.l . , & helfand , d.j . 1995 , , 450 , 559 boyle , b.j . 1986 , phd thesis , university of durham brandenburg , a. , enquist , k. , & olesen , p. 1996 , sissa preprint , asrto - ph/9602031 broten , n.w . , macleod , j.m . , & valle , j , p . 1988 , , 141 , 303 burn , b.j . 1966 , , 133 , 67 camilo , f. , nice , d.j . , & taylor j.h . 1996 , , 461 , 812 cheng , b.l . , olinto , a.v . , schramm , d.n . , & truran , j.w 1996 , , 54 , 4714 condon , j.j . , cotton , w.d . , greisen , e.w . , yin , q.f . , perley , r.a . , & broderick , j.j . 1994 , in _ astronomical data analysis software and systems iii , a.s.p . conference series _ , vol . dennis , r. crabtree , r.j . , barnes h. , & barnes , j. p. 155 cress , c.m . , helfand , d.j . , becker , r.h . , gregg , m.d . , white , r.l . 1996 , , 473 , 7 dolgov , a. & silk , j. 1993 , 47 , 3144 dunlop , j.s . & peacock , j.a . 1990 , 247 , 19 foster , r.s . , cadwell , b.j . , wolszczan , a. , & anderson , s.b . 1995 , , 454 , 826 grasso , d. & rubinstein h.r . 1995 , nucl . b , 43 , 303 grasso , d. & rubinstein h.r . 1996 , phys . b , 388 , 253 gunn , j.e . & peterson , b.,a . 1965 , , 142 , 1633 harrison , e.r . 1973 , , 30 , 18 kato , t. , tabara , h. , inoue , m. , & aizu , k. 1987 , , 329 , 223 kolb , e.w . & turner , m.s . 1990 , _ the early universe _ , ( usa : addison - wesley ) p. 84 kooiman , b.l . , burns , j.o . , & klypin , a.a . 1995 , , 448 , 500 kronberg , p.p . 1976 , _ int . union symp . _ 74 , 367 kronberg , p.p . 1994 , rep . phys . , 57 , 325 ( k94 ) kronberg , p.p . & simard - normandin , m. 1976 , , 263 , 653 laing , r.a . 1984 , in _ physics of energy transport in extragalactic radio sources _ , edited by bridle , a.h . & eilek , j.a . ( green bank w.va . usa : nrao ) p. 90 lang , k.r . 1978 , _ astrophysical formulae _ , ( new - york : springer - verlag ) p. 57 lanzetta , k.m . , wolfe , a.m. , turnshek , d.a . 1995 , , 440 , 425 lee , s. , olinto , a.v . , & sigl , g. 1995 , , 455 , l21 loan , a.j . , wall , j.v . , & lahav , o. 1997 , , 286 , 994 loeb , a. & kosowsky , a. 1996 , , 469 , 1 masden , m.s . 1989 , , 237 , 109 minter , a.h . & spangler , s.r . 1996 , , 458 , 194 mller , p. & jakobsen , p. 1990 , , 299 monin , a.s . & yaglom , a.m. 1975 , _ statistical fluid mechanics _ ( cambridge ; mit press ) , pp . 2952 nissen , d. & thielheim , k.o . 1975 , ap&ss , 33 , 441 oren , a.l . & wolfe , a.m. 1995 , 445 , 624 osmer , p.s . 1981 , , 247 , 762 ostriker , j.p . & thompson , c. 1987 , , 323 , l97 parker , e.n . 1979 , _ cosmical magnetic fields _ ( oxford ; clarendon press ) p. 616 pudritz , r.e . & silk , j. 1989 , , 342 , 650 quashnock , j.m . , loeb , a. , & spergel , d.n . 1989 , , 342 , 650 ratra , b. 1992a , , 45 , 1913 ratra , b. 1992b , , 391 , l1 rees , m.,j . 1987 , , 28 , 197 sicotte , h. , 1995 , ph.d . thesis , princeton university simard - normandin , m. & kronberg , p.p . 1980 , , 242 , 74 simard - normandin , m. & kronberg , p.p . , & button , s. 1980 , , 45 , 97 ( skb ) simonetti , j.h . , cordes , j.m . , & spangler , s.r . 1984 , 284 , 126 simonetti , j.h . & cordes , j.m . 1986 , 310 , 160 tabara , h. & inoue , m. 1979 , supp . , 39 , 379 tadros , h. & efstathiou , g. 1995 , , 276 , 45 tajima , t. , cable , s. , shibata , k. , & kulsrud , r.m . 1992 , , 390 , 309 taylor , j.h . & cordes , j.m . 1993 , , 411 , 674 taylor , j.h . , manchester , r.n . , & lyne , a.g . 1993 , , 88 , 529 thompson , c. 1990 , _ int . union symp . _ , 140 , 127 tribble , p.c . 1991 , , 250 , 726 turner , m.s . & widrow , l.m . 1988 , , 37 , 10 tytler , d. , fan , x. , & burles , s. 1996 , , 381 , 207 vachaspati , t. 1991 , phys . lett . , 265 , 258 valle , j.,p . 1990 , , 360 , 1 waxman , e. & miralda - escud , j. 1996 , , 472 , l89 weinberg , s. 1972 , _ gravitation and cosmology _ ( wiley & sons , ny ) p. 413 welter , g.l . , perry , j.j . , & kronberg , p.p . 1984 , , 279 , 19 zeldovich , y.b . & novikov , i.d . , 1975 , _ relativistic astrophysics _ , ( chicago ; university of chicago press ) zweibel , e.g. 1988 , , 329 , l1
this paper introduces the formalism which connects between rotation measure ( @xmath0 ) measurements for extragalactic sources and the cosmological magnetic field power spectrum . 
 it is shown that the amplitude and shape of the cosmological magnetic field power spectrum can be constrained by using a few hundred radio sources , for which faraday @xmath0s are available . 
 this constraint is of the form @xmath1 h $ ] nano - gauss ( ng ) on @xmath2 scales , with @xmath3 the average baryon density and @xmath4 the hubble parameter in @xmath5 km s@xmath6 mpc@xmath6 units . 
 the constraint is superior to and supersedes any other constraint which come from either cmb fluctuations , baryonic nucleosynthesis , or the first two multipoles of the magnetic field expansion . 
 the most adequate method for the constraint calculation uses the bayesian approach to the maximum likelihood function . 
 i demonstrate the ability to detect such magnetic fields by constructing simulations of the field and mimicking observations . 
 this procedure also provides error estimates for the derived quantities .    the two main noise contributions due to the galactic rm and the internal rm are treated in a statistical way following an evaluation of their distribution . 
 for a range of magnetic field power spectra with power indices @xmath7 in a flat cosmology ( @xmath8=1 ) we estimate the signal - to - noise ratio , @xmath9 , for limits on the magnetic field @xmath10 on @xmath11 scale . 
 employing one patch of a few square degrees on the sky with source number density @xmath12 , an approximate estimate yields @xmath13 . 
 an all sky coverage , with much sparser , but carefully tailored sample of @xmath14 sources , yields @xmath15 with the same scaling . 
 an ideal combination of small densely sampled patches and sparse all - sky coverage yields @xmath16 with better constraints for the power index . 
 all of these estimates are corroborated by the simulations . 
 epsf.sty = cmr10 # 1#2.5pt 2 degm^-2 2 radm^-2