Secondary CMB anisotropies from magnetized halos --I: Power spectra of the Faraday rotation angle and conversion rate
Nadège Lemarchand, Julien Grain, Guillaume Hurier, Fabien Lacasa, Agnès Ferté
AAstronomy & Astrophysics manuscript no. main c (cid:13)
ESO 2018October 23, 2018
Secondary CMB anisotropies from magnetized halos –I: Powerspectra of the Faraday rotation angle and conversion rate
N. Lemarchand (cid:63) , J. Grain , G. Hurier , F. Lacasa , and A. Ferté Institut d’Astrophysique Spatiale, CNRS (UMR8617) and Université Paris-Sud 11, Bâtiment 121, 91405 Orsay, France Centro de Estudios de Física del Cosmos de Aragón (CEFCA), Plaza de San Juan, 1, planta 2, 44001 Teruel, Spain Département de Physique Théorique and Center for Astroparticle Physics, Université de Genève, 24 quai Ernest Ansermet, CH-1211 Geneva, Switzerland Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California, USA
ABSTRACT
Magnetized plasmas within halos of galaxies leave their footprint on the polarized anisotropies of the cosmic microwave background.The two dominant e ff ects for astrophysical halos are Faraday rotation generating rotation of the plane of linear polarization, and Fara-day conversion inducing a leakage from linear polarization to circular polarization. We revisit these sources of secondary anisotropiesby computing the angular power spectra of the Faraday rotation angle and of the Faraday conversion rate by the large scale structures.To this end, we use the halo model and we pay special attention to the impact of magnetic field projections. Assuming magneticfields of halos to be uncorrelated, we found a vanishing 2-halo term, and angular power spectra peaking at multipoles (cid:96) ∼ . TheFaraday rotation angle is dominated by the contribution of thermal electrons. For the Faraday conversion rate, we found that boththermal electrons and relativistic, non-thermal electrons contribute equally in the most optimistic case for the density and Lorentzfactor of relativistic electrons, while in more pessimistic cases the thermal electrons give the dominant contribution. Assuming themagnetic field to be independent of the halo mass, the angular power spectra for both e ff ects roughly scale with the amplitude ofmatter perturbations as ∼ σ , and with a very mild dependence with the density of cold dark matter. Introducing a dependence of themagnetic field strength with the halo mass leads to an increase of the scaling with the amplitude of matter fluctuations, up to ∼ σ . for Faraday rotation and ∼ σ for Faraday conversion for a magnetic field strength scaling linearly with the halo mass. Key words. galaxy halos, CMB, cosmology, power spectrum, modeling
1. Introduction
One of the main challenges in observational cosmology is a com-plete characterisation of the Cosmic Microwave Background(CMB) polarization anisotropies, targetted by a large amount ofon-going, being deployed or planned experiments, either fromground or space-borne missions (see for examples Aguirre et al.2018; Suzuki et al. 2018). In full generality, polarized light (inaddition to its total intensity, I ) is described by its linear com-ponent encoded in the two Stokes parameters Q and U , andits circular component encoded in the parameter V . For CMBanisotropies, there is no source of primordial V in the standardcosmological scenario (see however e.g. Giovannini 2010, forpotential primordial sources), with upper bounds on its r.m.s. of ∼ µ K at ten degrees (Mainini et al. 2013; Nagy et al. 2017).Hence, the CMB polarization field is completely described onthe sphere by two Stokes parameters, Q and U . In the harmonicdomain, this field can be described either by using spin-(2) andspin-( −
2) multipolar coe ffi cients, or by using gradient, E andcurl, B coe ffi cients. From a physical point of view, the gradi-ent / curl decomposition is more natural as it is directly linked tothe cosmological perturbations produced in the primordial Uni-verse. For symmetry reasons, at first order, scalar perturbationscan produce E -modes only and the B -modes part of the po-larization field is thus a direct tracer of the primordial gravitywaves (Zaldarriaga & Seljak 1997; Kamionkowski et al. 1997).Though such a picture is partially spoilt by the presence of a sec- (cid:63) [email protected] ondary contribution generated by the gravitational lensing of the E -modes polarization (Zaldarriaga & Seljak 1998), its peculiarangular-scale shape and delensing techniques should allow for areconstruction of the primordial component.Lensing of the CMB anisotropies is however not the solesource of cosmological and astrophysical E -to- B conversion.During the propagation of CMB photons from the last scatteringsurface to our detectors, the plane of linear polarization could berotated. Such a rotation could be due to Faraday rotation inducedby interactions of CMB photons with background magnetizedplasmas, with magnetic fields of either cosmological origin(Kosowsky & Loeb 1996; Kosowsky et al. 2005; Campanelliet al. 2004; Scoccola et al. 2004) or astrophysical origin (Takadaet al. 2001; Ohno et al. 2003; Tashiro et al. 2008, 2009), ordue to interactions with pseudoscalar fields (Carroll 1998).Furthermore, even though primordial circular polarization isnot present in the CMB in the standard model of cosmology,secondary circular polarization could be produced by Faradayconversion (Sazonov 1969; Cooray et al. 2003; De & Tashiro2015) or e.g. by non-linear electrodynamics (Montero-Camacho& Hirata 2018).With the significant increase of sensitvity of the forthcom-ing observatories aiming at an accurate mapping of the CMBpolarization on wide ranges of angular scales, clear predictionsfor such additional secondary anisotropies is of importance formany reasons. Article number, page 1 of 18 a r X i v : . [ a s t r o - ph . C O ] O c t & A proofs: manuscript no. main
First, they contain some cosmological and / or astrophysicalinformations, and could thus be used to probe e.g. parity viola-tion in the Universe (Li & Zhang 2008; Lue et al. 1999; Pospelovet al. 2009; Yadav et al. 2009), or intrahalo magnetic fields or gasevolution at early epochs (Takada et al. 2001; Ohno et al. 2003;Tashiro et al. 2008, 2009).Second, such a signal should be known to be correctly takeninto account for identifying the primordial component of the B -mode from such secondary anisotropies (or at least shown to besubdominant at superdegree scales where primordial B -mode isexpected to peak above the lensing B -mode).Thirdly, these secondary anisotropies are of importance forlensing reconstruction using CMB polarized data, shown tobe more powerful than starting from temperature data in thecase of highly sensitive experiments (Okamoto & Hu 2003;Hirata & Seljak 2003; Ade et al. 2014). Secondary polarizedanisotropies in addition to lensing-induced anisotropies couldindeed mimick contributions from the lensing potential thusbiasing its reconstruction from E - and B -modes. This lastpoint is also of relevance for the delensing, either internal(Seljak & Hirata 2004; Carron et al. 2017), or based onexternal tracers of the lensing potential such as the CosmicInfrared Background (Sigurdson & Cooray 2005; Marian &Bernstein 2007; Smith et al. 2012; Sherwin & Schmittfull 2015).For any possible non-primordial sources of CMBanisotropies, we have first to quantitatively predict the in-duced CMB anisotropies. Second, one can further investigatethe amount of cosmological / astrophysical informations theycarry, and finally estimate how they may bias the reconstructionof the primordial B -mode and the lensing potential reconstruc-tion. In this article we are interested in magnetized plasmas inhalos of galaxies as a source of secondary polarized anisotropiesof the CMB, revisiting and amending first estimates in Tashiroet al. (2008); Cooray et al. (2003). Observations with e.g.Faraday rotation measurements from polarized point sourcessuggest that they are magnetized with a coherence length ofthe size of the halo scale and a typical strength ranging from1 − µ G (Kim et al. 1989; Athreya et al. 1998; Bonafede et al.2010, 2009). This implies that the CMB linear polarisation fieldis rotated - an e ff ect known as Faraday rotation - and convertedto circular polarisation - referred to as Faraday conversion. Thegoal of the present paper is to give an accurate computationof the angular power spectra of the Faraday rotation angleand Faraday conversion rate, the first mandatory step beforeestimating its impact on CMB secondary anisotropies.This article is organized as follows. We first briefly describein Sect. 2 the propagation of CMB photons through a magne-tized plasma. We show that for the specific case of halos, thetwo dominant e ff ects are Faraday rotation and Faraday conver-sion. This section is also devoted to a brief presentation of thephysics and the statistics of halos. Second in Sect. 3, we presentour calculation of the angular power spectra of the Faraday ro-tation angle and the Faraday conversion rate. This is done usingthe halo model, and we amend previous analytical calculationsgiving special attention to the statistics of the projected mag-netic fields of halos. Our numerical results are provided in Sect.4 where we discuss the dependence of the angular power spectrawith cosmological parameters. We finally conclude in Sect. 5.Throughout this article, we use the Planck Collaborationet al. (2016) (PlanckTTTEEE + SIMlow) best fit parameters,namely σ = . Ω CDM h = . Ω b h = . h = .
2. Physics of halos
Propagation of radio and millimeter waves in a magnetizedplasma has been studied in Sazonov (1969), and later reassessedin Kennett & Melrose (1998); Melrose & McPhedran (2005);Heyvaerts et al. (2013); Shcherbakov (2008). In Eq. (1.5) ofSazonov (1969), the radiative transfer equation for the fourStokes parameters, ( I , Q , U , V ), is provided in a specific refer-ence frame in which one of the basis vector in the plane or-thogonal to the direction of light propagation is given by themagnetic field projected in that plane. The Stokes parameters( Q , U , V ) are however reference-frame dependant, and it is thusimportant to get this equation in an arbitrary reference frame,for at least two reasons. First, we are interested in the Stokes pa-rameter of the CMB light and there is a priori no reason for thereference frame chosen to measure the Stokes parameter to bespecifically aligned with the magnetic fields of the many halosCMB photons pass through. One usually makes use of ( e θ , e ϕ , n )with n pointing along the line-of-sight and e θ , e φ the unit vectorsorthogonal to n associated to spherical coordinates, and there isno reason for e θ to be aligned with the projection of the manymagnetic fields. Second, we are here interested in computing thetwo-point correlation function and there is obviously no reasonfor the chosen reference frame to coincide at two arbitrary se-lected directions on the celestial sphere with the specific refer-ence frame used in Sazonov (1969), which clearly di ff ers fromdirections to directions on the celestial sphere.The radiative transfer equation is written in an arbitrary refer-ence frame by performing an arbitrary rotation of the basis vec-tors in the plane orthogonal to the light propagation, or equiv-alently an arbitrary rotation of the magnetic field projected insuch a plane (Huang et al. 2009). We denote here by θ B the anglebetween the magnetic field projected on the plane orthogonal tothe line-of-sight and the basis vector e θ . By further introducingthe spin-( ±
2) field for linear polarization, P ± = Q ± iU , thisgivesdd r IP P − V = (cid:20) M abs + M I → P + M P → P (cid:21) IP P − V , (1)with r labelling the path of light. The three matrices encodingthe di ff erent contributions to radiative transfer are M abs = ˙ τ τ τ
00 0 0 ˙ τ , (2) M I → P = φ I → P e − i θ B ˙ φ I → P e i θ B ˙ φ I → V ˙ φ I → P e i θ B φ I → P e − i θ B φ I → V , (3) M P → P = − i ˙ α − i ˙ φ P → V e i θ B i ˙ α i ˙ φ P → V e − i θ B i ˙ φ P → V e − i θ B − i ˙ φ P → V e i θ B , (4)where ˙ f means di ff erentiation with respect to r .The di ff erent coe ffi cients, ˙ τ , ˙ α and the ˙ φ i → j ’s are real andtheir expressions can be found in e.g. Sazonov (1969) bysetting θ B =
0, which basically corresponds to choosing thespecific reference frame adopted in Sazonov (1969). They are
Article number, page 2 of 18emarchand et al.: Secondary CMB anisotropies from magnetized halos (I) ˙ τ ˙ φ I → P ˙ φ I → V ˙ α ˙ φ P → V Thermal electrons n e / ( ν T / e ) 10 ( n e B ⊥ ) / ( ν T / e ) 10 ( n e B (cid:107) ) / ( ν T / e ) 10 ( n e B (cid:107) ) / ( ν ) 10 ( n e B ⊥ ) / ( ν ) ∼ − m − ∼ − m − ∼ − m − ∼ − m − ∼ − m − Relativistic electrons n ( r ) e B ⊥ /ν n ( r ) e B ⊥ /ν n ( r ) e B / n /ν / n ( r ) e B (cid:107) /ν n ( r ) e B ⊥ /ν ∼ − m − ∼ − m − ∼ − m − ∼ − m − ∼ − m − Table 1.
Scaling of the radiative transfer coe ffi cients for thermal electrons and relativistic electrons with the projection of the magnetic fields alongor orthogonal to the line-of-sight, the frequency of photons, and the density and temperature of free electrons (adapetd from Sazonov 1969). Forthermal electrons, the numerical constants in front of the reported scalings span a large range of values and we provide their value relative to theone for the parameter ˙ τ . These constants are all of the same order in the case of relativistic electrons. The corresponding values are obtained forthe case of halos with n e =
10 m − , T e = K for thermal electrons, and n ( r ) e =
10 m − for relativistic electrons. In both cases, the magnetic fieldis set to B = µ G, and the frequency to ν =
30 GHz. interpreted as follows. First, the coe ffi cient ˙ τ in M abs simplycorresponds to absorption of light by the medium. Second in M I → P , the coe ffi cients ˙ φ I → P and ˙ φ I → V amount the transfer fromtotal intensity to linear polarization and to circular polarizationrespectively. Finally in M P → P , the coe ffi cient ˙ α correspondsto Faraday rotation which mixes the two modes of linearpolarization, while ˙ φ P → V is Faraday conversion which transferslinear polarization in circular polarization.The expressions of the di ff erent coe ffi cients and their relativeamplitude depend on the nature of free electrons in the magne-tized plasma. Two extreme situations are either normal waves ofthe plasma are circularly polarized, or these normal waves arelinearly polarized. In the former case, Faraday rotation is dom-inant, which is the case for a plasma made of non-relativisticelectrons. In the latter, Faraday conversion is dominant. Thiscan occur for a population of relativistic and non-thermal elec-trons, with some restrictions on their energy distributions (seeSazonov 1969).For the case of astrophysical clusters and halos as consideredas magnetized plasmas, two populations of electrons are at play.First, the thermal electrons which are e.g. at the origin of thethermal Sunyaev-Zel’dovich (tSZ) e ff ect, and second, relativis-tic electrons generated by either AGN or shocks. For the caseof thermal electrons, the typical temperature of clusters is ∼ K, corresponding to about few keV’s, hence much smaller thanthe electron mass. This population of electrons is thus mainlynon-relativistic. A typical value of the number density of thermalelectrons for clusters is n e ∼ m − for a halo mass of 10 M (cid:12) .For the case of relativistic electrons, the coe ffi cients depend onthe energy distribution of the relativistic electrons in the injectedplasma via the minimal Lorentz factor, Γ min , and the spectral in-dex of the energy distribution, i.e. n ( r ) e ( Γ ) ∝ Γ − β E , as well as onthe spatial distribution of the energy distribution of the injectedrelativistic electrons in the plasma. We here follow Cooray et al.(2003); De & Tashiro (2015) by considering a spectral index ofthe energy distribution of relativistic electrons of 2, a minimalvalue of the Lorentz factor of Γ min = n ( r ) e =
10 m − (Colafrancesco et al. 2003).The expressions of these radiative transfer coe ffi cients fromSazonov (1969) are provided in Tab. 1 up to numerical constants.We highlight their scaling with the electron number density ( n e and n ( r ) e ), the magnetic field either projected on the line-of-sight, This population is dubbed "cold plasma" in Sazonov (1969). B (cid:107) , or in the plane perpendicular to it, B ⊥ , the frequency of theradiation light, ν , and, for the case of thermal electrons the tem-perature of electrons, T e . For thermal electrons, the numericalconstants in front of the reported scalings span a large range ofvalues and we provide their value relative to the one for the pa-rameter ˙ τ . These constants are all of the same order in the caseof relativistic electrons. The values reported are for a magneticfield of 3 µ G and a frequency of 30 GHz.For linear polarization, the dominant e ff ect is Faraday rota-tion by thermal electrons. Faraday rotation from relativistic elec-trons is 7 orders of magnitude smaller, and absorption, ˙ τ , is 13(thermal electrons) and 9 (relativistic electrons) orders of mag-nitude smaller than Faraday rotation. Faraday conversion from V to P ± is zero for CMB since there is no primordial circu-lar polarization. Intensity of the CMB is about 1 to 2 orders ofmagnitude higher than the E -mode of linear polarization, and atleast 3 orders of magnitude higher than the B -mode. Leakagesof I to P ± could thus rapidly become important because of thisgreat hierarchy. However, the transfer coe ffi cient ˙ φ I → P for ther-mal electrons and relativistic electrons is 32 and 9 (resp.) ordersof magnitude smaller than ˙ α . Hence leakages from intensity tolinear polarization is totally negligible as compared to Faradayrotation by thermal electrons.The dominant e ff ect for circular polarization is Faradayconversion from both thermal electrons and relativistic elec-trons. Absorption is vanishing for zero initial V . Leakages fromintensity to circular polarization remains smaller than Faradayconversion. In the most optimistic case for the number density ofrelativistic electrons ˙ φ I → V indeed remains 5 orders of magnitudesmaller than ˙ φ P → V , meaning that circular polarization generatedthrough leakages of intensity is about 3 orders of magnitudesmaller than the one generated through Faraday conversion. An important last comment is in order here. The terms e ± i θ B naturally appear for preserving the symmetry properties of thefour Stokes parameters.One reminds that these parameters are defined in the plane( e θ , e ϕ ) orthogonal to the line-of-sight and in a reference-frame-dependant manner. The total intensity I is independant of rota-tion and parity transformations of the reference frame (i.e. it isa scalar ). Linear polarization, P ± , are spin- ± ± θ ) by a rotation θ of the referenceframe, and spin-( +
2) and spin-( −
2) are interchanged by a par- Note that here Faraday conversion and Faraday rotation by relativis-tic electrons are of equal magnitude. Faraday rotation by this populationremains however much smaller than the one due to thermal electrons.Article number, page 3 of 18 & A proofs: manuscript no. main ity transformation. Finally, circular polarization V is unchangedthrough rotations but changes its sign via a parity transformationof the reference frame (i.e. it is a pseudo-scalar ).The coe ffi cients ˙ α and ˙ φ P → V are independent of the refer-ence frame. The angle θ B however is reference-frame dependentand the quantities e ± i θ B are spin-( ±
2) fields. One can then checkthat indeed all the symmetry properties are properly preservedthrough radiative transfer. For example, one obtains˙ V ( n ) = i ˙ φ P → V ( n ) (cid:104) e − i θ B ( n ) P ( n ) − e i θ B ( n ) P − ( n ) (cid:105) , (5)where the right-hand-side is an appropriate combination of dif-ferent spin-( ±
2) fields leading to a pseudo-scalar field, V . Wenote that this is in agreement with expressions used in Montero-Camacho & Hirata (2018); Kamionkowski (2018), reading˙ V ( n ) = φ U ( n ) Q ( n ) − φ Q ( n ) U ( n ) with φ Q = φ P → V cos(2 θ B ) and φ U = φ P → V sin(2 θ B ).It is also easily checked that by selecting the specific refer-ence frame adopted in Sazonov (1969), i.e. setting θ B =
0, theEq. (1.5) of Sazonov (1969) is recovered. In particular in thisreference frame one sees that I is transferred into Q only, while V receives contribution from U only, i.e. ˙ V = − φ P → V U . (Wenote that this last expression was used in Cooray et al. (2003);De & Tashiro (2015) which is however valid on a very specificreference frame.) The impact of radiative transfer within magnetized halos on theCMB is in theory obtained by integrating Eq. (1). Such radiative-transfer distortions of the CMB within halos are expected tomainly occur at low redshifts, z (cid:46)
1. One can thus take as initialconditions the lensed CMB fields.In full generality, the matrix (cid:20) M abs + M I → P + M P → P (cid:21) is toocomplicated to find a general solution of this.The dominant ef-fect is however the Faraday rotation by thermal electrons. Ne-glecting the other coe ffi cients, only linear polarization is modi-fied and the solution is P FR ± ( n ) = e ∓ i α (0 , r CMB ) P ± ( n ) , (6)with P ± the { primary + lensed } CMB linear polarization field,and α (0 , r CMB ) is the integral of ˙ α over the line-of-sight fromthe last scattering surface at r CMB , to present time at r = n ). The Faraday rotation re-mains a tiny e ff ect and one can Taylor expand the exponentialfor small α ’s.The next-to-leading order e ff ect is the Faraday conversionwhose impact on the CMB can be implemented with a pertur-bative approach to solve for Eq. (1). Since the initial V param-eter is vanishing, this leaves the solution for linear polarizationunchanged. Circular polarization generated should in principlebe generated by Faraday conversion of the rotated linear polar-ization, P rot ± , integrated over the line-of-sight, hence mixing therotation angle and the conversion rate. These e ff ects are howeverexpected to be small. Multiplicative e ff ect of rotation and con-version are thus of higher orders and it can be neglected. Thisgives for circular polarization V ( n ) = i (cid:2) φ − (0 , r CMB ) P ( n ) − φ (0 , r CMB ) P − ( n ) (cid:3) , (7) with φ ± (0 , r CMB ) the integral over the line-of-sight of˙ φ P → V e ± i θ B . Distortions of the CMB polarized anisotropies by Faraday rota-tion and Faraday conversion is a multiplicative e ff ect. Their im-pact on the CMB angular power spectra will thus be determinedby the angular power spectra of the Faraday rotation angle, α ,and the Faraday conversion rate, φ ± .We make use of the halo model (Cooray & Sheth 2002) in or-der to characterize the statistical properties of the radiative trans-fer coe ffi cients of the halos as magnetized plasmas. The basicelements in this theoretical framework are first the physics inter-nal to each halos, i.e. its gas and magnetic field distributions, andsecond the statistical properties of halos within our Universe. In the following, we will mainly need two characteristics of ha-los: their free electron density and magnetic field spatial profiles,which for simplicity are considered as spherically symmetric.For the profile n e of free electrons we choose to take the β -profile of Cavaliere & Fusco-Femiano (1978) as what was donein Tashiro et al. (2008): n e ( r ) = n ( c ) e (cid:32) + r r c (cid:33) − β/ (8)where r and r c are respectively the physical distance to thehalo centre and the typical core radius of the halo (note thatit could be comoving distances as only the ratio of these twodistances shows up in the expression). The physical halo coreradius r c is related to the virial radius by: r vir ∼ r c , with r vir = ( M / (4 π ∆ c ( z ) ¯ ρ ( z ) / / and ∆ c ( z ) = π Ω m ( z ) . is thespherical overdensity of the virialized halo, ¯ ρ ( z ) is the criticaldensity at redshift z (see Tashiro et al. 2008). The quantity n ( c ) e isthe central free electron density. For thermal free electrons, it isgiven by: n ( c ) e = . × − cm − (cid:32) M M (cid:12) (cid:33) (cid:32) r vir (cid:33) (cid:32) Ω b Ω m (cid:33) (9) × F − (3 / , β/
2; 5 / − ( r vir / r c ) ) , with F the hypergeometric function.The properties of relativistic electrons inside halos are notwell known. Hence we just take a constant value for the central A perturbative approach to solve for Eq. (1) keeping ˙ α at the leadingorder and ˙ φ P → V at the next-to-leading order gives P FR + FC ± = e ∓ i α (0 , r CMB ) P ± ∓ i (cid:34)(cid:90) r CMB d s ˙ φ P → V ( s ) e ± i θ B ( s ) e ∓ i α (0 , s ) (cid:35) V , and V FR + FC = V ( n ) + i (cid:34)(cid:90) r CMB d s ˙ φ P → V ( s ) e − i θ B ( s ) e − i α ( s , r CMB ) (cid:35) P − i (cid:34)(cid:90) r CMB d s ˙ φ P → V ( s ) e i θ B ( s ) e i α ( s , r CMB ) (cid:35) P − . P ± and V are the { primary + lensed } CMB polarization field. Solutionsgiven in the core of the text are obtained setting the initial circular po-larization to zero, V =
0, and keeping the leading order in a Taylorexpansion of e ± i α ( s , r CMB ) .Article number, page 4 of 18emarchand et al.: Secondary CMB anisotropies from magnetized halos (I) free electron density: n ( c ) e =
10 m − , which was the highest valuewe found in the literature (Colafrancesco et al. 2003).The magnetic field, denoted B , is in full generality a functionof both x and x i (respectively labelling any position within thehalo and the center of the halo), as well as a function of the massand the redshift of the considered halo. Because we have only apoor knowledge of the magnetic field inside halos, we allow our-selves to chose a model for B that will simplify a bit the calcula-tions of the angular power spectra. Therefore, the first of our as-sumptions is that the orientation of the magnetic field is roughlyconstant over the halo scale, though we still allow for potentiallyradial profile for its amplitude, i.e. B ( x , x i ) = B ( | x − x i | ) ˆ b ( x i ).The vector ˆ b ( x i ) is a unit vector labelling the orientation of themagnetic field of a given halo, thus depending on the halo posi-tion only and considered as a random variable. Here, we alsoassumed a spherically symmetric profile for the amplitude ofthe magnetic field. Observations suggest that the amplitude ofthe magnetic field scales radially as the halo matter content, i.e. B ∝ ( n gas ) µ (see e.g. Hummel et al. 1991; Murgia et al. 2004;Bonafede et al. 2010, 2009). We thus choose a form for the am-plitude of the magnetic field that corresponds to the β -profile: B ( r ) = B c ( z ) (cid:32) + r r c (cid:33) − βµ/ , (10)where B c is the mean magnetic field strength at the centre of thehalo. Its time evolution is given by (Widrow 2002): B c ( z ) = B exp (cid:32) − t − t ( z ) t d (cid:33) µ G (11)where t is the present time and t d = (cid:113) r / GM , and B is thefield strength at present time. The spatial distribution of halos and their abundance in massand redshift is described using the halo model (Cooray & Sheth2002). The abundance in mass and redshift is given by the halomass function, d N / d M , and their spatial correlation is derivedby the matter power spectrum plus halo bias. In this study, wemake use of the halo mass function derived in Despali et al.(2016) which is defined using the virial mass.The radiative transfer coe ffi cients introduced in Sect. 2.1 de-pend on the projection of the magnetic field either along the line-of-sight, or in the plane orthogonal to it. One thus needs to intro-duce some statistics for the orientation of halo’s magnetic fields.This statistics of the relative magnetic field orientations of halosis however poorly known. To motivate our choice (presented lat-ter), let us first birefly comment on previous results obtained inthe literature.The angular power spectrum of the Faraday rotation anglehas been firstly computed in Tashiro et al. (2008), using an ap-proach adapted from the study of the Sunyaev-Zel’dovich e ff ectdeveloped in Cole & Kaiser (1988); Makino & Suto (1993); Ko-matsu & Kitayama (1999). We however believe that this first pre-diction should be amended. This is motivated by the followingintuitive idea (most easily formulated using the 2-point correla-tion function).The Faraday rotation angle is derived from the projectionof the magnetic field on the light-of-sight followed by CMB photons, i.e. α ( n ) ∝ n · B , and the correlation function is thus ξ ( n , n ) : = (cid:104) α ( n ) α ( n ) (cid:105) ∝ (cid:68) ( n · B i ) (cid:16) n · B j (cid:17)(cid:69) , where the sub-scripts i , j label the halos which are respectively crossed by theline-of-sight n and n . A first case is that the line-of-sight aresuch that they cross two distinct halos, i.e. i (cid:44) j , correspondingto the so-called 2-halos term in the angular power spectrum. Onefurther assumes that magnetic fields in halos are produced by as-trophysical processes. Hence two di ff erent halos are statisticallyindependent (from the viewpoint of magnetic fields), leading to ξ ( n , n ) ∝ (cid:104) n · B i (cid:105) (cid:68) n · B j (cid:44) i (cid:69) . To be in line with a statisti-cally homogeneous and isotropic Universe, the orientation of themagnetic field of halos should be uniformly distributed leadingto (cid:104) n · B i (cid:105) = One thus expects the 2-halos term to be zero,which is however not the case in Tashiro et al. (2008) wheresuch a term is not vanishing. Considering then the 1-halo term, this reads ξ ( n , n ) ∝(cid:104) ( n · B i ) ( n · B i ) (cid:105) providing that both line-of-sight cross thesame halo. This is a priori non-zero since (cid:104) B i B i (cid:105) does not van-ish. There is however a subtlety which to our viewpoint, has notbeen considered in Tashiro et al. (2008). They consider that thestatistical average of the orientation of magnetic fields for the1-halo term is (cid:68) ( n · B i ) (cid:69) = / di ff erent lines-of-sight crossing the same halo, and there is a priori no reasonthat ( n · B i ) = ( n · B i ) for a randomly selected halo. As a con-sequence, this is (cid:104) ( n · B i ) ( n · B i ) (cid:105) which enters as a statisticalaverage on the1-halo term, and not (cid:68) ( n · B i ) (cid:69) .A similar argument applies for Faraday conversion exceptthat this the projection of the magnetic field on the planeorthogonal to n which is here involved.We will thus suppose that orientations are uniformly dis-tributed in the Universe, independant for two di ff erent halos, andindependant of the spatial distribution of halos. This can be un-derstood as follows: we assume no coherence of the magneticfield orientations of di ff erent halos or, to put it di ff erently, themagnetic field correlation length is smaller than the inter-haloscale. This assumption is clearly in line with the cosmologicalprinciple, and it is motivated by the idea that halos’ magnetism isa result of processes isolated from other halos. Thus, this orien-tation is a random variable which should be zero once averagedover halos.Orientations are given by the unit vector, b , which is thuslabelled by a zenithal angle, β ( x i ), and an azimuthal angle, α ( x i ).In the cartesian coordinate system, the three component are b ix = sin ( β ( x i )) cos ( α ( x i )) , (12) b iy = sin ( β ( x i )) sin ( α ( x i )) , (13) b iz = cos ( β ( x i )) . (14) Note that for two distinct halos having though the same mass andbeing at the same redshift, it may well be that they share the same am-plitude for B . This remains consistent with a statistically homogeneousand isotropic Universe as long as the orientations of the magnetic fieldsaverage down to zero. We mention that the 2-halos term may not be vanishing assumingsome correlations between the magnetic fields of two di ff erent halos(for example if these magnetic fields are seeded by a primordial mag-netic field). In this case however, the 2-halos term should be composedof a convolution of the matter power spectrum with the magnetic fieldpower spectrum, as one could expect from results obtained for the simi-lar case of the kinetic Sunyaev-Zel’dovich e ff ect induced by the peculiarvelocity of halos (Hernandez-Monteagudo et al. 2006).Article number, page 5 of 18 & A proofs: manuscript no. main
Any projection of the magnetic field orientation can be writtenas a function of the two angles, β and α . Our assumption of uni-formly distributed orientations translates into the following av-eraging (cid:68) f ( α i , β i ) (cid:69) = π (cid:90) f ( α i , β i ) d α i d(cos β i ) , (15)with β i and α i a shorthand notation for β ( x i ) and α ( x i ). Since weassume two halos to be independant, one does not need to furtherintroduce some correlations and the above fully described thestatistics of orientations of magnetic fields.
3. Angular power spectra of Faraday rotation andFaraday conversion
The Faraday rotation angle is given by the following integralover the line-of-sight α ( n ) = e π m e c ε (cid:90) r CMB a ( r )d r ν ( r ) (cid:88) i = halo [ ˆ n · B ( x , x i )] n e ( | x − x i | ) , (16)where r stands for the comoving distance on the line-of-sight, x = r n , r CMB is the distance to the last-scattering surface, and x i is the center of the i -th halo. With our assumption regarding themagnetic field, and further replacing the summation over halosby integrals over the volume and over the mass range, the abovereads α ( n ) = e π m e c ε (cid:90) r CMB a ( r )d r ν ( r ) (cid:34) d M i d x i (cid:104) n h ( x i ) (17) × b ( n , x i ) X ( | x − x i | ) (cid:105) , with n h ( x i ) the abundance of halos, b ( n , x i ) = n · b ( x i )the projection along the line-of-sight, and X ( | x − x i | ) = B ( | x − x i | ) n e ( | x − x i | ).Two simplifications result from the di ff erent assumptionsmade about the statistics of the orientation of the magnetic field.To this end, let us introduce the notation A i ( n ) = e π m e c ε (cid:90) r CMB a ( r )d r ν ( r ) X ( | x − x i | ) , where we stress that the impact of orientation is omitted in theabove. It can basically be interpreted as the maximum amountof rotation the halo i can generate. (We note that this is also afunction of the mass of the halo.)In the halo model first, the angular power spectrum, or equiv-alently the 2-point correlation function, is composed of a 1-haloterm and a 2-halo term. This gives for the 1-halo term (cid:104) α ( n ) α ( n (cid:105) = (cid:34) d M i d x i (cid:32) d N d M (cid:33) A i ( n ) A i ( n ) (18) × (cid:104) b ( n , x i ) b ( n , x i ) (cid:105) , where we use (cid:68) n h ( x i ) (cid:69) = d N / d M . The 2-halo term then reads (cid:104) α ( n ) α ( n ) (cid:105) = (cid:34) d M i d x i (cid:34) d M j d x j (cid:68) n h ( x i ) n h ( x j ) (cid:69) × A i ( n ) A j ( n ) (cid:68) b ( n , x i ) b ( n , x j ) (cid:69) , (19) We remind that abundances are given by a Poisson staistics for which (cid:68) n h (cid:69) = (cid:104) n h (cid:105) . where in the above the halo j is necessarily di ff erent fromthe halo i . The 2-halo term is however vanishing becauseof averaging over the orientation of magnetic field. Sincetwo di ff erent halos have uncorrelated magnetic fields, one has (cid:68) b ( n , x i ) b ( n , x j ) (cid:69) = (cid:68) b ( n , x i ) (cid:69) (cid:68) b ( n , x j ) (cid:69) , which is finallyequal to zero since magnetic orientations have a vanishing en-semble average.Second, the 2-point correlation function is described by anangular power spectrum, i.e. (cid:104) α ( n ) α ( n ) (cid:105) = (cid:88) (cid:96) C α(cid:96) Y (cid:96) m ( n ) Y (cid:63)(cid:96) m ( n ) . (20)As detailed in App. A, this angular power spectrum, C α(cid:96) , is givenby the convolution of two angular power spectra reading C α(cid:96) = π (cid:88) L , L (cid:48) (2 L + (cid:0) L (cid:48) + (cid:1) (cid:32) L L (cid:48) (cid:96) (cid:33) D AL D (cid:107) L (cid:48) (21)where D AL is the angular power spectrum associated to the 2-point functions of the maximum of the rotation angle, i.e. (cid:34) d M i d x i (cid:32) d N d M (cid:33) A i ( n ) A i ( n ) , and D (cid:107) L (cid:48) is the angular power spectrum associated to the corre-lation function of orientations, (cid:104) b ( n , x i ) b ( n , x i ) (cid:105) . Finally, theterm (cid:32) L L (cid:48) (cid:96) (cid:33) corresponds to Wigner-3 j ’s. The expression in Eq. (21) meansthat the total angular power spectrum is obtained as the angularpower spectrum for the maximum amount of the e ff ect, D AL ,modulated by the impact of projecting the magnetic field on theline-of-sight, hence the convolution with D (cid:107) L (cid:48) .It is shown in App. B that the angular power spectrum D AL reads using Limber’s approximation D AL = (cid:90) z CMB d z (cid:32) r ν ( r ) (cid:33) d r d z (cid:90) d M d N d M (cid:2) α ( c ) α L (cid:3) , (22)with α c ( M , z ) the rotation angle at the core of the halo given by α c = e m e c ε √ π n ( c ) e ( M , z ) B c ( B , z ) . (23)This core angle depends on the mass, the redshift and the mag-netic field amplitude of the considered halos. The projectedFourier transform of the profile is α (cid:96) = (cid:114) π r (phys) c (cid:96) c (cid:90) ∞ d x x U ( x ) j (( (cid:96) + / x /(cid:96) c ) , (24)with (cid:96) c = D ang ( z ) / r c the characteristic multipole for a halo ofsize r c at a redshift z , and D ang ( z ) the angular diameter dis-tance. The normalized profile U ( x ) for a β -profile is U ( x ) = (1 + x ) − β (1 + µ ) / where x = r / r c . Note that in the above (cid:68) n h ( x i ) n h ( x j ) (cid:69) = (cid:32) d N d M i (cid:33) (cid:32) d N d M j (cid:33) (cid:104) + b ( M i , z i ) b ( M j , z j ) ξ m ( x i − x j (cid:17) , with b ( M , z ) the bias and ξ m the 2-point correlation function of the mat-ter density field.Article number, page 6 of 18emarchand et al.: Secondary CMB anisotropies from magnetized halos (I) Similarly in App. C, the angular power spectrum for the ori-entation of the magnetic field projected on the line-of-sight is D (cid:107) L (cid:48) = π δ L (cid:48) , . (25)Using the triangular conditions for the Wigner-3 j (see e.g Var-shalovich et al. 1988), the angular power spectrum of the Fara-day rotation angle boils down to C α(cid:96) = (cid:34)(cid:32) (cid:96) (cid:96) + (cid:33) D A (cid:96) − + (cid:32) (cid:96) + (cid:96) + (cid:33) D A (cid:96) + (cid:35) . (26)We note that the above does not assume Limber’s approximationin the sense that the involved D A (cid:96) ’s can be either the oneobtained from the Limber’s approximation, Eq. (22), or thenon-approximated one as given in Eq. (B.6).The impact of projecting the magnetic fields on the line-of-sight translates into the modulation of the angular power spectrafor the maximum amount of rotations halos can generate. In thelimit of high values of (cid:96) , the two line-of-sights, n and n , canbe considered as very closed one to each other. This leads to (cid:104) b ( n , x i ) b ( n , x i ) (cid:105) (cid:39) (cid:68) b ( n , x i ) (cid:69) = / restricted tothe 1-halo term however. In this high- (cid:96) limit, Eq. (26) simpli-fies to C α(cid:96) = D A (cid:96) /
3. From the expression of D A (cid:96) using Limber’sapproximation, one can check that this is identical to the 1-haloterm derived in Tashiro et al. (2008). For the Faraday conversion, one first reminds that irrespectivelyof the nature of free electrons (either from a thermal distributionor from a relativistic, non-thermal distribution) the conversionrate is proportional to B ⊥ e ± i θ B with B ⊥ the norm of the projectedmagnetic field on the plane orthogonal to n , and θ B is the anglebetween the projected magnetic field and the first basis vectorin the plane orthogonal to n . This defines the spin-( ±
2) struc-ture of these conversion coe ffi cients which can be convenientlyrewritten using projections of the magnetic field on the so-calledhelicity basis in the plane orthogonal to n , i.e. B ⊥ e ± i θ B = B ( | x − x i | ) (cid:104) b ( x i ) · (cid:16) e θ ± i e ϕ (cid:17)(cid:105) , (27)where we remind in the above that the norm of the magneticfield is a radial function and its orientation depends on the haloslocation only. The radiative transfer coe ffi cients integrated over the line-of-sight is defined as φ ± ( n ) = (cid:82) a ( r )d r (cid:80) halos ˙ φ P → Vi ( n , r ) e ± i θ ( i ) B ( n , r ) .For thermal electrons, this explicitly reads φ ± ( n ) = e π m e c ε (cid:90) r CMB a ( r ) ν ( r ) d r (cid:34) d M i d x i (cid:104) n h ( x i ) (28) × b ± ( n , x i ) X ( | x − x i | ) (cid:105) , where now X ( | x − x i | ) = n e ( | x − x i | ) B ( | x − x i | ), and b ± ( n , x i ) = (cid:104) b ( x i ) · (cid:16) e θ ± i e ϕ (cid:17)(cid:105) .Apart from the spin-( ±
2) structure encoded in b ± , the abovehas exactly the same structure as the Faraday rotation angle, Eq. (17), and we adopt the same strategy as for the Faraday rotationangle. The key di ff erence for Faraday conversion lies in the spinstructure and one has to compute three correlations (2 autocorre-lations and 1 cross-correlation). One can either use spin fields ormore conveniently, E and B decompositions which is referenceframe independent (see e.g. Kamionkowski et al. 1997; Zaldar-riaga & Seljak 1997). Here we will first compute the correlationfor spin fields, defined as (cid:68) φ ± ,(cid:96) m φ (cid:63) ± ,(cid:96) (cid:48) m (cid:48) (cid:69) = C ± , ± (cid:96) δ (cid:96),(cid:96) (cid:48) δ m , m (cid:48) , (29) (cid:68) φ ,(cid:96) m φ (cid:63) − ,(cid:96) (cid:48) m (cid:48) (cid:69) = C , − (cid:96) δ (cid:96),(cid:96) (cid:48) δ m , m (cid:48) . (30)These angular power spectra are easily transformed into angularpower spectra for the E and B field associated to φ ± using φ E (cid:96) m = − ( φ ,(cid:96) m + φ − ,(cid:96) m ) / φ B (cid:96) m = i ( φ ,(cid:96) m − φ − ,(cid:96) m ) / (cid:68) b ± ( n , x i ) b ± ( n , x j (cid:44) i ) (cid:69) = (cid:28) b ± ( n , x i ) (cid:29) (cid:68) b ± ( n , x j (cid:44) i ) (cid:69) for two di ff erent halos, and for uniformly random orientationsone found that (cid:104) b ± ( n , x i ) (cid:105) = C ± , ± (cid:96) = π (cid:88) L , L (cid:48) (2 L + L (cid:48) + (cid:32) L (cid:48) L (cid:96) ∓ ± (cid:33) D Φ L D ⊥ L (cid:48) , (31)and C , − (cid:96) = π (cid:88) L , L (cid:48) (2 L + L (cid:48) + D Φ L D ⊥ L (cid:48) , × (cid:32) L (cid:48) L (cid:96) − (cid:33) (cid:32) L (cid:48) L (cid:96) − (cid:33) . (32)The above is interpreted in a very similar way to C α(cid:96) . It is thepower spectrum of the maximum of the e ff ect of Faraday conver-sion, D Φ L , which is further modulated by the impact of projectingthe magnetic field in the plane orthogonal to the line-of-sight,which is encoded in D ⊥ L (cid:48) .The angular power spectrum of the amplitude of the e ff ect isderived using the standard technique reminded in App. B and byselecting the appropriate profile, n E B instead of n E B . This giveswith the Limber’s approximation D Φ L = (cid:90) z CMB d z (cid:32) r ν ( r ) (cid:33) d r d z (cid:90) d M d N d M (cid:2) Φ ( c ) φ L (cid:3) , (33)with the amplitude of the conversion at the core of the halo givenby Φ ( c ) = (cid:32) e π ) / m e c ε (cid:33) n ( c ) e B c . (34)The Fourier-transformed normalized profile reads φ (cid:96) = (cid:114) π r (phys) c (cid:96) c (cid:90) ∞ d x x U ( x ) j (( (cid:96) + / x /(cid:96) c ) , (35)where the profile is now given by U ( x ) = (1 + x ) − β (1 + µ ) / . Theangular power spectrum for the orientation contribution is de-tailed in App. D. It is nonzero for a multipole of 2 only and itreads D ⊥ L (cid:48) = (32 π/ δ L (cid:48) , . Article number, page 7 of 18 & A proofs: manuscript no. main
The last step consists in deriving the angular power spec-trum in the E and B decomposition of the spin-( ±
2) of the Fara-day conversion coe ffi cients. This first shows that the (cid:104) EB (cid:105) cross-spectrum is vanishing, i.e. C φ E φ B (cid:96) =
0. The autospectra are givenby C φ E φ E (cid:96) = (cid:34) ( (cid:96) + (cid:96) + (cid:96) − (cid:96) + D Φ (cid:96) − + (cid:96) − (cid:96) + (cid:96) − (cid:96) + D Φ (cid:96) + (cid:96) ( (cid:96) − (cid:96) + (cid:96) + D Φ (cid:96) + (cid:35) , (36)and C φ B φ B (cid:96) = (cid:34)(cid:32) (cid:96) + (cid:96) + (cid:33) D Φ (cid:96) − + (cid:32) (cid:96) − (cid:96) + (cid:33) D Φ (cid:96) + (cid:35) . (37)In the above, we made use of the triangular conditions for theWigner-3 j ’s. (We note that the above angular power spectra arespin-( ±
2) and they are nonvanishing for (cid:96) ≥ (cid:96) limit, the two autospectra are identical and equal to C φ E φ E (cid:96) (cid:39) C φ B φ B (cid:96) (cid:39) (4 / D Φ (cid:96) . For relativistic electrons, the rate of Faraday conversion inte-grated over the line-of-sight reads φ ± ( n ) = e Γ min π m e c ε (cid:32) β E − β E − (cid:33) (cid:90) r CMB a ( r ) ν ( r ) d r (38) (cid:34) d M i d x i (cid:104) n h ( x i ) b ± ( n , x i ) X ( | x − x i | ) (cid:105) , where Γ min is the minimum Lorentz factor of the relativistic elec-trons, and β E is the spectral index of the energy distribution ofrelativistic electrons. The profile is X = n ( r ) e B , i.e. the same asfor thermal electrons replacing the number density of thermalelectrons by the number density of relativistic ones.The angular power spectrum for the Faraday conversion ratedue to relativistic electrons has exactly the same form as for ther-mal electrons, i.e. Eqs. (36) & (37) for the E and B autospectra.The expression for D Φ (cid:96) also reads the same. It is given by Eq.(33) where one just has to replace Φ ( c ) by Φ ( r ) = (cid:32) e Γ min π ) / m e c ε (cid:33) (cid:32) β E − β E − (cid:33) n ( r ) e B c . (39) Let us briefly comment on possible cross-correlation. The firstpoint is that in this approach, the cross-correlation between theFaraday rotation angle with any tracer of halos which is not cor-related with the projection of magnetic fields on the line-of-sightwill be vanishing. This is because the cross-correlation is pro-portional to either (cid:104) b · n (cid:105) or (cid:28)(cid:104) b · (cid:16) e θ ± i e ϕ (cid:17)(cid:105) (cid:29) , both of whichaverage down to zero. This will be indeed the case for cross-correlation with the thermal and relativistic Sunyaev-Zel’doviche ff ect, the lensing potential, or the CIB fluctuations. This willalso be the case for cross-correlation with the absorption coe ffi -cients, µ .Finally, we checked that the averages (cid:28) [ b · n ] (cid:104) b · (cid:16) e (2) θ ± i e (2) ϕ (cid:17)(cid:105) (cid:29) equals to zero. This yield avanishing cross-correlation between the Faraday rotation angleand Faraday conversion.
4. Numerical results
Our results are shown for a frequency of observation ν = B = µ G. We remindthat the angular power spectrum for the Faraday rotation anglescales as C α(cid:96) ∝ B /ν . The angular autospectra of the E and B modes of Faraday conversion scales as C φ E φ E ( φ B φ B ) (cid:96) ∝ B /ν .Unless specified, the parameters for the β -profile are β = µ = /
3, which would correspond to a magnetic field frozen into mat-ter. All the numerical results reported here are obtained usingthe universal mass function from Despali et al. (2016). For con-sistency, we checked that similar results are obtained using themass function of Tinker et al. (2008). In particular, we foundsimilar scaling with cosmological parameters, despite small vari-ation regarding the overall amplitude of the angular power spec-tra.
Fig. 1 shows the mass and redshift distributions of the Fara-day rotation angle power spectrum for di ff erent multipoles (cid:96) ,with on the left dln C α(cid:96) / dln M as a function of mass, and on theright dln C α(cid:96) / dln z as a function of redshift. Compared to Tashiroet al. (2008) (Fig. (4) and (3) respectively), we note that our dis-tributions are slightly shifted to higher masses and lower red-shifts. This results in the Faraday rotation e ff ect being more sen-sitive to higher mass values and lower redshift galaxy halos thantheir Faraday rotation angle, so that its power spectrum seems tobe slightly shifted to lower (cid:96) values as compared to the one inTashiro et al. (2008): indeed, low multipoles correspond to highangular scales, hence to high masses or low redshifts halos be-cause these halos appear bigger on the sky than low masses andhigh redshifts halos.Fig. 2 shows the angular power spectrum of the Faradayrotation angle for di ff erent values of the parameters β and µ of the spatial distribution profiles of the free electrons densityand magnetic field, respectively. First, one can note a shift ofpower to higher multipoles when increasing β or µ : indeedthe profile of free electrons and magnetic fields then becomessteeper so that the they are more concentrated in the centre ofthe halo which consequently appears smaller on the sky. Thisresult is consistent with Tashiro et al. (2008). We also see thatthe di ff erence in amplitudes is more significant when we change β rather than µ , because β appears both in the free electrons andmagnetic field profiles. However, the trend is di ff erent whenchanging β or µ . Indeed, when increasing µ , the amplitude de-creases, as expected from the magnetic field profile Eq. (10). Onthe contrary, when increasing β , the amplitude also increases:this is because as the profile of free electrons is steeper, keepingthe number of electrons constant, their concentration increasesEq. (10), so does the amplitude.Fig. 3 shows two di ff erent representations of the dependenceof the angular power spectrum of the Faraday rotation angleon the amplitude of density fluctuation σ : on the left is plot-ted the angular power spectrum for di ff erent values of σ andon the right we plot the logarithmic derivative of the angularpower spectrum with respect to σ as a function of (cid:96) . The lattergives the scaling of C α(cid:96) with σ , i.e. by writing C α(cid:96) ∝ σ n ( (cid:96) )8 then n ( (cid:96) ) = d ln( C α(cid:96) ) / d ln( σ ).The angular power spectrum C α(cid:96) is composed of the 1-haloterm only. Hence its scaling with σ is driven by the mass func- Article number, page 8 of 18emarchand et al.: Secondary CMB anisotropies from magnetized halos (I)
Fig. 1.
Left: the mass distribution of the Faraday rotation e ff ect for various (cid:96) modes. Right: the redshift distribution of the Faraday rotation e ff ectfor various (cid:96) modes. Fig. 2.
The angular power spectra of the Faraday rotation angle, C α(cid:96) , fordi ff erent values of the parameter β of the β -profile and di ff erent valuesof the parameter µ of the magnetic field profile. tion, d N / d M , and the rotation angle at the core of halos, α c . Thelatter does not explicitly depends on σ . However, the scaling ofd N / d M with σ is mass-dependent. Then the mass dependenceof α c probes di ff erent mass ranges of the mass function, and as aconsequence, di ff erent scaling of d N / d M with the amplitude ofmatter perturbations.We find a dependence as C α(cid:96) ∝ σ . − σ . for (cid:96) =
10 and (cid:96) = , respectively. The power spectrum of the Faraday rota-tion angle is more sensitive to σ for low (cid:96) values than for high (cid:96) values because as seen above, the angular power spectrum is sen-sitive to higher mass at low (cid:96) and in this mass regime the massfunction is more sensitive to σ . We noticed that reducing themass integration range from M = M (cid:12) to M = × M (cid:12) (where it was [10 M (cid:12) , × M (cid:12) ] before) slightly increases thepower in σ . This may be due both to the facts that the Faradayrotation e ff ect is mainly sensitive to galaxy halos with masses inthe range M = to M = M (cid:12) , see Fig. 1, and that ourmass function depends on σ more strongly from M = M (cid:12) . We note that the scaling in σ of the angular power spec-trum is di ff erent than the one for the tSZ angular power spectrumwhich scales with σ . (see for example Hurier & Lacasa 2017).The reason is a di ff erent scaling in mass of the rotation angleat the core of halos as compared to the tSZ flux (note that thetSZ angular power spectrum is dominated by the 1-halo contri-bution). Indeed, | α c | scales as M whereas the square of the tSZflux at the core scales as M . . This results in a di ff erent weight-ing of the mass function which is more sensitive to σ for highmass values, the tSZ e ff ect giving more weight to high massesthan the Faraday rotation angle.The dependence with σ found here is however di ff erentfrom the one reported in Tashiro et al. (2008), the di ff erencebeing mainly due to the presence of a 2-halo term in Tashiroet al. (2008). The mass range (cid:104) M = M (cid:12) , × M (cid:12) (cid:105) is firstconsidered in Tashiro et al. (2008) for which the angular powerspectrum is dominated by its 1-halo contribution. In this case,the obtained scaling is σ . The di ff erence with the scaling foundhere lies in the reduced mass range which gives more weight onthe total e ff ect to higher mass halos. Second the mass range is ex-tended in Tashiro et al. (2008) down to 10 M (cid:12) , leading then to ascaling as σ . . In the mass range [10 M (cid:12) , M (cid:12) ], the 2-haloterm present in Tashiro et al. (2008) is not negligible anymore.This 2-halo term then gives much more contribution to low masshalos as compared to ours (see Fig. 7 of Tashiro et al. 2008).However, the scaling of the 2-halo term with σ is not drivenanymore by ∼ (cid:90) d M d N d M α c , but instead by ∼ (cid:32)(cid:90) d M d N d M b ( M , z ) α c (cid:33) P m ( (cid:96)/ r , z ) , with P m ( k , z ) being the matter power spectrum (proportional to σ ). The steeper scaling with σ found in Tashiro et al. (2008) isthus mainly due to the non-negligible contribution of the 2-haloterm in their work. We remind that the angular power spectra derived in Tashiro et al.(2008) has a non zero 2-halo contribution.Article number, page 9 of 18 & A proofs: manuscript no. main
Fig. 3.
Left: the angular power spectra of the Faraday rotation angle, C α(cid:96) , for di ff erent values of the density fluctuations amplitude σ . Right:d ln C α(cid:96) / d ln σ as a function of (cid:96) We now want to study whether the Faraday rotation angle issensitive to the matter density parameters. Keeping other cosmo-logical parameters fixed, we have two possibilities to vary Ω m :either by varying the density of cold dark matter, Ω CDM , or thatof baryons, Ω b .We found that the Faraday rotation e ff ect is almost indepen-dent of Ω m , when Ω b is kept fixed while varying Ω CDM : C α(cid:96) ∝ Ω − . CDM − Ω − . CDM for (cid:96) =
10 and (cid:96) = respectively. This translates into a similarscaling with Ω m for a varying density of dark matter C α(cid:96) ∝ Ω − . m − Ω − . m for (cid:96) =
10 and (cid:96) = respectively.However, when keeping Ω CDM fixed and varying Ω b , the de-pendence is clearly di ff erent: C α(cid:96) ∝ Ω . b − Ω . b for (cid:96) =
10 and (cid:96) = respectively. The resulting scaling with Ω m by varying the density of baryons is then C α(cid:96) ∝ Ω m − Ω m for (cid:96) =
10 and (cid:96) = respectively. The dependence with Ω b and Ω CDM is simply understood by the fact the angular powerspectrum scales with the fraction of baryons to the square.The e ff ect is almost Ω m -independent when varying Ω CDM , ascompared to the thermal Sunyaev-Zel’dovich e ff ect which scalesas ∼ Ω m (Komatsu & Kitayama 1999). One can thus hope to usethe Faraday rotation as a cosmological probe, by combining itwith another physical e ff ect having a di ff erent degeneracy in the Ω m − σ plane, such as the thermal Sunyaev-Zel’dovich e ff ect.Finally, we also want to study the e ff ect of having a massdependence of the (central) magnetic field strength. Indeed, thestructure of magnetic fields in galaxy halos is poorly known andthis fact could lead to degeneracies in the dependence with cos-mological parameters and astrophysical ones. We chose the cen-tral magnetic field strength to scale as B = B p ( M / M p ) γ , with M p = × M (cid:12) , B P = µ G, and γ varying from 0 to 1. The left part of Fig. 4 shows C α(cid:96) for five di ff erent values of γ .When increasing γ , the power spectrum is increased and the peakis shifted to lower (cid:96) values. Increasing the value of γ indeed leadsto a higher contribution of massive halos, which appears largeronce projected on the sky, hence a peak at smaller multipoles.This shift to lower (cid:96) values and the di ff erence in amplitude ofthe Faraday rotation angle could give insight on the scaling ofthe magnetic field strength with mass.Fig. 4 right shows how this mass dependence a ff ects thedependence on σ of the Faraday rotation e ff ect by plottingd ln C α / d ln σ with respect to γ . For (cid:96) = γ = C α(cid:96) ∝ σ . and we recover C α(cid:96) ∝ σ . for γ =
0. In be-tween, one has C α(cid:96) ∝ σ . − σ . − σ . for γ = . , . , . ff erent scal-ing with σ of the angular power spectrum as compared to thethermal SZ e ff ect came from a di ff erent scaling in mass. In-deed, the angular power spectrum of our e ff ect scales as M ,where it scales as M . for the tSZ e ff ect, hence we recover thesame scaling in σ for γ = .
75. From this we also see thatif we could model the magnetic field with a power-law massdependence, the more it would depend on mass, the more thee ff ect would be sensitive to σ , allowing for a better determi-nation of this cosmological parameter. Hence there is a corre-lation between the uncertainty on σ and the mass dependenceof the magnetic field strength. The Faraday rotation angle stillalmost does not depend on Ω m (when varying Ω CDM only). In-deed, one has: C α(cid:96) ∝ Ω − . m − Ω − . m − Ω − . m − Ω − . m − Ω − . m for γ = , . , . , .
75 and 1 respectively.
First, we found no significative di ff erence between the twoangular power spectra of the Faraday conversion rate, C φ E φ E (cid:96) and C φ B φ B (cid:96) . Hence, for simplicity we will now show results for the C φ E φ E (cid:96) power spectrum only.On Fig. 5, we compare the angular power spectra of the Fara-day conversion rate for two populations of free electrons, eitherthermal or relativistic. In the case of the relativistic electrons, the Article number, page 10 of 18emarchand et al.: Secondary CMB anisotropies from magnetized halos (I)
Fig. 4.
Left: The angular power spectra of the Faraday rotation angle, C α(cid:96) , when adding a mass dependence for the magnetic field strength at thecentre scaling as B = B p ( M / M p ) γ , with M p = × M (cid:12) , B p = µ G, and for di ff erent values of γ . Right: d ln C α / d ln σ as a function of γ . Wechose to plot this e ff ect for (cid:96) =
100 as C α(cid:96) depends more strongly on σ for low (cid:96) values. Fig. 5.
The angular power spectra of the Faraday conversion rate C φ E φ E (cid:96) for thermal electrons (dotted-blue) and for relativistic electrons (solid-orange). For relativistic electrons, one set n ( r ) e =
10 m − and Γ min = central density in the halo is taken to be constant, contrary to thecold case which follows Eq. (10). We took the value n ( c ) e = − which is the highest value we could find in the literature(Colafrancesco et al. 2003), noting that the properties of rel-ativistic electrons in halos are not well known. We also tookthe spectral index of the energy distribution of these relativis-tic electrons to be β E = . Γ min = C φ E φ E (cid:96) is two orders of magnitude higher in the ther-mal electron case compared to the relativistic case, despite push-ing the relativistic electron density to the maximum value al-lowed by observations.We note however that some studies suggest that the mini-mum Lorentz factor could be as high as 10 . Since the angu-lar power spectrum scales in amplitude as Γ , C φ E φ E (cid:96) would betwo orders of magnitude higher than the one displayed in Fig. 5,hence reaching a similar amplitude as the contribution of thermal electrons to Faraday conversion. One thus expect the contribu-tion of relativistic electrons to be at most at the same amplitudeas the one from thermal electrons. The dependence of the angular power spectra of the Faraday con-version rate from thermal electrons on the density fluctuationamplitude σ is similar to the one of the Faraday rotation angle: C φ E φ E (cid:96) ∝ σ . − σ . for (cid:96) =
10 and (cid:96) = respectively, the dif-ference between low (cid:96) and high (cid:96) values having already been ex-plained. The small di ff erences with Faraday rotation come fromthe fact that the Faraday conversion rate angular power spectrascale as 1 / (1 + z ) , where it scales as 1 / (1 + z ) for the Faradayrotation angle.As for the Faraday rotation e ff ect, there is almost no variationof the Faraday conversion rate with Ω m , when varying Ω CDM and Ω b kept fixed, the dependence going as C φ E φ E (cid:96) ∝ Ω − . m − Ω − . m or C φ E φ E (cid:96) ∝ Ω − . CDM − Ω . CDM for (cid:96) =
10 and (cid:96) = , respectively.When varying Ω m via Ω b instead ( Ω CDM kept fixed), the de-pendence is not very di ff erent from Faraday rotation either: C φ E φ E (cid:96) ∝ Ω m − Ω m or C φ E φ E (cid:96) ∝ Ω . b − Ω . b for (cid:96) =
10 and (cid:96) = , respectively.Although these scalings are not very di ff erent from Faradayrotation, the same remark on the σ scaling di ff erences applieshere, which is that the two e ff ects scale di ff erently with redshift.We also investigate the degeneracy between a scaling in massof the magnetic field at the centre of the halo and the σ scaling,as what we have done for the Faraday rotation angle, see Fig. 6.The dependence in σ is C φ E φ E (cid:96) ∝ σ when γ = C φ E φ E (cid:96) ∝ σ when γ = . C φ E φ E (cid:96) ∝ σ . when γ = . C φ E φ E (cid:96) ∝ σ . when γ = .
25. When γ = . σ to the power Article number, page 11 of 18 & A proofs: manuscript no. main
Fig. 6.
Left: the angular power spectra of the Faraday conversion rate, C φ E / B φ E / B (cid:96) when adding a mass dependence for the magnetic field strengthat the centre scaling as ∼ ( M / M p ) γ , with M p = × M (cid:12) for di ff erent values of γ . Right: d ln C α / d ln σ as a function of γ . We chose to plotthis e ff ect for (cid:96) =
100 as C φ E / B φ E / B (cid:96) depends more strongly on σ for low (cid:96) values. γ =
1, so that our analysis is consistent.
Let us mention the case of Faraday conversion with relativisticelectrons because the dependence of the angular power spectrawith cosmological parameters is a bit di ff erent.Indeed, when varying Ω CDM and Ω b kept fixed, the depen-dence goes like: C φ E φ E (cid:96) ∝ Ω . m − Ω . m or C φ E φ E (cid:96) ∝ Ω . CDM − Ω . CDM for (cid:96) =
10 and (cid:96) = , respectively.When varying Ω b and Ω CDM kept fixed, the dependence goeslike: C φ E φ E (cid:96) ∝ Ω . m − Ω . m or C φ E φ E (cid:96) ∝ Ω . b − Ω . b for (cid:96) =
10 and (cid:96) = , respectively.This di ff erence in dependence compared to the thermalelectron case, is explained by the constant value for the densityof relativistic free electrons at the centre of the halo, whereasthe density of cold free electrons at the centre scales with thefraction of baryons f b as well as the critical density ρ c and thespherical overdensity ∆ of the virialized halos.
5. Conclusion
We revisited the derivation of the angular power spectrum ofthe Faraday rotation angle using the halo model and extendedit to the case of Faraday conversion, with an emphasis on theassumptions made for the statistics and orientations of magneticfields inside halos. Indeed, we first assumed the magnetic fieldof a halo to have a spherically symmetric profile but the sameorientation over the halo scale. Second, the orientations aresupposed to be uniformly distributed in the Universe, to beconsistent with the cosmological principle. Third, the orienta-tions of magnetic fields in di ff erent halos are here independentfrom each other, with the underlying idea that magnetism is produced within halo in a local physical process. We alsomade the hypothesis that the distribution of the orientationsof the magnetic fields inside halos are independent from theabundance in mass and the spatial distribution of halos. All ofthese hypotheses simplified the derivation of the angular powerspectra: in particular, only the 1-halo term remains becauseof the independence of the orientations from one halo to another.We then explored the dependence of the angular power spec-tra with astrophysical and cosmological parameters. In tab. 2, wereport the scaling of the angular power spectra of the Faraday ro-tation angle and the Faraday conversion rate with the parameters σ and Ω m (assuming here that Ω m would vary by a change of thecold dark matter density). We also reported in this table the scal-ing for three other probes of the large-scale structures, namelyhalo number counts as observed through the thermal Sunyaev-Zel’dovich e ff ect (tSZ), the tSZ angular power spectrum, and theangular power spectrum of CMB lensing potential.In particular, the angular power spectra of both Faradayrotation and Faraday conversion scale with the amplitude ofthe density fluctuations as σ while it scales with σ to σ forthe other probes. However, this scaling with σ is degeneratedwith a mass dependent magnetic field. Still di ff erent from thetSZ and lensing probes is the scaling with the matter densityparameter Ω m : while for the SZ one has a scaling with Ω m ,here it is almost independent of this parameter. Thus the twoe ff ects could be combined to lift the degeneracy in the σ − Ω m ,assuming nonetheless the magnetic field mass dependencemodel to be known. Conversely, a joint analysis could be usedso as to infer the scaling of the magnetic fields with the massesof halos.Although other physical e ff ects happen in these magnetizedplasmas being halos of galaxies, the dominant contributionsare from Faraday rotation and Faraday conversion with thermalelectrons as stated in Sect. 2. Indeed, an estimation of the angu-lar power spectra of secondary anisotropies suggests that at 1GHz, with a magnetic field of 10 µ G and a density of relativistic The impact of the scaling of B with the mass can be viewed as theequivalent of the mass bias in the analysis of the tSZ number counts andangular power spectrum.Article number, page 12 of 18emarchand et al.: Secondary CMB anisotropies from magnetized halos (I) σ Ω m Faraday rotation power spectrum at (cid:96) (cid:39) (cid:96) (cid:39) (thermal electrons) 2.1 -0.1Faraday conversion power spectrum at (cid:96) (cid:39) (relativistic electrons) 2.1 1Halo number counts from thermal SZ 9 3thermal SZ power spectrum at (cid:96) (cid:39) (cid:96) (cid:39)
30 2 0.5
Table 2.
Scaling of di ff erent large-scale-structure probes with σ and Ω m . The scaling reported here is to be understood as P ∝ σ n Ω pm with P anyof the considered probe. They are given at the peaking multipole of (cid:96) ( (cid:96) + C (cid:96) for the Faraday rotation angle, the Faraday conversion rate, and thetSZ flux, and the peaking multipole of (cid:96) ( (cid:96) + C φφ(cid:96) for the lensing potential. electrons of n rel =
10 m − for the absorption coe ffi cients, thereare eighteen orders of magnitude between the secondary linearlypolarized anisotropies produced thanks to Faraday rotation andthose produced by absorption of intensity. Similarly, there arenine orders of magnitude between the secondary circularlypolarized anisotropies produced thanks to Faraday conversionand those produced by absorption of intensity. These di ff erencesin orders of magnitude do not change significantly whenchanging the frequency to 30 GHz and the magnetic field to 3 µ G. Thus, we can safely conclude the secondary anisotropiesinduced by absorption would be negligible compared to theFaraday rotation and Faraday conversion induced anisotropies.
Note added:
During the completion of this work, another studyof Faraday conversion in the cosmological context has been pro-posed in Ejlli (2018). It however focuses on magnetic fields av-eraged over very large scales.
Acknowledgements.
The authors would like to thank G. Fabbian, S. Ilic, andM. Douspis for helpful discussions. Part of the research described in this paperwas carried out at the Jet Propulsion Laboratory, California Institute of Technol-ogy, under a contract with the National Aeronautics and Space Administration(NASA).
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Appendix A: Formal derivation of the angular power spectra for any effect considered
Appendix A.1: Radiative transfer coefficients
Looking at the expressions of the di ff erent radiative transfer coe ffi cients show that they can assume two possible forms. They arefirst of scalar type as the Faraday rotation angle, and they read as an integral over the line-of-sight and over halos distribution asfollows α ( n ) = (cid:90) r CMB a ( r )d r (cid:34) d M i d x i n h ( x i ) [ b ( x i ) · n ] A ( M i , | x − x i | ) . (A.1)In the above, the dependance over orientations is encoded in [ b ( x i ) · n ] which is a scalar function. The function A is the profile ofthe e ff ect which reads for Faraday rotation by thermal electrons A ( M i , | x − x i | ) = (cid:32) e π m e c ε ν ( r ) (cid:33) n e ( | x − x i | ) B ( | x − x i | ) . (A.2)Since projection e ff ects can only reduce the impact of the e ff ect, the quantity A can be interpreted as the maximum of the e ff ect agiven halo can generate. Finally, n h is the halo distribution. The above scalar-type of coe ffi cients are the Faraday rotation angle andthe conversion from intensity to circular polarization, φ I → V .The second type of coe ffi cients are the ones which are proportional to B ⊥ e ± i γ , with B ⊥ the amplitude of the projected magneticfield on the plane orthogonal to the line-of-sight, and γ the angle between the projected magnetic field with the basis vector e θ inthat plane. This is typical of the coe ffi cients for Faraday conversion, φ P → V , or conversion from intensity to linear polarization, φ I → P .In terms of the amplitude of the magnetic field, B , and its orientation b , the phase reads B ⊥ e ± i γ = B (cid:104) b · (cid:16) e θ ± i e ϕ (cid:17)(cid:105) . One thus havespin-( ±
2) coe ffi cients reading as an integral over the line-of-sight as follows i φ P → V ( n ) e ± i γ ( n ) = (cid:90) r CMB a ( r )d r (cid:34) d M i d x i n h ( x i ) (cid:104) b · (cid:16) e θ ± i e ϕ (cid:17)(cid:105) P ( M i , | x − x i | ) . (A.3)The spin structure of the above is entirely encoded in projection coe ffi cients (cid:104) b · (cid:16) e θ ± i e ϕ (cid:17)(cid:105) . The other terms are identical to theones in scalar coe ffi cients, except that the profile of the e ff ect, P ( r , | x − x i | ), admits a di ff erent explicit expression, e.g. for Faradayconversion by thermal electrons P ( M i , | x − x i | ) = (cid:32) e π m e c ν ( r ) (cid:33) n e ( | x − x i | ) B ( | x − x i | ) . (A.4)A part from the explicit expression of P , the formal expression of the coe ffi cient remains the same for relativistic electrons or forconversion from intensity to linear polarization.A formal expression for all the above radiative transfer coe ffi cients can be abstracted from the above. On denoting φ s any suchcoe ffi cients, with s = s = ± φ s ( n ) = (cid:90) r CMB a ( r )d r (cid:34) d M i d x i n h ( x i ) f s ( b i , n ) Φ ( M i , | x − x i | ) , (A.5)with Φ ( r , | x − x i | ) the profile amounting the maximum amount of the e ff ect, which is a scalar function, and f s ( b i , n ) the functionencoding the impact of projecting the magnetic field (the subscript i is to remind that the orientation is a priori a function of thehalos positions, x i ). This is this last function which contains the spin structure of the considered coe ffi cients. Appendix A.2: Angular power spectrum
To compute the angular power spectrum, the usual approach consists in first computing the multipolar coe ffi cients of φ s ( n ) thanksto φ s ,(cid:96), m = (cid:82) d n φ s ( n ) s Y (cid:63)(cid:96) m ( n ), and then to consider the 2-point correlation between these multipolar coe ffi cients, (cid:68) φ (1) s ,(cid:96) m φ (2) (cid:63) s (cid:48) ,(cid:96) (cid:48) m (cid:48) (cid:69) (superscripts 1 , ff erent radiative transfer coe ffi cients). The fields φ s being statistically homogeneous andisotropic, the 2-point correlation of multipolar coe ffi cients is entirely described by an angular power spectrum, i.e. (cid:68) φ (1) s ,(cid:96) m φ (2) (cid:63) s (cid:48) ,(cid:96) (cid:48) m (cid:48) (cid:69) = C (1 , (cid:96) δ (cid:96),(cid:96) (cid:48) δ m , m (cid:48) .Here we adopt a slightly di ff erent path (totally equivalent though) by first considering the 2-point correlation function on thesphere, denoted ξ , ( n , n ) = (cid:68) φ (1) s ( n ) φ (2) s (cid:48) ( n ) (cid:69) , which is further simplified thanks to our assumption about the statistics of themagnetic fields orientations. The 2-point correlation of the multipolar coe ffi cients is secondly derived from the 2-point correlationfunction via (cid:68) φ s ,(cid:96) m φ (cid:63) s (cid:48) ,(cid:96) (cid:48) m (cid:48) (cid:69) = (cid:82) d n (cid:82) d n ξ , ( n , n ) s Y (cid:63)(cid:96) m ( n ) s (cid:48) Y (cid:96) m ( n ).Assuming that orientations of the magnetic fields is not correlated to the spatial distribution of halos leads to ξ , ( n , n ) = (cid:90) r CMB [ a ( r )d r ] [ a ( r )d r ] (cid:34) (cid:104) d M i d x i (cid:105) (cid:104) d M j d x j (cid:105) Φ (1) ( M i , | x − x i | ) Φ (2) (cid:16) M j , (cid:12)(cid:12)(cid:12) x − x j (cid:12)(cid:12)(cid:12)(cid:17) × (cid:68) n h ( x i ) n h ( x j ) (cid:69) (cid:68) f s ( b i , n ) f s (cid:48) ( b j , n ) (cid:69) . (A.6) Article number, page 14 of 18emarchand et al.: Secondary CMB anisotropies from magnetized halos (I)
Since the orientation of the magnetic fields of two di ff erent halos is uncorrelated, this gives (cid:68) f s ( b i , n ) f s (cid:48) ( b j , n ) (cid:69) ∝ δ i , j , and onlythe 1-halo term contributes to the 2-point cross-correlation function. In addition, these orientations are statistically homogeneousand isotropic, meaning that (cid:68) f s ( b i , n ) f s (cid:48) ( b j , n ) (cid:69) is a function of (cid:12)(cid:12)(cid:12) x i − x j (cid:12)(cid:12)(cid:12) only. Because there is only the 1-halo term, this relativedistance is zero and (cid:68) f s ( b i , n ) f s (cid:48) ( b j , n ) (cid:69) is a function of n and n , i.e. (cid:68) f s ( b i , n ) f s (cid:48) ( b j , n ) (cid:69) = ξ Os , s (cid:48) ( n , n ) δ i , j . Hence the 2-pointcorrelation function boils down to ξ , ( n , n ) = ξ Os , s (cid:48) ( n , n ) × (cid:90) r CMB [ a ( r )d r ] [ a ( r )d r ] (cid:34) d M i d x i d N d M Φ (1) ( M i , | x − x i | ) Φ (2) ( M i , | x − x i | ) , (A.7)where the mass function arises from the 1-halo average of the abundance (cid:68) n h ( x i ) (cid:69) = d N / d M . The function ξ Os , s (cid:48) ( n , n ) is interpretedas the correlation function of orientations, while the remaining term is the 1-halo contribution of the 2-point correlation function ofthe amplitude of the radiative transfer coe ffi cient. Let us denote this second correlation function ξ Φ , .The full correlation function is thus a product of two correlation functions, one for the orientation and one for the amplitude ofthe coe ffi cient, i.e. ξ , ( n , n ) = ξ Os , s (cid:48) ( n , n ) × ξ Φ , ( n , n ) . (A.8)The correlation function of the amplitude, ξ Φ , , is formally identical to the 1-halo term of the correlation function of e.g. the thermalSunyaev-Zel’dovich e ff ect, which is well-known to be described by an angular power spectrum, i.e. ξ Φ , ( n , n ) = (cid:88) L , M D Φ L Y LM ( n ) Y (cid:63) LM ( n ) , (A.9)with D Φ L the angular power spectrum. Similarly, the correlation function of orientations is described by an angular power spectrum, D OL (cid:48) , since this is a statisically homogeneous and isotropic field, i.e. ξ Os , s (cid:48) ( n , n ) = (cid:88) L (cid:48) M (cid:48) D OL (cid:48) s Y L (cid:48) M (cid:48) ( n ) s (cid:48) Y (cid:63) L (cid:48) M (cid:48) ( n ) . (A.10)We note that in the above, spin-weighted spherical harmonics are used to take into account the nonzero spins of the projectedorientations.Plugging Eqs. (A.9) & (A.10) into Eq. (A.8), and then taking the spherical harmonic transforms of ξ , , one shows that (cid:68) φ (1) s ,(cid:96) m φ (2) (cid:63) s (cid:48) ,(cid:96) (cid:48) m (cid:48) (cid:69) can be expressed as a function of Gaunt integrals, the latter being defined as G (cid:96) m s (cid:96) m s ; (cid:96) m s = (cid:90) d ˆ n s Y (cid:96) m ( ˆ n ) × s Y (cid:96) m ( ˆ n ) × s Y (cid:96) m ( ˆ n ) . (A.11)Gaunt integrals can be casted as products of Wigner-3 j symbols. By then using triangular conditions and symmetries of the Wignersymbols (Varshalovich et al. 1988), one finds (cid:68) φ (1) s ,(cid:96) m φ (2) (cid:63) s (cid:48) ,(cid:96) (cid:48) m (cid:48) (cid:69) = √ (2 (cid:96) + (cid:96) (cid:48) + π (cid:88) L , L (cid:48) (2 L + L (cid:48) + D Φ L D OL (cid:48) (cid:32) L (cid:48) L (cid:96) − s s (cid:33) (cid:32) L (cid:48) L (cid:96) (cid:48) − s (cid:48) s (cid:48) (cid:33) (A.12) (cid:88) M , M (cid:48) (cid:32) L (cid:48) L (cid:96) M (cid:48) M − m (cid:33) (cid:32) L (cid:48) L (cid:96) (cid:48) M (cid:48) M − m (cid:48) (cid:33) . The last summation over M and M (cid:48) of two Wigner-3 j ’s is equal to (2 (cid:96) + − δ (cid:96),(cid:96) (cid:48) δ m , m (cid:48) . One thus finally obtains that the correlationmatrix of the multipolar coe ffi cients is diagonal (as expected for statistically homogeneous and isotropic process), i.e. (cid:68) φ (1) s ,(cid:96) m φ (2) (cid:63) s (cid:48) ,(cid:96) (cid:48) m (cid:48) (cid:69) = C (1 , (cid:96) δ (cid:96),(cid:96) (cid:48) δ m , m (cid:48) , (A.13)with the angular power spectrum of the Faraday e ff ect given by C (1 , (cid:96) = π (cid:88) L , L (cid:48) (2 L + L (cid:48) + (cid:32) L (cid:48) L (cid:96) − s s (cid:33) (cid:32) (cid:96) L (cid:96) − s (cid:48) s (cid:48) (cid:33) D Φ L D OL (cid:48) . (A.14)Since the 2-point correlation function is the product of two 2-point correlation functions, we consistently find that the angular powerspectrum is the convolution of the respective two angular power spectra D Φ L and D OL (cid:48) . Article number, page 15 of 18 & A proofs: manuscript no. main
Appendix B: Derivation of D AL We describe the derivation of the expression of D ?? (cid:96) . This is very reminiscent to the calculation of the angular power spectrumof e.g. the thermal Sunyaev-Zel’dovich e ff ect (see for example Cole & Kaiser 1988; Makino & Suto 1993; Komatsu & Kitayama1999), here simplified since one only need to derive the 1-halo term. To this end let us define A ( ˆ n ) such that: A ( ˆ n ) = e π m e c ε (cid:90) r CMB a ( r )d r ν ( r ) (cid:34) d M i d x i n h ( x i ) X ( | x − x i | ) . (B.1)Then D AL is the angular power spectrum of the above quantity restricted to its 1-halo contribution.The integral over x i in A ( n ) is the convolution of the halo abundance, n h , with the profile of the halo, X . This is then written asa product in Fourier space to get A ( n ) = e π m e c ε (cid:90) r CMB a ( r )d r ν ( r ) (cid:34) d M i d k (cid:101) n h ( k , M i ) (cid:101) X ( k ) e i k · x , (B.2)with ˜ f ( k ) meaning the 3D Fourier transform of f ( x i ). The radial profile being spherically symmetric, it only depends on the normof the wavevector k ≡ | k | and can be expressed using spherical Bessel functions (cid:101) X ( k ) = (cid:101) X ( k ) = (cid:114) π (cid:90) ∞ d R R X ( R ) j ( kR ) , (B.3)with R ≡ | x − x i | and j the spherical Bessel function at order (cid:96) =
0. We further make use of the Rayleigh formula to expressthe e i k · x using spherical Bessel functions and spherical harmonics. The multipolar coe ffi cients are then obtained through A LM = (cid:82) d n A ( n ) Y (cid:63) LM ( n ) leading to A LM = e π m e c ε (cid:90) r CMB a ( r )d r ν ( r ) (cid:34) d M i d k (cid:101) n h ( k , M i ) (cid:101) X ( k ) × (4 π ) (cid:88) L , M ( i ) L j L ( kr ) Y (cid:63) LM ( k / k ) . (B.4)The 2-point correlation of the above set of multipolar coe ffi cients will involve the auto-correlation of the Fourier transform of thehalo abundance. The Poisson part of the 2-point correlation of the halo density field reads (cid:68) n h ( x i , M i ) (cid:69) = (d N / d M i ) δ ( M i − M j ) δ ( x i − x j ) with d N / d M i the mass function. The corresponding power spectrum is constant (independent of scale) : (cid:68)(cid:101) n h ( k ) (cid:101) n (cid:63) h ( q ) (cid:69) = (d N / d M i ) δ ( M i − M j ) δ ( k − q ). Thanks to the scale independance of it, and to the fact that the Fourier-transformed profile ofthe angle depends on k only, one can perform the integral over ( k / k ) to get (cid:68) A LM A (cid:63) L (cid:48) M (cid:48) (cid:69) = D AL δ L , L (cid:48) δ M , M (cid:48) , (B.5)with the angular power spectrum D AL = (cid:32) e π m e c ε (cid:33) (cid:90) r CMB a ( r ) d r ν ( r ) (cid:90) r CMB a ( r ) d r ν ( r ) (cid:90) d M d N d M (cid:90) k d k (cid:12)(cid:12)(cid:12)(cid:101) X ( k ) (cid:12)(cid:12)(cid:12) j L ( kr ) j L ( kr ) . (B.6)The numerical evaluation of the angular power spectrum D L as derived above is still prohibitive due to the presence of the highlyoscillating Bessel functions. It is however built from expressions of the form (cid:34) d r d r H ( r ) H ( r ) (cid:90) k d k π P ( k ) j (cid:96) ( kr ) j (cid:96) ( kr ) , which can be simplified using the Limber’s approximation (LoVerde & Afshordi 2008). Using this approximation, we obtain D AL = e m e c ε √ π (cid:90) r CMB d r a ( r ) r ν ( r ) (cid:90) d M d N d M (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:101) X (cid:32) L + / r (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (B.7)We finalize our expression of the angular power spectrum D AL by introducing the projected Fourier transform of the profile. To thisend, we first note that X ( R ) = X ( c ) ( M , z , B c ) U ( R / R c ) with B c the mean magnetic field strength at the center of the halo (which canalso depend on M and z , see Tashiro et al. (2008)), and U a normalized profile which only depends on the ratio of the comovingdistance from the center, R , to the typical comoving radius of the halo, R c , which is also a function of z and M . For a β -profile, theyread X ( c ) = n ( c ) e B c and U ( R / R c ) = (1 + R / R c ) − β (1 + µ ) / .Introducing the variable x = R / R c and physical radius of the halo, r (phys) c = a ( z ) R c , one finds: (cid:101) X (cid:32) (cid:96) + / r (cid:33) = (cid:32) r a ( r ) (cid:33) X ( c ) × (cid:114) π r (phys) c (cid:96) c (cid:90) ∞ U ( x ) j (( (cid:96) + / x /(cid:96) c ) x d x , Article number, page 16 of 18emarchand et al.: Secondary CMB anisotropies from magnetized halos (I) with (cid:96) c = D ang ( z ) / r (phys) c the typical multipole associated to the typical size of the halo (the latter being also a function of M and z through r (phys) c ), and D ang ( z ) the angular diameter distance. By defining the projected Fourier transform of the profiles α (cid:96) ( M , z ) = (cid:114) π r (phys) c (cid:96) c (cid:90) ∞ U ( x ) j (( (cid:96) + / x /(cid:96) c ) x d x , (B.8)the angular power spectrum D L then writes: D AL = e m e c ε √ π (cid:90) d z ν ( z ) d r d z r (cid:90) d M d N d M (cid:12)(cid:12)(cid:12) X ( c ) (cid:12)(cid:12)(cid:12) α L . (B.9) Appendix C: Derivation of D (cid:107) L for Faraday rotation The angular power spectrum D (cid:107) L for Faraday rotation is obtained through the computation of the correlation (cid:68) b ( n , x i ) b ( n , x j ) (cid:69) .(We remind that b ( n , x i ) = n · b ( x i ).) We will work using the vector basis ( e z , e + , e − ) where e ± = ( e x ± i e y ) / √
2, and ( e x , e y , e z ) is thestandard cartesian basis of R . The components of the orientation of the magnetic field b and the line-of-sight direction n are givenby: b = cos( β ( x i ))1 √ β ( x i )) e i α ( x i ) − √ β ( x i )) e − i α ( x i ) , and , n = (cid:114) π Y ( n ) Y − ( n ) Y ( n ) . (C.1)We note that in the specific reference frame adopted here, the components of the line-of-sight unit vector are expressed using thespherical harmonics for (cid:96) =
1. This way of expressing the components of the unit vector of the line-of-sight is appropriate for furtherreading the angular power spectrum from the 2-point correlation function; see Eq. (A.10).For uniformly distributed unit vectors, one obtains the following average: (cid:68) b ( x i ) b ( x j ) (cid:69) = / − / − / δ i , j , (C.2)which is only nonzero for the same halos. This is also constant in space because, as explained in App. A, it results from anhomogeneous and isotropic process. The 2-point correlation function finally reads (cid:68) b ( n , x i ) b ( n , x j ) (cid:69) = π δ i , j (cid:88) m = − Y , m ( ˆ n ) Y (cid:63) , m ( ˆ n ) , (C.3)from which the angular power spectrum is easily obtained to be D (cid:107) L = (4 π/ × δ L , . Appendix D: Derivation of D ⊥ L for Faraday conversion In this appendix one computes the following 2-point correlation functions: (cid:68) b ± ( n , x i ) b ± ( n , x j ) (cid:69) and (cid:68) b ± ( n , x i ) b ∓ ( n , x j ) (cid:69) where we remind that b ± ( n , x i ) ≡ [ b ( x i ) · ( e θ ± i e ϕ )] . Working in the basis ( e z , e + , e − ) as used in App. B, squares of inner-dotproducts b ± ( n , x i ) are conveniently expressed as b ± ( n , x i ) = (cid:88) µ = b µ ( x i ) e ( ± ) µ ( n ) , (D.1)where the 5 coe ffi cients b µ depends on the orientations of the magnetic fields only (i.e. they are functions of β ( x i ) and α ( x i ) only).They are given by b µ ( x i ) = (cid:113) (2 cos( β ( x i )) − sin( β ( x i )) ) , − β ( x i )) cos( β ( x i )) e − i α ( x i ) , β ( x i )) cos( β ( x i )) e i α ( x i ) , sin( β ( x i )) e − i α ( x i ) , sin( β ( x i )) e i α ( x i ) . (D.2) Article number, page 17 of 18 & A proofs: manuscript no. main
The 5 coe ffi cients e ( ± ) µ ( n ) are functions of the line-of-sight only and with our choice of the reference frame, they are expressed usingspin-weighted spherical harmonics for s = ± (cid:96) = e ( ± ) µ ( n ) = (cid:114) π ± Y , ( ˆ n ) , ± Y , ( ˆ n ) , ± Y , − ( ˆ n ) , ± Y , ( ˆ n ) , ± Y , − ( ˆ n ) . (D.3)Ensemble averages are done for the b µ coe ffi cients which for a uniform distribution of orientations gives (cid:68) b µ ( x i ) b ν ( x j ) (cid:69) = δ µ,ν δ i , j .The di ff erent correlation functions are then given by (cid:68) b ± ( n , x i ) b ± ( n , x j ) (cid:69) = π δ i , j (cid:88) m = − ± Y , m ( n ) ± Y (cid:63) , m ( n ) , (D.4) (cid:68) b ± ( n , x i ) b ∓ ( n , x j ) (cid:69) = π δ i , j (cid:88) m = − ± Y , m ( n ) ∓ Y (cid:63) , m ( n ) . (D.5)All these correlation are thus described by the angular power spectrum (which is nonzero for the 1-halo term only) reading D ⊥ (cid:96) = (32 π/ δ (cid:96), ..