| { | |
| "J. Chluba⋆, L. Dai†and M. Kamionkowski‡": [ | |
| "Department of Physics and Astronomy, Johns Hopkins University, Bloomberg Center, 3400 N. Charles St., Baltimore, MD 21218, USA ", | |
| "Accepted 2013 September 30; Received 2013 August 26 " | |
| ], | |
| "ABSTRACT ": [ | |
| "Future high-resolution, high-sensitivity Sunyaev-Zeldovich (SZ) observations of individual clusters will allow us to study the dynamical state of the intra cluster medium (ICM). To exploit the full potential of this new observational window, it is crucial to understand the origin of di erent contributions to the total SZ signal. Here, we investigate the signature caused by multiple scatterings at lowest order of the electron temperature. Previous analytic discussions of this problem used the isotropic scattering approximation (ISA), which even for the simplest cluster geometries is rather rough. We take a step forward and consistently treat the anisotropy of the ambient radiation •eld caused by the •rst scattering. We show that the multiple scattering SZ signal directly probes line of sight anisotropies of the ICM, thereby delivering a new set of SZ observables which could be used for 3D cluster-pro•le reconstruction. The multiple scattering signal should furthermore correlate spatially with the cluster•s X-ray and SZ polarization signals, an e ect that could allow enhancing the detectability of this contribution. ", | |
| "Key words: Cosmology: cosmic microwave background • theory • galaxy clusters " | |
| ], | |
| "1 INTRODUCTION ": [ | |
| "A number of high-resolution Sunyaev-Zeldovich (SZ) experiments, including ALMA1 (<>), CARMA2 (<>), CCAT3 (<>), and MUSTANG4 (<>), are underway or planned, promising a dramatic increase in sensitivities, spatial resolution and spectral coverage over the next few years. These experiments will allow us to directly probe the dynamical state of the intra cluster medium (ICM) of individual clusters, opening a new window to understanding their cosmological formation and evolution. To realize the full potential of these observations, it is important to understand the SZ signal and its dependence on the cluster atmosphere with high precision. ", | |
| "Since the •rst considerations of the thermal SZ (thSZ) e ect ((<>)Zeldovich & Sunyaev (<>)1969) and kinetic SZ (kSZ) e ect ((<>)Sunyaev & Zeldovich (<>)1980), several related signatures have been discussed in the literature. For instance, because the electrons residing inside clusters are very hot [kTe ≃5 keV −10 keV (with k denoting the Boltzmann constant and Te the electron temperature)] relativistic temperature corrections need to be included ((<>)Wright (<>)1979; (<>)Rephaeli (<>)1995; (<>)Challinor & Lasenby (<>)1998; (<>)Itoh et al. (<>)1998; (<>)Sazonov & Sunyaev (<>)1998). Also, higher order kSZ terms, both caused by the clusters bulk motion ((<>)Sazonov & Sunyaev (<>)1998; (<>)Nozawa et al. (<>)1998; (<>)Challinor & Lasenby (<>)1999) and the motion ", | |
| "of the observer ((<>)Chluba et al. (<>)2005; (<>)Nozawa et al. (<>)2005) arise5 (<>). Furthermore, relativistic electrons can create a non-thermal SZ signature indicative of past AGN activity and merger shocks ((<>)Enßlin & Kaiser (<>)2000; (<>)Colafrancesco et al. (<>)2003). Additional effects are related to details of the cluster•s atmosphere. For example, internal bulk and turbulent gas motions introduce spatially varying SZ signatures (e.g., (<>)Chluba & Mannheim (<>)2002; (<>)Nagai et al. (<>)2003; (<>)Diego et al. (<>)2003). Line-of-sight variations of the electron temperature cause a non-trivial SZ morphology which depends on moments of the cluster temperature and velocity pro•les ((<>)Chluba et al. (<>)2013, CSNN hereafter). Extra pressure support from large-scale magnetic •eld deforms the electron distribution inside clusters ((<>)Koch et al. (<>)2003). And •nally, shocks and substructures in merging clusters leave distinct signatures in the SZ •ux (e.g., (<>)Komatsu et al. (<>)2001; (<>)Kitayama et al. (<>)2004; (<>)Colafrancesco et al. (<>)2011; (<>)Korngut et al. (<>)2011; (<>)Menanteau et al. (<>)2012; (<>)Mroczkowski et al. (<>)2012; (<>)Zemcov et al. (<>)2012; (<>)Prokhorov & Colafrancesco (<>)2012). ", | |
| "In this work we consider the e ect of multiple scattering on the SZ signal ((<>)Sunyaev & Zeldovich (<>)1980; (<>)Sazonov & Sunyaev (<>)1999; (<>)Molnar & Birkinshaw (<>)1999; (<>)Dolgov et al. (<>)2001; (<>)Itoh et al. (<>)2001; (<>)Colafrancesco et al. (<>)2003; (<>)Shimon & Rephaeli (<>)2004). As we show here, this signature depends on inhomogeneities in the clus-ter•s electron and temperature distribution. It therefore provides a new set of SZ observables that are valuable for the reconstruction ", | |
| "of 3D cluster pro•les6 (<>), potentially allowing geometric degeneracies to be broken. We restrict our analysis to the lowest order in the electron temperature. Higher order corrections will be discussed in (<>)Chluba & Dai ((<>)2013, CD13 hereafter). ", | |
| "The main new physical ingredient is that for the second scattering we include the full anisotropy of the singly-scattered radiation •eld. In previous studies (e.g., (<>)Itoh et al. (<>)2001; (<>)Colafrancesco et al. (<>)2003; (<>)Shimon & Rephaeli (<>)2004), this aspect was omitted; however, even for the simplest cluster geometries this approximation is very rough (see Sect. (<>)3). Some discussion of the scattering-induced anisotropy in the radiation •eld can be found in connection with SZ polarization e ects (e.g., (<>)Sunyaev & Zeldovich (<>)1980; (<>)Sazonov & Sunyaev (<>)1999; (<>)Lavaux et al. (<>)2004; (<>)Shimon et al. (<>)2006) and accretion •ows (e.g., (<>)Sunyaev & Titarchuk (<>)1985), but the SZ signature in the frequency dependence of the speci•c intensity that we consider here was previously not studied. We determine a new set of SZ observables related to the second scattering correction (Sect. (<>)3.4) and demonstrate how the signal depends on the cluster•s geometry (see Fig. (<>)5). We also •nd that the spatial pattern of the second scattering SZ signal can be used to enhance its detectability by cross-correlating with X-ray and future SZ polarization data (Sect. (<>)4.1.3). ", | |
| "For those readers less interested in the details of the derivation, we note that the central result of the paper is an expression, Eq. ((<>)12), for the total change I(x,\u001a ) to the specic intensity induced along a line of sight \u001a through the cluster in the rst and second scattering as a function of frequency x. The new parameters γT( \u001a ) and γE( \u001a ) that appear in that expression are explicitly given for a spherically symmetric cluster as a function of impact parameter b in Eq. ((<>)15). " | |
| ], | |
| "2 EQUATION OF RADIATIVE TRANSFER ": [ | |
| "We begin with the radiative transfer equation in an anisotropic, static medium7 (<>)(e.g., (<>)Mihalas (<>)1978; (<>)Rybicki & Lightman (<>)1979; (<>)Mihalas & Mihalas (<>)1984): ", | |
| "(1) ", | |
| "Here, n(x,r,\u001a ) denotes the photon occupation number at the location r, in the direction \u001a and at frequency where is the cosmic microwave background (CMB) temperature at redshift z with the present-day monopole T0 = 2.726 K ((<>)Fixsen et al. (<>)1996; (<>)Fixsen & Mather (<>)2002; (<>)Fixsen (<>)2009), and h denotes the Planck constant. The radiative transfer equation is solved along the photon path (parallel to \u001a ) through the ICM, parametrized by the Thomson optical depth, where Ne(r) is the free electron number density. " | |
| ], | |
| "2.1 Compton collision term at the lowest order in kTe/mec2 ": [ | |
| "The collision term C[n] in Eq. ((<>)1) accounts for Compton scattering of photons of frequency x and direction \u001a out of the beam and for scattering of photons of other directions and frequencies into the beam. To linear order in the electron-temperature parameter θe ≡kTe/mec2 (where me is the electron mass), the collision operator for anisotropic incoming radiation, as shown in Appendix (<>)A, is ((<>)Chluba et al. (<>)2012a) ", | |
| "where µsc = \u001a ·\u001a ′. Stimulated scattering and electron recoil were neglected, because for SZ clusters Tγ≪Te. ", | |
| "The •rst integral in Eq. ((<>)2) describes the e ect of Thomson scattering (no energy exchange or equivalently scattering of photons by resting electrons), leaving the photon spectrum of the incoming radiation •eld unaltered. If the incoming photon •eld is isotropic this term vanishes. ", | |
| "The second integral in Eq. ((<>)2) accounts for leading order temperature corrections to the total scattering cross-section. These again do not alter the shape of the photon spectrum (no energy exchange), and vanish for cold/resting electrons. For isotropic incoming photon •eld this term cancels [Klein-Nishina corrections, leading to contributions ≃θe(hν/mec2) are neglected here], but for the radiation dipole through octupole a rst-order temperature correction arises: although for resting electrons the Thomson cross-section only couples to the monopole and quadrupole anisotropy of the incoming photon eld, once electrons are moving, due to light aberration (see e.g., (<>)Chluba (<>)2011, for related discussion), also photons from the radiation dipole and octupole scatter into the line of sight at O(θe). Higher order multipoles couple at order O(θeℓ−2). ", | |
| "The last integral in Eq. ((<>)2) takes the energy exchange between electrons and photons into account. It depends only on the radiation monopole through octupole. If the incoming radiation monopole through octupole have identical energy distribution, then their spectra are all altered in the same way by the scattering event. The di erence is captured by an overall, frequency independent scattering eÿciency factor that depends on the multipole. This is the case relevant to the second scattering SZ correction, for which all multipoles of the singly-scattered radiation •eld have a spectrum ∝Y0(x), which represents the well-known thSZ spectrum ((<>)Zeldovich & Sunyaev (<>)1969) dened in Eq. ((<>)7). " | |
| ], | |
| "2.2 Rewriting the collision term using Legendre polynomials ": [ | |
| "The collision term can be further simpli•ed by expanding the radiation •eld in terms of Legendre polynomials, Pℓ(x): ", | |
| "(3) ", | |
| "Here, n(x,r,µsc) denotes the azimuthally averaged occupation number of the photon eld along the propagation direction \u001a . The Legendre coeÿcients can also be obtained from a spherical harmonic expansion of the radiation eld with respect to a general system, n(x,r,\u001a ′) = ℓm Yℓm( \u001a ′)nℓm(x,r), using the identity ", | |
| "nℓ(x,r) ≡m Yℓm( \u001a ) nℓm(x,r). Carrying out the integrals in Eq. ((<>)2), we nd (compare Eq. (C15) and (C19) of (<>)Chluba et al. (<>)2012a): ", | |
| "This expression captures all the relevant physics of the collision term, Eq. ((<>)2). Like Eq. ((<>)6), it can be directly obtained by computing the •rst and second moments of the frequency shift over the scattering kernel when accounting for the full angular dependence of the incoming photon •eld (see (<>)Chluba et al. ((<>)2012a) and CNSN). Only multipoles with ℓ≤3 really matter for the scattering problem, greatly simplifying the next steps of the calculation. This expressions also clearly shows that only the azimuthally symmetric part of the incoming photon distribution with respect to \u001a really matters, a property that due to symmetries of the scattering problem also holds at higher temperature (see CD13). " | |
| ], | |
| "2.3 Born approximation for multiple scattering ": [ | |
| "In the optically thin limit (τ≪1), relevant for galaxy clusters, repeated Compton scattering only introduces a small correction to the radiation eld. To solve Eq. ((<>)1), we can thus use a perturbative Ansatz (a Born-series), n(x,r,\u001a ) ≈n(0) +ǫn(1) +ǫ2n(2) +...+ǫkn(k) , yielding the iteration scheme ", | |
| "(5) ", | |
| "The Compton collision term is linear in n(x,r,\u001a ) since induced scattering terms cancel for Tγ≪Te. Without scattering nothing changes (∂τn(0)(x,r,\u001a ) = 0), and the rst scattering signal is given by ∂τn(1)(x,r,\u001a ) = C[n(0)(x,r,\u001a )]. For higher order scatterings, the local anisotropy of the radiation eld becomes important, even if the unscattered radiation eld was isotropic. Since the optical depth of typical clusters is τ. 0.01, we can restrict ourselves to the rst and second scattering e ect. " | |
| ], | |
| "3 MULTIPLE SCATTERING SZ CONTRIBUTIONS AT LOWEST ORDER IN ELECTRON TEMPERATURE ": [], | |
| "3.1 SZ signal after the •rst scattering ": [ | |
| "Assuming that the incoming, unscattered radiation •eld is given by the isotropic CMB blackbody, n(0)(x,r,\u001a ) = nPl(x) ≡1/[ex −1], the only non-vanishing incoming-radiation multipole is the monopole (ℓ= 0). The collision term in this case simplies to the well-known form (see (<>)Zeldovich & Sunyaev (<>)1969), ", | |
| "for an isotropic radiation eld. The coeÿcients, 4θe and θe, of the derivative terms are simply the rst and second moments of the scattered photons relative frequency shift, ν≡(ν′−ν)/ν, over the scattering kernel (e.g., see (<>)Sazonov & Sunyaev (<>)2000). ", | |
| "This then leads to the well-known thSZ e ect ((<>)Zeldovich & Sunyaev (<>)1969), with spectral shape, ", | |
| "Here, the Compton y parameter, for ", | |
| "Figure 1. Optical depth anisotropy of the local radiation •eld introduced by the •rst scattering of CMB photons for a constant density, isothermal sphere of free electrons. The dashed lines indicate the lines-of-sight for the two cases, and the dot marks the locus of the second scattering event. At the center (left panel), the total optical depth in any direction is the same because the total path through the medium is identical. Observing the singly-scattered radiation •eld around an o center position (right panel) on the other hand shows anisotropies because of optical depth variations. ", | |
| "a distant observer is computed along the line of sight through the ICM, and Io = (2h/c2)(kT0/h)3 ≈270 MJy sr−1 . We also assumed that the cluster is at rest in the CMB rest frame and internal motions are negligible (no kSZ). " | |
| ], | |
| "3.2 Second scattering correction ": [], | |
| "3.2.1 Simple illustration for the scattering-induced anisotropy ": [ | |
| "While the derivation given in this section applies to general cluster geometry, to illustrate the physics of the multiple scattering problem, let us consider a spherical, isothermal cloud of free electrons at rest in the CMB rest frame with constant density, Ne, and radius R as a simple example. At any location, r, inside the cluster we can compute the singly-scattered radiation •eld in di erent directions using Eq. ((<>)7), with isothermal y parameter, Here, the optical depth, is computed from r in the direction ˆ ′through the ICM, where s parametrizes the length along photon path. At the cluster center, the incident radiation eld is fully isotropic, even in the limit of innite number of scatterings; however, due to variations of the scattering optical depth in di erent directions, after one scattering the local photon eld becomes anisotropic elsewhere (see Fig. (<>)1). This scattering-induced anisotropy, generally being sourced by both electron temperature and density variations, has to be taken into account when calculating the second scattering correction to the SZ signal. " | |
| ], | |
| "3.2.2 Explicit form of the second scattering SZ correction ": [ | |
| "We now give the explicit expression of the second scattering correction to the thSZ e ect. After the •rst scattering, the correction to the radiation •eld at any position r inside the cluster is given by ", | |
| "(8) ", | |
| "where the spatial dependence is introduced only by the variation of the y parameter in di erent directions ˆ ′, with the scatterer centered at position r. This expression shows that for the second scattering SZ signal, spectral and spatial parts of the radiation eld separate. Since all multipoles of the singly-scattered radiation have the same ", | |
| "Figure 2. Spectral shape of the lowest order •rst and second scattering contributions to the thSZ signal. Note that we scaled Y0∗(x) = x−2∂x x4∂xY0 to make it more comparable in amplitude to Y0. ", | |
| "spectrum, they also have the same after the second scattering. When including higher order temperature corrections and variations of the electron temperature along di erent lines-of-sight, this separation becomes slightly more complicated and the spectra of each multi-pole di er after the second scattering (see CD13). ", | |
| "Inserting Eq. ((<>)8) into Eq. ((<>)4), and carrying out the line of sight average, we •nd the second scattering correction to the thSZ signal: ", | |
| "with G(x) = xex/[ex −1]2 . Here, z parametrizes the photon path along the line of sight \u001a . We also introduced the Legendre coeÿ-cients, yℓ, of y(r,\u001a ′), which are dened like in Eq. ((<>)3). ", | |
| "Equation ((<>)9) describes the full second scattering correction at lowest order in the electron temperature. The •rst two lines in Eq. ((<>)9b) provide corrections, I(2),T and I(2),σ, to the amplitude of the usual thSZ distortion, Eq. ((<>)7). The second is smaller by a factor θe ≪1 compared with the rst, and so we neglect it below. The third provides the new frequency dependence, proportional to Y0∗(x), induced in the second scattering. The results indicate that only radiation multipoles up to ℓ= 3 (the octupole) are required to describe the radiation eld obtained in second scattering. ", | |
| "Since the energy-exchange correction, I(2),E , has a frequency dependence that di ers from that of the usual lowest order thSZ effect, measurement of its amplitude can provide information about anisotropies in the electron temperature and density proles. As shown in Fig. 2, the function Y0∗(x) numerically reaches values ≃10 ", | |
| "times larger than the maximum of Y0(x). This makes this second-scattering correction roughly an order of magnitude larger than would be guessed from its θe ≃10−2 suppression relative to the lowest order SZ signal. ", | |
| "To calculate the observable emergent speci•c intensity, we only need to compute the line of sight averages of the y-parameter multipoles, weighted by the Thomson optical depth, ∝yℓ, and the y parameter itself, ∝θeyℓ. These moments and ", | |
| "(10) ", | |
| "can be obtained directly from cluster simulations. This greatly simpli•es the computation, since spatial and spectral terms have been separated by our method. Below we give some simple examples assuming spherical symmetry. " | |
| ], | |
| "3.3 Second scattering SZ correction in the isotropic scattering approximation (ISA) ": [ | |
| "In the ISA (e.g., (<>)Itoh et al. (<>)2001; (<>)Colafrancesco et al. (<>)2003; (<>)Shimon & Rephaeli (<>)2004), the singly-scattered photon •eld is assumed to be isotropic and the same as in the direction along the line of sight dened by \u001a . Explicitly this means y(r,\u001a ′) ≈y(r,\u001a ), where \u001a ′is the direction of the incoming photon. Therefore, none of the anisotropies are taken into account and even the variation of the local monopole is neglected, although generally y(r,\u001a ′) d2 \u001a ′, y(r,\u001a ). At lowest order in θe, with Eq. ((<>)9) we therefore have ", | |
| "where we used the identity θey(r,\u001a )≡y( \u001a )2/2, which can be deduced using Eq. ((<>)10). This expression neglects geometric form factors. In addition, no Thomson correction, I(2),T(x,\u001a ), is obtained in the ISA, although it usually is comparable to the energy-exchange correction, I(2),E(x,\u001a ), in Eq. ((<>)9) [see Fig. (<>)5]. " | |
| ], | |
| "3.4 Total SZ signal with second scattering corrections ": [ | |
| "The second scattering SZ signal gives rise to new observables that depend on the geometry of the cluster. The required line of sight moments scale like yl∝τ( \u001a )y( \u001a )/2, θeyl∝y2( \u001a )/2 and y(r,\u001a )∝τ( \u001a )y( \u001a )/2. With Eq. ((<>)7) and Eq. ((<>)9), it is therefore useful to rewrite the total SZ signal as ", | |
| "The correction due to Thomson scattering is determined by γT( \u001a ), while the energy-exchange correction depends on γE( \u001a ). For the ISA, we have γT( \u001a ) = 0 and γE( \u001a ) = 1. ", | |
| "Since the total SZ signal depends on two independent spectral functions, Y0(x) and Y0∗(x) [see Fig. (<>)2], from the observational point of view it in principle is possible to directly measure A( \u001a ) = y( \u001a ) 1 + τ( \u001a ) γT( \u001a )/2and B( \u001a ) = γE( \u001a ) y2( \u001a )/2 for different lines-of-sight. These observables depend on both the temperature and density proles of the electron distribution, and thus allow probing properties of the cluster atmosphere. " | |
| ], | |
| "4 SPHERICALLY SYMMETRIC SYSTEMS ": [ | |
| "For spherically symmetric systems, the computations of the required y-parameter moments can be further simpli•ed. Due to the symmetry of the problem, the y parameter in any direction ˆ ′around location r is given by ", | |
| "(13) ", | |
| "with r∗(s,µr ) = r2 + 2rsµr + s2 and µr = r\u001a·\u001a ′, where r\u001a denes the radial direction. To compute the Legendre coeÿcients of y(r,\u001a ′), we can use the addition theorem for spherical harmonics to simplify the calculation. This gives ", | |
| "(14) ", | |
| "where we introduced the impact parameter b of the line√of sight \u001a from the cluster center at the position z, so that r = b2 + z2 . The Legendre coeÿcient yℓ∗(r) ≡yℓ(b = 0,z) describes the simplest case, i.e., the line of sight through the center, and the factor Pℓ(z/r) provides the geometric transformation to the case b , 0. From the practical point of view, this means that for spherically symmetric systems we can compute the required line of sight moments yℓand θeyℓin two independent steps, rst giving y∗ℓ(r) and then averaging over z. Since yℓ(b,−z) = (−1)ℓyℓ(b,z), all odd moments vanish after averaging over the line of sight. For general cluster geometry this is not necessarily true but it greatly simplies the calculation for (quasi) spherically systems. ", | |
| "Performing the required integrals, the parameters γE(b) and γT(b) in Eq. ((<>)12), as a function of impact parameter b become (see Appendix (<>)B for more details) ", | |
| "in terms of yobs(b), the y parameter observed along a line of sight that passes√an impact parameter b from the center. In these expressions, r = z2 + b2 , and κ0 = 1 and κ2 = 1/2. We also assumed that the cluster extends only to a maximal radius R and introduced the pressure prole Pe(r) = kTe(r)Ne(r). To simplify the computation, the innermost integrals over dµcan be tabulated as a function of r. " | |
| ], | |
| "4.1 Optical depth moments for illustrative cases ": [ | |
| "The second scattering SZ signal, Eq. ((<>)12), directly depends on the geometry of the cluster atmosphere. General cases can be studied ", | |
| "Figure 3. Weighted average optical depth multipoles for an isothermal, constant density sphere with radius R as a function of impact parameter. All curves were normalized to their central value with and according to Eq. ((<>)C1). Odd moments vanish. ", | |
| "Figure 4. Weighted average optical depth multipoles for isothermal β-prole with core radius, rc, as a function of impact parameter. All curves where normalized to their central value with and Odd moments vanish by symmetry. ", | |
| "using cluster simulations, but to illustrate the main e ects we consider an isothermal, constant density sphere and an isothermal β-model (see Appendix (<>)C for denitions). Since for this case yℓ= θe τℓ, θeyℓ= θe2 τℓand y(r,\u001a )= θeτ( \u001a )2/2, we only have to compute line of sight moments of the optical depth eld, τℓ. Some details of the calculation for b = 0 are given in Appendix (<>)C, and the general case is obtained by averaging yℓ(b,z) ≡Pℓ(z/r) yℓ∗(r) along the line of sight. ", | |
| "The dependence of τℓon impact parameter b for the two cases is illustrated in Fig. (<>)3 and (<>)4. For the constant density sphere, the moments show signicant variation with b. They drop rapidly with ℓ, and in all cases the largest values of τℓare found for the line of sight directly through the center. This behavior depends critically on the geometry and density prole of the cloud, and for an ", | |
| "Figure 5. Total second scattering correction for the constant density, isothermal sphere and the isothermal β-model at zero impact parameter. For comparison, we show the single-scattering thSZ e ect (the same in all cases), and the second scattering signal obtained with the ISA. ", | |
| "isothermal β-prole the situations is already very di erent. In contrast to the constant density sphere, the proles τℓare monotonic, with higher order terms decreasing more gradually. It is evident that the amplitude and spatial morphology of the second scattering SZ correction di ers in the two cases. Consequently, the second scattering correction directly probes the geometry of the scattering medium, an e ect that is not captured by the ISA. " | |
| ], | |
| "4.1.1 Total second scattering signal and comparison to the ISA ": [ | |
| "We now compute the total correction caused by second scattering terms. In Fig. (<>)5 we show a comparison of the signal for the constant density, isothermal sphere and the isothermal β-model. The overall signal is very small, in both cases reaching no more than ≃0.2% of I(1) . However, as Fig. (<>)5 illustrates, the second scattering signal directly depends on the geometry of the scattering electron distribution, an e ect that is absent in the ISA. ", | |
| "Comparing the di erent contributions in Eq. ((<>)12) also shows that the dominant correction is caused in the Thomson limit, I(2),T . For the isothermal sphere and impact parameter b = 0, we nd I(2),T(x) ≈−0.06 τc I(1)(x). For some clusters one encounters central optical depth τc ≃0.01, so that this e ect gives rise to a correction ≃−0.06% relative to the singly-scattered SZ signal. For an isothermal β-prole (β= 2/3), we obtain a slightly larger e ect, giving I(2),T(x) ≈−0.15 τc I(1)(x) or a correction ≃−0.15% relative to I(1)(x) for τc = 0.01. Comparing the energy-exchange correction with the ISA, we nd I(2),E,iso ≃1.12I(2),E and I(2),E,iso ≃1.4I(2),E for the isothermal sphere and β-model, respectively. Hence, this part of the correction is overestimated when using the ISA. The total correction (I(2),T ) for the isother-+ I(2),E mal β-model is ≃3 −4 times larger than obtained with the ISA. " | |
| ], | |
| "4.1.2 E ect on the crossover frequency ": [ | |
| "Because the frequency dependence of I(2),E di ers from I(1) , the position of the crossover frequency (≡null of the SZ signal at ν≃217 GHz or xc ≃3.83) is a ected by the second scattering ", | |
| "Figure 6. Spatial dependence of the di erent contributions to the second scattering signal for an isothermal β-prole with β= 2/3. To represent the rst scattering we show y = τθe, renormalized by its central value. For the ISA, we presented κEiso = (τθe)2/2 again renor-malizing by its central value. The Thomson scattering term is κT = γT (θeτ2/2) = θe τ0+ τ2/10 −τ2/2 and was normalized by X0 = −θeτ2/2 at b = 0. Finally, the energy-exchange term is κE = γE (θe2τ2/2) = θe2 [τ0+ τ2/10] and was normalized by X0 = (θeτ)2/2 at b = 0. ", | |
| "contribution. From Eq. ((<>)12), we •nd ", | |
| "(16) ", | |
| "which for the isothermal β-model (γE ≃0.71 at b = 0) with total y parameter y ≃10−4 means xc ≃2.2 ×10−4 . For the constant density sphere (γE ≃0.89 at b = 0), we nd xc ≃2.7 ×10−4 , while the ISA gives xc ≃3.1 ×10−4 . ", | |
| "For comparison, at Te . 30 keV the shift in the position of the crossover frequency caused by higher order temperature corrections is xc ≃0.0421T\u001ae −3.29 ×10−4T\u001a e2 with T\u001ae = Te/5keV. For θe = 0.01(≡T\u001a e ≃1), this means xc ≃0.0418, which is roughly two orders of magnitudes larger. Similarly, the shift introduced by line of sight variations of the electron temperature is roughly given by xc ≃0.042T\u001ae[1−0.069T\u001ae]ω(1) , where ω(1) parametrizes the line of sight temperature variance (see CSNN). For θe = 0.01 and typical temperature-dispersion ω(1) ≃0.1 this means xc ≃3.9 ×10−3 , which is about one order of magnitude larger than the e ect caused by multiple scattering. Note that at lowest order in Te, Eq. ((<>)16) agrees with the expressions given by (<>)Dolgov et al. ((<>)2001) when setting γE ≡1, i.e., ignoring scattering-induced anisotropies of the radiation. " | |
| ], | |
| "4.1.3 Spatial dependence of the total second scattering signal ": [ | |
| "One interesting aspects of the second scattering correction is that it introduces both a new spatial and spectral dependence. At lowest order in temperature, the singly-scattered signal has a spectrum ≃x3Y0(x) and spatial variation y ≃τθe. In the ISA, the second scattering contribution [cf. Eq. ((<>)11)] has spectrum and spatial dependence On the other hand, when including the anisotropy of the singly-scattered radiation eld, two new contributions arise, each with individual spatial and spectral dependence [see Eq. ((<>)9)]. The Thomson correction term scales like with a spectrum ≃x3Y0(x), while the energy-exchange term has spectrum ", | |
| "and spatial dependence κE = γE (θe2τ2/2) ≃θe2 [τ0+ τ2/10] (τ1= τ3= 0 by symmetry). ", | |
| "In Fig. (<>)6 we illustrate the spatial dependence of di erent terms for an isothermal β-prole with β= 2/3. The second scattering correction is most important at b ≃0, exhibiting a Ne2-scaling like X-rays. Thus, the detectability of the second scattering signal may be enhanced by cross-correlating with X-ray maps, as the X-ray emission also scales with Ne2 . Similarly, part of the SZ polarization signature (e.g., (<>)Sunyaev & Zeldovich (<>)1980; (<>)Sazonov & Sunyaev (<>)1999; (<>)Lavaux et al. (<>)2004; (<>)Shimon et al. (<>)2006) caused by the second scattering should correlate spatially with the SZ intensity signal. " | |
| ], | |
| "4.2 Angle-averaged SZ signal ": [ | |
| "For spherical systems with •nite radius R, we can also compute the angle-averaged SZ signal, assuming that the cluster is •lling a circular beam with radius R. It is well known, that the average single-scattering signal depends on the volume integral over the electron pressure pro•le, Pe(r) = kTe(r)Ne(r) ", | |
| "For the second scattering signal, geometric form factors appear, which depend on the cluster atmosphere. This can be seen from ", | |
| "where we used r∗= r2 + 2r sµ′r + s2 ≡|r + r′|. In the ISA, we have y0ISA = 0 and θey0ISA = y¯2/2. Similar to Eq. ((<>)12), it thus makes sense to dene the scattering form factors ", | |
| "With Eq. ((<>)7) and ((<>)9), the total average SZ signal thus reads ", | |
| "(20) ", | |
| "The above expressions show that the average SZ signal just depends on the overall geometry of the electron pressure and density distribution, but local anisotropies, that were visible for a resolved cluster, no longer contribute. ", | |
| "The geometric form factors have to be computed on a case-by-case basis. For constant pressure and electron density pro•les, we •nd and where we de•ned and used This implies Assuming that the pressure and electron density pro•les scale like 1/r, we similarly •nd ", | |
| "and θey0= [ln(2)/R2] y2c = ln(2) y¯2 ≡y0/θe, or form factors γ¯ T + 1 = γ¯ E = 2 ln(2) ≈1.4. This shows that depending on the geometry of the cluster prole the ISA scattering approximation only provides a rough approximation for the second scattering signal. In particular it underestimates the average SZ signal related to Y0∗(x). Although the ISA overestimated the second scattering SZ signal for small impact parameters, at larger radii, which contribute strongly to the averaged signal, it underestimates the signal, explaining this net total e ect. " | |
| ], | |
| "5 CONCLUSION ": [ | |
| "We analyzed the e ect of multiple scattering on the SZ signal demonstrating how this small correction (order ≃0.1%) depends on the geometry of the ICM. Previous studies applied the ISA which does not capture the e ects discussed here, giving very di erent results for the second scattering correction (see Fig. (<>)5). The di er-ence has two main sources: (i) due to the anisotropy of the singly-scattered radiation eld, the second scattering already introduces a modication in the Thomson limit which is absent in the ISA; (ii) the e ective monopole of the radiation eld along the line of sight di ers from the one used in ISA. ", | |
| "Although, very small in comparison to the singly-scattered signal, our analysis shows that the second scattering SZ signal delivers a new set of observables (see Sect. (<>)3.4) which in the future might be useful for 3D cluster-pro•le reconstruction, allowing us to break geometric degeneracies. Since at lowest order in kTe/mec2 , the second scattering correction for isothermal clusters roughly scales as τ2 ∝Ne2 , it introduces an additional spatial dependence of the SZ signal, similar to the one of X-rays. Cross-correlating high-resolution X-ray maps with future SZ maps might thus allow enhancing the detectability of the second scattering signal. Similarly, combining with high-resolution SZ polarization measurements should allow extracting this contribution, potentially opening a path for more detailed studies of the clumpiness of the ICM. ", | |
| "The method developed here can in principle be directly applied to simulated clusters. This can be used to give more realistic estimates for the signi•cance of the second scattering signal. The computation is greatly simpli•ed with our approach, since spatial averages over the ICM can be performed independently of the spectral integrals, accelerating the calculation. ", | |
| "For the examples discussed here, we assumed spherical symmetry. In this case, all odd moments of the optical depth •eld vanish; however, in general this is an oversimpli•cation. More realistic cluster atmospheres might exhibit larger di erences in the second scattering signal and could thus provide a way to directly probe the asphericity and clumpiness of the ICM along the line of sight. From the observational point of view this is very intriguing, but given the low amplitude of the e ect this certainly remains challenging. For our estimates we also assumed isothermality. Variations of the electron temperature cause additional anisotropies in the singly-scattered radiation which might even exceed those introduced by optical depth e ects. The second scattering SZ contribution therefore could be used to directly probe the cluster•s temperature •eld. These aspects should, however, be addressed using realistic cluster simulations. ", | |
| "We mention that a similar analysis can be carried out for SZ polarization e ects. In particular, the scattering-induced quadrupole anisotropy of the radiation •eld sources a SZ polarization e ect, which is directly related to line of sight average of the Compton y parameter•s quadrupole components. In this case, the ", | |
| "projections perpendicular to the line of sight are most important (i.e., moments like y2,±2 ), dening both amplitude and direction of the polarization signature. Our separation of spatial and spectral dependence should again allow simplifying the calculation of the SZ polarization signature from cluster simulations. Similarly, our techniques can be applied to other scattering media at small and intermediate optical depth, providing a starting point for eÿcient numerical schemes, e.g., for cosmological simulations with Lyman-αradiative transfer (e.g., (<>)Maselli et al. (<>)2003; (<>)Zheng et al. (<>)2010). We plan to explore these possibilities in the future. " | |
| ], | |
| "ACKNOWLEDGEMENTS ": [ | |
| "JC thanks Donghui Jeong for stimulating discussions on the problem. The authors are also grateful to Anthony Challinor, Daisuke Nagai and Rashid Sunyaev for useful comments on the manuscript. This work was supported by the grants DoE SC-0008108 and NASA NNX12AE86G. " | |
| ] | |
| } |