The Polarization of Scattered Lyman Alpha Radiation Around High-Redshift Galaxies

The high-redshift Universe contains luminous Lyman Alpha (hereafter Lya) emitting sources such as galaxies and quasars. The emitted Lya radiation is often scattered by surrounding neutral hydrogen atoms. We show that the scattered Lya radiation obtains a high level of polarization for a wide range of likely environments of high-redshift galaxies. For example, the back-scattered Lya flux observed from galaxies surrounded by a superwind-driven outflow may reach a fractional polarization as high as ~40%. Equal levels of polarization may be observed from neutral collapsing protogalaxies. Resonant scattering in the diffuse intergalactic medium typically results in a lower polarization amplitude (<7%), which depends on the flux of the ionizing background. Spectral polarimetry can differentiate between Lya scattering off infalling gas and outflowing gas; for an outflow the polarization should increase towards longer wavelengths while for infall the opposite is true. Our numerical results suggest that Lya polarimetry is feasible with existing instruments, and may provide a new diagnostic of the distribution and kinematics of neutral hydrogen around high-redshift galaxies. Moreover, polarimetry may help suppress infrared lines originating in the Earth's atmosphere, and thus improve the sensitivity of ground-based observations to high-redshift Lya emitting galaxies outside the currently available redshift windows.

The interpretation of existing and future observations is complicated by the fact that Lyα photons are typically scattered both in the interstellar medium (ISM, e.g. Hansen & Oh 2006) as well as in the intergalactic medium (IGM, Santos 2004;Dijkstra et al 2007b). The impact of scattering on the observed Lyα flux and spectrum may be derived by careful modeling of the observed Lyα line profile, combined with constraints on the galaxy's stellar population derived from its broad band colors (Verhamme et al. 2006(Verhamme et al. , 2008. Also, observations of the Lyα line in local starburst galaxies can shed light on the processes that regulate the transfer and escape of Lyα from star forming galaxies (Kunth et al. 1998;Hayes et al. 2007). Of course, it is not obvious whether local starburst galaxies are representative of high-redshift Lyα emitters (LAEs), and detailed modeling of the line profile may be hampered by the quality of the data of high-redshift LAEs.
In this paper, we describe a complementary approach to studying the environments of high-redshift LAEs, which involves the polarization properties of the scattered Lyα radiation. In the context of the solar system, the polarization of resonantly scattered Lyα photons from the sun by neutral hydrogen in the interplanetary medium and in the Earth's atmosphere has been studied extensively (e.g. Brandt & Chamberlain 1959), occasionally by using a Monte-Carlo method to compute the Lyα radiative transfer (Modali et al. 1972;Keller et al. 1981). In the cosmological context, Lee & Ahn (1998) showed that Lyα emerging from an unresolved star-bursting galaxy may be polarized to a considerable level (∼ 5%), and  showed that scattering of Lyα photons from a point source embedded in the Hubble flow of a neutral IGM would produce a polarized Lyα halo around the source. Despite the emergence of several Monte-Carlo Lyα radiative transfer codes in recent years (e.g. Zheng & Miralda-Escudé 2002;Cantalupo et al. 2005;Tasitsiomi 2006;Dijkstra et al. 2006;Verhamme et al. 2006;Laursen & Sommer-Larsen 2007;Semelin et al. 2007), the polarization properties of scattered Lyα photons around galaxies have not been investigated in a broader context beyond this early work.
The goal of this paper is to demonstrate that Lyα radiation may be highly polarized around high-redshift galaxies in a broad range of cosmological circumstances. We will show that Lyα polarimetry may place constraints on the kinematics of the gas through which the photons propagate. This is important: understanding the impact of scattering in both the ISM and IGM on the observed Lyα properties is required to fully exploit Lyα observations as a probe of the high-redshift Universe. Furthermore, we discuss the possibility of separating Lyα sources from low-redshift line emitters (such as [OII], [OIII], Hα, Hβ emitters, and OH-skylines) based on polarimetry.
The outline of this paper is as follows: In § 2 we describe the basic principles of Lyα radiative transfer and polarization, and how polarization is incorporated in our radiative transfer calculations. In § 3 we present our main numerical results. In § 4 we discuss our results and their implications, before presenting our final conclusions in § 5. The parameters for the background cosmology used throughout this paper are (Ωm, ΩΛ, h) = (0.24, 0.76, 0.73) (Spergel et al. 2007).

Lyα Radiative Transfer Basics
We start by summarizing the basic principles of Lyα scattering. We express photon frequency ν in terms of a dimensionless variable x ≡ (ν − ν0)/∆νD, where ∆νD = v th ν0/c, and v th is the thermal velocity of the hydrogen atoms in the gas, given by v th = p 2kBT /mp, where kB is the Boltzmann constant, T = 10 4 K the gas temperature, mp the proton mass and ν0 = 2.47 × 10 15 Hz is the Lyα resonance frequency. For reference, the optical depth through a column of hydrogen, NHI, for a photon in the line center, τ0, is given by τ0 = 5.9 × 10 6 " NHI 10 20 cm −2 "" T 10 4 K The optical depth for a photon at a frequency x reduces to (e.g. Rybicki & Lightman 1979) where a is the Voigt parameter given by a = A21/4π∆νD = 4.7 × 10 −4 (v th /13 km s −1 ) −1 , where A21 = 6.25 × 10 8 s −1 is the Einstein A-coefficient for the transition. The transition between 'wing' and 'core' occurs at x ≡ xt ≈ 3.3 for T = 10 4 K. Under most astrophysical conditions, absorption of a Lyα photon is followed be re-emission of a photon of the same energy in the frame of the atom. However, in the observer's frame the Lyα photon's energy before and after scattering is modified by Doppler shift due to the thermal motion of the atom, and scattering is only 'partially' coherent in the observer's frame. These Doppler shifts are important as they cause the photon's frequency to change by an rms shift of ∆νD per scattering event (Osterbrock 1962). Therefore, Lyα scattering through an optically thick medium can be described as a sequence of random walks in both frequency and real space (e. g Harrington 1973;Neufeld 1990;. Frequency diffusion is very efficient and spatial diffusion occurs predominantly when photons are in the wing of the line profile (e.g. Adams 1972;Harrington 1973). Hence, the last scattering event occurs in the wing of the line profile for the majority of Lyα photons emerging from an optically thick medium 1 .
Other quantities of relevance to our discussion are: (i) the scattering phase function (also known as the anisotropy function) which gives the probability p(θ) that the scattered photon is re-emitted at an angle θ relative to the incoming photon; and (ii) The degree of polarization Π(θ) caused by scattering which is defined as Π(θ) ≡ I || −I ⊥ I || +I ⊥ , where I || and I ⊥ are the intensities parallel and perpendicular to the plane of scattering (defined by the wave vectors of the ingoing and outgoing photons), as a function scattering angle θ. Both p(θ) and Π(θ) are different for core and wing scattering, as we discuss in more detail below.

Phase Function & Degree of Polarization for Lyα Scattering
For resonant scattering the phase function and the degree of polarization are strongly dependent on the atomic levels involved in the scattering event, and must be calculated quantum-mechanically. For example, the sequence of scattering events 1S 1/2 → 2P 1/2 → 1S 1/2 result in an unpolarized isotropically re-emitted Lyα photon, while the scattering events 1S 1/2 → 2P 3/2 → 1S 1/2 produce Lyα with a maximum polarization of 3 7 (e.g. Hamilton 1947;Chandrasekhar 1960;Ahn et al. 2002). Here we used a notation nLJ, in which n, L and J are the principal quantum number, orbital angular momentum number, and total angular momentum number of the hydrogen atom involved in the scattering event, respectively. By summing over all possible Lyα transitions, the phase function and degree of 1 Note that Lyα photons typically scatter only ∼ τ 0 times before emerging from a medium of optical depth τ 0 , as opposed to the τ 2 0 -scaling, that is expected in the absence of frequency diffusion. For scattering events that occur in the wing, a so called 'restoring force' pushes the photon back towards the core by an average amount of −1/x (Osterbrock 1962;Adams 1972).  Figure 1. Schematic geometry of the last-scattering event of a Lyα photon that occurs at a radius vector r away from the source galaxy. Before last scattering the photon's propagation direction is n (which does not necessarily lie in the plane of the sky). After scattering, the photon travels in a direction n ′ which is perpendicular to the sky plane. The polarization vector e ′ after scattering lies in the plane of the sky and intersects the projected (on the sky plane) radius vector r at an angle χ. This photon contributes cos 2 χ and sin 2 χ to the intensity of the radiation field parallel (I l (α)) and perpendicular (Ir(α)) to r, respectively. Here α = r/d A (z) (see text for additional details).
As is shown in Appendix A, this corresponds to a case of a superposition of Rayleigh scattering plus isotropic scattering with corresponding weights of 1/3 and 2/3. The phase function p(θ) satisfies R p(θ)dΩ = 4π. In quantum mechanics a Lyα scattering event cannot go separately through either 1S 1/2 → 2P 1/2 → 1S 1/2 or 1S 1/2 → 2P 3/2 → 1S 1/2 . Instead, each scattering event is a superposition of scattering through both levels simultaneously. Stenflo (1980) has shown that this introduces quantum interference terms which cause Lyα wing-scattering to be described by Rayleigh scattering for which This remarkable result implies that Lyα wing scattering can produce three times more polarization than Lyα resonant scattering (Stenflo 1980).

Incorporating Lyα Polarization in Radiative Transfer Codes
The code we used for the Monte-Carlo Lyα radiative transfer is described in Dijkstra et al. (2006). The code follows individual Lyα photons through spherical concentric shells with user-specified density and velocity fields. The code accurately describes the process of frequency and spatial diffusion in Lyα radiative transfer as described in § 2.1 (including other effects that were not discussed such as for example energy loss due to recoil). For a detailed description of the code, the interested reader is referred to Dijkstra et al. (2006). The code was adapted to the context of this paper as follows: Figure 2. Schematic illustration of the polarization of scattered Lyα radiation. Arrows pointing away (towards) the galaxy represent a radiation field for which I l < Ir (i.e. P(α) > 0), while I l > Ir is represented by arrows tangential to spheres of constant radius. The magnitude of the polarization is represented by the size of the arrows. The arrows vanish for unpolarized radiation. In all cases considered in this paper, P(α) < 0 (this corresponds to a Stokes parameter Q < 0), and the polarization vectors form concentric shells surrounding the central source.
• The modified code uses different phase functions for scattering in the core and wing as described in § 2.2. The original code (which was constructed for a different purpose) assumed only the Rayleigh scattering phase function.
• In § 2.2 we showed that the polarization of scattered Lyα photons depends strongly on whether Lyα was resonantly scattered or not. We also mentioned in § 2.1 that the transition between resonant and wing scattering occurs at x ∼ 3.3. However, it is possible to determine whether a photon scattered resonantly for each scattering event directly from the Monte-Carlo simulation. While generating the velocity of the atom that scatters the Lyα photon, a scattering event is defined to be resonant if it occurred less than xcrit = 0.2 Doppler widths away from resonance in the frame of the atom involved in the scattering event (see Appendix A for a justification of this number). The precise choice of xcrit does not affect our main results.
• The code was modified to include polarization based on the scheme developed by . Although their method is applicable strictly to wing/Rayleigh scattering, only minor modifications are required to calculate polarization for resonantly scattered Lyα photons. This is mainly because resonant scattering is a superposition of isotropic and Rayleigh scattering. We will briefly discuss the  method, before describing our modifications.  assigned 100% linear polarization to individual photons by endowing each Lyα photon with a polarization vector e, which is perpendicular to the photon's propagation direction n, i.e. e · n = 0. In this formulation, the Stokes parameters result from binning together multiple independent photons.
The propagation direction (n ′ ) of a photon after scattering is obtained by means of a rejection method: A random direction n ′ , and a random number R ∈ [0, 1] are generated. The new propagation direction is accepted when R < 1 − (e · n ′ ) 2 (as shown in Appendix A3, this corresponds to a more general case of the phase-function p(θ) given in Eq 4). The polarization vector after scattering (e ′ ) is obtained by normalizing the vector g ′ , which is obtained by projecting the polarization vector prior to scattering (e) onto the plane normal to n ′ . Symbolically, e ′ = g/|g|, where g = e − (e · n ′ )n ′ .
The observed radiation is characterized by its intensities parallel (I l ) and perpendicular (Ir) to the radius vector to the location of last scattering, denoted by r, projected on the plane of the sky (see Figure 1), denoted byr. For the last scattering event n ′ is perpendicular to the plane of the sky, and the polarization vector e ′ lies in the plane of the sky (see Fig 1). We use χ to denote the angle between the polarization vector e ′ andr. A photon that is observed from impact parameter α ≡ r/dA(z) (with dA(z) being the angular diameter distance to redshift z) contributes cos 2 χ to I l and sin 2 χ to Ir for Rayleigh (or wing) scattering .
However, for resonant scattering a photon contributes pR cos 2 χ + 1 2 (1 − pR) to I l and pR sin 2 χ + 1 2 (1 − pR) to Ir. Here, pR = (1 + cos 2 θ)/( 11 3 + cos 2 θ) is the probability that a core photon was Rayleigh scattered in the last scattering event (see Appendix A1), in which θ is the angle between the propagation directions of the incoming and outgoing photons at last scattering (cos θ = n · n ′ ). The fact that pR depends on angle θ has a simple reason. Although resonant scattering may be viewed as a superposition of Rayleigh scattering and isotropic scattering of weights 1/3 and 2/3, these weights are averaged over solid angle. The actual weight of each process is angle dependent (see Appendix A). In other words, in the case of resonant scattering the intensities in radial and azimuthal directions are obtained by scaling down the intensities obtained from Rayleigh scattering by a factor of pR, and by adding an unpolarized intensity of magnitude 1 − pR.
The observed fractional polarization at impact parameter α is given by The fractional polarization can also be expressed in terms of the Stokes parameters Q and I as P = |Q|/I. In this paper, we always find that I l < Ir, and Q < 0. This may be represented by arrows drawn in concentric rings around the central Lyα emitter, where the size of an arrow indicates the magnitude of P (see Fig 2).
• Monte-Carlo calculations of Lyα transfer can be accelerated by skipping scattering events that occur in the line core (e.g. Ahn et al. 2002;Dijkstra et al. 2006). One has to be somewhat careful when applying this technique in calculations of polarization, as it reduces the number of scattering events that a Lyα photon encounters and hence distorts its resulting polarization state. However, we have verified explicitly that the polarization properties of Lyα emerging from optically thick clouds are correctly calculated by the accelerated scheme. Generally speaking, the accelerated Monte-Carlo scheme yields the correct polarization whenever this scheme reliably computes other observable quantities such as the spectrum and surface brightness profile.
Lastly, we point out that the expressions for p(θ) and Π(θ) in § 2.2 were derived under the assumption that the radiation field prior to the scattering event was unpolarized.
The precise functional forms of p(θ) and Π(θ) are different when the radiation field prior to scattering is polarized (see Appendix A1). The procedure of propagating the polarization vector in a single Rayleigh scattering event that was outlined above, naturally accounts for the polarization dependence of Π(θ) and p(θ) (see Appendix A3). For resonantly scattered Lyα the situation is more complex and the density matrix formalism should be used (e.g. Lee 1994; Lee & Ahn 1998;Ahn et al. 2002). However, the precise dependence of p(θ) and Π(θ) on initial polarization does not affect the calculations regarding resonant scattering. The maximum fractional polarization we find for resonantly scattered Lyα is 7% (see § 3.2). The typical polarization of the Lyα radiation field 'seen' by atoms involved in the scattering process is even lower. This only introduces changes in the phase function at the level of a few per cent, and we have verified that our results regarding resonantly scattered Lyα are insensitive to the precise form of the phase function.

LYα POLARIZATION AROUND HIGH-REDSHIFT SOURCES
Next we calculate the expected polarization of the Lyα line in a set of models which span the likely environments of high-redshift galaxies. Unless otherwise stated, we assume the redshift of Lyα sources to be z = 5.7 in all cases. The Monte-Carlo radiative transfer calculations rely on stacking individual photons into bins, and the errorbars shown in some figures were calculated assuming Poisson fluctuations in the number of photons within a given bin.

Description of the Model
A major fraction of Lyman break galaxies (LBGs) at high redshifts show evidence of being surrounded by outflowing enriched gas (Shapley et al. 2003). Scattering of Lyα photons by neutral hydrogen atoms in these outflows may cause the observed Lyα line to be redshifted systematically relative to the systemic velocity of the galaxy (Ahn et al. 2003). This redshift is attributed to the Doppler boost that Lyα photons experience when they scatter off the outflow on the far side of the galaxy back towards the observer (hence the term 'backscattering'). This redshifted Lyα flux is less prone to resonant scattering in the IGM, and is therefore more easily observable than Lyα photons that were not backscattered. The impact of the outflow on the observable properties of the Lyα line depends on various parameters including the outflow speed, vexp, the total column density of neutral hydrogen atoms in the outflow, NHI, and the dust content and distribution inside the outflow (Hansen & Oh 2006;Verhamme et al. 2006). We show results for a single expanding thin shell of gas with NHI = 10 19 cm −2 and NHI = 10 20 cm −2 , based on column densities that are observed in Lyα emitting galaxies (Kunth et al. 1998;Verhamme et al. 2008). We discuss the choice of our model in more detail after presenting our results below. The outflow speed is typically of order the circular velocity of the host dark matter halo, vcirc (e.g. Furlanetto & Loeb 2003). We assume the galaxy to be embedded in a halo of total  mass Mtot = 3 × 10 11 M⊙, which corresponds to vcirc = 200 km s −1 .

Results
The emitted Lyα spectrum prior to scattering is assumed to be a Gaussian with a velocity width σ = vcirc (Dijkstra et al. 2007b) and is denoted by the dotted line in the left panel of Figure 3. The dashed (solid) lines represent the observed spectrum for an outflow with NHI = 10 19 cm −2 (10 20 cm −2 ). The Figure shows that backscattering off the expanding out- Figure 5. The frequency dependence of the polarization at α = 0 (thin black histogram), 0.5αc (red dashed histogram), and αc (blue dotted histogram). The polarization increases with impact parameter. The thick solid line shows the observed spectrum which was also shown in Fig 3 (the units are arbitrary). The bulk of the photons have frequencies −50 < x < 10. Within this frequency range, the polarization increases towards the red. This is because the redder Lyα photons appear farther from resonance in the frame of the expanding gas; consequently they scatter less and achieve a higher polarization amplitude (see Fig 4). This generic frequency dependence of the polarization amplitude can be used as a fingerprint of outflows. The frequency dependence is reversed for infalling gas (see § 3.3).
flow results in a systematic redshift of the Lyα line (as mentioned above), with an increase in redshift as NHI increases. The top right panel shows that the surface brightness profile remains flat out to some value of α = αc after which it drops to zero (as there is no scattering outside the shell). The magnitude of αc and the surface brightness S(α) ≡ I l (α) + Ir(α) are determined by the radius of the expanding shell. For example, if the outflow is lo-cated at a distance d = 10 kpc from the central galaxy (and its thickness ≪ d), then the surface brightness profile drops to zero at αc = 1.6 ′′ (d/10 kpc) for a galaxy at z = 5.7. The mean surface brightness scales approximately 2 as S ≈ X × 10 −18 (αc/1.6 ′′ ) −2 (LLyα/10 43 erg s −1 ) erg s −1 cm −2 arcsec −2 , where X = 3 for NHI = 10 19 cm −2 , and X = 5 for NHI = 10 20 cm −2 .
The bottom right panel shows that the fractional polarization, P(α), increases towards larger values of α and reaches a maximum near αc. The polarization amplitude obtains values as high as ∼ 40% for NHI = 10 19 cm −2 , and ∼ 18% for NHI = 10 20 cm −2 .
The reason for the overall large values of the polarization is easy to understand. Although the line center optical depth through the outflow is very large for both models, τ0 = 5.9 × 10 5 (NHI/10 19 cm −2 ) (see Eq 1), most photons reside in the wings of the line profile when they reach the expanding shell. For example, in the model with NHI = 10 19 cm −2 , the optical depth is less than unity for 63% of the photons when they enter the outflow for the first time. Therefore, most photons scatter only once in the wing of the line profile before escaping to the observer.
The above interpretation is quantified by Figure 4, in which the thick solid line shows the probability distribution for the number of scattering events encountered by the Lyα photons, P (Nscat). The function P (Nscat) peaks at Nscat = 1 and rapidly decreases for increasing Nscat. Also shown as the histogram is the angle-averaged polarization of the photons as a function of Nscat. The polarization is largest for photons that scatter only once ( P (α) ∼ 30%), and nearly vanishes for Nscat 10. The decrease in the polarization amplitude with increasing Nscat reflects that the radiation field becomes increasingly isotropic as the number of scattering events increases. As NHI increases, the fraction of photons that escape after a single scatter decreases, and the overall polarization declines. Figure 5 shows the frequency dependence of the polarization at three different impact parameters, namely α = 0 (thin black histogram), 0.5αc (red dashed histogram), and αc (blue dotted histogram). As already illustrated in Figure 3, the polarization increases with increasing impact parameter. The thick solid line shows the observed spectrum which was previously shown in Fig 3. The bulk of photons have frequencies −50 < x < 10. Within this frequency range, the polarization increases towards the red. This is because the redder Lyα photons appear farther from resonance in the frame of the gas; they therefore scatter less and achieve a higher polarization amplitude (see Fig 4). This frequency dependence of the polarization can be used as a fingerprint to distinguish outflows from infall (see § 3.3).
One may wonder whether the above results depend sensitively on the assumed model: e.g. do the result change for a continuous, dusty, wind with a radial dependence of its outflow speed? A wind for which the outflow speed increases linearly with radius would resemble the IGM with a Hubble flow as discussed in , who found even higher levels of polarization. Furthermore, scat-tering through an optically thick collapsing gas cloud also results in a comparable polarization (Pmax ∼ 35%, see § 3.3). Changing the sign of the velocity of the gas only affects the frequency dependence of the polarization (see § 3.3). The column density of neutral hydrogen in both these models is NHI ≫ 10 20 cm −2 , which implies that polarized Lyα is also expected for outflow models with larger column densities and non-zero velocity gradients.
Furthermore, the presence of dust in the expanding wind can boost the polarization. The reason is that photons that scatter only once obtain the highest level of polarization, while it decreases for photons that scatter multiple times (Fig 4). However, photons that scatter multiple times traverse a longer path through the wind, which enhances the probability for absorption by dust. Hence, dust can preferentially quench the low polarization photons, which would strengthen our results.

Description of the Model
Even after reionization, residual neutral hydrogen in the IGM can scatter up to > ∼ 90% of the Lyα emitted by galaxies out of our line of sight (Santos 2004;Dijkstra et al. 2007b). This scattered Lyα radiation would appear as a faint Lyα halo surrounding the central Lyα emitting galaxy . We note that back-scattered Lyα radiation is much less prone to resonant scattering in the IGM. Therefore we now consider galaxies that are not surrounded by wind-driven outflows, and examine examples that complement the cases discussed previously.
We adopt the fiducial model described by Dijkstra et al. (2007b) to examine the impact of the IGM on the observed Lyα line profile of a z = 5.7 galaxy embedded in a dark matter halo with a total mass of Mtot = 10 11 M⊙. The halo has a virial radius rvir = 23 kpc and a circular velocity vcirc = 133 km s −1 . The galaxy is assumed to form stars at a rate ofṀ * = 10M⊙ yr −1 , and the escape fraction of its ionizing photons is taken to be fesc = 0.1. The resulting Lyα luminosity of the galaxy is LLyα= 2 × 10 43 erg s −1 for a gas metallicity of Z = 0.05Z⊙. Gas within the virial radius is assumed to be fully ionized (xH = 0) and no scattering occurs here. Just outside the virial radius the IGM density, ρ(rvir) = 20ρ, whereρ is the mean baryon density of the Universe. The density decreases as ρ(r) ∝ r −1 until it approachesρ. Just outside the virial radius, the IGM is assumed to be collapsing at vIGM ∼ vcirc. Only at sufficiently large radii (r > 10rvir) the IGM is comoving with the Hubble flow (see Barkana 2004 and Dijkstra et al 2007b for a more detailed description and a motivation of this model) 3 .
We also examine a case where we artificially boost the Figure 6. The observable properties of Lyα radiation that is resonantly scattered in the intergalactic medium (IGM). The cases considered here represent galaxies that are not surrounded by a superwind-driven outflow and complement the examples shown in Fig 3  (see text). The IGM is assumed to have a peculiar infall velocity (see text). The upper left panel shows the spectrum of Lyα radiation that was transmitted without scattering once as the red dashed line. This spectrum reflects the line-emission spectrum observed from the galaxy (see text). The spectrum of resonantly scattered Lyα radiation is shown as the solid line (both spectra are normalized). The upper right panels show that the fractional polarization is only P(α) ∼ 2%. The main reason is that each photon resonantly scatters ∼ 50 times, as shown by the dotted line. The lower panels show the results for the same calculation but with an increased level of the ionizing background. The boost in the ionizing background increases the transmission on the blue-side of the line (lower left panel). In this case each Lyα photon scatters on average only a few times, and the scattered Lyα is polarized up to ∼ 7% (lower right panels). Thus, the polarization of resonantly scattered Lyα photons in the IGM is low compared to that in § 3.1, and its level depends on the local level of the ionizing background.
photoionization rate in the IGM. This boost may represent: (i) an enhanced local ionizing background due to clustering of undetected surrounding sources (Wyithe & Loeb 2005;Dijkstra et al. 2007b); (ii) an enhanced ionizing background due to the presence of a nearby quasar; (iii) a galaxy at a lower redshift where the residual neutral hydrogen fraction in the IGM is lower; (iv) an enhanced local photoionization rate due to vigorous star formation in the galaxy itself, or due to an enhanced escape fraction of ionizing photons; (v) a lower neutral fraction because of a lower overall density of hydrogen nuclei along the line of sight (e.g. for a galaxy on the edge of a void.)

Results
In the top left panel of Figure 6 we show the spectrum of Lyα radiation that was transmitted without scattering once (dashed line), which represents the Lyα spectrum observed from the galaxy. This spectrum was the focus of the analysis of e.g Santos (2004) and Dijkstra et al. (2007b). The plot shows that the blue side of the Lyα line observed from the galaxy (x > 0) is eliminated by the IGM. The suppression extends into the red part of the line because of resonant scattering by residual neutral hydrogen gas that is falling onto the galaxy. Therefore, the peak of the observed spectrum is redshifted relative to the true line center by an amount which is set by the gas infall velocity. The redshift in velocity is vIGM(rvir) = −vcirc. The spectrum of the resonantly scattered Lyα halo is shown as the solid line.
The top right panel of Figure 6 shows the surface brightness profile of the scattered Lyα halo as a function of α. The observed surface brightness of the scattered Lyα photons is ∼ 10 −19 erg s −1 cm −2 arcsec −2 near the center, and decreases with increasing angular separation from the galaxy. Overall, the fractional polarization is small, P(α) ∼ 2%. Besides the fact that resonantly scattered Lyα radiation is always less polarized than the radiation in the line wings (see § 2.2), there is the fact that when the Lyα enters resonance, it typically scatters multiple times before escaping towards the observer. This is shown by the dotted line which gives the mean number of scattering events for the Lyα photons, Nscat , as a function of impact parameter α. In this model Nscat ∼ 50. Hence, the Lyα radiation field of resonantly scattered Lyα photons will locally be close to isotropic, resulting in a low polarization.
The lower panels in Figure 6 show the results for the model in which the ionizing background was boosted. The lower left panel shows that boosting the photoionization rate decreases the fraction of residual neutral hydrogen gas, which in turn increases the transmission on the blue-side of the line. The lower right panels show that the surface brightness profile falls slightly faster with a slight peak in Figure 7. The observable properties of Lyα radiation emerging from a neutral collapsing gas cloud. The model represents protogalaxies in the process of their assembly. The left panel shows that the spectrum emerging from such a cloud has a systematic blueshift. The right panel shows that the Lyα surface brightness profile (units are erg s −1 cm −2 arcsec −2 ) is rather flat (top), and that it is highly polarized. The fractional polarization increases roughly linearly towards the edge of the cloud, and reaches a maximum value of Pmax ∼ 35%.
the surface brightness profile near the virial radius. The reason for this peak is that in our model the IGM is densest right outside the virial radius, which results in the largest effective optical depth here (this also produces the dip in the spectrum). In this model each Lyα photon scatters on average only a few times which causes the scattered Lyα to be polarized up to ∼ 7%. The mean polarization decreases at α < αvir = 4 ′′ . This is because in our model no scattering occurs inside the virial radius, which extends out to αvir. Therefore, the relative number of photons that was scattered at almost-right-angles decreases at α < αvir, which in turn decreases the polarization. For example, scattered photons emerging from α = 0 ′′ were all scattered by θ = 180 • and are therefore unpolarized (Π(180 • ) = 0).
Our radiative transfer code does not account for gas clumping in the IGM (in our simulation the gas density is a smooth function of radius r). Gas clumping affects the predicted polarization: Lyα photons that resonate with gas inside denser clumps scatter more than calculated in our model, and are therefore polarized at lower levels (see Fig 4). On the other hand, photons that resonate with gas in the interclump medium scatter less and are expected to me more strongly polarized. Since this interclump medium has the largest volume filling factor, gas clumping results in net higher level of polarization. The impact of gas clumping on the predicted polarization properties -at a fixed level of the ionizing background -can therefore be mimicked by increasing the level of the ionizing background somewhat.
In summary, the polarization of resonantly scattered Lyα photons in the IGM is low compared to the values derived in § 3.1, with the maximum polarization level reaching values of Pmax < ∼ 7%. The polarization amplitude depends strongly on the local level of the ionizing background. This is because the polarization decreases with increasing effective optical depth, τ eff , which in turn increases with a decreasing local photoionization rate.  . The level of anisotropy of the Lyα radiation field, quantified by n · er (see text), is shown. For an isotropic radiation field n · er = 0. Clearly, n · er > 0 and increases with radius. Photons that are scattered towards the observer were therefore preferentially moving radially outward prior to their last scattering event. This Figure shows clearly that the Lyα radiation field becomes increasingly anisotropic with radius, which results in an increasingly large fractional polarization with impact parameter (Fig 7).

Scattering in Optically
Lyα radiative transfer through optically thick, spherically symmetric, collapsing gas clouds, which represent simplified models of protogalaxies in the process of their assembly. Here we compute the Lyα polarization properties for one of the fiducial models presented by Dijkstra et al. (2006), in which a dark matter halo of mass Mtot = 5.2 × 10 11 M⊙ collapses at z = 3 while it continuously emits Lyα over a spatially extended region (see 'model 1.' of Dijkstra et al 2006).

Results
The left panel of Figure 7 shows that the Lyα emerges with a systematic blueshift, which is due to energy transfer from the gas to the photons, combined with a reduced escape probability for photons in the red wing of the line profile (see Dijkstra et al. 2006, for a detailed discussion). The right panels of Figure 7 show the surface brightness profile (top), and the fractional polarization P(α) (bottom) as a function of α. The fractional polarization again reaches values as high as Pmax = 35%. This may be surprising given the fact that each photon scatters numerous times before emerging from the cloud. As mentioned in § 2.1, Lyα photons generally escape from an optically thick medium in the line wing, where they acquire larger polarizations than in the line core. Furthermore, each spherical shell inside the cloud must see a net outward flow of Lyα photons, or else the cloud would not cool. This is quantified in Figure 8 which shows the radial dependence of the average value of n · er. Here n denotes a photon's propagation direction prior to the last scattering event (as in Fig 1), and er denotes a unit-vector that is pointing radially outward from the center of the cloud. In this convention n · er = 1 (n · er = −1) for a photon that is moving radially outward (inward). For an isotropic radiation field n · er = 0. Clearly, n · er is positive at all radii, and increases with radius. In other words, photons that are scattered towards the observer were preferentially moving radially outward prior to the scattering event. Furthermore, this anisotropy in the radiation field 4 increases with radius, which results in an increasingly large fractional polarization with impact parameter (Fig 7).
In Figure 9 we show the frequency dependence of the polarization at 3 different angular separations: α ∼ 0, 4.0, and 8.0 arcseconds. The thick solid line is the spectrum shown previously in Figure 7, indicating that the bulk of the photons emerge with frequencies −10 < x < 80. Within this frequency range, the polarization increases towards the blue. The reason for this wavelength dependent polarization is that Lyα photons that are far in the blue wing of the line profile, appear even farther in the wing in the frame of the infalling gas. Therefore, the escape probability of Lyα photons increases, and the radiation field becomes increasingly anisotropic towards higher photon frequencies. This in turn increases the polarization towards higher frequencies. Lastly, it should be pointed out that the frequency dependence of the polarization applies mostly to the case α ∼ 8.0 arcseconds, and similarly to the case α ∼ αc that was shown in Figure 5. Hence, the frequency dependence of the polarization is evident only where the overall (frequency averaged) polarization is significant (P > ∼ 10%).

Constraints on EoR and High-z Galaxy Formation
The Lyα emission needs to be spatially resolved in order for its polarization to be measured. Over the past decade, a large number of spatially extended Lyα emitters, also known as Lyα 'blobs', were discovered (Steidel et al. 2000;Matsuda et al. 2004;Saito et al. 2006). In fact, the size distribution of Lyα sources is continuous and Lyα blobs simply 4 We point out that the actual Lyα radiation field that is 'seen' by each atom in the cloud is dominated by photons that are resonantly scattering, and is almost completely isotropic. However, these photons have a negligible probability of escaping towards the observer and do not affect the observable Lyα polarization properties of the cloud.  Fig 7). The thick solid line shows the spectrum (arbitrary units, previously shown in Figure 7). The bulk of the photons emerge with frequencies −10 < x < 80. Within this frequency range, the polarization increases towards higher frequencies, in sharp contrast with the outflow model discussed in § 3.1.
represent the largest and rarest objects in this distribution ). This implies that there is no shortage of spatially extended Lyα sources in the Universe. As was shown in § 3, the polarization properties of spatially resolved Lyα emission encode information about the medium through which the Lyα photons propagate, especially in combination with spectral information (Fig 5 and  Fig 9). For example, large fractional polarizations (tens of percent) are expected for Rayleigh scattering. An increase in the polarization towards longer wavelengths would be indicative of backscattering off an expanding outflow. Such an inference could be particularly important because (i) it provides evidence that the extended Lyα emission results from scattering (ii) it allows to better gauge the impact of resonant scattering in the IGM on the observed Lyα line profile, since back-scattered Lyα radiation is much less prone to resonant scattering in the IGM. Understanding the impact of resonant scattering on the Lyα line profiles is essential for (iia) deriving constraints on the ionization state of the IGM (Dijkstra et al. 2007a), and (iib) interpreting Lyα emitters with large equivalent widths (Malhotra & Rhoads 2002;Dijkstra & Wyithe 2007).
In this paper we have shown numerical examples for the polarization properties of Lyα radiation that is either (i) resonantly scattered in the IGM; (ii) backscattered in an outflow; or (iii) associated with cooling radiation from a collapsing protogalaxy. There are other classes of spatially extended Lyα sources that have not been discussed: • Gas in close proximity to a bright source of ionizing radiation (e.g. a quasar) may 'glow' in fluorescent Lyα emission (Haiman & Rees 2001;Weidinger et al. 2004). If the recombining Lyα emitting gas is sufficiently close to the quasar or the quasar is particularly bright, then this gas is optically thin to Lyα photons and the emergent Lyα radiation is not expected to be polarized. However, fluorescent Lyα emission is also expected from Lyman limit systems and Damped Lyα systems (Gould & Weinberg 1996;Cantalupo et al. 2005;Adelberger et al. 2006;Cantalupo et al. 2007), which are optically thick to Lyα. Especially in cases where the gas is illuminated anisotropically, scattering could lead to a detectable polarization.
• Spatially extended Lyα emission may be generated by gas that was shock heated by a superwind-driven outflow (e.g. Taniguchi & Shioya 2000;Mori et al. 2004). The backscattering mechanism described in § 3.1 may also operate here, and this could lead to a measurable polarization amplitude.
• Extended luminous Lyα emission is common around radio galaxies, and is thought to be powered by a jet-IGM interaction (e.g. Chambers et al. 1990). Little is known about the source of the Lyα emission and to what degree the Lyα is scattered by residual neutral gas. Polarimetry may provide new insights on the nature of these sources.
We have focused on spatially extended Lyα sources, as the polarization averages out to zero for a symmetric unresolved source. However, as was shown by Lee & Ahn (1998), anisotropic outflows (e.g biconical outflows) may still result in observable polarization, even if the Lyα source is completely unresolved. Hence, a measurable polarization may not be restricted to spatially resolved Lyα sources.
Lastly, we point out that accurate polarization measurements of faint high-redshift sources is difficult. It would therefore be prudent to obtain polarimetry of the brightest sources first. Polarization measurements of one of the brightest known Lyα Blobs (at z = 2.7, Dey et al. 2005) have already been attempted (Prescott et al. 2008). This approach will likely lead to insights about the processes that regulate the transfer of Lyα in and around high-redshift galaxies.

Comparison to Polarization of Foregrounds
The dominant foregrounds in the infrared sky (at wavelengths λ < 2µm), are Rayleigh scattered moonlight and starlight (e.g. Glass 1999, p.33), and are therefore polarized. However, line emitters can easily be extracted from this scattered foreground for which the spectrum is continuous. Here, we investigate whether Lyα emitters can be distinguished from foreground line emitters, such as [OII], [OIII], Hα, and Hβ emitters, based on their polarization properties.
In practice it has been possible to effectively discriminate between these low-redshift interlopers and actual highredshift Lyα emitters based on broad-band colors and on the shape of the Lyα line (e.g. Kashikawa et al. 2006). However, the continuum spectrum of some Lyα emitters with a large equivalent width is difficult to detect, and in some cases the Lyα line is not asymmetric (see e.g the lower left panel of Fig 6).
The possibility that high-redshift Lyα emitters are highly polarized may provide another diagnostic that distinguishes them from low-redshift interlopers. A low level of polarization (P < ∼ 2%) is expected for any extragalactic line emitter due to scattering by dust in the Milky-Way (Schmidt & Miller 1985). Furthermore, spatially resolved [OIII] line emission at higher levels of polarization (P ∼ 4%) has been observed in a small fraction of Seyfert 2 galaxies (Goodrich 1992). This polarization arises when the narrow line region is obscured. In this case, the only [OIII] photons that are observed are scattered towards the observer, either by dust or by electrons. This same mechanism could produce polarized [OII], Hα and Hβ line emission. This suggests it is risky to distinguish high-redshift Lyα emitters from low-redshift interlopers on the basis of polarimetry. On the other hand, the observed levels of polarization in low-redshift interlopers are small compared to the values we find in our models, which suggests that high levels of polarization (P ∼ 10%) are indicative of scattered Lyα.
Raman scattering of Lyβ wing photons may produce highly polarized Hα emission in a radiative cascade of the form 3p Hα → 2s 2γ → 1s (e.g. Lee & Yun 1998;Yoo et al. 2002). This mechanism requires the presence of large HI column densities NHI > ∼ 10 20 cm −2 , and has been observed to operate around symbiotic stars (Ikeda et al. 2004), and potentially in close proximity to quasars (Lee & Yun 1998). In distant galaxies, however, these emission regions would be completely unresolved, and the polarization averages out to zero. On the other hand, columns in excess of NHI > ∼ 10 20 cm −2 can easily exist in the neighborhood of galaxies, and it is not possible to rule out altogether the possibility of highly polarized Hα emission as a contaminant of polarized Lyα emitters. Nevertheless, polarized Hα lines produced by this mechanism are typically broad (FWHM ∼ 20 − 40Å) and symmetric, with wings extending into both the longer and shorter wavelengths (see e.g Yoo et al. 2002, their Fig 4-8); these features clearly distinguish them from Lyα line emitters.
Ground-based searches for high-redshift Lyα emitters are complicated by the presence of bright OH emission lines, which are produced by the excited radical OH * in the ionosphere, following the reaction H+O3 → OH * +O2 (e.g. Glass 1999). These lines make up the Meinel bands (Meinel 1950a,b) and are typically much brighter than Lyα sources. Furthermore, these lines exhibit large temporal and spatial variations in intensity due to the passage of density waves through the ionosphere (Ramsay et al. 1992). However, the wavelengths of these lines are well known, and future instruments (such as e.g. the Dark Age Z Lyα Explorer, Horton et al, 2004) intend to search for redshifted (z = 6.5 − 12) Lyα lines between these lines using high-resolution (R = 1000) spectrographs.
Polarimetry may provide an interesting alternative method of suppressing OH-emission lines. Although the OHlines have been studied in detail (e.g Ramsay et al. 1992;Maihara et al. 1993), we are unaware of any polarization measurements. If OH-lines are unpolarized, then polarimetry may be used to remove them efficiently 5 . For example, consider an unpolarized OH sky line that is detected at the N -σ level after an integration time tint. Suppose we repeat the same observation, but we create a first image by passing the incoming radiation through a linear polarizer for 0.5tint. Next, we create a second image by observing the remaining 0.5tint through a linear polarizer that is rotated by 90 degrees compared to the first observation. If one subtracts the two images, then any unpolarized emission is removed down the level of ∼ σ.

CONCLUSIONS
The high-redshift Universe is known to contain luminous Lyα emitting sources such as galaxies and quasars. The Lyα photons that are emitted by these sources are typically scattered both in the interstellar medium and in the intergalactic medium. In this paper we have calculated polarization properties of this scattered Lyα radiation.
We used a Monte-Carlo Lyα radiative transfer code, and endowed each Lyα photon with a polarization vector which is perpendicular to the photon's wave vector. In this formulation the Stokes parameters resulted from binning together multiple independent photons. We differentiated between resonant and wing scattering, as these are described by different scattering matrices (see Appendix A). Wing scattering is described by classical Rayleigh scattering and in this case we advanced the photon's polarization vector in a scattering event following the well-tested approach of . However, resonant scattering is described by a superposition of Rayleigh scattering and isotropic scattering with corresponding weights of 1/3 and 2/3 ( § 2.2). In this case, minor modifications to the approach of  are required when advancing the photon's polarization vector ( § 2.3).
We have applied our code to three classes of models which represent the diverse sets of environments around high-redshift galaxies: • First, we computed the polarization of the backscattered Lyα radiation observed from galaxies surrounded by a superwind-driven outflow. We have found that the fractional polarization may reach values as high as Pmax ∼ 40%, where the maximum polarization depends on parameters such as the speed of the outflow, vexp, and the column density of neutral hydrogen atoms, NHI ( § 3.1). In this case we have found the polarization to increase towards longer wavelengths.
• Second, we considered resonant scattering in the intergalactic medium (IGM) after reionization. Residual intergalactic hydrogen can scatter up to > ∼ 90% of the Lyα emitted by galaxies out of the line of sight. (This high fraction does not apply to galaxies with superwind-driven outflows where the Lyα photons are scattered out of resonance before they encounter the IGM.) We have found the polarization of the scattered Lyα halo around galaxies to be polarized at lower levels (Pmax < ∼ 7%) than the halos around galaxies with outflows. This follows from the fact that Lyα photons scatter multiple times when they enter resonance, which isotropizes ters that search for the circular polarization pattern shown in Figure 2 around the sources of interest. the local Lyα radiation field and reduces its net polarization. The polarization decreases with the mean number of scattering events the photons experience (i.e. the polarization decreases with an increasing effective optical depth in the Lyα line). For this reason, we expect the polarization of resonantly scattered Lyα photons in the IGM to be largest around low redshift sources or around sources where the IGM is more highly ionized than average (e.g. in the vicinity of a bright quasar, see § 3.2).
The overall lower levels of polarization of resonantly scattered Lyα radiation in a reionized IGM are significantly lower than that of pre-reionization Lyα halos around galaxies embedded in a fully neutral IGM . This suggests that the polarization properties of Lyα halos may help constrain the neutral fraction of the IGM during the epoch of reionization.
• Third, we considered neutral collapsing protogalaxies that are emitting Lyα cooling radiation. Despite the fact that these clouds are extremely optically thick to Lyα photons, and photons typically scatter ∼ τ times, we found the polarization to linearly increase toward the edge of the cloud, reaching a maximum amplitude of Pmax ∼ 35%. ( § 3.3). The resulting polarization increases towards shorter wavelengths in contrast to the trend found in outflow models.
Our results indicate that resolved high-redshift Lyα emission may be highly polarized under a variety of likely circumstances. High-redshift Lyα emitters are usually distinguished from low-redshift line emitters on the basis of their broad-band colors and their asymmetric spectral line shape. The work presented here implies that polarization may provide an additional diagnostic. Moreover, polarimetry has the potential to better remove the glow of infrared lines in the Earth's atmosphere, which would improve the sensitivity of ground-based observations to high-redshift Lyα emitting galaxies outside the currently available redshift windows.
The polarization properties of Lyα radiation encode information about the distribution and kinematics of neutral gas in and around galaxies. Polarimetry therefore complements the constraints that are derivable from spectroscopy. Hence, from a theorist's perspective it is well worth to include polarization in Monte-Carlo calculations of Lyα radiative transfer.
The redshifted Lyα line has provided us with an important window into the high-redshift Universe. This work suggests that in order to fully exploit the observations, one should focus on both Stokes parameters I and Q, rather than just I. Polarization measurements to better than 1% accuracy can be carried out by existing facilities such as the FOcal Reducer/low dispersion Spectrograph (FORS1) 6 on the VLT, the LRIS imaging spectropolarimeter at the W.M. Keck Observatory (Goodrich & Cohen 2003), and the CIAO polarimeter on Subaru (Tamura et al. 2003). The results of this paper imply that Lyα emitting sources would provide excellent future targets for these instruments.
A2 Quantum Interference as in Stenflo (1980) In § 2.2 the phase function and the degree of polarization caused by resonant and wing scattering were given. Here we discuss the transition between these two regimes. Stenflo (1980) has calculated the value of E1 as a function of frequency (but note that E1 = W2 in the notation of Stenflo 1980, Eq. 3.25). We have plotted E1 as a function of wavelength (bottom label) and frequency (top label) in Figure A1. Figure A1 shows that E1 = 1 2 at the H resonance frequency and E1 = 0 at the K resonance frequency, which correspond exactly to the values quoted above. The Figure  also shows that at sufficiently large separations from both resonances, E1 → 1. In dimensionless frequency units, this asymptotic value is practically reached when |xat| > ∼ 0.2. This was the motivation for using the threshold xcrit = 0.2 to separate core from wing scattering in § 2.3.
Interestingly, E1 < 0 for a range of frequencies between the H and K resonance frequencies. However, scattering at these frequencies does not occur often enough to leave an observable imprint. The reason for this is simple: the natural width of the line for both the H and K transitions is much smaller than their separation, i.e γH,K ∼ 10 8 Hz ≪ νH − νK = 1.1×10 10 Hz. Since the absorption cross-section in the atom's rest-frame scales as, σ(ν) ∝ [(ν − νH,K) 2 + γ 2 H,K /4] −1 , a Lyα photon is much more likely absorbed by an atom for which the photon appears exactly at resonance, than by an atom for which the photon has an energy corresponding to a negative W2. Quantitatively, the Maxwellian probability P that a photon of frequency x is scattered at frequency xat in the atom's rest-frame is given by where H(a, x) is the Voigt function (H(a, x) ≡ τx/τ0, see Eq 2). This probability is shown in Figure A2 for 8 x = 3.3 (solid line) and x = −5.0 (dashed line). Figure A2 shows that for x = 3.3, photons are scattered either when they are exactly at resonance or when they appear ∼ 3 Doppler widths away from resonance. The inset of Figure A2 zooms on the region near xat = 0.0. Clearly, if a photon is resonantly scattered, then it is more likely to be scattered when it is exactly at resonance then when it is, say, 0.05 Doppler widths away from resonance. For frequencies |x| > 3.3 there are not enough atoms moving at velocities such that the Lyα photon appears at resonance in the frame of the atom. Instead, the photon is scattered while it is in the wing of the absorption profile. This is illustrated by the dashed line which shows the case x = −5.0, for which resonant scattering is less likely by orders of magnitude. From the above we conclude that (i) when a photon is scattered in the wing of the line profile, then E1 = 1 and (ii) when a photon is scattered resonantly, then it is scattered almost exactly at resonance, and therefore E1 = 1