COSMIC MICROWAVE BACKGROUND CONSTRAINTS ON THE DURATION AND TIMING OF REIONIZATION FROM THE SOUTH POLE TELESCOPE

Scattering of CMB photons into the line of sight introduces temperature anisotropy through the Doppler effect if there are perturbations in the baryon density, ρ_b, or ionization fraction, x_e. These scatterings slightly change the temperature of the CMB blackbody. The CMB Doppler effect is often called the kSZ effect (Sunyaev & Zel'dovich 1970, 1980) when referring to reionization or the nonlinear regime of structure formation, and the Ostriker–Vishniac effect (Ostriker & Vishniac 1986) when referring to the linear regime, but we will call it the kSZ effect throughout. The total contribution to the temperature anisotropy from a redshift interval [z₁, z₂] is given by

$$\begin{equation} \frac{\Delta T_{\rm kSZ}}{T_{\rm CMB}} ({\bf \hat{n}}) = \frac{\sigma _T}{c} \int _{z_1}^{z_2}\frac{dx}{dz}\frac{dz}{(1+z)} \bar{n}_e(z) e^{-\tau (z)}{\bf \hat{n}}\cdot \mathbf {q}, \end{equation} \tag{ 1 }$$

where σ_T is the Thomson scattering cross section, dx/dz is the comoving line element, and ${\bf \hat{n}}$ is the line-of-sight unit vector. $\bar{n}_e(z)$ is the mean free electron density,

$$\begin{equation} {\bar{n}}_e(z) = \frac{\bar{x}_e(z)\bar{\rho }_b(z)}{\mu _e m_p} \,, \end{equation} \tag{ 2 }$$

where $\bar{x}_e(z)$ and $\bar{\rho }_b(z)$ are the mean free electron fraction and mean baryon density of the universe as functions of redshift and μ_em_p is the mean mass per electron. We set μ_e = 1.22, appropriate for singly ionized helium. Due to their similar ionization potential, we assume that helium is singly ionized alongside hydrogen, that is, when all hydrogen is ionized and all helium is singly ionized ${\bar{x}}_e=1$. Note that ${\bar{x}}_e= 1.07$ when helium is fully ionized; our model for the post-reionization kSZ signal assumes that this occurs at z = 3.

The optical depth, τ(z), from the observer to redshift z is given by

$$\begin{equation} \tau (z)=\sigma _T \int _0^z \frac{{\bar{n}}_e(z^\prime)}{1+z^\prime }\frac{dx}{dz^\prime }dz^\prime. \end{equation} \tag{ 3 }$$

The WMAP large-scale polarization signal constrains the optical depth to the end of recombination, τ(z ∼ 1000) = 0.088 ± 0.014.

Finally, inhomogeneities in the ionization fraction and baryon density enter through

$$\begin{equation} \mathbf {q}=\left(1+\delta _x\right) \left(1+\delta _{\rm b}\right) \mathbf {v}, \end{equation} \tag{ 4 }$$

where $\delta _x=x_e/{{\bar{x}}_e}-1$, $\delta _b=\rho _b/{\bar{\rho }_b}-1$, and v is the bulk motion of free electrons with respect to the CMB.

We discuss the post-reionization kSZ signal in Section 4.1.1. Reionization produces kSZ power primarily through local changes in the ionization fraction. To predict the amplitude of this signal we require a prescription for calculating the evolution of the ionization morphology during the epoch of reionization.

Our ability to infer the properties of the first ionizing sources from observations hinges on the accuracy with which the ionization morphology during reionization can be modeled. A number of groups have developed three-dimensional (3D) radiative transfer codes (e.g., Gnedin 2000; Sokasian et al. 2001; Razoumov et al. 2002; Ciardi et al. 2003; Mellema et al. 2006; McQuinn et al. 2007b; Semelin et al. 2007; Trac & Cen 2007; Altay et al. 2008; Aubert & Teyssier 2008; Finlator et al. 2009; Petkova & Springel 2009). However, the mass resolution and volume requirements for simulations of reionization are daunting. Simulations must resolve the low-mass galaxies (down to the atomic cooling threshold corresponding to a host-halo mass ∼10⁸ M_☉ at z ∼ 7–10) that are expected to dominate the ionizing photon budget. They must also be large enough to statistically sample the distribution of ionized regions. These regions can span tens of comoving megaparsecs in size toward the end of reionization (Furlanetto & Oh 2005; Zahn et al. 2005, 2007; Mesinger & Furlanetto 2007; Shin et al. 2008). Recently, some groups have come close to achieving this dynamic range in a single simulation (Iliev et al. 2006; McQuinn et al. 2007b; Trac & Cen 2007; Shin et al. 2008; Trac et al. 2008; Zahn et al. 2011; see Trac & Gnedin 2009 for a recent review).²⁶

Given the large uncertainties in the production and escape rates of ionizing photons in high-redshift galaxies (and also in how the photons are absorbed by dense systems), a large parameter space must be explored to interpret observations. These concerns have prompted several groups to develop much more CPU efficient, approximate algorithms (Furlanetto et al. 2004; Zahn et al. 2005, 2007, 2011; Mesinger & Furlanetto 2007; Geil & Wyithe 2008; Alvarez et al. 2009; Choudhury et al. 2009; Thomas et al. 2009). The basic idea is that semi-analytic models can be applied in a Monte Carlo fashion to large-scale realizations of the density and velocity fields. The resulting complex reionization morphology can be compared side by side with simulations based on the same initial conditions combined with the same source and sink prescriptions. We use a modification of the Zahn et al. (2011) model to set up our reionization parameter space.²⁷

The model associates ionized regions with the ionizing sources they contain. The size distribution of ionized regions is related to the halo mass function through the ansatz, M_ion = ζM_gal. M_gal is the mass in collapsed objects. ζ is the efficiency factor for ionization, and can be decomposed as ζ = f_escf_*N_γ/bn⁻¹_rec. Here, f_esc is the escape fraction of ionizing photons from the object, f_* is the star formation efficiency, N_γ/b is the number of ionizing photons produced per baryon converted into stars, and n_rec is the typical number of times a hydrogen atom has recombined. Although the efficiency factor is a rough combination of uncertain source properties, it can encapsulate a wide variety of reionization scenarios.

We calculate M_gal according to the extended Press–Schechter model (Bond et al. 1991). In this model, the collapsed fraction (or the fraction of baryons that lie in galaxies) in a region of size r depends on the mean overdensity of that region, $\overline{\delta }(r)$, as

$$\begin{equation} f_{\rm coll}(r,M_{\rm min})={\rm erfc}\left[\frac{\delta _c(z)-\overline{\delta }(r)}{\sqrt{2[\sigma ^2(r_{\rm min})-\sigma ^2(r)]}}\right] \,. \end{equation} \tag{ 5 }$$

Here, δ_c(z) is a numerical factor from linear theory equal to 1.686 today, and σ²(r) is the linear-theory rms fluctuation smoothed on scale r. r_min is defined as the radius that encloses the mass M_min (at average density $\overline{\rho }$) corresponding to a virial temperature of 10⁴ K, above which atomic hydrogen line cooling becomes efficient. The redshift dependence of the minimum mass (see Barkana & Loeb 2001) is given in terms of the virial temperature:

$$\begin{equation} M_{\rm min} \simeq 10^8 \frac{M_\odot }{h} \left[\frac{T_{\rm vir}}{2 \times 10^{4}\,{\rm K}} \frac{10}{1+z} \right]^{3/2} \left[\frac{\Omega _0}{\Omega _m(z)} \frac{\Delta _c(z)}{18 \pi ^2}\right]^{-1/2}, \end{equation} \tag{ 6 }$$

with the fitting function Δ_c(z) = 18π² + 82d − 39d² (Bryan & Norman 1998), where d ≡ Ω_m(z) − 1 is evaluated at the collapse redshift. The luminous mass in galaxies required to fully ionize all hydrogen atoms is inversely proportional to the ionizing efficiency, so ionization requires

$$\begin{equation} f_{\rm coll} \ge \zeta ^{-1} \,. \end{equation} \tag{ 7 }$$

Hence, we can define a barrier δ_x(r, z) that fluctuations have to cross for their baryonic content to become ionized:

$$\begin{equation} \delta (r) \ge \delta _x(r,z) \equiv \delta _c(z)-\sqrt{2} {\rm erfc}^{-1}(\zeta ^{-1}) [\sigma ^2(r_{\rm min})-\sigma ^2(r)]^{1/2} \,. \end{equation} \tag{ 8 }$$

The algorithm then proceeds as follows. For every position, the linear matter overdensity is calculated within a range given by the smoothing kernel (initially set to a large value comparable to the simulation size). Zahn et al. (2011) show that good agreement with radiative transfer simulations is achieved when the smoothing kernel is a top hat in harmonic space. The halo collapse fraction is estimated from this overdensity, and translated with the efficiency factor ζ into an expected number of ionizing photons based on the number of collapsed halos above the atomic cooling threshold mass. The number of ionizing photons is then compared to the number of hydrogen atoms at each point. If there are sufficient ionizing photons, a point is labeled ionized. If not, a smaller radius is set and the algorithm repeated until the resolution of the simulation box is reached.

We optimize the algorithm for mock patchy kSZ spectra at the SPT angular resolution. First, we simulate the density and velocity fields in a rectangular box with a high dynamic range, 1² × 3 Gpc h⁻¹ with 512² × 1536 volume elements. The simulation box subtends 8.5 deg on the sky at the central z = 9 plane, yielding arcminute resolution and low sample variance on the scales of interest. Along the z-axis, the comoving distance is translated to redshift. The volume thus extends from z ≃ 4 to z ≃ 27. The large redshift extent allows us to capture most reasonable reionization scenarios. The densities, velocities, expansion rate, minimum virial mass for star formation, and excursion set barrier (Equation (8)) are adjusted with redshift. The large angular and line-of-sight coverage are important because the ionized regions and large-scale velocity streams that source the kSZ signal appear on scales of tens to hundreds of megaparsecs.

In addition to the global ionization efficiency parameter, ζ, which sets the timing of reionization, it is important to allow for an additional degree of freedom to capture the influence of physical processes that may impact the duration, such as feedback effects and recombinations. To do this, we add a "feedback" parameter, α, that makes the effective ionization efficiency an inverse power law of the expected ionization fraction in the absence of feedback, x*, which is a measure of the ionizing photons available at a given time:²⁸

$$\begin{equation} \zeta =\zeta _0 \left(\frac{1}{x^*}\right)^\alpha , \quad x^*=\zeta _0 f_{\rm coll}(r=\infty ,M_{\rm min}) \,. \end{equation} \tag{ 9 }$$

A database of simulated models is set up for ζ ∈ [10, 1000] and α ∈ [ − 50, 1]. This leads to ∼10, 000 simulations that have reionization durations of Δz ∈ [0.1, 12], and ending redshifts of z_end ∈ [4, 16].²⁹ Here we have defined Δz = z_beg − z_end, $z_{\rm end}=z_{{\bar{x}}_e=0.99}$, and $z_{\rm beg}=z_{{\bar{x}}_e=0.20}$. The choice of a relatively large ionization fraction of ${\bar{x}}_e=0.20$ for z_beg is due to the fact that lower ionized fractions are largely made up of ionized regions too small to be probed by the current SPT data, hence we do not attempt to constrain the reionization state preceding this stage. An example of a 20 Mpc h⁻¹ deep projection of an extended reionization simulation with ζ = 12, α = 0.8, and Δz ≃ 9 is shown in Figure 1.

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For each reionization simulation, we interpolate rays from constant comoving positions to constant angular positions, produce a map of the kSZ with a size of ≃ 7 × 7 deg, and measure its angular power spectrum. Throughout this paper, power spectra are given in terms of D_ℓ ≡ C_ℓℓ(ℓ + 1)/2π (μK²). For each simulation, we also calculate the total integrated optical depth assuming helium is singly ionized alongside hydrogen and doubly ionized at z = 3 (the latter assumption has a negligible influence on the results).

The spatial resolution and box size used lead to convergence of the patchy kSZ amplitude on the relevant angular scale (see the next section). However, the ionized fraction is underestimated during the early stages of reionization, when ionized bubbles smaller than the spatial resolution of the simulation contribute significantly. To correctly predict the duration and integrated optical depth we therefore scale the ionized fraction at each fraction to the analytical value ζf_coll(r = ∞, M_min), the mean ionization degree of the universe at that redshift.

In these simulations, we assume a ΛCDM cosmology with Ω_ch² = 0.111, Ω_bh² = 0.0222, Ω_Λ = 0.736, n_s = 0.96, τ = 0.085, and σ₈ = 0.8, consistent with Komatsu et al. (2011). We discuss the cosmological scaling of the kSZ predictions in Section 4.1.1 and the effects of uncertainty in the cosmological parameters in Section 5.2.

The amplitude and shape of the patchy kSZ power spectrum depend on the redshift evolution of the ionized fraction and the reionization morphology. Here we discuss the dependence of the kSZ power spectrum on the physics of reionization and motivate the template shape used to fit to the SPT data in order to constrain the ionization history.

The reionization model outlined in Section 2.2 has been shown to agree well with full radiative transfer simulations on scales up to 100 Mpc h⁻¹ (see, e.g., Zahn et al. 2007, 2011). As a generic consequence of the increase in source collapse fraction with time, the size distribution of ionized regions and volume filling factor evolve more slowly at the beginning of reionization than at the end. Also, the roles of feedback effects and recombinations depend on the degree of ionization. If we instead used a model with a parameterized yet physically unmotivated analytic form of ${\bar{x}}_e$, the relation between the redshift corresponding to various ionization fractions $z_{{\bar{x}}_e}$, the kSZ power, and integrated optical depth τ would be different, potentially biasing constraints on ${\bar{x}}_e$.

Using radiative transfer simulations, it has been found that the size distribution of ionized regions at fixed ${\bar{x}}_e$ is relatively robust to redshift translations z₁ → z₂ (e.g., Zahn et al. 2007; McQuinn et al. 2007b). In addition, the angular size corresponding to a given comoving scale varies little across the redshifts of interest. The change in the patchy kSZ power spectrum due to a translation from z_mid ≃ 7.5 to z_mid ≃ 11.5, where $z_{\rm mid} \equiv z_{{\bar{x}}_e=0.5}$, for similar Δz is shown by the black solid and dot-dashed lines in Figure 2. The shapes of both power spectra are very similar, with the main difference being in their normalization. The amplitude is increased in the z_mid ≃ 11.5 model due to the higher mean density of the universe at higher redshift, which is counteracted somewhat by linear growth of density and velocity fields toward later times.

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When moving to higher redshifts, rarer sources have to yield enough ionizing photons. Such sources are more heavily biased, and their lower number density yields larger and more spherical ionized regions (Zahn et al. 2007; McQuinn et al. 2007b). The same is true at low redshifts if feedback effects limit the role of low-mass sources. For instance, low-mass sources might suffer from thermal feedback onto their host galaxies during early stages of reionization (Hultman Kramer et al. 2006; Iliev et al. 2007a). The effect on the patchy kSZ power spectrum of increasing the minimum source mass to 10¹⁰ M_☉ (an increase of two orders of magnitude over the fiducial M_min, discarding >90% of sources, e.g., Trac & Cen 2007) is shown by the black solid and dashed lines in Figure 2. The impact of source bias on the shape of the patchy kSZ template is found to be small compared to our experimental uncertainty.

Furthermore, recombinations in dense systems might effectively stall bubble growth early on in reionization, leading to a slowly percolating web of similarly sized ionized regions (Furlanetto & Oh 2005). To estimate the effect of such a scenario on the kSZ power spectrum, we impose a relatively extreme maximum bubble scale of r_max = 5 Mpc h⁻¹. The result is shown by the blue solid line in Figure 2. Again, the changes are modest when compared to a model with the same duration but no limit on the bubble scale (dotted curve).

We conclude that the shape of the kSZ power spectrum due to patchy reionization is relatively robust to the effects of redshift translation, feedback, and recombinations. Most importantly for this work, the amplitude is expected to be proportional to the duration of reionization for uncorrelated ionized regions and fixed z_mid (e.g., Gruzinov & Hu 1998). If the duration doubles in this Poisson process, there are twice as many independently moving ionized regions along the line of sight, which doubles the power. We find this to be a good approximation to the results from our simulations on the angular scales of interest (compare solid versus dotted lines in Figure 2).

Given the robustness of the shape of the patchy kSZ power spectrum to all these changes, we will assume a fixed template shape to derive our reionization results. As base template we choose a model with an efficiency of ζ = 20, making this model extend from $z_{{\bar{x}}_e=0.2}\simeq 11.0$ to $z_{{\bar{x}}_e=0.99}\simeq 7.8$; however, as explained above the results are robust to variations in this choice. The base template used to compare to the SPT data was run at eight times higher spatial resolution than the simulations in the database. The resolution of the latter was chosen to lead to convergence at ℓ = 3000 to allow comparison to the data-derived constraint on the power on that scale; see Section 5.2.

We adopt two main templates for the kSZ power spectrum. The first template includes only the contribution from patchy reionization (see Section 2). The second and more realistic template includes roughly equal contributions from the reionization and post-reionization epochs. We henceforth refer to the post-reionization, homogeneously ionized component as the "homogeneous kSZ" signal, and that from reionization as the "patchy kSZ" signal. We discuss the homogeneous kSZ model next.

4.1.1. Homogeneous Kinetic Sunyaev–Zel'dovich Effect

For the homogeneous kSZ signal, we adopt the cooling plus star formation (CSF) model presented by Shaw et al. (2011). This model is constructed by calibrating an analytic model for the homogeneous kSZ power spectrum with a hydrodynamical simulation including metallicity-dependent radiative CSF. Shaw et al. (2011) measured the power spectrum of gas density fluctuations in the simulation over a range of redshifts, and used this to calculate the homogeneous kSZ power spectrum. The approximate scaling for this model with cosmological parameters is given by

$$\begin{eqnarray} D^{\rm homog}_{3000}&\simeq & 1.9\,\mu {\rm K}^2 \left(\frac{h}{0.71}\right)^{1.7} \left(\frac{\sigma _8}{0.80}\right)^{4.5} \left(\frac{\Omega _b}{0.044}\right)^{2.1} \nonumber \\ &&\times \left(\frac{\Omega _m}{0.264}\right)^{-0.44} \left(\frac{n_s}{0.96}\right)^{-0.19}. \end{eqnarray} \tag{ 10 }$$

This scaling (but not amplitude) is also a good approximation to the cosmological dependence of the patchy kSZ component. We vary the predicted post-reionization D^homog₃₀₀₀ with z_end when interpreting the results in Section 6. The CSF model kSZ component for z_end = 8 is shown by the red line in Figure 2. The model utilized here assumes helium is singly ionized between 3 ⩽ z ⩽ z_end and doubly ionized for z < 3.

The CIB is produced by thermal emission from dusty star-forming galaxies (DSFGs) over a very broad range in redshift (Lagache et al. 2005; Marsden et al. 2009). The dust grains, ranging in size from a few molecules to 0.1 mm, absorb light at wavelengths smaller than their size, and re-radiate it at longer wavelengths. Sufficient absorption occurs to account for roughly equal amounts of energy in the CIB and in the unprocessed starlight that makes up the optical/UV background (Dwek & Arendt 1998; Fixsen et al. 1998).

The power spectrum of these DSFGs will have Poisson and clustered components. Both CIB components have more power than the kSZ power spectrum on the angular scales and photon frequencies of interest (except at 95 GHz where the powers are comparable). The Poisson component shows up as D_ℓ∝ℓ² since the galaxies are small (effectively point sources) compared to the angular scales probed here. We model the clustered component by a power law of the form D_ℓ∝ℓ^0.8, a shape motivated by recent observations (Addison et al. 2012; R12). We have explored allowing freedom in the angular shape via freeing the power-law exponent or adding a linear-theory template, and find that the additional shape parameter increases the kSZ uncertainties by ∼10% without changing the mean.

The frequency dependence of the CIB is a key question for the kSZ measurement. Essentially, the CIB is well measured at ⩾220 GHz by SPT, Planck, and other experiments, but the signal to noise falls off toward longer wavelengths. The CIB power at 95 GHz is an extrapolation based on a fit to the modeled frequency dependence. Following previous work, we adopt the phenomenological modified blackbody model:

$$\begin{equation} \eta _{\nu } = \nu ^\beta B_\nu (T), \end{equation} \tag{ 11 }$$

where ν is the observing frequency and B_ν(T) is the blackbody spectrum for temperature T. T and β are free parameters with uniform priors T ∈ [5, 35 K] and β ∈ [0, 2]. R12 also examined the single-spectral energy distribution (SED) model presented by Hall et al. (2010). The resulting kSZ power upper limits were similar when allowing for the possibility of correlation between the CIB and tSZ and 20% tighter when any correlation was assumed to be negligibly small.

We assume the clustering and Poisson components have the same frequency scaling in this work. Implicitly this is equivalent to assuming that the source populations and redshift distributions are similar. Hall et al. (2010) argued that any difference in the frequency scaling should be small based on CIB simulations. As mentioned in R12, the data have some preference for differing spectral indices when considering the 95–220 GHz alone, however this preference vanishes when Planck/HFI 353 GHz data are added.

An important component of our model is the potential for a spatial correlation between the DSFGs that produce the CIB and the groups and clusters that generate the tSZ power spectrum. At ℓ = 3000, most of the tSZ power is predicted to originate from clusters of mass greater than 5 × 10¹³ h⁻¹ M_☉ and redshift less than 1.5 (Shaw et al. 2009; Battaglia et al. 2011; Trac et al. 2011). If the galaxies residing in these halos contribute a significant fraction of the CIB power at the SPT frequencies, then the spatial correlation between the tSZ and CIB signal will be non-negligible. Below 220 GHz, the tSZ effect produces temperature decrements; we thus expect an anti-correlation between the CIB and tSZ fluctuations. Note that cross-spectra involving the 220 GHz band—the tSZ null frequency—may still contain power due to tSZ–CIB correlation as the CIB signal couples with the tSZ at the second frequency.

Following R12, we model the contribution of a tSZ–CIB correlation to the measured power spectrum as

$$\begin{equation} D^{\rm{tSZ\hbox{--}CIB}}_{\ell ,\nu _i \nu _j} = \xi _\ell \Big[ \sqrt{\big(}D^{\rm tSZ}_{\ell ,\nu _i \nu _i} D^{\rm CIB}_{\ell ,\nu _j \nu _j}\big) + \sqrt{\big(}D^{\rm tSZ}_{\ell ,\nu _j \nu _j} D^{\rm CIB}_{\ell ,\nu _i \nu _i}\big) \Big] . \end{equation} \tag{ 12 }$$

Here $D^{\rm {tSZ\hbox{--}CIB}}_{\ell ,\nu _i \nu _j}$ is the power due to correlations, $D^{{\rm tSZ}}_{\ell ,\nu _i\nu _i}$ is the tSZ power spectrum at frequency i, and $D^{{\rm CIB}}_{\ell ,\nu _i\nu _i}$ is the sum of the Poisson and clustered CIB components. ξ is the correlation coefficient; we define ξ₃₀₀₀ as the amplitude of this correlation at ℓ = 3000. We consider two templates for the spatial dependence of the correlation, one flat and one rising with ℓ. The reasoning behind these templates is discussed later in this section.

R12 measures the magnitude of the tSZ–CIB correlation by introducing ξ as a free parameter in their model fit to the SPT bandpowers, assuming an ℓ-independent template for D^tSZ–CIB_ℓ. They find ξ₃₀₀₀ = −0.18 ± 0.12, consistent with a fraction of the CIB emission being spatially coincident with the tSZ signal. However, George et al. (2011) extrapolate the 24 μm flux of galaxies in their (X-ray-selected) group catalog to 2 mm and find a very small signal. Even allowing for uncertainties in this extrapolation, they conclude that the contamination of DSFGs to the tSZ signal is no more than a few percent. This would imply only a very weak spatial correlation between these signals. A robust measurement of the total DSFG emission in groups and clusters at millimeter/submillimeter wavelengths would place tight observational constraints on ξ_ℓ.

There currently does not exist any published theoretical estimates of the magnitude of the tSZ–CIB correlation. However, we find that the publicly available simulations of Sehgal et al. (2010) predict a large anti-correlation. Taking the cross-spectrum of their simulated CIB-only and tSZ-only maps at 148 GHz, we find ξ₃₀₀₀ = −0.37.

We make new predictions for the tSZ–CIB correlation by combining the tSZ model of Shaw et al. (2010) with the halo-model-based CIB calculations of Shang et al. (2012). In brief, the Shang et al. (2012) model populates halos and sub-halos with DSFGs utilizing the halo mass function of Tinker et al. (2008) and the sub-halo mass function of Wetzel & White (2010). It is assumed that the relation between galaxy luminosity and sub-halo mass is lognormal, with free parameters for the overall normalization, characteristic mass, and redshift evolution. These parameters are calibrated by comparing the model predictions with the recent Planck measurements of the CIB power spectrum (Planck Collaboration 2011). Shang et al. (2012) explore model parameter constraints for various scenarios in which the characteristic dust temperature of DSFGs, the redshift evolution of the sub-halo mass—DSFG luminosity relation, or the shape of their SED are alternatively fixed or allowed to vary. For each of these scenarios (labeled "cases 0–5" in Shang et al. 2012), we calculate the correlation coefficient, ξ.

We find that the predicted value of ξ₃₀₀₀ varies between −0.02 > ξ₃₀₀₀ > −0.34, well within the range found by R12. ξ depends sensitively on the assumed redshift evolution of DSFG luminosity (or equivalently, the evolution of their star formation rate), which always increases toward higher redshift. For models in which the luminosity evolves rapidly, a larger fraction of the CIB power is contributed by high-redshift (z > 2) objects and thus the magnitude of ξ decreases.

The ℓ-dependence of ξ also varies between our CIB models. In Figure 3, we plot ξ_ℓ for each of the model cases presented by Shang et al. (2012). The values of ξ₃₀₀₀ are given in the caption. The shaded region represents the 1σ and 2σ constraints on ξ obtained by R12. We find a general trend that the correlation coefficient is a stronger function of ℓ for models in which the overall amplitude is closer to zero. This is straightforward to understand; models with a lower value of |ξ| are those for which high-redshift galaxies contribute a larger fraction of the total CIB power. Most of the tSZ–CIB correlation then comes from higher redshift halos which principally contribute to the tSZ power at small scales. Therefore, the correlation coefficient increases with ℓ.

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We will explore the dependence of constraints on the kSZ power spectrum on the shape of ξ_ℓ, focusing on the model shown by the solid curve, which has an average correlation coefficient of 0.04, but letting the amplitude freely vary. Note that this is conservative in that models with steeper amplitude have correlations close to zero; however, we will not impose a prior.

WMAP measures the optical depth to reionization to be τ = 0.088 ± 0.014 (Larson et al. 2011). Figure 4 provides intuition on how this constrains the epoch of reionization. In the left panel, we show ionization histories for several models from our simulation database. The right panel plots the large-scale polarization signal as calculated by CAMB (Lewis et al. 2000) from the simulated ionization histories along with the WMAP7 bandpowers. The near instantaneous model at z_mid = 10.5 shown by the purple dotted curve, as well as the more extended green dot-dashed curve (Δz ≃ 3) and very extended blue dashed curve (Δz ≃ 8) are good fits to the WMAP data, as shown in the right-hand panel of Figure 4. The large-scale polarization data do not allow one to discern between shorter and longer duration models. However, the red dot-dot-dashed and orange solid curves with z_mid ≃ 7 and 12.5 are poor fits to the WMAP data. For the data used in this paper (CMB+BAO+H₀), the optical depth constraint is τ = 0.085 ± 0.014. This corresponds to a ∼0.25 shift to lower redshifts compared to using the WMAP-only optical depth constraint; we use τ = 0.085 ± 0.014 for all reionization constraints in this work.

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The SPT constraints on the kSZ power spectrum are summarized in Figure 5 and Table 1. The table contains 95% confidence upper limits on the patchy kSZ power at ℓ = 3000 for different homogeneous kSZ and CIB models.

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The kSZ contributions from various redshifts have the same frequency dependence, and only small differences in angular dependence (as shown in Figure 2). We find that the 2008 and 2009 SPT data presented by R12 do not yet significantly discriminate between the patchy and homogeneous kSZ template shapes. As a result, the modeled homogeneous kSZ power affects the level of patchy kSZ allowed by the data. With more homogeneous kSZ power, less patchy kSZ power is allowed. We note, however, that the patchy kSZ likelihood curves in Figure 5 shift by relatively little when changing the homogeneous kSZ model by ∼2 μK². This is because the likelihood curves peak at or below zero kSZ power (−1.2σ without tSZ–CIB correlation), and a positivity prior is applied. As will be discussed below, the SPT data prefer a non-negligible tSZ–CIB anti-correlation together with an increased kSZ amplitude.

We consider two models for the homogeneous kSZ signal. The first, zero homogeneous kSZ (dashed lines in Figure 5), is the most conservative when determining upper limits on the patchy kSZ signal. However, it is physically unmotivated. The second case (solid lines) is the CSF model described in Section 4.1.1 which is our best estimate for the homogeneous kSZ signal. We scale the homogeneous kSZ signal according to the cosmological parameters at each step in the chain following Equation (10). We tabulate the patchy kSZ upper limits with these two homogeneous kSZ models in the first and second rows of Table 1.

To constrain the epoch of reionization, we use kSZ power constraints based on a fixed template set to the sum of the CSF and patchy kSZ models in the fiducial cosmology. Constraints on the total kSZ power with this template are found in the third row of Table 1. The small error caused by our choice of a fixed ratio between patchy and homogeneous kSZ templates amounts to, at most, a fraction of the observed 15% difference between the kSZ upper limits for the CSF and patchy templates.

A significant uncertainty in the current kSZ constraints is related to the unknown tSZ–CIB correlation. As discussed in Section 4.3, the amount of correlation is highly uncertain; we therefore explore three cases for the correlation. For no correlations, shown by the blue lines of Figure 5 and left column of Table 1, the inferred 95% confidence upper limits on the patchy kSZ power at ℓ = 3000 are 2.5 and 2.1 μK² with the zero and CSF homogeneous kSZ models, respectively. Our second patchy kSZ constraint (black lines in Figure 5 and center column of Table 1) uses a rising tSZ–CIB correlation shape based on the modeling in Section 4.3. This correlation shape is the solid line in Figure 3, and corresponds to the ξ₃₀₀₀ = −0.04 Shang model case. However, the amplitude of the tSZ–CIB correlation is allowed to vary freely in the MCMC. The third case has an ℓ-independent tSZ–CIB correlation of unknown amplitude. The data prefer a larger anti-correlation in this final case; the 95% confidence range is ξ₃₀₀₀ ∈ [ − 0.28, 0.03] for the rising correlation shape and ξ₃₀₀₀ ∈ [ − 0.43, −0.01] for the ℓ-independent correlation shape. As a result, the third case yields the most conservative constraints on the kSZ effect. The 95% confidence upper limits on the patchy kSZ power weaken to 5.5 and 4.9 μK² with the zero and CSF homogeneous kSZ models, respectively.

Figure 6 shows the degeneracy between the patchy kSZ contribution, and the tSZ–CIB correlation, ξ. The figure assumes no homogeneous kSZ contribution and an ℓ-independent tSZ–CIB correlation; qualitatively, the result is independent of these assumptions. It is evident that the kSZ power increases with the anti-correlation (more negative values of the correlation, ξ). Anti-correlation has the opposite effect on the tSZ power. The decrease in tSZ power with increasing anti-correlation is shown in the color-coding of the independent samples, ranging from values of 4 μK² for the tSZ with zero tSZ–CIB correlation, to small values of 2 μK² for a −0.5 correlation.

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Decreasing tSZ power with increasing anti-correlation contradicts our naive expectations. In single-frequency bandpowers (e.g., only 150 GHz), increasing tSZ–CIB anti-correlation would increase the allowed tSZ contribution for any observed power level. The anti-correlation term is negative in this band, thereby allowing more tSZ power. However, this picture changes when we consider the SPT multi-frequency data. In Figure 7, we show relevant model components across the three deepest SPT frequency combinations (95 × 150 GHz, 150 × 150 GHz, and 150 × 220 GHz from left to right). We have normalized each model component to 2 μK² at 150 GHz and ℓ = 3000. Note that a tSZ–CIB anti-correlation would lead to a negative power rather than the positive signal (shown by the blue dot-dashed line) in the plot. As shown, a tSZ–CIB anti-correlation reduces the power at 150 × 150 GHz (which could be compensated by increasing tSZ power, shown by the black dashed line) but it also has ∼50% larger effect at 150 × 220 GHz (where there is effectively zero tSZ power). Conversely, a tSZ–CIB anti-correlation reduces the power at 95 × 150 GHz by a similar amount to 150 × 150 GHz, whereas there is more tSZ power in the 95 GHz band. Adding kSZ power (shown as the combined template in the black solid line) more effectively compensates for the tSZ–CIB correlation term than adding tSZ power in the most sensitive combinations of the three frequencies. A combination of the kSZ and tSZ does even better, as qualitatively shown by the red dotted line for 1.2D^kSZ_ℓ–0.2D^tSZ_ℓ.

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Here we present constraints on the duration of reionization, defined as $\Delta z_{\rm rei}\equiv z_{{\bar{x}}_e=0.20}-z_{{\bar{x}}_e=0.99}$. These constraints are primarily due to the SPT kSZ measurement; WMAP data alone do not discriminate between reionization models of varying durations. However, by constraining the integrated optical depth, the WMAP data indirectly affect the duration inferred from the SPT-derived kSZ constraint since models with given duration have a larger signal at higher redshift due to the higher mean density at higher redshift (see Figure 2).

In Figure 8, we show constraints on Δz for the three tSZ–CIB correlation cases described in Section 5.2. Under the assumption of no tSZ–CIB correlation, we find a 95% upper limit on the duration of Δz < 4.4. The rising correlation shape template leads to an intermediate limit of Δz < 6.2. With the ℓ-independent correlation, we find the weakest 95% upper limit of Δz < 7.9. Note from the left panel of Figure 4 that a shorter redshift interval follows the midpoint of the epoch of reionization than precedes it. As a result, only a fraction of the duration in Figure 8 can lie at lower redshift than the midpoint z ∼ 10.25.

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In this section, we show that we can use the CMB data to constrain the history of reionization, ${\bar{x}}_e$, by combining the constraints on the integrated optical depth and duration of the epoch. To this end, we compute the posterior distributions of $z({\bar{x}}_e)$ for all values of ${\bar{x}}_e$ using our simulation grid given the constraints on τ and the amplitude of the kSZ power spectrum, D^patchy₃₀₀₀.

As previously mentioned, the integrated optical depth constraint alone cannot constrain the evolution of the ionized fraction as a function of time. Figure 9 shows the 68%/95% confidence intervals in the z_end–Δz plane. There is a perfect degeneracy in the large-scale WMAP polarization data between the duration and end of reionization; a low redshift for the end of reionization can be made consistent with the data by increasing the duration and the redshift of the starting point of reionization. The SPT constraint on the patchy kSZ contribution breaks this degeneracy by constraining the duration, as shown for the case of no tSZ–CIB correlations in the inner contours of Figure 9.

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To obtain constraints on the evolution of the ionized fraction with redshift, we integrate the posterior distributions at each ${\bar{x}}_e$ to obtain the 68% and 95% likelihood intervals for that ionization fraction to fall within a certain redshift range. Again, we assume the CSF homogeneous kSZ model for the post-reionization kSZ power. As indicated by Table 1, the effect of different homogeneous kSZ power is modest given our tight upper limits. The constraints on ${\bar{x}}_e$ from the addition of SPT to WMAP are shown for the three tSZ–CIB correlation variants in Figure 10. The case with arbitrary, ℓ-independent correlation (right panel, green lines) shows the weakest constraints. The case with rising correlation shape (center panel, black lines) gives intermediate constraints. The case without correlations (left panel, blue lines) shows the tightest constraint on the ionized fraction, with the error on the timing dominated by the WMAP optical depth uncertainty. The bulk of the epoch of reionization is confined to a narrow redshift interval from z ≃ 12.5 to z ≃ 7 at 95% confidence level (CL).

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We next consider the end of reionization which has been the focus of most current reionization observations. Recall that we define the end of reionization as ${\bar{x}}_e=0.99$. Constraints on z_end are shown in the left panel of Figure 11 and tabulated in Table 2. This is equivalent to taking a slice through Figure 10 at ${\bar{x}}_e= 0.99$. The SPT data enable the first CMB constraint on z_end. The SPT data disfavor a late end to reionization due to their preference for short durations. For ξ = 0, we find that having reionization end at z_end < 7.2 is disfavored at 95% CL (inner blue contour, dotted line). The constraint weakens considerably when an arbitrary, ℓ-independent correlation is allowed (wider green contour, solid line). However, in this most conservative interpretation of our data, reionization still concluded at z_end > 5.8 at 95% CL. The rising-ℓ correlation shape yields an intermediate constraint of z_end > 6.4 (cyan contour, dashed line). Note that while WMAP does not place a lower limit on the end of reionization, it trivially places an upper limit (the WMAP contour lies directly beneath the SPT contours on this side) by limiting the integrated optical depth. Note also that the innermost contour is shifted slightly to the left compared to the cases with tSZ–CIB correlation. This is because the tighter SPT upper limit on the combined kSZ means that short models ending at lower redshift are preferred because they entail a smaller homogeneous kSZ contribution. In other words, there is a slight tension between the SPT requirement for a small kSZ amplitude and the WMAP requirement for a relatively large integrated opacity.

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Finally, we present constraints on when the first ionizing sources turn on and begin reionizing the universe. Again, we have defined the beginning of reionization by ${\bar{x}}_e=0.20$. The likelihood function for z_beg is shown in the right panel of Figure 11. This is equivalent to taking a slice through Figure 10 at ${\bar{x}}_e= 0.20$. We find that the combination of WMAP and SPT data rule out an early onset of reionization at z > 12.1 at 95% confidence in the ξ = 0 case. When allowing for a free, ℓ-independent tSZ–CIB correlation, the 95% confidence upper limit increases to z_beg ⩽ 13.1. Again, the WMAP optical depth constraint leads only to a lower limit on the beginning of reionization; see above. Again the ξ = 0 likelihood peaks at slightly lower redshift, for the same reason as in the z_end case.

In the near future, we expect improved measurements of the kSZ power from the full SPT survey, optical depth (from Planck), and CIB (from Herschel  and Herschel/SPT cross-correlation analyses). The SPT survey of 2500 deg² (three times the area used in this work) was completed in 2011 November. The Planck survey is ongoing and the first power spectrum results including the optical depth constraint from the large-scale E-mode polarization feature should be released in 2013. Herschel observations of the deepest 100 deg² of the SPT survey will conclude in 2012, and should enable a detailed study of the tSZ–CIB correlation as well as the CIB in general.

We estimate the improvement in the kSZ power constraint from the full SPT survey with 100 deg² of Herschel overlap by running an MCMC with simulated bandpowers and uncertainties. We assume a 1% temperature calibration uncertainty and a 5% beam FWHM uncertainty in the SPT frequency bands; we assume that the Herschel data are used to create a CIB template map which will be subtracted from the SPT bands to reduce the CIB contribution. Given the wide frequency range spanned, we introduce one new parameter to allow for decorrelation in the CIB between frequency bands. Instead of a single β in Equation (11), we assume a distribution of β in the galaxies of N(β, σ_β). We set a uniform prior of σ²_β ∈ [0, 0.35]; the upper edge is chosen such that the correlation between 150 and 220 GHz is at least 95%. For the full SPT survey without Herschel, we find that the kSZ constraint improves proportionally to the reduced SPT bandpower uncertainties; the uncertainties on the bandpowers and kSZ power are reduced by a factor of $\sqrt{3}$. The combination of SPT and Herschel leads to a factor of six improvement in the kSZ constraints. Assuming that the post-reionization kSZ contribution is known, we find that the combination of SPT and Herschel data should be able to positively identify extended reionization at 95% confidence if Δz ⩾ 2. Using the Planck published sensitivity numbers leads to a forecasted constraint on the optical depth of Δτ ≃ 0.005 (Zaldarriaga et al. 2008). This is almost three times better than the current WMAP7+SPT τ = 0.085 ± 0.014 constraint (Keisler et al. 2011). Figure 12 shows constraint forecasts for the end of reionization. Our current constraints with an uncertain, ℓ-independent, tSZ–CIB correlation is reproduced for comparison in the green contour, solid line. We show three variations on the predicted z_end likelihood function. The cyan contour (dotted line) shows constraints centered on the WMAP7+SPT optical depth if the combined future data set continues to set only a stricter upper limit on the kSZ power. The predicted z_end tightens significantly around z_end ≃ 10.25 and models that end at z < 8 would be ruled out at 95% confidence. At the same τ value, a measured patchy kSZ amplitude of 2 μK², corresponding to a reionization duration of Δz ≃ 4 and in good agreement with the flat correlation at the preferred SPT value of ξ = −0.18 leads to the blue contour (dashed line). Reducing the central optical depth to the ∼1σ lower bound of the WMAP measurement with zero tSZ–CIB correlation leads to the red contour (dot-dashed line). The latter two cases have similar effects on the z_end constraints. The predicted z_end tightens around z_end ≃ 8–8.5 and models that end at z ≲ 6.5 would be disfavored at ⩾95% confidence.

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