Draft version January 31, 2020 Typeset using LATEX twocolumn style in AASTeX63

Beyond optical depth: Future determination of ionization history from the CMB

D. J. Watts ,1 G. E. Addison ,1 C. L. Bennett ,1 and J. L. Weiland 1

1JHU Department of Physics and Astronomy, 3701 San Martin Drive, Baltimore MD, 21218, USA

(Received October 1, 2019; Revised December 3, 2019; Accepted January 31, 2020) Submitted to ApJ

ABSTRACT We explore the fundamental limits to which reionization histories can be constrained using only large-scale cosmic microwave background (CMB) anisotropy measurements. The redshift distribution of the fractional ionization x z affects the angular distribution of CMB polarization. We project e ( ) constraints on the reionization history of the universe using low-noise full-sky temperature and E- ˆ TE mode measurements of the CMB. We show that the measured TE power spectrum, C` , has roughly ˆ EE one quarter of the constraining power of C` on the reionization optical depth τ, and its addition ˆ EE improves the precision on τ by 20% over using C` only. We also use a two-step reionization model min with an additional high-redshift step, parameterized by an early ionization fraction xe , and a late reionization step at zre. We find that future high signal-to-noise measurements of the multipoles min 10 6 ` < 20 are especially important for breaking the degeneracy between xe and zre. In addition, we show that the uncertainties on these parameters determined from a map with sensitivity 10 µK arcmin are less than 5% larger than the uncertainties in the noiseless case, making this noise level a natural target for future large sky area E-mode measurements.

Keywords: Cosmic background radiation (317); Reionization (1383); Cosmological parameters (339); Cosmology (343)

1. INTRODUCTION zation at high signal-to-noise can be used to constrain Cosmic reionization is a poorly understood part of ionization histories. standard ΛCDM cosmology. Reionization, when neu- It is observationally known that after the universe be-  tral hydrogen and helium in the intergalactic medium came neutral at the epoch of recombination, by z 6 (IGM) become ionized, creates a plasma that scatters it was ionized once again (e.g., Gunn & Peterson 1965; cosmic microwave background (CMB) photons (Rees Becker et al. 2001; Fan et al. 2006). Determinations of 1968; Basko & Polnarev 1980; Bond & Efstathiou 1984). the ionization fraction of the IGM have been made at This reduces the amplitude of the CMB anisotropy at redshifts 6 . z . 8 (Bouwens et al. 2015; McGreer et al. angular scales ` & 10 and creates additional polarized 2015; Greig et al. 2017; Bañados et al. 2018; Davies et al. power that dominates at scales ` . 10 (Zaldarriaga 2018; Mason et al. 2018; Greig et al. 2019; Mason et al. arXiv:1910.00590v2 [astro-ph.CO] 30 Jan 2020 1997). We illustrate the separate effects of reionization 2019) by probing the epoch of reionization via measure- and recombination on the E-mode power spectrum in ments of Lyman α emission, but these data are sparse, Figure 1. Because the temperature and E-mode po- and do not yet constrain the free electron fraction during TT TE EE the epoch of reionization (see, e.g., Planck Collaboration larization angular power spectra (C` , C` , and C` ) depend on the redshift of scattering, their characteri- I 2018, Figure 36). Commonly, CMB constraints on the reionization his- tory of the universe are derived assuming a sharp tran- Corresponding author: Duncan J. Watts sition from a neutral to fully ionized IGM. Measure- [email protected] ments of the large-scale CMB polarization constrain the ionization history by inferring the optical depth to the ∫ t0 last scattering surface of the CMB, τ cσTne t dt, ≡ tlss ( ) 2 Watts et al.

1 where c is the speed of light, σT is the Thomson scat- 10− tering cross section, n t is the free electron number Reionization e ( ) density, t0 is the current age of the universe, and tlss 2 10− is the last time photons interacted with matter during ] the epoch of recombination. Determining the free elec- sr 2 3 Recombination 10− tron density ne t is then an inverse problem that relies K

( ) µ on assumptions and priors. For example, a tanh-like [ 4 reionization history (e.g., Lewis 2008, Equation (B3)) 10− ` EE

with a transition from neutral to ionized at a single C  5 reionization redshift zre with width δzre 0.5 has been 10− used (e.g., Hinshaw et al. 2013; Planck Collaboration VI 2018, Section 3.3). Observations from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite were used 101 102 to make a measurement of the optical depth from the ` surface of last scattering τ  0.089 0.014 (Hinshaw ±  et al. 2013), although this decreases to τ 0.067 0.013 Figure 1. Effect of reionization on the CEE power spectrum. ± ` when using Planck 353 GHz data as a template to remove We take the difference between an E-mode signal with τ  0  2τ Galactic dust emission (Planck Collaboration XI 2016). and one with τ 0.06 with fixed As e− to demonstrate the EE Planck Collaboration I(2018) increased the precision of effects of tanh-like reionization on C` versus those from re- this measurement to τ  0.0544 0.0073. Pagano et al. combination. The black dashed line is the total CEE spectrum ± ` (2019) claim to have further reduced large-scale Planck when τ  0.06. The E-mode signal from recombination dom- systematics, reporting τ  0.059 0.006. inates above ` & 20, whereas the reionization signal emerges ± As a cross-check, it is possible to obtain competitive at multipoles ` . 20. constraints without using CMB polarization. Planck temperature measurements combined with Planck weak high-band experiment (Monsalve et al. 2017) measures lensing and baryon acoustic oscillation (BAO) data give the brightness temperature of the global 21 cm emission τ  0.067 0.016 (Planck Collaboration XIII 2016), con- from neutral hydrogen, T21, at frequencies 90–190 MHz ± sistent with results using WMAP temperature, Planck (redshifts 6.5–14.8). Monsalve et al.(2017) assume a hot  weak lensing, and BAO data, τ  0.066 0.020 (Weiland IGM with spin temperature Ts 28 K TCMB and find ±  et al. 2018). Weiland et al.(2018) include a compilation a lower limit of δzre & 0.5 with 95% confidence over the of τ measurements, and conclude that the measured redshift range 6.6 . z . 11. The brightness tempera- values are all consistent with τ  0.07 0.02. Unlike the ture has an rms scatter of 17 mK, which corresponds to ± Hubble constant H0, (e.g., Bernal et al. 2016, Freedman a xHi rms scatter of & 0.5, assuming a fixed spin temper-  2017, Addison et al. 2018, and Riess et al. 2019), the issue ature Ts 28 mK. With a future rms sensitivity 1 mK ∼ with reionization is not tension between measurements, measurement of T21, a percent-level determination of but a lack of desired precision. xHi z will be possible. ( ) Using the one-to-one mapping of τ zre in tanh-like It is typically assumed that the universe was ionized ↔ reionization, Planck Collaboration I(2018) use the low- by ultraviolet photons from massive stars escaping from ` polarization power spectra to infer zre  7.67 0.73 galaxies. To confirm this, it is necessary to characterize ± (Planck likelihood Plik best fit), while measurements the emissivity of galaxies at the onset of star forma- of the kinetic Sunyaev–Zel’dovich effect at arcminute tion, to see whether enough ionizing photons are gen- scales by the South Pole Telescope (SPT) and the At- erated and can escape from galaxies on a short enough acama Cosmology Telescope (ACT) can be used to timescale (Madau & Dickinson 2014; McCandliss et al. limit the duration of inhomogeneous reionization to 2019). This requirement is fulfilled if there is a steep δzre < 2.8 at the 95% C.L. with the prior that reioniza- luminosity function with galaxies contributing down to tion ends by z  6 (Zahn et al. 2012; Sievers et al. 2013; an absolute magnitude MUV 13 (Bouwens et al. 2015; ∼ − Planck Collaboration Int. XLVII 2016). Finkelstein et al. 2019) and that a volume-averaged es- While the CMB probes the reionization process cape fraction of 5–20% is achieved (Madau & Dickinson through the scattering of photons by free electrons, 2014; Khaire et al. 2016; Finkelstein et al. 2019). Deter- direct observations of the neutral hydrogen fraction minations of the UV luminosity function through the would constrain the ionization state of the universe onset of reionization is limited by sensitivity to galaxies during the epoch of reionization directly. The EDGES up to redshift z . 10, and will be enabled by deep ob- Reionization Histories 3 servations by the James Webb Space Telescope (Robertson 2. MAXIMIZING INFORMATION USED et al. 2015). IN POWER SPECTRUM ANALYSIS Other potential mechanisms with different redshift In this section, we develop a formalism for extract- dependence have also been put forward. In particular, ing reionization information from a full-sky map of the binary black hole collisions can be a source of X-rays at intensity and linear polarization of the CMB. In Sec- z & 30, which can raise the ionizing fraction with less tion 2.1, we define the three likelihoods we use for dif- fractional contribution from star formation (Inayoshi ferent subsets of data; Wishart (for Cˆ TT + Cˆ TE + Cˆ EE), et al. 2016). Quasars and annihilating particles have ` ` ` χ2 (for Cˆ EE), and variance-gamma (for Cˆ TE). In Sec- also been proposed as ionizing mechanisms (Mapelli & ` ` tion 2.2, we characterize these likelihoods for the case Ripamonti 2008; Madau & Haardt 2015; Khaire et al. of instantaneous tanh-like reionization. 2016; Mitra et al. 2018). As we look to the future with more sensitive data, 2.1. Likelihoods for power spectra we would like to make quantitative statements about In standard ΛCDM, the CMB Stokes parameters a more detailed physical model for reionization. We m  I, Q, U are a realization of a Gaussian random explore the potential to make these constraints in this ( ) process. The spherical harmonic transforms of these paper. In this work, we explore potential future CMB maps a  aT , aE , aB are therefore also Gaus- constraints on the reionization history as parameterized `m ( `m `m `m) sian distributed. Neglecting B-modes, the a are dis- by both instantaneous and extended redshift scenar- `m tributed as a complex Gaussian a 0, C with ios. We focus specifically on a reionization history that `m ∼ N ( `) mean 0 and covariance consists of the usual instantaneous reionization and a second early high-redshift period of reionization that TT TE! C` C` partially ionizes the universe. C`  . (1) CTE CEE This paper is organized as follows. In Section 2, we ` ` quantify the relative constraining power for parame- As demonstrated in Hamimeche & Lewis(2008), the ter likelihoods based on Cˆ EE alone, Cˆ TE alone, and ` ` sample covariance matrix of measured power spectra Cˆ TT + Cˆ TE + Cˆ EE. We define the different likelihoods ` ` ` Cˆ ` drawn from a theory covariance matrix C` is given in Section 2.1 and obtain constraints on the nearly in- by a Wishart distribution, stantaneous tanh-like reionization model in Section 2.2. Õ In Section 3 we explore a toy reionization history model 2` + 1 Cˆ a† a W 2` + 1, C (2) ( ) ` ≡ `m `m ∼ n( `) that consists of the usual instantaneous reionization and m a second early (high-redshift) period of reionization that partially ionizes the universe. We then quantify the where n is the number of dimensions in a`m. A Wishart projected limits the CMB can impose on a reionization distribution is a multivariate gamma distribution. A history of this type with free parameters of reionization gamma distribution is a two-parameter probability dis- min tribution of which the χ2 distribution is a special case. redshift zre and high-redshift ionization fraction xe . When considered as a likelihood C P Cˆ C , this We describe this modification to the standard reioniza- L( `) ≡ ( ` | `) is often normalized such that χ2 2 ln C  0 tion history in Section 3.1. We then forecast sensitiv- eff,` ≡ − L( `) ity to this model’s parameters as a function of noise when C`  Cˆ `, i.e., and multipole range in Section 3.2, and demonstrate h i  + ˆ 1 ˆ 1 that most of the parameter space can be precisely con- 2 ln C` 2` 1 Tr C`C−` ln C`C− n . (3) 1 2 − L( ) ( ) [ ] − | | − strained with the map sensitivity w−p / . 10 µK arcmin using the multipole range 10 . ` . 20. We summarize In the single-dimensional case, this reduces to the more 2 our findings in Section 4. familiar χ distribution, Throughout this paper our default model is   Cˆ ` Cˆ ` flat ΛCDM with the Planck Collaboration VI 2 ln C`  2` + 1 ln 1 , (4) (2018) Plik TT,TE,EE+lowE+lensing mean parame- − L( ) ( ) C` − C` − 2  2   ters Ωb h 0.02237, Ωc h 0.1200, 100θMC 1.04092, in agreement with Equation 8 of Hamimeche & Lewis 10 2τ   ln 10 As e− 2.9352, and ns 0.9649. When τ is (2008) when normalized such that ln  0 when ( 10) varied, ln 10 As is set to 2.9352 + 2τ.  ˆ L ( ) C` C`. ˆ TE We also use the distribution of C` , i.e., the mean of the product of correlated Gaussian random variables. This was derived in Mangilli, Plaszczynski, & Tristram 4 Watts et al.

(2015) and independently in Nadarajah & Pogány(2016) 1.0 CTT + CTE + CEE and Gaunt(2018), and is given by a variance-gamma dis- ˆ ` ˆ ` ˆ ` CEE tribution (also called a generalized Laplace distribution 0.8 ˆ ` CTE or a Bessel function distribution) with functional form ˆ ` )i `

  ˆ 0.6 N+1 2 N 1 2 Nρcˆ ξ N cˆ C N( )/ cˆ ( − )/ e / K` ξ| | | P Cˆ TE θ  | | , (5) τ ( ` | ) N 1 2 Γ √ TT EE N 2 0.4 2( − )/ √π N 2 ξ σ` σ` /

( / ) ( ) hL( 0.2 where θ  CTT, CTE, CEE , cˆ  Cˆ TE, ρ  CTE σEEσTT { ` ` ` } ` ` /( ` ` ) is the correlation coefficient between the two noisy vec- q 0.0 XX  XX + XX tors, σ` C` N` is the total uncertainty on the XX XX 0.02 0.04 0.06 0.08 power spectrum C` , N` is the noise power spectrum, N  2` + 1 is the number of modes per multipole, τ ξ  1 ρ2 σTTσEE is a useful auxiliary variable, Γ is the ( − ) ` ` gamma function, and K is the modified Bessel function Figure 2. Normalized product of 50 000 likelihood distribu- ν ˆ  of the second kind of order ν. tions of C` realizations with input τ 0.06. We plot the likeli- hood from the variance-gamma distribution for Cˆ TE (red), the To better understand the variance-gamma distribu- ` likelihood from the χ2 distribution for Cˆ EE (orange), and the tion, we show how it reduces to the χ2 distribu- ` ˆ TT + ˆ TE + ˆ EE tion when taking a cross spectrum of identical vec- likelihood from the Wishart distribution for C` C` C` (blue). The standard deviations of these distributions for in- tors, i.e., ρ 1. This distribution P x is propor-   →   ( ) put τ 0.06 are στ 0.0072, 0.0021, 0.0017 respectively. N 1 2 Nρx ξ N x 1 2 { } tional to x e K | | ξ . The modi- | |( − )/ / ` ξ − / fied Bessel function of the second kind decays expo- 2.2. Likelihood for instantaneous reionization nentially, and its zeroth order expansion is given by To demonstrate the relative constraining power of the K x p π e x (Abramowitz & Stegun 1965). In the ν( ) ≈ 2x − Wishart, χ2, and variance-gamma likelihoods, we start limit of large x, the functional form of the variance- with the theoretical power spectra as a function of the gamma distribution goes to reionization optical depth τ in the case of instantaneous TT TE EE r / /  2τ     reionization, C` f τ, As , with As e− fixed. N 1 2 Nρx N x πξ 1 2 ( ) P x x ( − )/ exp exp | | ξ− / Additionally, we include a white noise component that ( ) ∝ | | ξ − ξ 2N x | | is uncorrelated between I, Q, and U and whose ampli- (6) 1 2   tude w−p / varies between 0–230 µK arcmin. Using this N 2 1 ρx x x / − exp − | |) . (7) formalism allows us to make predictions for the best- ∝ | | 1 ρ2 − case constraining power on τ for future experiments, assuming instantaneous reionization.  For perfectly correlated variables, the correlation ρ 1 We characterize the likelihood of τ by evaluating and the data are positive definite with x > 0, giving TT TE EE ˆ / / N 2 1 x 2 2 τ C` for many realizations of the CMB sky. P x x / − e− / , the χ distribution with N degrees L( |{ }) ( ) ∝ We create 50 000 realizations of a with 2 6 ` 6 100 of freedom. `m to test this formalism using the HEALPix1 routine This parameterization of the variance-gamma distri- synalm. In Figure 2, we show the averaged likeli- bution has mean and variance per multipole hood of these different spectra in the case of a full- ˆ TE  TE sky cosmic variance-limited measurement, and obtain C` C` (8) TT+TE+EE  EE  TE  h i στ 0.0017, στ 0.0021, and στ 0.0072. TE 1  TE 2 TT EE var Cˆ  C + C C , (9) The TE-only constraint is comparable to the uncertainty ( ` ) 2` + 1 ( ` ) ` ` Planck  from Planck, στ 0.007, which only includes E- mode data. The distribution for ˆ TE in Figure 2 is vis- in agreement with the mean and variance of the off- C` diagonal component of the Wishart distribution and ibly skewed. This is a manifestation of the underlying T E the Gaussian distribution of a`m and a`m. We have also 4 validated the functional form using 10 realizations of 1 Hierarchical Equal Area isoLatitude Pixelation ˆ TE https://healpix.sourceforge.io/ a`m vectors, and find that the distribution of C` agrees with Equation 5. Reionization Histories 5

TT TE EE 3.1. A simple model for early reionization στ from C + C + C 0.012 ˆ ` ˆ ` ˆ ` EE We explore the constraining power of low-multipole στ from C ˆ ` CMB polarization data using a specific parameteriza- 0.010 σ from CTE τ ˆ ` tion of the reionization history. We parameterize the 0.008 global reionization history xe z using the the ratio of τ ( )

σ free electrons to hydrogen nuclei as a function of time, 0.006 x n nH,2 and write the contribution to the reion- e ≡ e / 0.004 ization optical depth between two redshifts z1 and z2 as

¹ t z2 0.002 ( )     τ z1, z2 cσTxe z t nH z t dt. (10) ( ) ≡ t z1 ( ) ( ) 0.000 ( ) 1 10 100 We parameterize the reionization history using a similar 1 2 model to that used in Equation A3 of Heinrich & Hu w− / µK arcmin p [ ] (2018), + min    Figure 3. Uncertainty on τ as as a function of white noise 1 fHe x yre y x z  − e 1 + tanh − amplitude in polarization for a full-sky measurement. Using e ( ) 2 δy ˆ TE ˆ EE C alone is always less constraining than C by a factor of min rec    ` ` x x yt y 3.5. Including Cˆ TE and Cˆ TT data improves the precision of + e − e 1 + tanh − + xrec ∼ ` ` 2 δy e a τ measurement by 20% over using Cˆ EE alone. ` (11)

TT TE EE 3 2 3 1 2 ˆ / / where 1 + and δ  1 + δ . The skewed distributions that the C` are themselves y z z / y 2 z / zre ( ) ≡ ( ) ( ) rec drawn from. ionization fraction from recombination alone is xe , the ˆ TE second transition step is given at the redshift , the Since the uncertainty on C` in Equation 9 is a func- zt tion of CEE,th + NEE CTT,th + NTT , it is reasonable to amplitude of reionization from the second transition is ( ` ` )( ` ` ) min ask whether there is a combination of uncertainties that xe , and the fraction of electrons from singly ionized TE EE helium is given by fHe nHe nH. We use this form makes στ competitive with στ . Figure 3 demonstrates ≡ / 1 2 because it parameterizes a small but nonzero early ion- that at a given polarized white noise level w−p / , the con- ization fraction. An upper limit on xe 15 z 30 was straining power on τ from the Cˆ TE alone is a factor of ( ≤ ≤ ) ` first inferred by Millea & Bouchet(2018) and further ˆ EE 3.5 weaker than the C` constraint. This means that constrained by Planck Collaboration VI(2018). Fig- ∼ ˆ TT + ˆ TE + ˆ EE using C` C` C` results in an approximately 20% ure 45 of Planck Collaboration VI(2018) shows that ˆ EE increase in precision compared to using C` data alone. above z & 10, Planck measurements do not rule out min The white noise temperature component is function- xe 10%. Motivated by this result, we choose a fidu- ≈ min  ally negligible for this analysis. We can see this by cial value of xe 0.05 to demonstrate the potential looking at the components of Equation 9 contributing effects this ionization fraction can have on CMB mea- TE TT,th TT EE,th EE  to the white noise in Cˆ , C + N C + N . surements. We also choose zre 6.75 so that the total ` ( ` ` )( ` ` ) The theory-noise cross-terms are comparable when optical depth τ  0.06 is consistent with the Planck NTTCEE,th NEECTT,th. Since CTT,th CEE,th 104 for Collaboration VI(2018) values. We highlight the pa- ` ` ` ` ` ` min ≈ / 1 2 ' rameters of this model in Figure 4, with x set to 0.2 ` . 100, the polarization sensitivity w− / would have e p for visibility purposes. to be 10 2 times that of temperature for the temper- O( − ) There are several measurements of the neutral hy- ature spectrum’s white noise component to noticeably drogen fraction, which we summarize in the inset of contribute to the Cˆ TE variation. ` Figure 4. These direct measurements are in general agreement with the reionization constraints from CMB 3. THE CMB’S SENSITIVITY TO VARYING REIONIZATION HISTORIES measurements, but do not yet provide strong evidence We begin discussing a specific simple model for early reionization in Section 3.1, then discuss quantitative 2 The free electron fraction xe is greater than one at low red- forecasts in Section 3.2. shifts because of the free electrons corresponding to helium. When helium is singly ionized, the electron number density is ne  nH + nHe  1 + fHe nH, and when it is doubly ionized n  1 + 2 f n . ( ) e ( He) H 6 Watts et al.

 1.25 of these transitions to δzre 0.5 because it is weakly 1.0 constrained by E-mode power spectra for a reionization  1.00 history that is complete by z 0. 0.5 3.2. Constraints on high-redshift reionization

) 0.75 z δzre ( In the parameterization of Equation 11, it is natural e

x 0.0 to compare to Equation 10 and constrain the parame- 0.50 6 8 ters τ τ 0, z and τ τ z , z . We choose lo ≡ ( split) hi ≡ ( split dark) zdark  100 as a redshift sufficiently far removed from 0.25 both recombination and reionization effects. We define min x δzre z zre + 1. This parameterization essentially allows e split ≡ 0.00 a one-dimensional mapping such that τlo  f zre and z min ( ) zre zt τ  g x . In the case of standard tanh-like reioniza- hi ( e ) tion, τ τ and τ 0, or equivalently xmin xrec. lo → hi → e → e Figure 4. Visualization of our toy model for early reionization. The primary effect of adding a second component to We indicate the central redshift of late reionization zre (red), xe z is an increase in the total reionization optical depth min ( ) the amplitude of the early reionization fraction xe (blue), τ, and therefore the rough amplitudes of the polarized the redshift zt where early reionization begins (cyan), and the EE 2 TE min power spectra, specifically C` τ and C` τ at width δzre of these transitions (brown). We inflate x ∝ ∝ e → the lowest multipoles ` . 10. The second and more 0.2 to illustrate this parameter’s effect in the model. Inset: comparison of toy model with direct measurements of IGM distinguishing effect is that both of these power spectra ionization fraction. We plot measurements with upper and change shape due to the different angular sizes of local lower limits in blue, upper limits in orange, and lower limits quadrupoles at the primary and secondary reionization in red. These data do not rule out our toy model evaluated redshifts. This provides an opportunity to go beyond  min  at our fiducial parameters of zre 6.75 and xe 0.05. The τ in probing the nature of reionization. We demon- data include the compilation by Bouwens et al.(2015) and the min strate the effects of varying x and zre on the polar- measurements from McGreer et al.(2015), Greig et al.(2017), e ized power spectra (see Figure 5) using the Boltzmann Bañados et al.(2018), Mason et al.(2018), Davies et al.(2018), CLASS Greig et al.(2019), and Mason et al.(2019). code (Blas et al. 2011). For every reionization his- 2τ tory, we compute τ and vary As such that As e− is held constant. for or against reionization models more complex than Using the CLASS code, we set reio_parameterization the instantaneous tanh-like model. More concretely, equal to reio_many_tanh with δzre  0.5, fixing zt  30, these direct measurements do not rule out a free elec-  min  fHe 1.324 using the fiducial helium mass fraction tron fraction xe 0.05 at redshifts 7 . z . 8.3 rec 4 min Yp  0.25, and x  2 10− . We vary zre and x to We show the dependence of CEE and CTE on these e × e ` ` write the cosmological power spectrum as a function of reionization histories in Figure 5. We choose the ranges TT TE EE two parameters, C / /  f z , xmin . of the parameters such that they induce roughly equiva- ` re e 2 ( ) lent changes in the amplitude of the output power spec- In Figure 6, we plot χeff,` using Equation 3. By vary- min trum. The equivalent white noise powers are labeled ing zre and xe separately, we can observe a few note- on the right-hand side of Figure 5. We also vary δzre worthy features. First, although there is more varia- to show that although this parameter does affect the tion in the power spectra at the very largest scales, the power spectra, unphysically large widths δ & 5 are constraining power peaks at ` 10, corresponding to zre ' needed to affect the power spectra as much as zre and fluctuations on scales of tens of degrees. Second, the min two different reionization histories have notably differ- xe . Variations in δzre induce a high-redshift tail, sim- 2 ilar to the empirical extended phenomenological mod- ent χeff,`, demonstrating that the partial degeneracy be- els described in Lapi et al.(2017), Roy et al.(2018), and tween these two modifications to reionization history Kulkarni et al.(2019). The rightmost column of Fig- can be broken with high signal-to-noise measurements ure 5 shows these models are not strongly constrained across this range of angular scales. The very largest by large-scale CMB polarization data. We fix the width scales ` < 10 are much more constraining for zre than min xe , whereas the 10 6 ` < 20 range is very sensitive to both parameters. 3 NB: The highest-redshift points in Fan et al.(2006) were incorrectly We quantify this multipole dependence by perform- transcribed as measurements and not upper limits in Bouwens et al. (2015). This has been corrected in Figure 4. ing Fisher forecasts on subsets of cosmic variance- limited data in Figure 7. As expected, there is relatively Reionization Histories 7

min zre xe δzre 6 8 10 0.0 0.1 0.2 0.5 5.0 10.0

1.0 e x 0.5

0.0 0 10 z 20 30 0 10 z 20 30 0 10 z 20 30

1 w 10− − 1000 p 1 / ]

500 2

sr 2 10 [

− µ 2 300 arcmin K K

µ 150

[ 3 10− 100 ` EE 50 C 4 10− 30 ] w − p

5000 1 / ]

1 3000 2 sr [ µ 2 1500

1 arcmin K K 10− 1000 µ [ 500 2 ` TE 10− 300 C 3 150 10− 100 ] 2 5 10 20 50 2 5 10 20 50 2 5 10 20 50 ` ` `

Figure 5. Reionization histories and their corresponding power spectra. Negative values are plotted as dashed lines. Each column EE TE has curves with each color corresponding to a reionization history xe z , and its corresponding spectra C` and C` . In addition, ( ) min the left- and right-hand axes share limits and tick values across each row. Each column varies zre, xe , and δzre independently, min with a baseline black curve zre, x , δzre  6, 0, 0.5 , corresponding to τ  0.039. Variations of the reionization history of the ( e ) ( ) universe create corresponding variations in the polarization of the CMB at large angular scales. Changing the reionization history EE TE changes both the C` and C` power spectra, and the information from these two power spectra can be used both as a better constraint on the reionization history and as a consistency check. While varying these different parameters has similar effects on the low-` amplitude, the variation as a function of multipole can be used to eliminate degeneracies between the similar parameters. We include for comparison the width of reionization parameter δzre to point out that while it is possible for this parameter to affect the CMB on large angular scales, the impact of δzre is reduced given that the universe is ionized today, i.e., xe z  6 > 1. ( ) 8 Watts et al.

. 25 9 0 2 6 ` < 100 . 20 8 5 2 6 ` < 10 10 6 ` < 20 8.0 15

,` 20 6 ` < 30 ff re

2 e 7.5 z

χ 30 6 ` < 100 10 7.0

5 6.5 0.15 0 6.0 2 10 20 30 40 ` 0.1 e min x 25 0.05 0.05 20 0.04

5 6 7 8 9 0.05 0.10 0.15 15 0.03

,` min

ff zre xe min e 2 e x χ 10 0.02 Figure 7. Fisher forecasts for cosmic variance-limited ˆ TT + ˆ TE + ˆ EE 5 0.01 C` C` C` data over subsets of multipole ranges. The high opacity and low opacity ellipses are 1σ and 2σ contours, respectively. When considering the two-parameter reioniza- 0 0.00 2 10 20 30 40 tion model, the 10 6 ` < 20 range is most important for distin- guishing between alternate models of reionization. This range ` 2 has the maximum χeff variation due to its relatively strong model dependence compared to higher multipoles and rela- Figure 6. Effective goodness-of-fit as a function of multipole. min tively small cosmic variance compared to lower multipoles. The top subplot varies zre and the bottom varies xe . Using an observed set of power spectra Cˆ TT, Cˆ TE, Cˆ EE that are { ` ` ` } TT TE EE  identical to their theory values C` , C` , C` with zre 6 min  { } and xe 0, we calculate the global goodness-of-fit while min 0.010 varying zre and xe independently of each other. We note στlo that the 2 . ` . 20 range of angular scales contains most of the στhi effective constraining power of polarized CMB measurements. 0.008 στtot

στ (tanh-like) little constraining power in the 30 . ` . 100 multipole 0.006 τ

range, but the majority of constraining power comes σ from the 10 . ` . 20 range, in agreement with the 0.004 2 shape of the χeff,` curves in Figure 6. While there is significant constraining power in the 2 . ` . 10 and 0.002 20 . ` . 30 ranges, the 10–20 range is by far the most important for quantitative assessment for this reioniza- 0.000 tion scenario. 1 10 100 We show the uncertainty on the optical depth param- 1 2 + eters τlo, τhi, τtot τlo τhi, and τ from tanh-like reion- w−p / [µK arcmin] ≡ ization as a function of white noise level in Figure 8. This demonstrates that the optical depth from high- Figure 8. Constraints on τ uncertainty as a function of white redshift reionization can be meaningfully constrained noise. In all cases, the uncertainty saturates at white noise 1 2 with relatively high white noise levels, and that the un- level w− / 10 µK arcmin. Here we display the uncertainty p ∼ certainty on any additional optical depth from high- on the optical depth from various components of reionization, redshift sources can be improved by an order of mag- as well as the total reionization optical depth τtot. Reionization Histories 9 nitude above current measurements with white noise 100 µK arcmin levels as high as 10 µK arcmin. The constraining power of low-` polarization data can 60 µK arcmin most clearly be seen in the Fisher contours in Figure 9. 10 µK arcmin At current noise levels, the constraints on the reion- 0 µK arcmin ization redshift are relatively weak, and the presence of high-redshift reionization cannot be distinguished from instantaneous reionization. As expected, there is min a negative degeneracy between xe and zre that is most pronounced at high noise levels where only the low- 0.15 est multipoles contribute to the variation of the power spectra, while the degeneracy becomes less severe as the noise level decreases. 0.1 e min Figure 9 demonstrates the possible advances in x our understanding of reionization from the CMB. 0.05 min The ultimate sensitivity to zre, x  6.75, 0.05 — ( e ) ( ) equivalent to τtot  0.06—from the CMB is shown in blue, using a Fisher forecast with zero instrumen- 5 6 7 8 9 0.05 0.10 0.15 min tal noise. This noiseless measurement represents the zre xe fundamental limits for constraining these reionization min parameters with large-scale CMB polarization mea- Figure 9. Fisher forecasts for xe and zre as a function of surements. We plot Fisher contours for noise levels white noise level. The high opacity and low opacity ellipses 1 2 w− /  10, 60, 100 µK arcmin. The smallest number are 1σ and 2σ contours, respectively. A noiseless measure- p { } corresponds to the projected Cosmology Large Angu- ment (blue) represents the fundamental limits of constraining these reionization parameters with large-scale CMB polariza- lar Scale Surveyor (CLASS) white noise level over 70% tion measurements. For comparison, a 10 µK arcmin white of the sky in Essinger-Hileman et al.(2014) using all noise level is shown (orange) and is almost completely hid- four of its observing bands. This value is a bench- den under the blue 0 µK arcmin contour. We also plot the mark for detecting primordial gravitational waves with projected white noise contribution for a CLASS Q-band fore- a tensor-to-scalar ratio r 0.01, a goal for the current ground cleaned map (red), and the white noise contribution ∼ ˆ 100 143 generation of ground-based CMB measurements. The in the Planck 2018 C` × data (cyan). 1 2  w−p / 60 µK arcmin white noise level corresponds to the sensitivity of the CLASS Q-band (40 µK arcmin) With our fiducial parameters, the ultimate sensitivity cleaned using the WMAP K-band (280 µK arcmin) as a to this model with large-scale CMB anisotropy mea- synchrotron template. The 100 µK arcmin value cor- surements is σ  0.3 and σ min  0.005. Remarkably, zre xe responds to the geometric mean of the 100 GHz and this constraint does not weaken appreciably either when 143 GHz white noise levels reported in Table 4 of Planck examining only the multipole range 10 6 ` < 20, or Collaboration I(2018). when the data are contaminated by white noise at the We transform the contours in Figure 9 to the inte- 10 µK arcmin level. grated quantities τlo and τhi, both approximately single- min 4. CONCLUSIONS variable functions of zre and xe , respectively, in Fig- ure 10. We also plot lines of constant τ  τlo+τhi to show In this work we have explored the constraining power the total integrated contribution of this two-parameter of the CMB temperature and E-mode polarization on reionization model. reionization history. We summarize the results of this section in Table 1. • We have demonstrated the potential for a 20% im- We highlight data rows that are particularly constrain- provement on the precision of the reionization op- ing. This includes the full resolution cosmic vari- tical depth τ by using Cˆ TT and Cˆ TE in addition ance measurement, the noiseless measurement with ` ` 1 2  to Cˆ EE when the white noise level drops below 10 6 ` < 20, and the w−p / 10 µK arcmin measure- ` ments. We highlight these to emphasize the relative 10 µK arcmin. importance of future data with these properties to con- • We have shown that in the case of an early 5% strain reionization histories. ionization fraction, a scenario allowed by mea- surements in Planck Collaboration VI(2018), the 10 Watts et al.

100 µK arcmin that a key multipole range for this scenario is 10 6 ` 6 20, where σz  1.0 and σ min  0.005. 60 µK arcmin re xe 10 µK arcmin Future measurements of the large-scale polarized 0 µK arcmin CMB will be made by CLASS (Essinger-Hileman et al. 2014, currently observing) and LiteBIRD4 (Hazumi et al. 2019, expected launch late 2020s). LiteBIRD’s goal of measuring primordial B-modes with σr  0.001 with 1 2  w−p / 2 µK arcmin observations of the whole sky will be able to constrain the reionization model presented in 0.03 this paper to its cosmic variance limit. Now operating, 1 2  CLASS’s projected sensitivity of w−p / 10 µK arcmin

hi 0.02 will yield observations 70% of the sky with lower sen- τ sitivity, but as we have demonstrated here, this will be 0.01 more than sufficient to constrain a period of early reion- ization to its cosmic variance limit. 0 This work has focused on the ultimate sensitivity to 0.03 0.04 0.05 0.06 0.01 0.02 0.03 a specific toy model of reionization with an early high- τlo τhi redshift contribution. Processes that ionize the IGM across different epochs of cosmic time will generate dif- Figure 10. Fisher forecasts for τ and τ as a function of white lo hi ferent xe z profiles. Constraints on our model can noise level. The color of the contours is the same as in Figure 9. ( )  + therefore help discriminate between physical mecha- Additionally, we plot dashed lines of constant τ τlo τhi in five equal steps from τ  0.04 to τ  0.08, denoted by nisms that ionized the IGM. Reionization history con- straints from CMB measurements will both inform increasing dash length. The optimal τhi uncertainty is 0.001, while Fisher forecasts using 100 µK arcmin white noise project and complement future tomographic measurements of  an uncertainty of στhi 0.008, which is degraded further by 21 cm emission and of the first generation of galax- its strong negative degeneracy with τlo. ies designed to characterize x z . Knowledge of the e ( ) ionization history is also important for understanding the large-scale B-modes in the reionization peak, whose min Table 1. Fisher Forecasts for zre and xe when Varying fluctuations are created during the same epoch as large- Multipole Range Analyzed and White Noise Levels, Using scale E-modes. Fluctuations attributed to deviations min the Fiducial Model zre, x  6.75, 0.05 . ( e ) ( ) from single-field slow roll inflation can also be induced 1 2 by deviations from the standard tanh-like reionization w− / Multipole Range σ σ min στ στ p zre xe lo hi model, and these effects must be taken into account 0 2 6 ` < 100 0.3 0.005 0.003 0.001 when analyzing large angular scale B-modes. 0 2 6 ` < 10 0.8 0.020 0.007 0.005 0 10 6 ` < 20 1.0 0.005 0.009 0.001 0 20 6 ` < 30 5.9 0.024 0.054 0.006 0 30 6 ` < 100 10.8 0.152 0.991 0.038 10 2 6 ` < 100 0.4 0.005 0.003 0.001 60 2 6 ` < 100 0.7 0.018 0.006 0.004 100 2 6 ` < 100 1.0 0.031 0.009 0.008 Note—We have highlighted the three forecasts that are most constraining in bold text.

maximum precision from CMB large-scale mea- surements is σ  0.3, and σ min  0.005. We zre xe also show that this constraint is very nearly met when the white noise level is 10 µKarcmin, with 4 Lite (Light) satellite for the studies of B-mode polarization and inflation from cosmic background radiation detection σ  0.4 and σ min  0.005. We have also shown zre xe Reionization Histories 11

adstex ACKNOWLEDGMENTS Software: (https://github.com/yymao/ adstex), camb (Lewis et al. 2000, 2019b), CLASS (Blas This research was supported in part by NASA grants et al. 2011), getdist (Lewis et al. 2019a; Lewis NNX16AF28G, NNX17AF34G, and 80NSSC19K0526. 2019), healpy (Górski et al. 2005; Zonca et al. 2019), We acknowledge the National Science Foundation IPython (Pérez & Granger 2007), matplotlib (Hunter Division of Astronomical Sciences for their support 2007), numpy (van der Walt et al. 2011), scipy (Virtanen of CLASS under Grant Numbers 0959349, 1429236, et al. 2019) 1636634, and 1654494. Calculations for this paper were conducted using computational resources at the Mary- land Advanced Research Computing Center (MARCC). We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA), part of the High Energy Astrophysics Science Archive Center (HEASARC). HEASARC/LAMBDA is a service of the Astrophysics Science Division at the NASA Goddard Space Flight Center. We also acknowledge use of the Planck Legacy Archive. Planck is an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada. Some of the results in this paper have been derived using the healpy and HEALPix packages. This research has made use of NASA’s Astrophysics Data System. We thank Tobias A. Marriage, Zhilei Xu, Matthew Petroff, and Thomas Essinger-Hileman for discussions that improved this work. 12 Watts et al.

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