Mitigating the optical depth degeneracy using the kinematic Sunyaev-Zel'dovich effect with CMB-S4
Marcelo A. Alvarez, Simone Ferraro, J. Colin Hill, Renée Hložek, Margaret Ikape
MMitigating the optical depth degeneracy using the kinematic Sunyaev-Zel’dovich effectwith CMB-S4
Marcelo A. Alvarez,
1, 2, ∗ Simone Ferraro,
2, 1, † J. Colin Hill,
3, 4, ‡ Ren´ee Hloˇzek,
5, 6, § and Margaret Ikape
5, 6, ¶ Berkeley Center for Cosmological Physics, Department of Physics,University of California, Berkeley, CA 94720, USA Lawrence Berkeley National Laboratory, One Cyclotron Road, Berkeley, CA 94720, USA Department of Physics, Columbia University, New York, NY, USA 10027 Center for Computational Astrophysics, Flatiron Institute, New York, NY, USA 10010 Dunlap Institute for Astronomy and Astrophysics, University of Toronto,50 St George Street, Toronto ON, M5S 3H4, Canada David A. Dunlap Department of Astronomy and Astrophysics,University of Toronto, 50 St George Street, Toronto ON, M5S 3H4, Canada
The epoch of reionization is one of the major phase transitions in the history of the universe, andis a focus of ongoing and upcoming cosmic microwave background (CMB) experiments with im-proved sensitivity to small-scale fluctuations. Reionization also represents a significant contaminantto CMB-derived cosmological parameter constraints, due to the degeneracy between the Thomson-scattering optical depth, τ , and the amplitude of scalar perturbations, A s . This degeneracy subse-quently hinders the ability of large-scale structure data to constrain the sum of the neutrino masses,a major target for cosmology in the 2020s. In this work, we explore the kinematic Sunyaev-Zel’dovich(kSZ) effect as a probe of reionization, and show that it can be used to mitigate the optical depthdegeneracy with high-sensitivity, high-resolution data from the upcoming CMB-S4 experiment. Wediscuss the dependence of the kSZ power spectrum on physical reionization model parameters, aswell as on empirical reionization parameters, namely τ and the duration of reionization, ∆ z . Weshow that by combining the kSZ two-point function and the reconstructed kSZ four-point function,degeneracies between τ and ∆ z can be strongly broken, yielding tight constraints on both param-eters. We forecast σ ( τ ) = 0 .
003 and σ (∆ z ) = 0 .
25 for a combination of CMB-S4 and
Planck data,including detailed treatment of foregrounds and atmospheric noise. The constraint on τ is nearlyidentical to the cosmic-variance limit that can be achieved from large-angle CMB polarization data.The kSZ effect thus promises to yield not only detailed information about the reionization epoch,but also to enable high-precision cosmological constraints on the neutrino mass. I. INTRODUCTION
The epoch of reionization (EoR) is a source of bothsignals and foregrounds in cosmic microwave background(CMB) observations. The EoR is the period in cosmichistory in which the baryonic contents of the Universetransitioned from a neutral to an ionized state, as a re-sult of the ionizing radiation emitted by the first galax-ies and quasars. Along with the preceding dark agesand cosmic dawn, it is one of the least well-measuredepochs in observational cosmology. Fortunately, thissituation is set to change with the advent of power-ful new facilities that observe the EoR in myriad dif-ferent ways, including CMB experiments (e.g., SimonsObservatory [1], CMB-S4 [2], LiteBIRD [3]), 21 cm in-terferometers (e.g., Hydrogen Epoch of Reionization Ar-ray [4], Square Kilometer Array [5]) and monopole exper-iments (e.g., EDGES [6], SARAS [7], LEDA [8]), high-redshift galaxy surveys (e.g., Hyper Suprime-Cam [9], ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] James Webb Space Telescope [10], Roman Space Tele-scope [11]), and many others.In CMB measurements to date, the most relevantEoR signature has been the Thomson-scattering opticaldepth, τ = (cid:82) t t ∗ ¯ n e ( t ) σ T dt , where the integral runs fromthe surface of last scattering ( t ∗ ) to today ( t ), ¯ n e is thecosmic-average free electron number density, and σ T isthe Thomson cross-section. In fact, τ is one of the six freeparameters of the standard model of cosmology, ΛCDM,although unlike the others it is not a “fundamental” pa-rameter of the Universe. The optical depth predomi-nantly influences the CMB angular power spectra in twoways: (i) the overall amplitude of the temperature andpolarization power spectra on small scales is proportionalto A s e − τ , where A s is the primordial amplitude of scalarfluctuations; and (ii) the large-scale ( (cid:96) (cid:46) E -modepolarization auto-power spectrum is proportional to τ .These effects arise due to the scattering of CMB photonsoff free electrons during the EoR, which scatters photonsout of the line-of-sight (suppressing the temperature andpolarization anisotropies), and generates new polariza-tion anisotropies due to the scattering of the tempera-ture quadrupole (analogous to the generation of E -modepolarization at the surface of last scattering) (e.g., [12–16]). The latter effect is a unique signal of the EoR in theCMB power spectra, while the former effect is essentially a r X i v : . [ a s t r o - ph . C O ] J un a foreground, due to the degeneracy introduced between A s and τ , which weakens constraints on the primordialamplitude. Importantly, this also weakens constraintson beyond-ΛCDM parameters for which the sensitivityis dominated by their effect on the growth of structurebetween recombination and the present day, such as thesum of the neutrino masses ( M ν ≡ (cid:80) m ν ). Effectively,the increased error bar on A s due to the τ degeneracybecomes the limiting factor preventing a detection of M ν through massive neutrinos’ suppression of the growth ofstructure (e.g., [17–19]). Similar degeneracies are presentfor dark energy and modified gravity parameters.This situation strongly motivates measurements of τ at higher precision. The standard method of infer-ring τ is via the large-angle E -mode power spectrum.The current constraint from the Planck τ = 0 . ± .
007 [20], although some re-analyses haveclaimed error bars ≈
30% smaller than this [21]. Theultimate cosmic variance (CV) limit on τ from the pri-mary CMB power spectra is σ ( τ ) ≈ . Planck error bar. Becausethis signal requires measurements on the largest angu-lar scales, it is a primary target for a next-generationsatellite mission (e.g., LiteBIRD [3] or PICO [22]), al-though the Cosmology Large Angular Scale Surveyor(CLASS) is aiming to get close to this precision fromthe ground [23, 24]. This gain would be significant. Ifone considers CMB lensing measured by the Simons Ob-servatory (SO) as a late-time structure growth probe,then the current
Planck τ constraint limits the neutrinomass precision to σ ( M ν ) ≈ . .
04 eV, i.e., a (cid:46) σ de-tection of the minimal mass allowed by oscillation datain the normal hierarchy (0.059 eV) [1]. If the CV limiton τ is achieved, then the identical SO CMB lensing(and CMB high- (cid:96) primary anisotropy) data would yield σ ( M ν ) ≈ .
02 eV, i.e., a 3 σ detection of the minimalmass. Even more significant improvements would be seenwith data from CMB-S4 [2].Unfortunately, proposed satellite experiments thatwould reach the CV limit on τ are at least several yearsaway from launch. Thus, it is worth considering alterna-tive methods with which to constrain the optical depth,which is the primary motivation for this paper. In [19],it was suggested that 21 cm reionization measurementscould be used to constrain τ . The idea is that the 21 cmpower spectrum, which traces the spatial distribution ofneutral hydrogen as a function of redshift, can be used toconstrain a physical model of reionization. This modelcan then be used to predict τ . If the model constraintsare sufficiently precise, then τ can in principle be pre-dicted sufficiently well so as to improve on the current Planck constraints, and eventually surpass even the CVlimit from the primary CMB.We adopt a similar approach here, but instead of the21 cm line, we consider the kinematic Sunyaev-Zel’dovich(kSZ) effect as a probe of reionization. The kSZ effect isthe Doppler boosting of CMB photons as they Compton-scatter off free electrons moving with a non-zero veloc- ity along the line-of-sight [25–28]. The signal receivescontributions from both the EoR, often called “patchy”kSZ (e.g., [13, 15, 16, 29]), and from galaxies, groups,and clusters at late times (sometimes called the “homoge-neous” kSZ because the ionization fraction is essentiallyuniform after reionization). The EoR kSZ signal dependssensitively on the astrophysical details of reionization, asit directly probes the distribution of free electrons. Itis effectively the complement of the 21 cm field, whichdirectly probes the distribution of neutral hydrogen.In this work, our primary focus is not on extracting as-trophysical information from reionization kSZ measure-ments — although this is a very worthwhile pursuit —but rather on using these measurements to constrain τ and thereby resolve the parameter degeneracy problemdiscussed above. We consider two statistical probes ofthe EoR kSZ signal: (i) the angular power spectrum(two-point function) and (ii) a particular configurationof the trispectrum (four-point function), first pointedout in [30], with forecasts for τ presented in [31]. InSec. II, we describe the reionization model used in thiswork and present the relevant two-point and four-pointsignals. In Sec. III, we present the CMB experiment set-up and sky modeling used in this work, including a de-tailed treatment of foregrounds and component separa-tion. Sec. IV presents our primary science results, in-cluding constraints on τ and the duration of reionizationfrom the combination of these kSZ statistics. We discussthese results and future challenges for this program inSec. V. II. REIONIZATION KSZ
While the kSZ effect has long been recognized as oneof the most promising probes of the intergalactic mediumduring and after reionization (e.g., [32–34]), it has begunto be used only recently to provide constraints on reion-ization through the analysis of the angular power spec-trum of the CMB temperature at (cid:96) ≈ ζ ; the minimum mass of halos host-ing ionizing sources, M min ; and the mean free path of [] / = ./ = . / = ./ = . [] = .= . = .= . FIG. 1: Dependence of kSZ power spectrum on reionization model parameters. The shaded bars show 1 σ uncertainties on the powerspectrum, including instrumental noise and residual foregrounds for a combination of CMB-S4 and Planck data (see Sec. III), and samplevariance in the primary CMB and kSZ temperature for our fiducial model. The solid and dotted lines show variation of input modelparameters, M min and ζ (left panel), and the resulting reionization history parameters, τ and ∆ z (right panel). ionizing photons, λ mfp . The absorption systems that de-termine the mean free path are the dominant sinks ofionizing photons, limiting the size of HII regions in thepercolation phase.The derivative of the power spectrum with respect tomodel parameters used in our Fisher forecasts are ob-tained by running a series of simulations with differentparameters. Each simulation generates a realization ofthe ionization and density field on the observer’s pastlight cone, from which we generate a map of the temper-ature fluctuation field, (∆ T /T ) kSZ , over 1600 square de-grees, corresponding to the optical-depth-weighted line-of-sight velocity for z > .
5. The reionization historyfor each simulation on the grid of physical parameters isused to determine the Thomson scattering optical depth, τ , and the duration of reionization, ∆ z ≡ z − z , theredshift interval over which the volume filling factor ofionized regions evolves from 25 to 75 percent, for eachof these parameters. We also compute the power spec-trum for each of these maps which, together with themapping from physical parameters to τ and ∆ z , is usedfor both the two-point and four-point Fisher forecasts,as described in subsequent sections. The fiducial modelvalues we adopt are M min , = 3 × M (cid:12) , ζ = 70, and λ mfp , = 300 Mpc /h , for which τ (cid:39) .
06 and ∆ z (cid:39) . τ and ∆ z in ourFisher forecast. Fig. 1 illustrates the effect of varyingthe optical depth and duration of reionization on the kSZpower spectrum, in terms of D (cid:96) ≡ (cid:96) ( (cid:96) + 1) C (cid:96) / (2 π ). A. kSZ from the Two-Point function
As described above, we map the physical parameterscontrolling reionization to the empirical parameters τ and ∆ z . Compared to template-based approaches, thismodel generates a kSZ power spectrum with an (cid:96) depen- dence. The sensitivity of upcoming CMB experimentsto the temperature power on small scales, the improvedability to remove foregrounds from the power spectrumbased on multi-frequency maps, and the ability of CMBpolarization data to independently constrain the primarycosmological parameters all allow one to exploit this (cid:96) de-pendence fully. In particular, Fig. 1 shows the residualerrors from foreground cleaning and instrumental noise(see Sec. III for details) as shaded bars with simulatedspectra varying the physical reionization parameters aslines. The spectral shape between 2000 < (cid:96) < M min and ζ are the most closelyrelated to the empirical parameters considered here,namely the optical depth τ and the duration of reioniza-tion ∆ z, and we fix λ mfp = 300 Mpc /h for this analysis.The dependence of the power spectrum on the parame-ters is illustrated in Fig. 1. We compute the spectrumderivatives for the reionization parameters by varying themodel parameters M min and ζ and then use the chain ruleto compute ∂C (cid:96) ∂τ = ∂ζ∂τ ∂C (cid:96) ∂dζ + ∂M min ∂τ ∂C (cid:96) ∂M min , (1)and similarly for ∂C (cid:96) /∂ ∆ z . These derivatives are shownin Fig. 2.In the Fisher analysis, we adopt a conservative modelfor our prior knowledge about the late-time “homo-geneous” kSZ contribution. We use a template forthe homogeneous component from [40], normalized to D (cid:96) ( (cid:96) = 3000) = 2 . µ K . The homogeneous term can beestimated from simulations, but it is subject to astro-physical and cosmological uncertainties [41, 42]. Giventhe degeneracy between the homogeneous and patchycomponents, we do not impose strong priors on the ho-mogeneous component. We modify the [40] template asa power law with a pivot at (cid:96) = 3000 , with an amplitude / FIG. 2: Response of kSZ power spectrum to variations in τ and ∆ z .Increase in either τ or ∆ z results in more power at (cid:96) > and slope with fiducial values of A homKSZ = 1 , α homKSZ =0. We marginalize over both terms with a flat, non-informative prior. The constraints are not strongly de-pendent on the choice of prior for the homogeneous pa-rameters; the homogeneous parameters are constrainedby the data to σ ( A homKSZ ) = 0 . , σ ( α KSZ ) = 0 . τ CMB , as our goal here is to isolate the reion-ization information coming from the kSZ signal alone.For the two-point forecasts we include the temperature,polarization, and cross-power spectra (TT, EE, TE). Tobe conservative as to any residual foregrounds that per-sist after multi-frequency cleaning, we restrict the TTpower spectrum to 30 < (cid:96) < < (cid:96) < , followinga similar treatment to that presented in [1]. Details re-garding the foreground and noise models are given inSec. III. Pushing to higher (cid:96) in TT could significantlyimprove the constraints derived from the two-point func-tion, but would require a very accurate model for theCIB, tSZ, and other small-scale foregrounds in order toavoid biases. B. kSZ from the Four-Point function
In addition to being the largest blackbody componentin the high- (cid:96)
CMB, the kSZ signal is also significantlynon-Gaussian. This is because small-scale fluctuationsin the ionization fraction are modulated by slowly vary-ing velocity fields, meaning that the locally-measured kSZ power spectrum varies significantly between differentsightlines with different realizations of the velocity. Sucha modulation can be detected by a four-point functionestimator [30, 31], in close analogy to the one used toreconstruct the CMB lensing power spectrum. Anotherfeature of this estimator is that the shape of the mea-sured four-point function is determined by the proper-ties of the velocity field, which is well described by lineartheory. The velocity coherence length acts as a “stan-dard ruler”, allowing us to separate the late-time andreionization contributions to the kSZ signal in a model-independent way [30, 31].In addition to the derivatives of C (cid:96) with respect tothe parameters ( τ, ∆ z ), for the four-point function anal-ysis, we also need to assume a redshift distribution of thesource of the reionization kSZ signal. Following [31], wetake (cid:18) dC (cid:96) dz (cid:19) rei ( z, l, τ, ∆ z ) = C (cid:96), rei ( τ, ∆ z ) e − ( z − ¯ z ) / σ z (cid:112) πσ z (2)where we take the duration ∆ z to be approximately thefull-width at half maximum (FWHM) of the distribution,such that σ z ≈ ∆ z/ √ z ( τ ) is the mean redshiftof reionization.In this paper, we use the forecasting formalism of [31],marginalizing over an arbitrary amplitude and shape forthe late-time kSZ (with no prior) as well as a white noisecontribution. For the purpose of this paper, we definethe reionization contribution as being all of the kSZ sig-nal coming from z >
6. Since the bulk of the late-timekSZ originating from galaxies and clusters originates frommuch lower redshift, we find that our results are insensi-tive to this particular choice. In addition to providing ro-bustness in separating the late-time component, the kSZfour-point function has a different parameter dependencethan the power spectrum, allowing for very effective de-generacy breaking, which is the main result of this work(see Sec. IV).Gravitational lensing of the CMB creates a four-pointfunction that could potentially mimic that of reioniza-tion. In [30] it was shown that using lensing reconstruc-tion from polarization (which is not affected by kSZ), thelensing contribution can be reduced to a white noise com-ponent, which we marginalize over in the forecast here.We use temperature modes from 2000 < (cid:96) <
III. EXPERIMENTAL ASSUMPTIONS ANDSKY MODELING
In this work, we consider forecasts for the future CMB-S4 experiment [2] (first light ∼ ∼ µ K · arcmin at 93 GHz and 2 µ K · arcmin at 145GHz. This survey will encompass 70% of the sky, but weassume an effective sky area of 45% for the high-precisionCMB blackbody temperature map reconstruction that isa necessary first step for the kSZ analyses considered be-low (our forecasts are thus somewhat conservative).To forecast an effective post-component-separationnoise power spectrum for the reconstructed CMB black-body temperature map, we employ the methodology de-scribed in Ref. [2] (see Appendix A.3; see also Sec. 2 ofRef. [1]). Planck data from 30–353 GHz are also assumedto be used in the CMB blackbody component separa-tion; these data are particularly crucial on large angularscales where atmospheric noise is significant for CMB-S4and other ground-based experiments. Nevertheless, weemphasize that the forecasts here are driven by the high-sensitivity, multi-frequency data of CMB-S4 on small an-gular scales, where the kSZ signal dominates the black-body sky. In total, we consider thirteen frequency chan-nels, six from CMB-S4 and seven from
Planck . Our com-ponent separation analysis includes realistic models of allmajor sky signals and foregrounds for every
Planck andCMB-S4 frequency channel, combined with the CMB-S4noise modeling mentioned above and white noise for the
Planck channels (with noise levels from [45, 46]). Wethen analyze these sky models with a harmonic-space in-ternal linear combination (ILC) [e.g., 47] code to computepost-component-separation noise power spectra for thecleaned CMB blackbody temperature map, N T T(cid:96) . Thesepower spectra thus capture the contributions of residualforegrounds and noise due to the detectors and atmo-sphere.For simplicity, we use “standard” ILC noise powerspectra here, in which the total variance of the final ‘ − ‘ ( ‘ + ) C TT ‘ / ( π ) [ µ K ] Reionization kSZ Power Spectrum [ τ = 0 .
06, ∆ z = 1 . Planck )Na¨ıve Noise Power Spectrum (CMB-S4 +
Planck ) FIG. 3: Signal and noise power spectra. The thick black curveshows our fiducial reionization kSZ power spectrum, computed ina model with λ mfp = 300 Mpc/ h , ζ = 70, and M min = 3 × M (cid:12) ,which yields τ = 0 .
06 and ∆ z = 1 .
2. The dashed black curveshows our fiducial late-time kSZ power spectrum, while the thinblue curve shows the lensed primary CMB temperature power spec-trum. Both of these contributions are spectrally degenerate withthe reionization kSZ signal. The thick green curve shows the effec-tive noise power spectrum determined from CMB-S4 and
Planck data using an ILC method, while the dashed green curve shows thena¨ıve noise power spectrum in the absence of foregrounds. blackbody map is minimized (subject to a constraint thatpreserves the signal), but in which no particular contami-nant is explicitly required to vanish. Future analyses maynecessitate the use of CMB blackbody ILC maps withparticular component SEDs nulled (e.g., tSZ or approxi-mate CIB SEDs) via a constrained ILC procedure [e.g.,48, 49] so as to mitigate possible biases from these con-taminants. It may also be the case that the kSZ powerspectrum will be inferred through an analysis directlyat the power spectrum level, i.e., without first construct-ing a foreground-cleaned blackbody map. (Measuring thekSZ four-point function will almost certainly require con-structing a foreground-cleaned map first.) Both of theseanalysis choices could modestly increase the error barson the forecasts presented here. High-frequency mapsfrom, e.g., CCAT-prime [50] could be useful in mitigat-ing foreground contamination effects, particularly due tothe cosmic infrared background (CIB). Due to currentuncertainties in CIB modeling, we defer detailed consid-eration of this issue to future work employing an end-to-end map-based simulation framework.Fig. 3 shows the final post-component-separation noisepower spectrum used in this analysis, as well as the CMBblackbody signal comprised of the lensed primary tem-perature power spectrum and the kSZ power spectrum.The latter includes contributions from both reionizationand the late-time universe, as labeled in the figure. Forcomparison, the figure also shows a na¨ıve noise powerspectrum that would result if all of the frequency mapswere co-added with inverse-noise-variance weighting only,and no foregrounds were present in the sky. This high-lights the importance of fully modeling all signals in themm-wave sky in such forecasts.
IV. RESULTS
We show the individual and joint constraints on ∆ z and τ from the kSZ two-point and four-point functions inFig. 4. The two-point function is weakly constraining onthe optical depth compared to the standard constraintsfrom the primary CMB, but tightly constrains the dura-tion of reionization. Conversely, the four-point functionis more sensitive to the optical depth than to the dura-tion of reionization. When combined together, and alsofolding in the Planck primary CMB constraint on τ , thejoint forecast yields a (marginalized) covariance matrixCov( τ, ∆ z ) = (cid:18) . × − − . × − − . × − . (cid:19) so that σ ( τ ) = 3 × − and σ (∆ z ) = 0 .
25. The error on σ (∆ z ) reduces to σ (∆ z ) = 0 . τ is nearly as tight as a CV-limited constraint fromthe primary CMB ( σ ( τ ) = 2 × − ), as targeted bynext-generation satellite missions.The different degeneracy direction between the two-and four-point estimators is straightforward to explain:while changing parameters such as the duration of reion-ization changes the power spectrum, it will also changethe amount of non-Gaussianity in a different way. For ex-ample, a shorter reionization epoch would lead to a morenon-Gaussian kSZ field, and enhance the four-point func-tion compared to the two-point function. By measuringboth, we can effectively break the parameter degeneracyand obtain tighter limits on reionization.Since the four-point estimator involves four powers ofthe map noise, one may wonder whether it would performbetter in a deeper but smaller survey (e.g., the “delens-ing” survey planned for the CMB-S4 primordial gravita-tional wave search [2]), rather than in the shallower widesurvey considered here. A simple estimate indicates thatbecause of foregrounds, the reduction in effective noiseis not large enough to compensate for the decreased skyarea, and thus the wide survey considered here is ex-pected to yield better performance. V. DISCUSSION AND CHALLENGES
Fig. 4 clearly illustrates the power of combining thefour-point and two-point constraints due to their com-plementary degeneracy directions in the reionization pa-rameter space. Given this statistical power, a carefulconsideration of potential biases and systematics of theseprobes is necessary, which we briefly outline here.
Foreground cleaning:
Multi-frequency coverage is cru-cial for isolating the blackbody kSZ signal from other, non-blackbody foregrounds in the high- (cid:96)
CMB, such asthe thermal SZ effect and the CIB. At the power spec-trum level, these contributions can be simultaneously fitin a multi-component analysis, although accurate mod-eling will be needed. To be conservative in this work,we have not used modes at (cid:96) >
Reionization modelling:
Perhaps the largest source ofuncertainty is the physical modelling of the reionizationprocess in a standard UV-dominated scenario, and morespecifically in the parameter dependence of the mean freepath, efficiency, and mass. Alternative reionization sce-narios involving very high-redshift sources would involvedifferent values of the parameters and model assumptionsthan those considered here. We leave the investigationof the sensitivity to these models to future work; how-ever, the precision of the model parameter constraints inthis analysis implies that we will indeed be able to ruleout other models including reionization from early X-raybinaries, population III sources, rare quasars, or otherexotic reionization scenarios. Also, our ability to pindown the exact model of reionization will be enhancedthrough cross-correlations of CMB-S4 data with exter-nal data sets such as 21-cm and Lyman- α emitter sur-veys (e.g., [51]). Finally, we note that independent CMB-based constraints on the reionization history and opticaldepth τ from the large-scale EE power spectrum mea-surements will further break the degeneracy by removinguncertainty on one axis. Non-patchy optical depth:
Here we have assumed thatall the patchiness in the ionization field will be resolvedby the measurements. However, certain scenarios, suchas reionization due to dark matter decay or annihilationor very hard X-ray sources, allow for additional contribu-tions to the optical depth that do not contribute patchi-ness (or the bubbles are too small to be resolved given thefinite beam size). Thus, these constraints are technicallya lower limit on the optical depth.
Covariance:
In the analysis above, we assumed thatthe kSZ two-point and four-point functions had zero co-variance. This assumption holds if the patches in whichthese signals are measured are non-overlapping on thesky, in which case the noise covariance is clearly zeroand we can straightforwardly combine them. If they areoverlapping, the calculation involves computing ∼ kSZ from CMB-S4 LATs ( ) = . , ( ) = . Planck primary Planck primary + +
FIG. 4: Constraints on the duration of reionization and optical depth. The vertical shaded contours are 68% and 95% confidence regionsfrom the primary CMB anisotropies measured by
Planck , which constrain the optical depth to an error of σ ( τ ) = 0 .
007 (the primary CMBdoes not constrain ∆ z ). The angled contours show forecast reionization constraints from the kSZ power spectrum (pink) and the kSZfour-point function (blue), as derived from CMB-S4 and Planck data. The black contours show forecast constraints from the combinationof all three probes. The complementary degeneracy directions of the two-point and four-point functions effectively break the degeneracybetween the reionization parameters, yielding tight constraints on both τ and ∆ z : σ ( τ ) = 0 .
003 and σ (∆ z ) = 0 . six-point functions, and we leave it to future work (asimulation-based analysis may be more tractable). Theuncorrelated assumption also holds if we use two differ-ent experiments on the same patch of sky, since our con-straints are dominated by high- (cid:96) information where theprimary CMB is negligible.While overcoming the challenges mentioned above willrequire significant effort, this is well justified by the kSZreionization constraints forecast here. The tight con-straint on τ will enable neutrino mass constraints fromupcoming surveys that utilize the full statistical poweravailable from large-scale structure data, including CMBlensing. The reionization constraints will yield rich astro-physical information about the nature and distributionof the ionizing sources, in particular when kSZ data arejointly analyzed with 21 cm data, intensity mapping sur-veys, high-redshift galaxy and quasar studies, and otherprobes of the EoR. ACKNOWLEDGMENTS
We thank Tom Crawford, Emmanuel Schaan, BlakeSherwin and Kendrick Smith for useful discussions. SF is supported by the Physics Division of Lawrence BerkeleyNational Laboratory. JCH thanks the Simons Founda-tion for support. RH is a CIFAR Azrieli Global Scholar,Gravity & the Extreme Universe Program, 2019, and a2020 Alfred P. Sloan Research Fellow. RH is supportedby Natural Sciences and Engineering Research Council ofCanada. The Dunlap Institute is funded through an en-dowment established by the David Dunlap family and theUniversity of Toronto. We acknowledge that the land onwhich the University of Toronto is built is the traditionalterritory of the Haudenosaunee, and most recently, theterritory of the Mississaugas of the New Credit First Na-tion. We are grateful to have the opportunity to work inthe community, on this territory. All authors contributedequally to the preparation of this manuscript. [1] P. Ade, J. Aguirre, Z. Ahmed, S. Aiola, A. Ali, D. Alonso,M. A. Alvarez, K. Arnold, P. Ashton, J. Austermann,et al., JCAP , 056 (2019), 1808.07445.[2] K. Abazajian, G. Addison, P. Adshead, Z. Ahmed, S. W.Allen, D. Alonso, M. Alvarez, A. Anderson, K. S. Arnold,C. Baccigalupi, et al., arXiv e-prints arXiv:1907.04473(2019), 1907.04473.[3] M. Hazumi, J. Borrill, Y. Chinone, M. A. Dobbs, H. Fuke,A. Ghribi, M. Hasegawa, K. Hattori, M. Hattori, W. L.Holzapfel, et al.,
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