Cosmic microwave background constraints on the tensor-to-scalar ratio
aa r X i v : . [ a s t r o - ph . C O ] J un RAA 2014
Vol. No. , 635–647 doi: 10.1088/1674–4527/14/6/003 R esearchin A stronomyand A strophysics Cosmic microwave background constraints on thetensor-to-scalar ratio
King Lau, Jia-Yu Tang and Ming-Chung Chu
Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong,Hong Kong, China; [email protected]
Received 2013 October 22; accepted 2013 December 27
Abstract
One of the main goals of modern cosmic microwave background (CMB)missions is to measure the tensor-to-scalar ratio r accurately to constrain inflationmodels. Due to ignorance about the reionization history X e ( z ) , this analysis is usu-ally done by assuming an instantaneous reionization X e ( z ) which, however, can biasthe best-fit value of r . Moreover, due to the strong mixing of B-mode and E-modepolarizations in cut-sky measurements, multiplying the sky coverage fraction f sky bythe full-sky likelihood would not give satisfactory results. In this work, we forecastconstraints on r for the Planck mission taking into account the general reionizationscenario and cut-sky effects. Our results show that by applying an N-point interpo-lation analysis to the reionization history, the bias induced by the assumption of in-stantaneous reionization is removed and the value of r is constrained within errorlevel, if the true value of r is greater than about 0.1. Key words: cosmology: cosmic microwave background — cosmology: cosmologicalparameters — cosmology: early universe — gravitational waves
Inflation (Guth 1981; Linde 1982; Albrecht & Steinhardt 1982) is now the leading paradigm in cos-mology. The inflation scenarios have been proposed to solve the problems of horizon, flatness andmagnetic monopoles and explain the generation of primordial perturbations in the early Universe.Most inflation models predict two types of initial perturbations: scalar and tensor. The scalar per-turbations are adiabatic, nearly Gaussian and close to being scale-invariant, which are consistentwith a series of observations (Hu & White 1996; Spergel & Zaldarriaga 1997; Hu et al. 1997;Peiris et al. 2003; Spergel et al. 2007; Hinshaw et al. 2013). The tensor perturbations producedduring inflation, also known as gravitational waves, can be quantified as a tensor-to-scalar ratio r .Therefore, a non-zero r is considered to be important evidence of inflation if it is observed. Sincetensor perturbations can be detected in a large-scale temperature power spectrum and should haveleft an imprint on the B-mode polarization of the cosmic microwave background (CMB) (Seljak &Zaldarriaga 1997; Kamionkowski et al. 1997), constraining r is one of the main goals of modernCMB surveys. Recent data from the nine year result of the Wilkinson Microwave Anisotropy Probe(WMAP9) and South Pole Telescope give the latest constraints of r < . and r < . at the95% confidence level (CL) respectively without a measurement of the B-mode polarization (Storyet al. 2013; Hinshaw et al. 2013; Bennett et al. 2013). Although Planck ’s results were released in
36 K. Lau, J. Tang & M. C. Chu
March 2013 (Planck Collaboration 2013a), its polarization data, which are crucial for constraintson r , are not yet available. Their current approach combines Planck ’s measurements of temperatureanisotropy with the WMAP large-angle polarization to constrain inflation, giving an upper limit of r < . as a 95% CL (Planck Collaboration 2013b).These constraints for r were obtained by assuming an instantaneous model for reionization his-tory, but X e ( z ) , the average ionized fraction at redshift z , is rather uncertain. Various sources suchas star formation (Springel & Hernquist 2003; Bunker et al. 2004), massive black holes (Sasaki &Umemura 1996) and dark matter decay (Mapelli et al. 2006; Belikov & Hooper 2009) have beensuggested to provide the energy flux necessary for reionizing hydrogen. CMB and quasar observa-tions show that recombination occurs at redshift z ∗ ∼ and the Universe must have been fullyreionized at z ∼ (Becker et al. 2001; Fan et al. 2002), but there is no detailed knowledge about theevolution of X e ( z ) between these two eras. The constraint imposed on X e ( z ) by current CMB mea-surements is also poor. The CMB temperature power spectrum C TT l only gives a strong constrainton A s e − τ , where A s and τ are the scalar amplitude and optical depth respectively. Even if we canbreak the degeneracy between these two parameters, we can only get the information from τ , where τ ∝ Z z ∗ (cid:0) X e ( z ) √ z (cid:1) dz , (1)but not X e ( z ) itself. The CMB E-mode polarization spectrum C EEl only has a weak dependence onthe reionization history, and thus an attempt at constraining X e ( z ) by the current E-mode polariza-tion measurement does not give any satisfactory result (Lewis et al. 2006).Several studies have considered the effects of uncertainties in X e ( z ) on cosmological param-eter estimation. To parameterize the reionization history, Lewis et al. assume a constant ionizationfraction in finite redshift bins and join the bins using a tanh function (Lewis et al. 2006). Mortonsonet al. propose that the reionization history can be expressed as a linear combination of finite num-bers of principal components S µ extracted from the Fisher information matrix that describes thedependence of E-mode polarization on reionization, so that the amplitudes of S µ are parameters for X e (Mortonson & Hu 2008b). To consider the general reionization scenario, Pandolfi et al. applythese two methods in their analysis to constrain the inflation parameters by WMAP7 data (tem-perature and E-mode polarization only) (Pandolfi et al. 2010) while the PLANCK Collaboration(Planck Collaboration 2013b) only adopts the method by Mortonson et al. A recent study investi-gates how instantaneous-like reionization models affect the estimation of all cosmological parame-ters from Planck -quality CMB data except for r (Moradinezhad Dizgah et al. 2013). To account forthe fact that parts of the sky are masked to eliminate foreground contaminations, they multiply thesky coverage fraction f sky ( f sky = 0 . for Planck ) by their full-sky likelihood.In this paper, we explore how well r can be constrained by the Planck mission with a generalparametrization of the reionization history. In Section 2, we discuss the degeneracy between thereionization history X e ( z ) and r in the full-sky CMB power spectra and the necessity of using bothtemperature and polarization power spectra for constraining r . Then we make a full-sky forecast andconclude that a bias is possibly introduced in r if an incorrect reionization assumption is appliedin the Markov Chain Monte Carlo (MCMC) analysis (Kosowsky et al. 2002). The N-point linearinterpolation method for reionization is also introduced. We then discuss the significance of strongmixing of E-mode and B-mode polarizations and show that the simple f sky modification is unrealisticin constraining r using cut-sky power spectra in Section 3. Thus we apply the Hamimeche and Lewislikelihood approximation which can handle the CMB temperature-polarization correlation for high l ’s in the cut-sky. In Section 4, we present the forecast of Planck ’s constraint on r using a generalreionization representation. osmic Microwave Background Constraints on the Tensor-to-scalar Ratio 637 n T , r AND n s In this study, we consider single-field inflation with the slow-roll approximation. Conventionally, thepower spectra of scalar perturbations P R and tensor fluctuations P h have the functional form k P R ( k ) ∝ k n s − , (2) k P h ( k ) ∝ k n T , (3)in which n s and n T are the spectral index and tensor tilt, respectively. Hence, the tensor-to-scalarratio r is defined as r ≡ P h ( k ) P R ( k ) , (4)where k is the pivot scale. Our choice for it is k = 0 . Mpc − . Moreover, in the simple slow-rollinflation model, there is a well-known consistency relation (Kinney 1998) n T = − r , (5)which is correct at first order in the slow-roll parameters. These parameters are of interest for study-ing the CMB because they give an accurate measurement of r and spectral index n s can discriminateamong inflation models (Dodelson et al. 1997; Kinney 1998).Previous analysis of CMB data usually assumes the reionization history to be instantaneous, X e ( z ) ∝
12 + 12 tanh (cid:18) z re − z ∆ z (cid:19) , (6)where ∆ z is a width parameter and z re is the redshift at which reionization occurs. Here, we take ∆ z = 1 . . Combining Equation (6) with Equation (1), we can utilize CMB data to make an infer-ence for the parameters z re and τ . However, it has been pointed out that C EE l and C TE l , unlike C TT l ,depend not only on τ but also on the detailed evolution of X e ( z ) (Lewis et al. 2006), especially for l < . To compare the impacts of the reionization history on CMB polarization power spectra, weconsider the instantaneous model and two other physically acceptable reionization models, doublereionization and two-step reionization, as illustrated in Figure 1. The former model has two instan-taneous reionizations occurring at z = 7 , and a sudden, midway recombination, while the latterdescribes a reionization process with a long, intermediate pause. All of them give τ = 0 . andare assumed to reach full H ionization at z ∼ , as well as a late time He reionization at z ∼ .The corresponding C EE l , C TE l and C BB l are shown in Figure 2. The distinct differences in the threecurves for l < indicate the dependences of CMB polarization power spectra on the reionizationhistory X e ( z ) . Although double reionization is rather disfavored by current observations (Zahn etal. 2012), it helps to demonstrate the bias on r if an incorrect reionization model is used.To examine the possible bias of parameters by an incorrect reionization model, we run a simpletest on two sets of full-sky CMB power spectra that have quality comparable to Planck . These twosets of power spectra are generated using the standard WMAP9 best-fit cosmological parameters butwith the two-step reionization and double reionization model respectively. Meanwhile, we still usean instantaneous reionization model and perform MCMC analysis to estimate τ from these powerspectra (refer to Sect. 4 for details about MCMC fitting).Figure 3 shows the probability functions of τ and z re after marginalizing over all other parame-ters. The dashed and dashed-dotted lines indicate the results using two-step reionization and doublereionization, respectively. Both probability functions show a sharp convergence, but a bias in τ isintroduced relative to the fiducial value of the optical depth τ = 0 . , indicated by the verticalline. Moreover, the bias with the double reionization is larger, which reflects the larger differencebetween the instantaneous model and the double reionization model. As there is a τ − n T degeneracy
38 K. Lau, J. Tang & M. C. Chu z X e Fig. 1
The three fiducial reionization models considered in this paper: instantaneous reion-ization ( solid line ), two-step reionization ( dashed line ) and double reionization ( dashed-dotted line ). All of them are assumed to reach full H ionization at z ∼ and have late timeHe reionization at z ∼ . All of them give τ = 0 . . l l ( l + ) CBB ll ( l + ) C TE ll ( l + ) C EE l Fig. 2
The CMB E-mode polarization power spectrum, temperature-E cross spectrum andB-mode polarization power spectrum (all generated by standard Λ CDM best-fit WMAP9parameters with r =0.1) for three reionization histories: instantaneous ( solid line ), two-step( dashed line ) and double reionization ( dashed-dotted line ) as shown in Fig. 1. They aresensitive to the reionization history for l ≤ . osmic Microwave Background Constraints on the Tensor-to-scalar Ratio 639 P / P m a x τ z re
10 11 12 130.20.40.60.81.00.0 P / P m a x Fig. 3
Probability functions P (relative to the peak value P max ) of the MCMC fittingresults for the optical depth τ ( left ) and reionization redshift z re ( right ) from two sets ofexpected Planck full-sky CMB power spectra (including temperature and polarization).We calculate these power spectra using the two-step reionization ( dashed line ) and doublereionization ( dashed-dotted line ), while assuming the instantaneous reionization history tomake the MCMC fitting. Both probability functions show a sharp convergence but a biasis introduced relative to the fiducial value of the optical depth ( τ = 0 . ) indicated bythe vertical line.on the B-mode polarization spectrum (Mortonson & Hu 2008a), if τ is biased to a larger value, thecorresponding n T becomes less negative; because r and n T are correlated in CMB constraints, abias in the estimation of r is introduced.Therefore, to avoid any assumption on reionization history, we modify an N-point parametriza-tion of reionization which was initially proposed by Lewis et al. (2006). We fix X e ( z = 6) = 1 . (with He reionization) and X e ( z = 22) = 0 , and we insert N floating point values { X e ( z i ) } Ni =1 atredshift z i = 6 + i (22 − / ( N + 1) , i = 1 , · · · , N. (7)Then the whole reionization history in ≤ z ≤ is the linear interpolation among these points.This method introduces N reionization points, which are extra parameters in the MCMC analysistogether with the cosmological parameters, and they are free to vary in the range [0,1] (the physicalrange of X e ( z ) ). We assume a late-time He reionization occurred at z ∼ . To make a forecast from fiducial CMB power spectra, we perform a TT-TE-EE-BB joint analysiscovering both small and large scales with the N-point method. It is important to include the C TT l power spectrum to constrain the baryonic density Ω b , dark matter density Ω c , Hubble parameter H and A s e − τ well. As tensor perturbations contribute to the CMB temperature power spectrumat large scales, its effect is degenerate with the change in n s in large-scale CMB measurements.Therefore, the C TT l power spectrum at small scales can help break this degeneracy by constraining n s well. It is also necessary to include C EE l and C BB l power spectra at large scales, and drop theassumption of instantaneous reionization to break the τ − r degeneracy as we discussed in Section2. Although C TE l has a weak dependence on reionization history, as it is a cross spectrum, its noiseis much reduced, and thus it helps to break this degeneracy in a joint analysis.Assuming that the CMB field is Gaussian and isotropic, the full-sky maximum likelihood L forthe measured TEB correlation spectra ˆ C l is − ln L = l max X l min (2 l + 1)[ Tr ( ˆC l C − l ) − ln | ˆC l C − l | − , (8)
40 K. Lau, J. Tang & M. C. Chu
500 1000 1500 2000 l l ( l + ) C TT l
500 100000.010.020.030.04 l l ( l + ) CBB l Fig. 4
The CMB temperature ( left ) and B-mode polarization power spectrum ( right ) forfull-sky ( solid lines ) and their pseudo power spectra for cut-sky ( dashed lines ), for thefiducial model of r = 0 . and standard cosmological parameters. P ( C BB l ) is increased bythe mask due to mixing with C EE l , making the simple f sky modification unrealistic. Themasks applied by WMAP are used in this illustration.where C l = C TT l C TE l C TB l C TE l C EE l C EB l C TB l C EB l C BB l (9)is the matrix of theoretical power spectra and ˆC l is defined similarly.In practice, masks are applied for both temperature and polarization data (Jarosik et al. 2011), toexclude the part of the sky map contaminated by the astrophysical foreground, mainly the Galacticplane. To have a cut-sky forecast, the sky coverage fraction f sky is usually included as a factor inEquation (8) to account for the fractional loss of ˆ C l power spectra in the cut-sky. While f sky modifi-cation works well for the temperature power spectrum, it is not sufficient for the polarization spec-tra, but the C BBl power spectrum is crucial for the constraint of r . Burigana et al. have considereda toy model to include the impact of the foreground contamination in full-sky likelihood analysis(Burigana et al. 2010). This method, however, will introduce uncertainties in modeling the noiseresiduals and therefore r and X e . With this consideration, in this work, we use the pseudo- C l as theestimators and apply the likelihood approximation introduced by Hamimeche and Lewis (hereafterH.L. likelihood) (Hamimeche & Lewis 2008). The pseudo power spectrum P ( C l ) , defined as thepower spectrum over a masked sky map, is related to the full-sky CMB spectrum by the relation(Kogut et al. 2003) * P ( C TT l ) P ( C TE l ) P ( C EE l ) P ( C BB l ) + = X l ′ M TT ll ′ M TE ll ′ M EE ll ′ M EB ll ′ M BE ll ′ M BB ll ′ C TT l ′ C TE l ′ C EE l ′ C BB l ′ , (10)where h ... i denotes the expectation value and { M XYll ′ } are the coupling mask matrices. Details aboutthis likelihood are described in the Appendix.Before we proceed, we illustrate in Figure 4 why a simple f sky modification of the full-skylikelihood would not give a realistic constraint on r . Figure compares the full-sky C TT l and C BB l (solid lines) and their corresponding pseudo power spectrum in the cut-sky (dashed lines), for r =0 . . We apply the same masks released by the WMAP team , which are equivalent to f sky = 0 . .As expected, P ( C TT l ) is approximately reduced by a factor of f sky compared with C TT l . However, P ( C BB l ) is significantly increased, due to the mixing between C EE l and C BB l and the coupling amongdifferent l -modes. The masks for WMAP are available at http://lambda.gsfc.nasa.gov/product/map/dr4/masks get.cfm osmic Microwave Background Constraints on the Tensor-to-scalar Ratio 641
PLANCK
To forecast the constraints on r by the Planck survey with a general reionization scenario, we performMCMC analysis with the H.L. likelihood discussed in the Appendix. The expected
Planck pseudopower spectra ˆ C l from the fiducial power spectra C fid l are calculated as ˆ C XYl = P ( b l C fid XYl + N XYl ) (11)for X , Y = T , E or B , where b l = exp (cid:18) − l ( l + 1)( θ fwhm / rad ) ln (cid:19) , (12)the beam width θ fwhm = 7 . ′ and the noises N XYl for the C XYl spectrum are N TT l = 1 . × − µ K , N EEl = N BBl = 3 . N TT l (PLANCK Collaboration 2006) and N T El = 0 . We use the frequencyband centered at 143 GHz and assume that there is no correlation among the random noise fields. C fid l is computed using CAMB (Lewis et al. 2000). The fiducial model used to compute C fid l is Ω b h = 0 . , Ω c h = 0 . , n s = 0 . , θ = 1 . and A s = 2 . × − , usingstandard notations for the cosmological parameters. Two fiducial models of reionization, the doublereionization and two-step reionization, are considered in our analysis, which are shown in Figure 1.In addition, we investigate three cases for r : r = 0 . , . and . and their corresponding tensortilt is taken as n T = − r/ . Thus, six sets of pseudo power spectra ˆ C XYl are generated.We perform the MCMC analysis using the modified version of CosmoMC (Lewis &Bridle 2002) by Mortonson and Hu (Mortonson & Hu 2008a). To study the impact of reion-ization history on estimation as discussed in Section 2, two treatments on X e are applied. Forthe first, we use the instantaneous reionization modeled as in Equation (6). The varied pa-rameters are { Ω b h , Ω c h , θ, n s , n T , log (10 A s ) , r, τ } . The second treatment uses the N-pointmethod defined in Equation (7); we further modify the CosmoMC program to vary parameters { Ω b h , Ω c h , θ, n s , n T , log (10 A s ) , r, { X e ( z i ) } Ni =1 } for N = 7 . The likelihood is summed over l min = 2 and l max = 2200 . We are aware that the H.L. likelihood is less reliable for pseudo- C l atlow- l range, and Hamimeche & Lewis (2008) state that exact likelihood calculation is feasible atlow- l when realistic foreground contamination is carefully considered. However, this investigationis beyond the scope of this paper. In addition, the CMB polarization power spectra at low- l are es-sential for breaking the degeneracy between the reionization history and inflation parameter r . Ourresults show that if l min is taken as , only τ is biased by ∼ σ by the choice of l min (smaller l min gives a better constraint), while similar best-fit values of r and other cosmological parameters withslightly larger uncertainties are obtained. Therefore, we extend the application of H.L. likelihood to l min = 2 and focus on the results based on this condition.Figure 5 and Figure 6 compare the constraints on r by the N-point method and the instantaneousmodel. The left and right panels of these plots show the results for the fiducial r = 0 . and r = 0 . ,respectively. The statistics describing them, including best-fit, mean ¯ r , standard deviation σ andCL, are shown in Tables 1 and 2. It can be seen that for a complex reionization history, the simpleinstantaneous assumption generally biases r to a smaller value. In our case, the true value of r iseven ruled out at the 68% CL for both double-step reionization and two-step reionization models.The application of the N-point method can correct for these biases and the best-fit value of r isconstrained to within the 5% error level if the true value of r & . . The N-point method also givesa better inference on ¯ r . CAMB is available at http://camb.info/ CosmoMC is available at http://cosmologist.info/cosmomc/ Mortonson and Hu’s program is available at http://background.uchicago.edu/camb rpc/
42 K. Lau, J. Tang & M. C. Chu
Table 1
Statistics of 1D marginalized probability of r for MCMC analysis on the expected Planck pseudo power spectra with double reionization.
Fiducial r Reionization model Best-fit ¯ r ± σ
68% CL 95% CL0.1 Instantaneous 0.048 0.074 ± ± ± ± Table 2
Same as Table 1, but for the expected
Planck pseudo power spectrawith two-step reionization.
Fiducial r Reionization model Best-fit ¯ r ± σ
68% CL 95% CL0.1 Instantaneous 0.048 0.070 ± ± ± ± r P / P m a x r P / P m a x Fig. 5
1D marginalized probability P of r (relative to the peak value P max ) in MCMCfitting of the expected Planck pseudo power spectra with double reionization as the fiducialmodel. The left and right panels show fiducial r = 0 . and r = 0 . ( indicated by verticallines ), respectively. The solid and dashed lines stand for results of the instantaneous andN-point parameterizations respectively. r P / P m a x r P / P m a x Fig. 6
Same as Fig. 5, but with two-step reionization as the fiducialmodel.Figures 7 and 8 show the corresponding 2D contours for r vs. n s . In these 2D cases, the truevalues of r and n s are not ruled out at the 68% CL and 95% CL for both the N-point method andinstantaneous reionization assumption applied in the analysis. osmic Microwave Background Constraints on the Tensor-to-scalar Ratio 643 r n s r s Fig. 7
2D marginalized probability contours of r vs. n s (68% CL and 95% CL) in theMCMC fitting of the expected Planck pseudo power spectra with double reionization asthe fiducial model. The left and right panels show the results with the fiducial r = 0 . and r = 0 . , respectively. The solid and dashed lines stand for results of the instantaneousand N-point parameterizations respectively. The asterisks indicate the fiducial values. r n s r s Fig. 8
Same as Fig. 7, but with the two-step reionization as the fiducial model. r P / P m a x Fig. 9
1D marginalized probability P of r (relative to the peak value P max ) in the MCMCfitting of the expected Planck pseudo power spectra with two-step reionization as the fidu-cial model, with r = 0 . ( indicated by a vertical line ). The solid and dashed lines standfor results of the instantaneous and N-point parameterizations respectively.However, the N-point method has its own limitation when r is small. Figure 9 shows the 1Dmarginalized probability distribution of r with the fiducial r = 0 . . It can be seen that the marginal-ized probability of the fitted r still prefers r = 0 , which indicates that if the N-point method isapplied, the Planck data cannot by themselves be used to detect r if its value is close to .
44 K. Lau, J. Tang & M. C. Chu Ω b h θ log[10 A s ] τ r Ω c h n s P / P m a x n T -0.6 0 3.15 3.20 3.25 0 0.1 0.2 0.08 0.09 0.10 0.110.20.40.60.81.00.0 P / P m a x -0.4 -0.2 r n s -0.4 -0.2 0 n T r Fig. 10
Left : 1D marginalized probability P of cosmological parameters (relative to thepeak value P max ) using the N-point method in the MCMC fitting of two sets of the ex-pected Planck pseudo power spectra with fiducial r = 0.1 but with two-step reionization( dashed lines ) and double reionization ( dashed-dotted lines ). The vertical lines indicatethe fiducial values. Right : The corresponding 2D marginalized probabilities of r vs. n s and r vs. n T . The asterisks indicate the fiducial values. Ω b h Ω c h θ n s n T log[10 A s ] r τ P / P m a x P / P m a x -0.6 -0.4 -0.2 s n r n T r Fig. 11
Same as Fig. 10, but for r = 0 . .Figures 10 and 11 show the fitting result of cosmological parameters applying the N-pointmethod for the fiducial r = 0 . and . respectively (we do not repeat the similar results from r = 0 . ). It shows that the N-point method with H.L. likelihood also gives reasonable results forthe fitting of other cosmological parameters ( Ω b h , Ω c h , θ, n s and log (10 A s ) ). By contrast, τ isstill biased with its true value being ruled out at the 68% CL. This is because τ is sensitive to theMCMC fitting result for the reionization points ( { X e ( z i ) } Ni =1 for N = 7 ) as it is obtained by com- osmic Microwave Background Constraints on the Tensor-to-scalar Ratio 645 puting an integration over X e ( z ) (Eq. (1)), but our results show poor convergence for them, whichmeans the constraints on the reionization history itself are poor based on Planck ’s power spectraalone. However, the constraint on r does give satisfactory results in the general reionization sce-nario by marginalization since C BB l is not as sensitive as τ to the N reionization points, as shown inFigure 2.The 2D contours for r vs. n s and r vs. n T are also shown in the same graphs as they can beused to discriminate among inflation models and check for the consistency relation. The true valuesof r , n s and n T are not ruled out at 68% CL and 95% CL for both two-step reionization and doublereionization. These contours also show that even with Planck -quality data, a large degeneracy stillexists among r , n s and n T . It has been pointed out that r can be biased by an incorrect assumption about reionization history. Inthis paper, we consider the general reionization scenario by studying double reionization and two-step reionization models. We apply the H.L. likelihood approximation to give an idealized constraintof r for the expected Planck cut-sky power spectra. We found that the estimation of r is possiblybiased by an overly simplistic instantaneous reionization model. The N-point linear interpolationmodel of reionization can correct this bias if r & . . On the other hand, if r . . , not eventhe N-point method can produce an accurate inference of r from the Planck data, given currentuncertainties in X e ( z ) .To calculate the expected Planck power spectrum, we simply add the fiducial power spectrumwith the appropriate beam factor and noise spectrum, following the multiplication of mask matrices,as shown in Equation (11). We did not extract the power spectrum from a CMB sky-map generatedrandomly according to a Gaussian distribution, as our focus is on the analysis method applied toaccount for the reionization history. Unlike the sky-map approach, which should have ∼ σ scattersfrom the input in the inference on cosmological parameters even if the correct cosmic model isapplied, we found that the outcome of r is sharply peaked at its fiducial value up to r = 0 . whenthe assumption on reionization is correct. Thus, our approach highlights that deviations shown bythe inference on r , if present, are due to the incorrect assumption on cosmic reionization rather thanrandomness in the sky-map simulation.Our conclusion is based on the results using N = 7 in the N-point method. In principle, betterresults may be obtained if N is further increased so as to improve the modeling of the reionizationhistory. However, it will also largely increase the convergence time for the chains in MCMC sincethe total numbers of varying parameters are N . We choose N = 7 in order to have a reasonablebalance between computation time and good representation of the reionization history.In our method, we focus on the cut-sky estimators P ( b l C l + N l ) instead of trying to recover thefull-sky CMB power spectra from the cut-sky spectra by imposing the inverse of the WMAP maskmatrices { M XYll ′ } . This is because the matrices { M XYll ′ } are almost singular and imposing theirinverse on the cut-sky power spectra amplifies the noise in them and thus worsens the forecast. Oneof the advantages of applying the H.L. likelihood is that we can change the CMB full-sky estimatorsin it to be the cut-sky ones.We omit the band-power, multiple frequency estimators and the anisotropic noises here.Moreover, we extend the application of the H.L. likelihood to l min = 2 for making the forecast,while the likelihood computed from the Internal Linear Combination maps is usually applied in therange l ≤ in practice (Dunkley et al. 2009; Larson et al. 2011). These factors may further limitthe accuracy of the constraints on r . We also omit the contamination of the B-mode power spectrumby the gravitational lensing effect as it is expected to be removed (Seljak & Hirata 2004). In thefuture, we would like to explore this problem using a more realistic method, for instance, using thefull likelihood in the low- l range and model the Planck -quality data from a simulated CMB sky-map.
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Acknowledgements
This work is partially supported by a grant from the Research Grant Councilof the Hong Kong Special Administrative Region, China (Project No. 400910). J. TANG is gratefulfor the support of a postdoctoral fellowship by The Chinese University of Hong Kong. We alsothank the ITSC of The Chinese University of Hong Kong for providing clusters that were used forcomputations.
Appendix A: HAMIMECHE-LEWIS LIKELIHOOD
The H.L. likelihood approximation gives the maximum likelihood function (Hamimeche &Lewis 2008) as − ln L = X Tg M − f X g = X ll ′ [ X g ] Tl [ M − f ] ll ′ [ X g ] l ′ , (A.1)where M f is the covariance block matrix for X l = vecp ( C l ) (a vector of all distinct elements ofmatrix C l ) and is evaluated for some specific fiducial model { C l } = { C fl } . It contains ( l max − l min +1) × ( l max − l min + 1) blocks labeled by l and l ′ , and X g is generally a ( l max − l min + 1) × rowblock vector: [ M f ] ll ′ = h ( ˆ X l − X l )( ˆ X l ′ − X l ′ ) T i f (A.2) [ X g ] l = vecp (cid:16) C / fl g [ C − / l ˆC l C − / l ] C / fl (cid:17) (A.3) C l = C TT l C TE l C TB l C TE l C EE l C EB l C TB l C EB l C BB l (A.4) C fl = C TT fl C TE fl C TB fl C TE fl C EE fl C EB fl C TB fl C EB fl C BB fl (A.5)for l, l ′ = l min , · · · , l max , where g ( x ) ≡ sign ( x − p x − ln x − (A.6) [ g ( A )] ij = ( g ( A ii ) δ ij A is diagonal [ U g ( D ) U T ] ij A is symmetric positive-definite (A.7)(then A = UDU T for some diagonal matrix D ). The assumption C TB l = C EB l = 0 is applied inour study. Equation (A.1) gives the exact results for the full sky C l . Moreover, it has been tested tobe reliable in the range < l < when used on the masked-sky spectra P ( C l ) . To deal withthe cut-sky effect, all of the full-sky power spectra C l described above are changed such that C l → P ( b l C l + N l ) . (A.8)For the computation of the covariance matrix M f , the fiducial model we applied is based on thesame cosmological parameters as the input, since they are well constrained by present cosmologicalsurveys, but we fixed r = 0 . and used the instantaneous reionization model with τ = 0 . and z re = 10 . . Using HEALPix (G´orski et al. 2005), we can compute M f by generating randomsamples from the same C l power spectra. As there are p = 6 × ( l max − l min + 1) estimators in { X l } l , we generate . × random samples in order to have a good convergence, following the N sample ∼ p ln p rule (Vershynin 2012). HEALPix is available at http://healpix.sourceforge.net/ osmic Microwave Background Constraints on the Tensor-to-scalar Ratio 647