All-sky reconstruction of the primordial scalar potential from WMAP temperature data
PPrepared for submission to JCAP
All-sky reconstruction of theprimordial scalar potential fromWMAP temperature data
Sebastian Dorn, a,b, Maksim Greiner, a,b and Torsten A. Enßlin a,b a Max-Planck-Institut für Astrophysik,Karl-Schwarzschild-Str. 1, D-85748 Garching, Germany b Ludwigs-Maximilians-Universität München,Geschwister-Scholl-Platz 1, D-80539 Munich, GermanyE-mail: [email protected], [email protected],[email protected]
Abstract.
An essential quantity required to understand the physics of the early Universe, inparticular the inflationary epoch, is the primordial scalar potential Φ and its statistics. Wepresent for the first time an all-sky reconstruction of Φ with corresponding σ -uncertaintyfrom WMAP’s cosmic microwave background (CMB) temperature data – a map of the veryearly Universe right after the inflationary epoch. This has been achieved by applying aBayesian inference method that separates the whole inverse problem of the reconstructioninto many independent ones, each of them solved by an optimal linear filter (Wiener filter).In this way, the three-dimensional potential Φ gets reconstructed slice by slice resulting in athick shell of nested spheres around the comoving distance to the last scattering surface. Eachslice represents the primordial scalar potential Φ projected onto a sphere with correspondingdistance. Furthermore, we present an advanced method for inferring Φ and its power spectrumsimultaneously from data, but argue that applying it requires polarization data with highsignal-to-noise levels not available yet. Future CMB data should improve results significantly,as polarization data will fill the present (cid:96) − blind gaps of the reconstruction. Keywords:
Primordial Density Perturbations - The Early Universe – Bayesian InferenceMethod – Cosmic Microwave Background – Primordial Power Spectrum Corresponding author. a r X i v : . [ a s t r o - ph . C O ] M a r ontents The cosmic microwave background radiation (CMB) is presently one of the most informativedata sets for cosmologists to study the physics of the early Universe. Of actual interestis in particular the verification of the existence of an inflationary phase of the Universe andinvestigations of the physical properties of the involved inflaton field(s). An essential quantityis thereby the primordial adiabatic scalar potential Φ . Its statistic, especially the two-pointfunction, was determined during inflation, when the quantum fluctuations of the inflationaryfield were frozen during their exit of the Hubble horizon. This statistic is conserved on super-horizon scales during the epoch of reheating until the individual perturbed modes re-enterthe horizon. Therefore, significant information on the inflationary phase is encoded in theobservable quantity Φ . The processes translating the initial modes after their horizon re-entry into the observed CMB fluctuations are described by the so-called radiation transferfunctions, see Refs. [1, 2]. As a consequence, many inference methods aim at constrainingparameters of the early Universe involve Φ or their statistics. Therefore the CMB fluctuationsprovide a highly processed view on the primordial scalar potential. In this work, we attempt,however, their direct reconstruction and visualization via Bayesian inference. Once theyare reconstructed a direct investigation of their statistics is possible, e.g., the inference ofthe primordial power spectrum, their connection to large scale structure [3], or primordialmagnetic fields [4, 5].The Planck observation, Ref. [6], of the almost homogeneous and isotropic CMB haveshown that the statistical deviations from Gaussianity of the primordial modes/perturbationsare still consistent with zero. Therefore, the two-point correlation function of Φ seems todescribe nearly fully the statistics of the early Universe up to high accuracy. This factsimplifies the inference of these modes significantly (see, e.g., Ref. [7, 8]), and enables a welljustified all-sky reconstruction of the primordial scalar potential from real data.– 1 –his work is organized as follows. In Sec. 2 we present a Bayesian inference approachto reconstruct the primordial scalar potential. This method, initially proposed by Ref. [2],requires the knowledge of the primordial power spectrum. We show further how Φ andits spectrum can be inferred (unparametrized) even without such an a priori knowledge orassumption. In Sec. 3, we reconstruct the primordial scalar potential with corresponding σ -uncertainty from WMAP temperature data [9] and partially its initial power spectrum.In Sec. 4, we summarize our findings. Exact derivations of all used reconstruction methodscan be found in appendices A-C. We derive the inference methods within the framework of information field theory (IFT) [10],where Φ is considered to be a physical scalar field, defined over the Riemannian manifold R . Since there is no solid evidence that Φ is non-Gaussian, we assume its statistics to beGaussian with a covariance matrix determined by its power spectrum , i.e., Φ ← (cid:45) G (Φ , P Φ ) with P Φ ( k, q ) ≡ (cid:68) ΦΦ † (cid:69) (Φ) = (2 π ) δ ( k − q ) P Φ ( k ) . (2.1)Thereby we introduced the notation G ( a, A ) ≡ (cid:112) | πA | exp (cid:18) − a † A − a (cid:19) and (cid:104) . (cid:105) ( a ) ≡ (cid:90) D a . G ( a, A ) , (2.2)with corresponding inner product a † b ≡ (cid:90) R d x a ∗ ( x ) b ( x ) (2.3)for the fields a, b . Here, † denotes a transposition, t , and complex conjugation, ∗ . The CMBdata, on the other hand, are of discrete nature, i.e., d ≡ ( d , . . . , d n ) t ∈ R n , n ∈ N . To set up a Bayesian inference scheme for the primordial scalar potential Φ we have to knowhow the data d are related to Φ . In the case of the data being the WMAP CMB temperaturemap this relation is well known, given by [11] d (cid:96)m ≡ ( R Φ) (cid:96)m + n (cid:96)m = M (cid:96)m(cid:96) (cid:48) m (cid:48) B (cid:96) (cid:48) π (cid:90) dk k (cid:90) dr r Φ (cid:96) (cid:48) m (cid:48) ( r ) g T(cid:96) (cid:48) ( k ) j (cid:96) (cid:48) ( kr ) + n (cid:96)m , (2.4)where g T(cid:96) ( k ) denotes the adiabatic radiation transfer function of temperature, j (cid:96) ( kr ) thespherical Bessel function, n ∈ R n the additive Gaussian noise, and B (cid:96) the beam transferfunction of the WMAP satellite. Repeated indices are implicitly summed over unless theyare free on both sides of the equation. We assume the noise to be uncorrelated to Φ . Theoperator R , which transforms Φ into the CMB temperature map, is assumed to be linearconsisting of an integration in Fourier space as well as over the radial (comoving distance)coordinate plus the instrument’s beam convolution and a foreground mask, M . Since thereis currently no hint for isocurvature modes [17] we exclude them from all calculations. Here we assume that Φ is also statistically homogeneous and isotropic. – 2 –he next logical step, the construction of an optimal linear filter within the frameworkof IFT, e.g. the Wiener filter [12] (see, e.g., Ref. [10]), is straightforward. Given the actual,very high resolution of current CMB data sets this, however, turns out to be extremelyexpensive.Fortunately, there is a way to split this single computation of reconstructing the pri-mordial scalar potential into multiple. Instead of reconstructing the three-dimensional Φ ina single blow, one can reconstruct it spherically slice by slice, each slice corresponding to aspecific radial coordinate starting from r = 0 to beyond the surface of last scattering (LSS), r LSS . To understand this procedure we want to recall the definition of the response stated inRef. [10], where R is the part of the data which correlates with the signal, R Φ = (cid:104) d (cid:105) ( d | Φ) . Itis straightforward to show that this is equivalent to R ≡ (cid:68) d Φ † (cid:69) (Φ ,d ) (cid:68) ΦΦ † (cid:69) − ,d ) . (2.5)To obtain the response acting on a sphere with corresponding comoving distance r it can nowalso be defined as the expectation value of the data given Φ restricted to a sphere insteadof over the three-dimensional regular space, i.e., R (2) Φ( r = const . ) = (cid:104) d (cid:105) ( d | Φ( r =const . )) . Theexact derivation of this modification can be found in App. A and yields R (2) (cid:96)m(cid:96) (cid:48) m (cid:48) ( r ) = M (cid:96)m(cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) B (cid:96) (cid:48)(cid:48) (cid:82) dk k P Φ ( k ) j (cid:96) (cid:48)(cid:48) ( kr ) g T(cid:96) (cid:48)(cid:48) ( k ) (cid:82) dk k P Φ ( k ) j (cid:96) (cid:48)(cid:48) ( kr ) δ (cid:96) (cid:48)(cid:48) (cid:96) (cid:48) δ m (cid:48)(cid:48) m (cid:48) ≡ M (cid:96)m(cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) B (cid:96) (cid:48)(cid:48) R (cid:96) (cid:48)(cid:48) δ (cid:96) (cid:48)(cid:48) (cid:96) (cid:48) δ m (cid:48)(cid:48) m (cid:48) , (2.6)with superscript “ (2) ” indicating that this response acts on the (two-dimensional) sphere Φ (cid:96)m ( r = const . ) . Initially, we assume P Φ to be known (see Sec. 2.3 if not), i.e. that it isdetermined via the primordial power spectrum of comoving curvature perturbations R , givenby P R ( k ) ≡ π k A s ∗ (cid:18) kk ∗ (cid:19) n s ∗ − , (2.7)with k ∗ the pivot scale with related primordial scalar amplitude A s ∗ and scalar spectral index n s ∗ . During matter domination, the relation R = −
53 Φ (2.8)is valid. Hence, the primordial power spectrum of Φ is given by P Φ ( k ) = 925 2 π k A s ∗ (cid:18) kk ∗ (cid:19) n s ∗ − . (2.9)Figure 3 shows the predicted data power spectrum using R (2) ( r = const . ) without instru-mental beam, noise, or mask. Having this response, we are able to construct the (data-spaceversion of the) Wiener filter formula (see App. B.1 for details), m (2) ( r ) = P Φ (cid:96) ( r ) R (2) † ( r ) (cid:104) ˜ C T T + N (cid:105) − d, (2.10) Optimal with respect to the L − error norm. – 3 –ith P Φ (cid:96) ( r ) the primordial power spectrum projected onto the sphere at comoving distance r and ˜ C T T = RP Φ R † = M BC
T T B † M † where C XY(cid:96) = 2 π (cid:90) dk k P Φ ( k ) g X (cid:96) ( k ) g Y (cid:96) ( k ) . (2.11) X, Y can denote temperature T or polarization E − mode. Equation (2.10) provides an optimalestimator of Φ (cid:96)m ( r ) and was stated first in Ref. [13]. The huge advantage of this methodis the reduction of computational time, by separating the whole inverse problem into manyindependent distance-dependent ones. This method permits an easy parallelization of theWiener filter in the three-dimensional space. The σ uncertainty of this estimate, ∆ m (2) ( r ) ,is given by [10] ∆ m (2) ( r ) ≡ ± (cid:112) diag [ D ]= ± (cid:115) diag (cid:20) P Φ (cid:96) − P Φ (cid:96) R (2) † (cid:16) ˜ C T T + N (cid:17) − R (2) P Φ (cid:96) (cid:21) , (2.12)where we have introduced the posterior covariance D in data space. A proxy of this formula,used in our numerical calculations, can be found in App. B. With future data releases of current experiments like Planck [14], it should be possible to in-clude polarization data (P) with acceptable signal-to-noise level into considerations. Includingpolarization measurements, parametrized by the Stokes parameters Q ,and U , the data aregiven by d = d T d Q d U = R Φ + n T n Q n U (2.13)with corresponding response R = M T B M P B
00 0 M P B W T,ET,Q,U (cid:124) (cid:123)(cid:122) (cid:125) ≡ R T,ET,Q,U R T R E , (2.14)where R T,E captures the radiation transfer, i.e., (cid:0) R T,E Φ (cid:1) (cid:96)m ≡ π (cid:90) dk k (cid:90) dr r Φ (cid:96)m ( r ) g T,E(cid:96) ( k ) j (cid:96) ( kr ) . (2.15)The adiabatic radiation transfer functions are g T,E for temperature and E-mode polarization,respectively. For the formal definition of g T,E see, e.g., Refs. [15, 16]. The operator W T,ET,Q,U transforms a vector, containing temperature and E-mode polarization, into Stokes
I, Q, U For a detailed derivation see App. A and B. The matrix inversion within Eq. (2.10), often solved by Krylov subspace methods like the conjugategradient method, is often computationally (very) expensive. – 4 –arameters, which are directly measured by experiments like WMAP or Planck. Thereforethe generalized data-space version of the Wiener filter equation reads m (2) ( r ) = P Φ (cid:96) ( r ) (cid:18) R (2) T † ( r ) R (2) E † ( r ) 0 (cid:19) (cid:16) R T,ET,Q,U (cid:17) † × R T,ET,Q,U C T T(cid:96) C T E(cid:96) C T E(cid:96) C EE(cid:96)
00 0 0 (cid:16) R T,ET,Q,U (cid:17) † + N − d T d Q d U , (2.16)where R (2) X = T,E denotes the two-dimensional version of R X , analogous to Eq. (2.6). Theuncertainty is given analogously to Eq. (2.12).The inclusion of polarization data will result in a significant improvement of reconstruc-tion quality not least because g T(cid:96) ( k ) and g E(cid:96) ( k ) are out of phase and thus compensating the (cid:96) − blind spots of each other, which was also noticed by Ref. [13] and can be observed in theirFig. 1. This is, however, only correct if the polarization data are not highly dominated bynoise. Once a signal estimate (optimally with uncertainty) is available the power spectrum of thestochastic process underlying the signal generation might be inferred. Usually, however, aninitial guess of the signal power spectrum is required to obtain a Wiener filter signal in thefirst place. This initial guess spectrum can affect the spectrum estimate and therefore mightact as a hidden prior. In order to forget the initial guess, the procedure of signal and spectruminference should be iterated until it has converged onto a spectrum that is then independentof the initial starting value. Fortunately, the primordial power spectrum is constrained wellby the existing CMB data-sets so that this process should converge rapidly. This iterative,unparametrized method was derived in Refs. [19, 20] and named critical filter. It can beregarded as a maximum a posteriori estimate of the logarithmic power spectrum and theassumption of a scale invariant Jeffreys prior of its amplitudes. The power spectrum on thesphere is written as P Φ (cid:126)(cid:96)(cid:126)(cid:96) (cid:48) = δ (cid:126)(cid:96)(cid:126)(cid:96) (cid:48) P Φ (cid:96) with (cid:126)(cid:96) ≡ ( (cid:96), m ) . (2.17)The iterative critical filter formula including a spectral smoothness prior is then given byEq. (2.10) and P Φ (cid:96) = (cid:80) { (cid:126)(cid:96) (cid:48) | (cid:96) (cid:48) = (cid:96) } (cid:16) m (2) (cid:126)(cid:96) m (2) † (cid:126)(cid:96) (cid:48) + D (cid:126)(cid:96)(cid:126)(cid:96) (cid:48) (cid:17) ρ (cid:96) + 2( S ln P Φ ) (cid:96) , (2.18)where ρ (cid:96) = (cid:80) { (cid:126)(cid:96) (cid:48) | (cid:96) (cid:48) = (cid:96) } is the number of degrees of freedom on the multipole (cid:96) and S anoperator that enforces smoothness (for details see Ref. [20]). We analyze the full resolution ( nside = 512 ) coadded nine-year WMAP (foreground-cleaned)V-band frequency temperature map, masked with the primary temperature analysis mask See section 7 of Ref. [17] and Ref. [18] for an overview of the literature on such methods. – 5 – =0 . r LSS -0.00018 0.00018 r =0 . r LSS -0.00018 0.00018 r =0 . r LSS -0.00018 0.00018 r =1 . r LSS -0.00018 0.00018 r =1 . r LSS -0.00018 0.00018 r =1 . r LSS -0.00018 0.00018 r =1 . r LSS -0.00018 0.00018 r =1 . r LSS -0.00018 0.00018
Figure 1 . (color online) All-sky maps of the reconstructed primordial scalar potential at differentcomoving distances according to Eq. (2.10) in the vicinity of the recombination sphere with r = r LSS .A Mollweide projection is used. (KQ85: 74.8% of the sky). The data as well as the corresponding beam transfer functionand noise properties (see App. C) we used can be found at http://lambda.gsfc.nasa.gov/product/map/dr5/m_products.cfm [9, 21]. We did not take polarization data into consider-ations due to the suboptimal signal-to-noise levels. To be consistent with the WMAP team’s– 6 – =0 . r LSS r =0 . r LSS r =0 . r LSS r =1 . r LSS r =1 . r LSS r =1 . r LSS r =1 . r LSS r =1 . r LSS
Figure 2 . (color online) σ uncertainty maps of the corresponding all-sky maps of of Fig. 1 accordingto Eq. (B.7) in the vicinity of the recombination sphere with r = r LSS . A Mollweide projection isused. Note that the color bar for r = 0 . r LSS is a different one, showing the natrual bounds of theuncertainty map. All uncertainty maps share this morphology. measurements we use the cosmological parameters obtained by their data analysis to com-pute the radiation transfer function as well as the primordial power spectrum. In particular– 7 –
00 400 600 800 1000 1200 1400 ‘ ‘ ( ‘ + ) C ‘ / ( π ) [ µ K ] r =0 . r LSS C ‘ = C TT‘ C ‘ = R ‘ P Φ ‘
200 400 600 800 1000 1200 1400 ‘ ‘ ( ‘ + ) C ‘ / ( π ) [ µ K ] r =0 . r LSS C ‘ = C TT‘ C ‘ = R ‘ P Φ ‘
200 400 600 800 1000 1200 1400 ‘ ‘ ( ‘ + ) C ‘ / ( π ) [ µ K ] r =1 . r LSS C ‘ = C TT‘ C ‘ = R ‘ P Φ ‘
200 400 600 800 1000 1200 1400 ‘ ‘ ( ‘ + ) C ‘ / ( π ) [ µ K ] r =1 . r LSS C ‘ = C TT‘ C ‘ = R ‘ P Φ ‘
200 400 600 800 1000 1200 1400 ‘ ‘ ( ‘ + ) C ‘ / ( π ) [ µ K ] r =1 . r LSS C ‘ = C TT‘ C ‘ = R ‘ P Φ ‘
200 400 600 800 1000 1200 1400 ‘ ‘ ( ‘ + ) C ‘ / ( π ) [ µ K ] r =1 . r LSS C ‘ = C TT‘ C ‘ = R ‘ P Φ ‘ Figure 3 . (color online) Predicted power spectra of data simulated with the estimator response R (2) compared to the CMB data power spectrum. The (cid:96) − blind spots move from large scales atdistances r < r LSS to small scales at r > r
LSS . The amplitude of the predicted power spectra getsmaximal at r = r LSS . For clarity and comprehensibility we exclude the instrumental beam, noise, andobservational mask. this has been done by using gTfast , which is based on CMBFAST [22]. We used the follow-ing settings: pivot scale k ∗ = 0 .
002 Mpc − , spectral index n s ∗ = 0 . , spectral amplitude http://lambda.gsfc.nasa.gov/toolbox/tb_cmbfast_ov.cfm – 8 – s ∗ = 2 . × − , noise level σ V − band0 = 3 . × − K, CMB temperature T CMB = 2 . K, optical depth τ = 0 . , density parameters Ω b = 0 . , Ω c = 0 . , Ω Λ = 0 . ,Hubble constant H = 70 . / s / Mpc , helium abundance Y He = 0 . , and the effectivenumber of massless neutrino species N eff ν = 3 . . The resulting distance to the LSS amounts . × Mpc. ‘ ‘ ( ‘ + ) P Φ ‘ / ( π ) P Φ ‘ recon P Φ ‘ recon smooth P Φ ‘ theory r q m i n ( σ ( r )) / q m a x ( σ ( r )) Figure 4 . (color online) Left: Estimated primordial power spectrum of Φ( r = r LSS ) according toEq. (2.18). Masking effects as well as the estimators power loss are compensated. At scales smallerthan (cid:96) ≈ the reconstruction fails due to sub-horizon physics [13] and noise-dominance. Right:Relative σ -uncertainty along the radial coordinate. Minimal values of σ correspond to Eq. (B.6)with “no mask”, maximal values to the same equation with “all mask”. With the parameters defined in the previous paragraph, we have reconstructed a shell aroundthe last scattering surface ( . × r LSS to . × r LSS ) in slices as well as additional 6slices within the range (50% − × r LSS from real data, see Fig. 1. For all reconstructions σ -uncertainty maps are provided, see Fig. 2 as well as the relative σ -error along the radialcoordinate, see Fig. 4 (Right). A detailed description of the calculation of these uncertaintymaps can be found in App. B. The respective data files of the reconstruction can be foundat . For the most interesting sphere at r = r LSS we also provide a power spectrum estimate, see Fig. 4 (Left). This power spectrumestimate has been obtained with the critical filter formula with smoothness prior but withoutiterations and D set to zero (defined in Eq. (2.12)).We also phenomenologically corrected for the effect of masking and power-loss in thepredicted power spectra of data simulated with the estimator response R (2) in comparison to With the correct application of the critical filter (iterative) one might be able to detect features in theprimordial power spectrum [8]. This, however, would require a highly resolved data set including polarizationto compensate for the (cid:96) − bind spots (one cannot get rid of with temperature data only) with a high signal-to-noise level in T -, Q -, and U -data maps. Perfect candidates for such data sets are future CMB experimentsand Planck polarization data releases. The power-loss is corrected by convolving the reconstructed Φ with α (cid:96) ≡ (cid:112) C TT(cid:96) / ( R (cid:96) P Φ (cid:96) ) ∀ (cid:96) : R (cid:96) P Φ (cid:96) (cid:54) = 0 before performing the power spectrum estimation. We also investigated how the mask affects the powerspectrum of Φ , by calculating β l ≡ (cid:10) power (cid:2) R mask (Φ) (cid:3) /P Φ (cid:96) (cid:11) where power[ . ] denotes the application of thecritical filter formula with smoothness prior. We re-scaled the inferred power spectrum with /β (cid:96) . – 9 –he power spectrum of Eq. (2.11). Therefore our spectrum estimate should rather be regardedas providing a consistency check of the algorithm than to necessarily provide precisely thecosmological power spectrum. Having stated these caveats, we like to note that a deviationfrom the power-law primordial power spectrum is not apparent over roughly one order ofmagnitude in Fourier space.Some of the reconstructed slices of the primordial scalar potential might look suspiciouslycrumby at first. The reason for this property are the (cid:96) − blind spots in the response R (cid:96) .Figure 3 shows the noiseless data power spectrum, C T T(cid:96) = RP Φ R † , as well as the powerspectrum R (2) P Φ (cid:96) R (2) † expected from noiseless, distance dependent data obtained with theestimator response, d (2) = R (2) Φ . The (cid:96) − blind spots are clearly recognizable, which movefrom large scales at distances r < r LSS to small scales at r > r
LSS , where the amplitude ofthis power spectrum gets maximal at r = r LSS .The numerical and computational effort to reconstruct one slice by one CPU amountsto roughly 45 minutes, which simultaneously represents the time for reconstructing the wholethree-dimensional primordial scalar potential at full parallelization. In our numerical imple-mentation we used the conjugate gradient method to solve Eq. (2.10).
We have presented a reconstruction of the primordial scalar potential Φ with corresponding σ -uncertainty from WMAP temperature data. This has been achieved by setting up aninference approach that separates the whole inverse problem of reconstructing Φ into manyindependent ones, each corresponding to the primordial scalar potential projected onto asphere with specific comoving distance. This way the reconstruction is done sphere by sphereuntil one obtains a thick shell of nested spheres around the surface of last scattering. Thisresults in a significant reduction of computational costs (since the reconstruction equation(Wiener filter) parallelizes fully), if only the small region around the last scattering surface isreconstructed, which is accessible through CMB data.We did not include polarization information yet due to the suboptimal signal-to-noiseratios of the WMAP polarization data. Hence we do not expect a huge improvement whenadditionally including WMAP Stokes Q and U parameters into the Wiener filter equation.This, however, will definitely change when the polarization data of Planck will be available inthe near future. Once one uses simultaneously temperature and polarization data, the (cid:96) − blindspots in the reconstructions will disappear and with it the crumbliness of the maps. At thispoint it also might be more rewarding to apply the critical filer equations to simultaneouslyobtain the power spectrum of the primordial scalar potential. Acknowledgments
We gratefully acknowledge Vanessa Boehm and Marco Selig for useful discussions and com-ments on the manuscript, as well as Eiichiro Komatsu for numerical support concerning gTfast , to be found at . All calculations have been done using
NIFTy [23] to befound at , in particular involving
HEALPix [24] to be found at http://healpix.sourceforge.net/ . We also acknowledge the supportby the DFG Cluster of Excellence “Origin and Structure of the Universe”. The calculationshave been carried out on the computing facilities of the Computational Center for Particleand Astrophysics (C2PAP). – 10 –
Response projected onto the sphere of LSS
The data are given by d (cid:96)m ≡ M (cid:96)m(cid:96) (cid:48) m (cid:48) a CMB (cid:96) (cid:48) m (cid:48) + n (cid:96)m = ( R Φ) (cid:96)m + n (cid:96)m = M (cid:96)m(cid:96) (cid:48) m (cid:48) B (cid:96) (cid:48) π (cid:90) dk k (cid:90) dr r Φ (cid:96) (cid:48) m (cid:48) ( r ) g T(cid:96) (cid:48) ( k ) j (cid:96) (cid:48) ( kr ) + n (cid:96)m . (A.1)Considering Gaussian statistics for the primordial curvature perturbations, Φ , the responseis defined by R ≡ (cid:68) d Φ † (cid:69) (Φ ,d ) (cid:68) ΦΦ † (cid:69) − ,d ) . (A.2)Instead of using the full three-dimensional response R , we introduce a two-dimensional re-sponse, R (2) , which acts on the primordial potential projected onto the last scattering surface(LSS), Φ (2) ≡ Φ ( r = r LSS ) = ˜ T Φ , where ˜ T denotes the projection operator: R (2) = (cid:68) R Φ( ˜ T Φ) † (cid:69) (Φ ,d ) (cid:68) ˜ T Φ( ˜ T Φ) † (cid:69) − ,d ) = (cid:16) RP Φ ˜ T † (cid:17) (cid:16) ˜ T P Φ ˜ T † (cid:17) − . (A.3)To derive the denominator at the distance of the LSS, we first transform it into position-space, (cid:16) ˜ T P Φ ˜ T † (cid:17) ˆn , ˆn (cid:48) = (cid:90) d x (cid:90) d y δ ( x − r LSS ˆn ) δ (cid:0) y − r LSS ˆn (cid:48) (cid:1) × (cid:90) d k (2 π ) (cid:90) d q (2 π ) (2 π ) δ ( k − q ) P Φ ( k ) e − i k · x e i q · y = (cid:90) d k (2 π ) P Φ ( k ) e − ir LSS k · ˆn e ir LSS k · ˆn (cid:48) . (A.4)Vectors are printed in bold for reasons of clarity and comprehensibility; unit vectors aredenoted by ˆ . Subsequently we use the Rayleigh expansion, e i k · r = 4 π ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) i (cid:96) j (cid:96) ( kr ) Y m ∗ (cid:96) ( ˆk ) Y m(cid:96) ( ˆr ) , (A.5)as well as the transformation rules f (cid:96)m ≡ (cid:73) d ˆn Y m ∗ (cid:96) ( ˆn ) f ( ˆn ) ,f ( ˆn ) = ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) f (cid:96)m Y m(cid:96) ( ˆn ) , and (cid:73) d ˆn Y m(cid:96) ( ˆn ) Y m (cid:48) ∗ (cid:96) (cid:48) ( ˆn ) = δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) , (A.6)– 11 –o obtain the final corresponding expression in the spherical harmonic space, (cid:16) ˜ T P Φ ˜ T † (cid:17) (cid:96)m(cid:96) (cid:48) m (cid:48) = (cid:73) d ˆn (cid:73) d ˆn (cid:48) Y m(cid:96) ( ˆn ) Y m (cid:48) ∗ (cid:96) (cid:48) ( ˆn (cid:48) ) (cid:90) d k (2 π ) P Φ ( k ) × (cid:88) (cid:96) (cid:48)(cid:48) (cid:96) (cid:48)(cid:48)(cid:48) m (cid:48)(cid:48) m (cid:48)(cid:48)(cid:48) (4 π ) i (cid:96) (cid:48)(cid:48)(cid:48) − (cid:96) (cid:48)(cid:48) j (cid:96) (cid:48)(cid:48) ( kr LSS ) j (cid:96) (cid:48)(cid:48)(cid:48) ( kr LSS ) Y m (cid:48)(cid:48) (cid:96) (cid:48)(cid:48) ( ˆk ) Y m (cid:48)(cid:48)(cid:48) ∗ (cid:96) (cid:48)(cid:48)(cid:48) ( ˆk ) Y m (cid:48)(cid:48) ∗ (cid:96) (cid:48)(cid:48) ( ˆn ) Y m (cid:48)(cid:48)(cid:48) (cid:96) (cid:48)(cid:48)(cid:48) ( ˆn (cid:48) )= 2 π (cid:90) dk k (cid:73) d ˆk P Φ ( k ) i (cid:96) (cid:48) − (cid:96) j (cid:96) ( kr LSS ) j (cid:96) (cid:48) ( kr LSS ) Y m (cid:48) ∗ (cid:96) (cid:48) ( ˆk ) Y m(cid:96) ( ˆk )= 2 π (cid:90) dk k P Φ ( k ) j (cid:96) ( kr LSS ) δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) ≡ P Φ (cid:96) δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) . (A.7) P Φ (cid:96) denotes the primordial power spectrum projected onto the sphere of LSS.To determine the numerator we fist have to transform P Φ ˜ T † into the basis of sphericalharmonics. Analogous to the calculation above we obtain (cid:16) P Φ ˜ T † (cid:17) (cid:96)m(cid:96) (cid:48) m (cid:48) ( r ) = 2 π (cid:90) dk k P Φ ( k ) j (cid:96) ( kr LSS ) j (cid:96) ( kr ) δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) , (A.8)and thus (cid:16) RP Φ ˜ T † (cid:17) (cid:96)m(cid:96) (cid:48) m (cid:48) = M (cid:96)m(cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) B (cid:96) (cid:48)(cid:48) π × (cid:90) dk k (cid:90) dr r (cid:26) π (cid:90) dk (cid:48) k (cid:48) P Φ ( k (cid:48) ) j (cid:96) (cid:48)(cid:48) ( k (cid:48) r LSS ) j (cid:96) (cid:48)(cid:48) ( k (cid:48) r ) (cid:27) × g T(cid:96) (cid:48)(cid:48) ( k ) j (cid:96) (cid:48)(cid:48) ( kr ) δ (cid:96) (cid:48)(cid:48) (cid:96) (cid:48) δ m (cid:48)(cid:48) m (cid:48) . (A.9)Using the identity (cid:90) ∞ dr r j (cid:96) ( kr ) j (cid:96) ( k (cid:48) r ) = π k δ ( k − k (cid:48) ) (A.10)finally yields (cid:16) RP Φ ˜ T † (cid:17) (cid:96)m(cid:96) (cid:48) m (cid:48) = M (cid:96)m(cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) B (cid:96) (cid:48)(cid:48) π (cid:90) dk k P Φ ( k ) j (cid:96) (cid:48)(cid:48) ( kr LSS ) g T(cid:96) (cid:48)(cid:48) ( k ) δ (cid:96) (cid:48)(cid:48) (cid:96) (cid:48) δ m (cid:48)(cid:48) m (cid:48) . (A.11)Putting the results together, the two-dimensional response is given by R (2) (cid:96)m(cid:96) (cid:48) m (cid:48) = M (cid:96)m(cid:96) (cid:48)(cid:48) m (cid:48)(cid:48) B (cid:96) (cid:48)(cid:48) (cid:82) dk k P Φ ( k ) j (cid:96) (cid:48)(cid:48) ( kr LSS ) g T(cid:96) (cid:48)(cid:48) ( k ) (cid:82) dk k P Φ ( k ) j (cid:96) (cid:48)(cid:48) ( kr LSS ) δ (cid:96) (cid:48)(cid:48) (cid:96) (cid:48) δ m (cid:48)(cid:48) m (cid:48) . (A.12)The response for arbitrary comoving distances r (cid:48) can be obtained by replacing r LSS by r (cid:48) . B Wiener filter formula and uncertainty estimate in data space
The Wiener filter in data space is defined by m (2) w ≡ (cid:68) Φ (2) (cid:69) (Φ | d ) = ˜ T (cid:104) Φ (cid:105) (Φ | d ) = ˜ T (cid:68) Φ d † (cid:69) (Φ ,n ) (cid:68) dd † (cid:69) − ,n ) d = ˜ T P Φ R † (cid:104) RP Φ R † + N (cid:105) − d = ˜ T P Φ R † (cid:104) ˜ C T T + N (cid:105) − d Eq. (A.12) = P Φ (cid:96) R (2) † (cid:104) ˜ C T T + N (cid:105) − d. (B.1)– 12 –ormally, the corresponding posterior covariance matrix is constructed as D = P Φ (cid:96) − P Φ (cid:96) R (2) † (cid:16) ˜ C T T + N (cid:17) − R (2) P Φ (cid:96) . (B.2)The square root of its position space diagonal would give us the 1 σ uncertainty map. However,as the operator is not directly accessible to us, but is only defined as a sequence of linearfunctions, calculating the diagonal requires very expensive probing routines which need toevaluate the covariance matrix several thousand times before converging.However, the covariance matrix becomes diagonal in spherical harmonic space under twoconditions : We assume that there is no masking in the data and the noise covariance N isa multiple of the identity. The noise covariance matrix for T T data is already diagonal anddominated by white uncorrelated noise. So this approximation seems appropriate given thebenefits in computational costs. The assumption that there is no masking is more drastic ofcourse. We therefore construct our uncertainty map out of the limiting cases of having nomasking and masking the whole sky. Both scenarios make the posterior covariance matrixdiagonal in spherical harmonic space.The constant approximation to the noise covariance is constructed as ˜ N ˆ n ˆ n (cid:48) = tr N tr δ (ˆ n − ˆ n (cid:48) ) . (B.3)The response with no mask is diagonal in spherical harmonic space, ˜ R (cid:96)m(cid:96) (cid:48) m (cid:48) = B (cid:96) R (cid:96) δ (cid:96)(cid:96) (cid:48) δ mm (cid:48) , (B.4)and the response with an all-sky mask is zero. Therefore the covariance matrix is diagonal ineither case. Since a diagonal matrix in spherical harmonic space results in a constant diagonalin position space, we can exploit the invariance of the trace to get the position space diagonalof the covariance matrix, D ˆ n ˆ n = tr D π , (B.5)where the trace is easily calculated in spherical harmonic space, where D is diagonal.In a region that is fully masked and where the edges of the mask are further away thanthe correlation length of Φ the uncertainty approaches the limiting case of an all-sky mask. Ina region that is fully exposed and more than a correlation length away from a masked regionthe uncertainty approaches the limiting case of no mask. We therefore combine the two casesinto one map by setting the uncertainty to the “all-sky masked” value in regions which aremasked and to the “no mask” value in regions which are not masked, i.e. σ n = (cid:40) D all maskˆ n ˆ n if M ˆ n ˆ n (cid:48) = 0 D no maskˆ n ˆ n otherwise . (B.6)The interpolation between these two regions is dictated by the prior covariance. It describesprecisely how information is correlated between masked and unmasked regions. Our finaluncertainty map is therefore the result of a smoothing of σ with the normalized square rootof the prior covariance, σ smooth = 1 N (cid:113) P Φ (cid:96) σ, (B.7) Note that this procedure is only valid for a temperature-only analysis. Once polarization data are includedthe 1 σ uncertainty must be calculated by the square root of the diagonal of Eq. (B.2). – 13 –here N = (cid:73) d ˆ nd ˆ n (cid:48) (cid:18)(cid:113) P Φ (cid:96) (cid:19) ˆ n ˆ n (cid:48) δ (ˆ n (cid:48) ) . (B.8) C WMAP noise characterization
The pixel noise level (in units mK) of a single map can be determined by σ = σ / √ N obs ,where σ can be found at http://lambda.gsfc.nasa.gov/product/map/dr5/skymap_info.cfm and the effective number of observations N obs , which can vary from pixel to pixel, isstored in the FITS file of a map, see http://lambda.gsfc.nasa.gov/product/map/dr4/skymap_file_format_info.cfm . Thus, the noise covariance matrix of a single map is givenby N ˆn , ˆn (cid:48) = σ N obs ( ˆn (cid:48) ) δ ˆnˆn (cid:48) . (C.1)Including polarization data, the noise covariance matrix in position space has to begeneralized by N − = N T T obs /σ T N QQ obs /σ P N QU obs /σ P N QU obs /σ P N UU obs /σ P , (C.2)where σ T,P is the respective noise level of temperature and polarization as given by WMAP.
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