Tomographic analyses of the CMB lensing and galaxy clustering to probe the linear structure growth
PPrepared for submission to JCAP
Tomographic analyses of the CMBlensing and galaxy clustering toprobe the linear structure growth
Gabriela A. Marques, a,b, Armando Bernui a a Observatório Nacional, Rua General José Cristino 77, São Cristóvão, 20921-400 Rio deJaneiro, RJ, Brazil b Department of Physics, Florida State University, Tallahassee, Florida 32306, USAE-mail: [email protected], [email protected]
Abstract.
In a tomographic approach, we measure the cross-correlation between the CMBlensing reconstructed from the Planck satellite and the galaxies of the photometric redshiftcatalogue based on the combination of the South Galactic Cap u-band Sky Survey (SCUSS),Sloan Digital Sky Survey (SDSS), and Wide-field Infrared Survey Explorer (WISE) data. Weperform the analyses considering six redshift bins spanning the range of . < z < . . Fromthe estimates of the galaxy-galaxy and galaxy-CMB lensing power spectrum, we derive thegalaxy bias and the amplitude of the cross-correlation for each redshift bin. We have finallyapplied these tomographic measurements to estimate the linear structure growth using thebias-independent ˆ D G estimator introduced by [1]. We find that the amplitude of the structuregrowth with respect to the fiducial cosmology is A D = 1 . ± . , closely consistent with thepredictions of the Λ CDM model ( A Λ CDM D = 1 ). We perform several tests for consistency ofour results, finding no significant evidence for systematic effects. Corresponding author. a r X i v : . [ a s t r o - ph . C O ] J un ontents ˆ D G Estimator 94.3 Covariance Matrix 9 ˆ D G Progress in the sensitivity of astronomical photometric surveys dedicated to the study thelarge-scale structure (LSS) has been providing valuable information about the features of theUniverse at several scales and redshifts [2–6]. The prospects of using LSS data to constraincosmology are very promising. Several upcoming astronomical surveys will produce extensivephotometric data covering a wide area of the sky such as the Large Synoptic Survey Telescope(LSST) [7] and the Wide-Field Infrared Survey Telescope (WFIRST). On the other hand, thecosmic microwave background radiation (CMB) allow us to test the primordial characteris-tics of the Universe [8, 9]. However, before reaching us, the CMB photons are affected byinhomogeneities along their path producing a range of secondary effects, beyond the primaryCMB temperature fluctuations at the last scattering surface [10–12]. One of these secondaryeffects is the gravitational deflection of the CMB photons by the mass distribution along theirpath, namely weak gravitational lensing.The CMB lensing has been investigated by several methods and experiments in thepast e.g, [13–19]. Recently, through observations of the Planck satellite, it was possible notonly to detect the lensing effect with high statistical significance but also to robustly recon-struct the lensing potential map in almost full-sky [20–22]. Such a reconstructed map containsunique information of the LSS since it is related to the integral of the photon deflections fromus until the last scattering surface. – 1 –lthough the CMB lensing signal covers a broad redshift range, from local to highredshifts, it is not possible to obtain the evolution of the LSS along the line of sight usingonly the CMB lensing data. However, the cross-correlation technique with another tracerof matter provides additional astrophysical and cosmological information. Several galaxycatalogs, such as those from the Wide Field Survey Infrared Explorer (WISE) and derivedcatalogues [20, 23], NRAO VLA Sky Survey (NVSS) [24], Canada-France-Hawaii Telescope(CFHT) [25], Sloan Digital Sky Survey (SDSS) [26–28], 2MASS [29, 30], Dark Energy Survey(DES) [1, 31, 32] and WISExSuperCOSMOS [30, 33] have already been cross-correlated withthe CMB lensing potential. Also there have been significant detections of the cross-correlationbetween CMB lensing and galaxy lensing shear maps e.g., [34–37]. In addition, analysis ina deeper Universe has been extended through the cross-correlation of CMB lensing withdensity tracers at high redshifts, e.g. quasars [38–40] and sub-mm galaxies from HerschelH-ATLAS survey [41, 42]. Particularly, the galaxy auto-correlation and cross-correlationwith the CMB lensing provides the opportunity to constrain the linear growth of the densityfluctuations. By probing the evolution of perturbations over time it is possible to understandthe mechanism that sources the late-time accelerated expansion of the Universe and shed lightto distinguish a variety of gravity models [43–45]. Analysis of the Redshift-Space Distortions[46] from spectroscopic surveys is a traditional way to measure the linear growth rate since itis commonly parameterized on large-scales by f σ , where f ≡ d ln D/d ln a is the logarithmicderivative of the growth factor with respect to the scale factor a , and σ is the linear mattervariance in a spherical shell of radius 8 Mpc h − . However, this quantity can not be accuratelymeasured for photometric surveys. Besides the photometric data be complementary withrespect to spectroscopic studies, many photometric surveys are currently in operation orplanned for the near future, with the obvious advantages of being lower-cost surveys andcapable of mapping large areas of the sky with optimal observation strategies. Therefore,the ˆ D G statistic introduced by [1] establishes an alternative to measure the linear growth forphotometric redshift surveys. This estimator combines properly the auto and cross-correlationof the galaxy clustering and CMB lensing in such a way that it is bias-independent on linearscales.The aim of the present work is to constrain the linear growth factor D in a tomographicapproach and, therefore, to measure the evolution of the linear growth function. For that,we consider the ˆ D G statistic using the CMB lensing map reconstructed by the P lanck team[21] with the galaxy overdensities from a multi-band photometric data released by [47], basedon imaging from South Galactic Cap u-band Sky Survey (SCUSS), SDSS and WISE. Theanalyses are performed in six redshift bins spanning . < z < . , being complementary tothe linear growth measures previously found for others photometric catalogs [1, 29, 30, 32].Although the ˆ D G estimator is galaxy bias independent, additionally, we use the measuredgalaxy-CMB lensing cross-correlations and galaxy auto-correlation to infer the correlationamplitude and the linear bias over the redshift bins.This paper is structured as follows : Firstly, we introduce the theoretical formalismin sec. 2. In sec. 3 we summarize the data used in the analysis. In sec. 4 we describe themethodology. We then present the results and explore possible systematics and null tests insec. 5, and our conclusions in sec. 6. – 2 – Background
The gravitational lensing effect remaps the CMB temperature anisotropies by a angular gra-dient of the lensing potential, α ( ˆn ) = ∇ ψ ( ˆn ) , where ∇ is the 2D gradient operator on thesphere and ψ ( ˆn ) is the lensing potential. The Laplacian of the lensing potential is relatedto the convergence κ ( ˆn ) , which can be written as a function of the three-dimensional matterdensity contrast δ (see e.g. [48]) κ (ˆ n ) = (cid:90) ∞ dzW κ ( z ) δ ( χ ( z ) ˆn , z ) , (2.1)where the CMB lensing kernel W κ is W κ ( z ) = 3Ω m c H H ( z ) (1 + z ) χ ( z ) χ ∗ − χ ( z ) χ ∗ . (2.2)Here we are considering a flat universe, c is the speed of light, H ( z ) is the Hubble parameter atredshift z , and Ω m and H are the present-day parameters of the matter density and Hubble,respectively. The comoving distances χ ( z ) and χ ∗ are set to the redshift z and to the lastscattering surface at z ∗ (cid:39) , respectively.On the other hand, the galaxy overdensity δ g from a galaxy catalogue with normalizedredshift distribution dn/dz also provides an estimate of the projected matter density contrast,given by δ g ( ˆn ) = (cid:90) ∞ dzW g ( z ) δ ( χ ( z ) ˆn , z ) , (2.3)where the galaxy kernel W g for a linear, deterministic and scale-independent galaxy bias b ( z ) [49] is W g ( z ) = b ( z ) dndz . (2.4)Under the Limber approximation [50], the two-point statistics in the harmonic space of thegalaxy-galaxy and galaxy-CMB lensing correlations become C gg(cid:96) = (cid:90) ∞ dzc H ( z ) χ ( z ) [ W g ( z )] P (cid:18) k = (cid:96) + χ ( z ) , z (cid:19) ,C kg(cid:96) = (cid:90) ∞ dzc H ( z ) χ ( z ) W κ ( z ) W g ( z ) P (cid:18) k = (cid:96) + χ ( z ) , z (cid:19) , (2.5)where P ( k, z ) is the matter power spectrum. The Limber approximation is quite accuratewhen (cid:96) is not too small ( (cid:96) > ) [50], which is the regime considered in this work. Moreover,is possible to rewrite the equations 2.5 in terms of the linear growth function D ( z ) , since P ( k, z ) = P ( k, D ( z ) . Therefore, C gg(cid:96) ∝ b ( z ) D ( z ) ,C kg(cid:96) ∝ b ( z ) D ( z ) . (2.6)Thus, by properly combining the two quantities of the equation 2.6, it is possible to break thedegeneracy between the galaxy bias and the linear growth through the estimator introducedby [1]: ˆ D G ≡ (cid:28) ( C κg(cid:96) ) obs ( /C κg(cid:96) ) th (cid:115) ( /C gg(cid:96) ) th ( C gg(cid:96) ) obs (cid:29) (cid:96) . (2.7)– 3 –n the above equation, the slashed quantities, /C gg(cid:96) and /C κg(cid:96) denote the theoretical functionsevaluated at z = 0 such that the growth function dependency is removed and ˆ D G is normalizedto z = 0 ( ˆ D G ( z = 0) = 1 ).In order to obtain the theoretical predictions for the matter power spectrum P ( k, z ) ,we use the public Boltzmann code CAMB [51] with the Halofit [52] extension to nonlin-ear evolution. Throughout the paper, we use the Planck 2018 cosmology [53] derived by the
TT,TE,EE+lowE+lensing data, with parameters { Ω b h , Ω c h , Ω m , τ, n s , A s , h } = { . , . , . , . , . , . × − , . } . In this study, we use the photometric redshift catalogue publicly released by [47]. This catalogis based on multi-band data from three independent surveys: the South Galactic Cap u-bandSky Survey (SCUSS; [54]), Sloan Digital Sky Survey (SDSS; [55]), and Wide-field InfraredSurvey Explorer (WISE; [56]). Below, we briefly describe the properties of each survey andof the final catalog used in our analysis.The SCUSS is a u-band (354 nm) imaging survey using the 2.3m Bok telescope locatedon Kitt Peak, USA. The data products were released in 2015 containing calibrated single-epoch images, stacked images, photometric catalogs, and the star proper motions. The surveycovers an area of approximately 5000 deg of the South Galactic Cap and overlaps roughly of the area covered by the SDSS [57]. The detailed information about the SCUSS andthe data reduction can be found in [57] and [58].The SDSS is a multi-spectral imaging and spectroscopic redshift survey, encompassingan area of about 14000 deg . The SDSS uses a wide-field camera that is made up of 30CCDs. The survey is carried out imaging in five broad bands u, g, r, i, z , with limit-magnitudewith completeness 22.0, 22.2, 22.2, 21.3 and 20.5 mag, respectively. The data have beenreleased publicly in a series of roughly annual data releases. Specifically, the photometric datafrom the SDSS Data Release 10 (DR10) [59] was considered to obtain the galaxy catalogueused in this paper, as detailed in [47].WISE is an infrared astronomical space telescope that scanned all-sky at 3.4, 4.6, 12and 22 µ m, known as W1, W2, W3, and W4, respectively. In September 2010, the frozenhydrogen cooling the telescope was depleted and the survey continued as NEOWISE, withthe W1 and W2 bands. In order to match properly the official all-sky WISE catalogs with theSDSS data, is considered a technique to measure model magnitudes of the SDSS objects innew coadds of WISE images, called as forced photometry, providing an extensive extragalacticcatalogue of over 400 million sources [60, 61].The catalogue we use has been built by combining the 7 photometric bands ranging fromthe near-ultraviolet to near-infrared. A local linear regression algorithm [6] is adopted with aspectroscopic training set composed mostly of galaxies from the SDSS DR13 spectroscopy, inaddition to several other surveys. The model magnitudes utilize the shape parameters fromSDSS r-band and also the SDSS star/galaxy separation to characterize the source type. Aftercorrecting for galactic extinction [62], the final catalogue contains ∼ with ∼ of the sources spanning the redshift interval of z ≤ . [47]. The use of a https://camb.info/ Available for download from http://batc.bao.ac.cn/ zouhu/doku.php?id=projects:photoz:start – 4 –eeper SCUSS u-band and the multi-band information allowed the photo-z estimate moreaccurately and less biased than the SDSS photometric redshifts, with the average bias of ∆ z norm = 2 . × − and standard deviation of σ z = 0 . . Such a relatively deep catalog,with a remarkable galaxy number density, accurate photo-z, and considerable sky area, allowsto use it to perform tests of the structure growth.In order to apply a tomographic approach, we split the full catalogue into six redshiftbins of width ∆ z = 0 . over . < z < . . We ignore the extreme redshift bins wherethe fractional photo-z errors become large and the galaxy density became small. We use theposition of the sources to create a pixelized overdensity map, δ g ( (cid:126)x ) = n g ( (cid:126)x ) − ¯ n ¯ n , (3.1)where n g is the number of observed galaxies in a given pixel and ¯ n is the mean number ofobjects per pixel in the unmasked area. We use the HEALPix scheme [63] with a resolutionparameter N side = 512. The figure 1 shows the overdensity map in these six redshift bins,where the gray area indicates the masked regions. However, we discard the stripes locatedin the galactic longitude range ◦ < l < ◦ due to the low density, remaining about f sky = 0 . in each map for analyses. The specifics of each bin are summarized in the table1. As discussed in the section 2, we need the overall redshift distribution dn/dz and thegalaxy bias to connect the galaxy overdensity δ g to the underlying matter overdensity δ .However, we need take into account the effect of the photometric redshift errors [64, 65].We can accurately reconstruct the true dn/dz distribution by the convolution of the sam-ple’s photometric redshift distribution dn/dz ( z ph ) with the catalog’s photo-z error function p ( z | z ph ) : dndz = (cid:90) ∞ dz ph dn ( z ph ) dz p ( z | z ph ) W ( z ph ) , (3.2)where p ( z | z ph ) is parameterized as a Gaussian distribution with zero mean and dispersion σ z so that p ( z | z ph ) ∝ exp ( − . z/σ z (1 + z )) ) , where σ z = 0 . [47] and the W ( z ph ) is thewindow function, such that W = 1 for z ph in the selected interval and W = 0 otherwise. Theredshift distribution for the total catalogue is shown as the solid black line in figure 2, whilethe distribution to each tomographic bin is shown as the dashed lines.Redshift range N tot ¯ n [gal sr − ]0.1 - 0.2 2,208869 . × . × . × . × . × . × Total 20,680257 . × Table 1 : The number of sources and the galaxy number density of each tomographic bins.– 5 – .1 < z < 0.2 -0.885592 6.66535 z < 0.3 -0.920227 6.02007 z < 0.4 -0.931241 4.29448 z < 0.5 -0.950678 2.79781 z < 0.6 -0.94148 2.68674 z < 0.7 -0.879447 2.97824 Figure 1 : All-sky projections of the galaxy overdensity of the six photo-z bins adopted in theanalysis. The maps are constructed in the
HEALPix pixelization scheme, with the resolutionparameter N side = 512 . The gray areas correspond to the masked regions. We consider the latest CMB lensing products from
Planck . The lensingconvergence map has been reconstructed based on the quadratic estimators that exploit thestatistical information introduced by weak lensing in the CMB data [66]. Specifically, weuse the convergence field κ obtained from the minimum-variance (MV) combination of theestimators applied to temperature (T) and polarization (P) of the SMICA foreground-reducedmap [22].The spherical harmonic coefficients of the convergence is band-limited to the multipole (cid:96) max = 4096 . Jointly to the released lensing products, it is available a set of realistic simu- http://pla.esac.esa.int/pla – 6 – . . . . . . . . . . z . . . . . . . d n / d z [ a r b . un i t s ] Figure 2 : Unnormalized redshift distributions for the total galaxy catalogue (black solidline) and for the six tomographic bins used in the analysis (dashed lines). These are obtainedby convolving the photo-z distribution in each redshift interval with a photometric errordistribution.lations, which accurately incorporate the Planck noise levels and the κ statistical properties[67]. In order to attenuate the foreground contamination in the lensing data, we apply thecorresponding confidence mask that leaves a total unmasked sky fraction of f sky = 0 . . In this work, we use the angular power spectrum of the galaxy overdensity and the angularcross-power spectrum between the galaxy overdensity and the CMB convergence map toestimate the cosmic growth information at several redshifts bins. In this section, we describethe procedure followed in the analysis of these two datasets.
The power spectra estimates for incomplete sky coverage are affected by the mask, whichintroduces coupling between different modes [68]. Therefore, we use a pseudo- C (cid:96) estimatorbased on the MASTER approach [69], that provides a very good approximation to this issue,mainly on larger scales which is the regime we are considering, as detailed below.Let us denote the two fields X and Y with the auto-power spectrum when X = Y andthe full sky cross-(auto-)spectrum denoted as C XY(cid:96) ( C XX(cid:96) ). The pseudo- ˜ C XY(cid:96) measured in afraction of the sky is ˜ C XY(cid:96) = 12 (cid:96) + 1 (cid:96) (cid:88) m = − (cid:96) ˜ X (cid:96)m ˜ Y ∗ (cid:96)m , (4.1)where ˜ X (cid:96)m and ˜ Y ∗ (cid:96)m are the spherical harmonic coefficients of the maps. The mask acts as aweight modifying the underlying harmonic coefficients so that the pseudo- C (cid:96) measured from– 7 –he data can be related to the true spectrum by the mode-mode coupling matrix M (cid:96)(cid:96) (cid:48) as ˜ C XY(cid:96) = (cid:88) (cid:96) (cid:48) M (cid:96)(cid:96) (cid:48) C XY(cid:96) , (4.2)where M (cid:96)(cid:96) (cid:48) is inferred by the geometry of the mask [70], given by M (cid:96)(cid:96) (cid:48) = 2 (cid:96) (cid:48) + 14 π (cid:88) (cid:96) (cid:48)(cid:48) (2 (cid:96) (cid:48)(cid:48) + 1) W (cid:96) (cid:48) (cid:18) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) (cid:19) . (4.3)Here W (cid:96) (cid:48) is the angular power spectrum of the mask when X = Y , while in the cross-correlation corresponds to the two joint masks. In the cross-correlation analysis, we multiplythe corresponding κ and δ g masks for each redshift bin.Depending on the size of the sky cut, the relation 4.2 cannot be inverted to obtain C XY(cid:96) because in general, the coupling matrix becomes singular. To mitigate the coupling effectand also to reduce the errors in the results, it is appropriate to bin the power spectrum in (cid:96) .An unbiased estimator of the true-bandpowers ˆ C XYL is given in terms of the binned couplingmatrix K LL (cid:48) ˆ C XYL = (cid:88) L (cid:48) (cid:96) K − LL (cid:48) P L (cid:48) (cid:96) ˜ C XY(cid:96) , (4.4)where K LL (cid:48) = (cid:88) (cid:96)(cid:96) (cid:48) P L(cid:96) M (cid:96)(cid:96) (cid:48) B X(cid:96) (cid:48) B Y(cid:96) (cid:48) p (cid:96) (cid:48) F (cid:96) (cid:48) Q (cid:96) (cid:48) L (cid:48) . (4.5)Here L denotes the bandpower index, P L(cid:96) is the binning operator and Q (cid:96) (cid:48) L (cid:48) is its reciprocalcorresponding to a piece-wise interpolation. The B (cid:96) (cid:48) is a beam function for each X and Y observed field, p (cid:96) is the pixel window function and F (cid:96) (cid:48) is the effective filtering function [69].One of the advantages of the cross-correlation is that we do not need to debias the noise,since the CMB lensing and the galaxy data are completely independent measurements andtherefore have, in principle, uncorrelated noise signals. However, we correct the galaxy powerspectrum ˆ C gg(cid:96) by subtracting the shot noise term: N gg(cid:96) = 1 / ¯ n , where ¯ n is the average numberdensity of galaxies per steradian.The analytical errors on the estimated auto-(cross-)spectrum are determined by [71] ∆ ˆ C XYL = (cid:115) L + 1) f sky ∆ (cid:96) (cid:2) ( ˆ C XYL ) + ˆ C XXL ˆ C Y YL (cid:3) / , (4.6)where we assume in this equation that both fields behave as Gaussian random fields and theauto-power spectrum incorporates the associated noise, N gg(cid:96) and N κ(cid:96) for the galaxy and CMBlensing, respectively.We bin the power spectrum in (cid:96) in a linearly spaced band powers of width ∆ (cid:96) = 10 inthe range < (cid:96) < . We test different bin width values, however, we find no significantimpact on the results. Due to the accuracy of the Limber approximation and the limited areacovered by the survey, the power spectrum for (cid:96) < is poorly estimated, and we did not useit in our analysis. However, we include the first bin to perform the inversion of the binnedcoupling matrix of equation 4.5 and the pseudo- C (cid:96) calculation, to prevent the bias from thelowest multipole.While we set the lowest value of (cid:96) on the (cid:96) min = 20 , we impose a conservative cut in (cid:96) max for each tomographic redshift bin to avoid several effects significant at small scales that– 8 –ould affect our results. From the theoretical galaxy power spectrum calculated by equation(2.5), we have therefore limited our analysis to the scales where the percent deviation betweenthe linear and non-linear models were smaller than , corresponding to limit the analysis tomodes k max (cid:46) . h Mpc − . We consider the same (cid:96) max for the galaxy-galaxy and galaxy-CMB lensing analyses. ˆ D G Estimator
We need to take into account the errors associated with the power spectrum measurementsto obtain the ˆ D G properly [1, 29]. Thus, we use the weighted average in the ˆ D G calculation, ˆ D G = (cid:80) L w L ˆ D G,L (cid:80) L w L , (4.7)where the weights takes into account the variance in the ˆ D G estimator w − = ˆ D G,L (cid:20)(cid:18) ∆ ˆ C κgL ˆ C κgL (cid:19) + 14 (cid:18) ∆ ˆ C ggL ˆ C ggL (cid:19) (cid:21) , (4.8)and the ˆ D G per bandpower L is written as ˆ D G,L = ˆ C κgL /C κgL (cid:115) /C ggL ˆ C ggL . (4.9) In order to check the possible impact of the off-diagonal contributions on the covariancematrix produced by non-linear clustering and by the mask as well as to verify the consistencyof the error bars, we estimate the covariance with three approaches: analytical, jackknife(JK), and Monte-Carlo (MC) realizations.In the analytical approach, we assume Gaussianity of the fields and the covariance issimply diagonal with its elements computed by equation (4.6). Although this method maybe a good approximation when dealing with scales that are in the linear or mildly nonlinearregimes, it may be unrealistic to neglect the off-diagonal components of the covariance matrix[72]. Hence, we calculate the JK technique by dividing the footprint covered by the masksinto N JK regions, defined by the HEALPix pixelization scheme. We remove each region in turnand compute the power spectrum using the remaining subsample, such that the covariance isdetermined by
Cov JK ( ˆ C XYL , ˆ C XYL (cid:48) ) = N JK − N JK N JK (cid:88) n =1 ( ˆ C XYL,n − ¯ C XYL )( ˆ C XYL (cid:48) ,n − ¯ C XYL (cid:48) ) , (4.10)where the ˆ C XYL,n is the power spectrum when removing the n − th jackknife region and the ¯ C XYL is the power spectrum averaged over all the jackknife regions. This method providesa covariance estimate in a model-independent way. We use N JK = 233 regions defined as HEALPix pixels with resolution N side = 16 . To establish this number we have considered ascriteria the minimal number of N JK patches as those in which the scatter in the number ofunmasked pixels deviates by less than from the mean, as described by [71]. However, the– 9 –esults may depend on the number of masked-out regions and the region size. We have testedthe results with different choices of the N side and we have found that starting at N side = 8 ,the diagonal elements of the associated covariance matrix and the mean ¯ C XYL among the N JK regions are stable.Finally, we also have exploited MC simulations to build the covariance. For that, wegenerate 500 correlated Gaussian realizations [73, 74] of the galaxy and CMB lensing maps,considering their noise properties. To simulate the Gaussian convergence noise maps, we usedthe convergence noise power spectrum N κκ(cid:96) provided by the Planck team [22]. From the galaxymock, we generated a galaxy number count map assuming the galaxy number density of thedata, where the value of each pixel is drawn as a Poisson distribution with the mean numberof sources per pixel. We then, transform each number count map into a galaxy overdensitymap and calculate the corresponding auto-power spectrum and cross-power spectrum usingthe lensing mock. The covariance evaluated from these measurements is Cov MC ( ˜ C XYL , ˜ C XYL (cid:48) ) = 1 N sims − N sims (cid:88) j =1 ( ˜ C XYL,j − (cid:104) ˜ C XYL (cid:105) )( ˜ C XYL (cid:48) ,j − (cid:104) ˜ C XYL (cid:48) (cid:105) ) . (4.11)In the figure 3 we show the covariance matrix elements estimated for the JK method (leftcolumn) and for the MC method (right column), both normalized by their diagonal elements.In the first row are displayed the results for the galaxy ˆ C ggL and in the second row for the cross ˆ C κgL , both for the galaxy catalogue with the redshift range . < z < . . In the galaxy powerspectrum, some off-diagonal elements are observed in both methods, although the amplitudeof the off-diagonal elements is less than of the diagonal elements for the MC and for the JK matrix, considering the multipole range of < (cid:96) < . Regarding the cross-power spectrum covariance, the amplitude of the off-diagonal terms corresponds ∼ ofthe diagonal elements amplitude in the MC method and ∼ in the JK method. Whilefor the MC these off-diagonal elements are assigned only due to the non-trivial correlationsbetween angular multipoles added by the mask, in the JK matrices may incorporate also thenon-Gaussian variance produced on small scales by the nonlinear evolution, the result is thatthe off-diagonal terms in the JK matrices are slightly larger than in the MC matrices.To illustrate the comparison of the diagonal elements of the covariance matrices, we showin figure 4 the errors from each matrices divided by the errors from the analytical estimateto the galaxy auto-power spectrum (left) and the cross-power spectrum (right). As expected,the amplitude of the MC errors is very similar to that of the Gaussian analytical errors asshown in the blue lines. In contrast, the JK errors are slightly larger than the analyticalprediction, as shown in the orange lines. Therefore, in our analysis we adopt the JK methodto be realistic in the error estimation and consistent in taking the off-diagonal terms intoaccount, despite its low amplitude. We demonstrate in Appendix A that this choice doesnot have a significant impact on our main results for the six tomographic redshift bins of thegalaxy catalogue. We present here the measurements of the galaxy auto-correlation and the cross-correlationbetween the CMB convergence and galaxy overdensity. We also explore a number of checkscarried out to ensure that our analysis is accurate and robust to observational and astrophys-ical systematic effects. Finally, we use our measurements to obtain the growth factor.– 10 – igure 3 : Covariance matrices normalized by their diagonal elements, obtained from differentmethods: Jackknife (left column) and Monte Carlo simulations (right column). In the firstrow, we show the covariance among the C L band-powers for the galaxy power spectrumand in the second row, we show the corresponding for the galaxy-CMB lensing cross-powerspectrum. In both cases, we consider the full galaxy catalogue, spanning the redshift range . < z < . .
50 100 150 200 250 / G a u ss ( C gg ) MCJK
50 100 150 200 250 / G a u ss ( C g ) MCJK
Figure 4 : Comparison between the amplitude of the error bar estimated from the Jackknifecovariance matrix and the analytical Gaussian (orange line) and from the Monte-Carlo co-variance matrix and the analytical Gaussian (blue line). The results in the left column referto the galaxy auto-power spectrum and, in the right, to the cross-power spectrum.– 11 – .1 Galaxy bias and lensing amplitude
We show the measurements of the (cid:96) -binned galaxy power spectrum (left panel) and the cross-power spectrum (right panel) in Figure 5. The six panels represent, from top to bottom,the estimates to each redshift bin. The error bars are calculated using the JK covariance, asdescribed in the previous section.Although the ˆ D G estimator is bias-independent for a narrow redshift bin, we can use theobserved galaxy-galaxy and the galaxy-CMB lensing power spectra to respectively estimatethe best-fit bias b and the amplitude of the cross-correlation A = bA lens , where the later isintroduced motivated by phenomenological reasons, and A lens is the CMB lensing amplitude[75]. Therefore, if the underlying cosmology conforms to the fiducial model, on average, it isexpected A lens = 1 and consequently, the amplitude A should assume the same value as thegalaxy bias b determined from the auto-correlation.We assume that the bias does not evolve within each redshift bin so that A and b arefree parameters obtained employing Bayesian analysis with an uninformative flat prior and aGaussian likelihood, L ( x | θ ) ∝ exp (cid:20) −
12 ( x − µ ( θ )) T C − ( x − µ ( θ )) (cid:21) , (5.1)where x is the extracted ˆ C gg(cid:96) or ˆ C κg(cid:96) , µ is the correspondent binned theoretical prediction forthe parameters θ , and C − is the inverse of the covariance matrix of section 4.3. Following theconclusions of that section, we use the JK covariance matrix. When inverting the covariancematrix, we take into account the effect of having a finite number of realisations multiplyingthe inverse covariance by ( N JK − p − / ( N JK − [76], where p is the number of bins used.In order to efficiently sample the parameter space, we use the Markov chain Monte Carlo(MCMC) method, employing emcee package [77]. We perform this analysis for each redshiftbin and as a comparison, also for the full sample, spanning the redshift range . < z < . .Our results are stable against the length of the chain as well as the initial walker positions.The best-fit bias and the cross-correlation amplitude with their σ errors are reportedin the captions of the Figure 5. The best-fit model with its σ uncertainties are shown asthe solid lines and the gray shaded region, respectively. The vertical dashed line shown inthe left panel indicates the (cid:96) max multipole used in the analysis, where the nonlinear powerspectrum differs from the linear theory by less than . For the cross-power spectrum we usethe same multipole range, however, we don’t show it in the right panel for the sake of clarity.The significance of the parameter detection is calculated as S/N = (cid:113) χ null − χ min ( θ ) , wherethe χ null is the χ ( θ = 0) and χ min ( θ ) is the value for the best-fit. The parameter values, the S/N , the χ min , and the probability-to-exceed (PTE) for each redshift bin are summarized inTable 2.In the tomographic approach, we have found the best-fit bias in agreement up to σ with the cross-correlation amplitude, indicating the lensing amplitude consistent with unity.For all the redshift bins, the best-fit bias has S/N greater than 11. Clearly, the cross-powerspectrum constraints are less significant than those using the galaxy power spectrum, althoughwe found
S/N ∼ . − . . Finally, we see that in most cases the reduced χ is generallyclose to (or below) unity, indicating consistency of the fit, except for the two highest redshiftbins in the galaxy-galaxy estimate, where we found a poor PTE. http://dfm.io/emcee/current/ – 12 –he theoretical power spectrum defined by equation 2.5 and consequently the qualityof our constraints rely on the robustness of the distribution of the galaxies as a function ofphotometric redshift, dn/dz . Any incompatibility between the true and the assumed redshiftdistribution can potentially influence the inferred value of the parameters. Indeed, accordingto [47], for the galaxies with spectroscopic redshifts z spec (cid:38) . , the photometric redshifts ofthe catalogue tend to be underestimated mainly due to a lack of high-z galaxies in the trainingset and therefore, more susceptible to uncertainties in the modeling of the theoretical powerspectrum in our analysis. As an example, we repeat the parameter constraints for the twohighest redshift bins, . < z < . and . < z < . , considering a broader and narrower dn/dz distribution, taking σ z = 0 . and σ z = 0 . , respectively. We find that the galaxy-CMB lensing cross-correlations are extremely robust to the change of σ z due to the broadnessof the CMB lensing kernel. However, the galaxy bias inferred from the galaxy-power spectrumis affected up to ∼ , getting lower PTE and values when σ z = 0 . , being b = 0 . ± . (PTE= . ) and . ± . (PTE= . × − % ) for . < z < . and . < z < . ,respectively. In contrast, the best-fit galaxy bias assumes a higher value for σ z = 0 . , b = 1 . ± . (PTE= 2.96 % ) and b = 1 . ± . (PTE = 0.11 % ) for . < z < . and . < z < . , respectively. It is important to emphasize that, an erroneous σ z was insertedonly to exemplify how the fit of the galaxy bias may be affected if the photo-z’s estimatesare more biased at z (cid:38) . than the considered in our dn/dz distribution. Therefore, theseresults don’t necessarily imply the presence of any systematic.The figure 6 shows the ˆ C κg(cid:96) (right panel) and the ˆ C gg(cid:96) (left panel) when consideringthe sample of the galaxy covering the redshift from 0.1 to 0.7. We found that the galaxyauto-power is best fitted by our fiducial cosmology with linear galaxy bias b = 1 . ± . .In contrast, we found that the cross-correlation with CMB lensing is best fitted by a loweramplitude value, A < b in more than σ (including only statistical errors). The results aresummarized also in table 2. The reduced χ reveals that our estimate of the covariance isrealistic and the model provides a good fit to the data.To explore the possible reason for the trend A < b found when using the whole catalog,but not found in the tomographic approach, we examine the variations of the result afterremove galaxies located at high and low redshifts. When we discard the galaxies at z = 0 . and z = 0 . , in the bins . < z < . and . < z < . , the difference between the bias b and the cross-correlation amplitude A is significantly reduced. Specifically, the inconsistencyof ∼ . σ found in . < z < . decrease to ∼ . σ (including only statistical errors) whenwe remove the galaxies in the bins . < z < . and . < z < . , being b = 1 . ± . and A = 0 . ± . . For the redshift range . < z < . , the A value is in agreement with b within σ , with b = 0 . ± . and A = 0 . ± . . However, when we discard only thegalaxies at the lower redshifts, i.e., galaxies in the bins . < z < . and . < z < . ,the trend A < b remains, with b = 1 . ± . and A = 0 . ± . for the redshift range . < z < . and b = 1 . ± . and A = 0 . ± . for . < z < . , respectively.Therefore, these results strongly suggest that the tension is driven by high redshift galaxies inthe analyses. As a sanity check, we proceed with the analysis verifying the constraints of thegrowth function ˆ D G ( z ) in section 5.3 with and without including these high redshift z-bins.In addition to the photo-z uncertainties, the inferred parameters may be changed bysome effects, including the reddening and/or extinction and the foregrounds in the CMBmaps. We consider in detail the impact of a variety of systematic effects and the null tests inSection 5.2. – 13 – .51.01.5 b = 0.79±0.03Linear b = 0.79±0.02Linear b = 0.84±0.02Linear b = 0.86±0.02Linear b = 0.85±0.02Linear
100 200
Multipole b = 0.81±0.02Linear C gg ( × ) z med = 0.165 A = bA lens = 1.01±0.44 z med = 0.260 A = bA lens = 0.92 ± 0.25 z med = 0.363 A = bA lens = 0.80±0.22 z med = 0.465 A = bA lens = 0.76±0.18 z med = 0.557 A = bA lens = 0.76 ± 0.16
100 200
Multipole z med = 0.655 A = bA lens = 0.82±0.13 C g ( × ) Figure 5 : The galaxy auto-power spectrum (left panel) and the cross power spectrum (rightpanel) of the six tomographic redshift bins. The panels refer to the photo-z bins, from low tohigh redshift (top to bottom). The median redshift of each bin is reported on each sub-panelof the cross power spectrum. The points are the direct estimates while the solid (dashed)line is the fiducial cosmology including (excluding) the non-linear corrections, rescaled by thebest-fit galaxy bias b (for the auto-spectra) and by the cross-correlation amplitude A = bA lens (for the cross-spectra). The best-fit parameters are reported in the captions with their σ error. The shaded gray region represents the σ values around the best-fit model. The best-fitwas inferred using up to the multipole (cid:96) max , represented as the vertical dashed line in the leftpanel .– 14 – Multipole C gg ( × ) b = 1.22±0.02Linear
50 100 150 200 250
Multipole C g ( × ) A = bA lens = 0.79±0.10 Figure 6 : The galaxy auto-power spectrum (left panel) and the cross power spectrum (rightpanel) for the full galaxy sample in the redshift range . < z < . . As the Figure 5, thecross-correlation amplitude and the galaxy bias are reported in the captions with their σ error. The gray shaded region denote the σ values around the best-fit model (solid line).Correlation Photo-z bin b ± σ b S/N χ /d.o.f PTE ( % )Gal-Gal . < z < . . ± . . < z < . . ± . . < z < . . ± . . < z < . . ± . . < z < . . ± . . < z < . . ± . . < z < . . ± . A ± σ A S/N χ /d.o.f PTE( % )Gal- CMB lensing . < z < . . ± . . < z < . . ± . . < z < . . ± . . < z < . . ± . . < z < . . ± . . < z < . . ± . . < z < . . ± . Table 2 : Summary of the results obtained from the galaxy-galaxy and galaxy-CMB lensingpower spectra for the six redshift bins and for the full galaxy sample with redshift range . < z < . : the top half table shows the best-fit galaxy linear bias b estimated fromthe galaxy auto-correlation, while the lower half shows the best-fit for the cross-correlationamplitude A = bA lens . The signal-to-noise ( S/N ) and the best-fit χ and the correspondingprobability-to-exceed (PTE) are also shown. In this section, we summarise a number of tests carried out to ensure that our analysis isaccurate and robust against systematic effects.– 15 – .2.1 Foreground contamination
Although the CMB lensing map is reconstructed based on the
SMICA foreground-reduced CMBmap, there may still be residual contributions from galactic and extragalactic foregrounds andit can bias the lensing reconstruction and consequently the cross-correlation results. One ofthe largest potential contaminants is the thermal Sunyaev-Zel’dovich (tSZ) effect, which willalso correlate with the galaxy density [78–81].We investigated the impact of the tSZ contamination by cross-correlating the galaxyoverdensity map with the Planck CMB lensing reconstructed from the tSZ deprojected
SMICA
CMB map. Additionally, this test also allows us to check the Cosmic Infrared Background(CIB) bias since it is significantly different in the case of the tSZ-deprojected weighting [22].In the figure 7 we show the difference between the cross power spectrum estimation withand without the tSZ-deprojection, in units of the statistical error. For all redshift bins, weindeed see non-zero residuals coming from tSZ/CIB, and the removal of the tSZ contaminationinduces only sub-percentage changes in the cross-correlation, by less than ∼ . σ . Therefore,the consistency between the tSZ-free and the fiducial cross-correlations provides additionalconfidence that this foreground contamination does not affect significantly our overall resultssince it is subdominant with respect to other sources of uncertainty.
50 100 150 200 250
Multipole C g , S Z / C g z < 0.20.2 < z < 0.30.3 < z < 0.40.5 < z < 0.60.4 < z < 0.50.6 < z < 0.7 Figure 7 : The difference of the galaxy-CMB lensing power spectrum when considering theCMB lensing map with the tSZ deprojected, in units of the statistical error. The severalmarkers denote the result for each redshift bin. For all cases, the result does not deviatesignificantly compared to the statistical error amplitude.We also have tested the stability of the results with respect to the extinction of distantsources by dust in our galaxy. Although the galaxy magnitudes have already been correctedby the reddening map of [62], we additionally use the extinction data from the Planck colourexcess map [82]. We create a mask in a
HEALPix scheme, excluding all pixels for which theextinction map has E ( B − V ) > . mag, removing about of the pixels. We set thislimit by looking at the contribution of E ( B − V ) along the pixels within the galaxy footprint,verifying that the most dusty areas are in this range. We then, use this mask jointly withthe galaxy and CMB lensing masks to calculate the power spectrum. In the figure 8 we showfor the galaxy power spectrum (upper panel) and the cross power spectrum (lower panel) thedifference of the results when we consider the extinction mask, in units of the statistical error.– 16 – ultipole C gg , E ( B V ) / C gg z < 0.20.2 < z < 0.30.3 < z < 0.40.5 < z < 0.60.4 < z < 0.50.6 < z < 0.7
50 100 150 200 250
Multipole C g , E ( B V ) / C g Figure 8 : Stability of the measured galaxy power spectrum (top) and the galaxy-CMBlensing power spectrum (bottom), when applied the extinction correction, in units of thestatistical error. We applied the extinction cut so that the most affected regions were removed,corresponding to E ( B − V ) > . of the Planck colour excess map.For both cases, we can see that the results are stable against the extinction correction, withvariations within the statistical errors by less than . σ . In order to check the validity of the cross-correlation against the possibility of residual sys-tematics or spurious signals in the data, we perform a null hypothesis test of no correlationbetween the CMB lensing and the galaxy density maps. We do this by considering thecross-correlation of these two fields, being one of them the real map and the second onefrom simulations. As these maps do not contain a common cosmological signal, the meancorrelation is expected to be consistent with zero.For each redshift bin, we cross-correlate the real galaxy maps with the convergencesimulations from the Planck 2018 data release [22]. In addition, we cross-correlate the PlanckCMB convergence map with galaxy simulations constructed considering the same prop-erties of the real galaxy data such as the best-fit bias, masks, and galaxy number density.The Figure 9 shows the cross-power spectrum estimated in both cases, where the errors barswere computed by the standard deviation of the simulated cross-power spectra divided by the √ N sim , with N sim = 300 . – 17 – Multipole C g ( × ) z < 0.20.2 < z < 0.30.3 < z < 0.40.5 < z < 0.60.4 < z < 0.50.6 < z < 0.70.1 < z < 0.7 Simulations × Real Galaxy Overdensity
50 100 150 200 250
Multipole C g ( × ) z < 0.20.2 < z < 0.30.3 < z < 0.40.5 < z < 0.60.4 < z < 0.50.6 < z < 0.70.1 < z < 0.7 Real × Galaxy Overdensity Simulations
Figure 9 : Null tests for the cross-power spectrum to the six redshift bins. In the rightpanel, is the mean correlation between the Planck CMB convergence map and 300 galaxyoverdensity simulations obtained considering the respective features of each redshift bin. Inthe left panel, is the mean correlation between the galaxy overdensity and the 300 simulatedPlanck CMB lensing maps. The errors bars are given by the square root of the covariancematrix diagonal derived from the set of simulations and divided by √ .Correlation Photo-z bin χ / d.o.f PTE ( % ) κ Planck Sims × Gal . < z < . . < z < . . < z < . . < z < . . < z < . . < z < . . < z < . χ /d.o.f PTE ( % ) κ Planck × Gal Sims . < z < . . < z < . . < z < . . < z < . . < z < . . < z < . . < z < . Table 3 : Summary of χ and the PTE for the null tests. The top half of the table showsthe results for the cross-correlation between the Planck CMB lensing simulations and the realgalaxy map, while the lower half shows the corresponding values for the real CMB lensingmap correlated with the galaxy density simulations.Considering the covariance matrices obtained from these simulations, we calculate the χ and the PTE. The results are displayed in the Table 3. We conclude that no significantsignal is detected in either case and therefore, our cross power spectrum measurements arerobust. – 18 – .3 Constraints of ˆ D G We apply the ˆ D G estimator, defined in the equation 4.7, to the extracted bandpowers of thedatasets. The figure 10 shows the growth factor for each redshift bin with the corresponding σ error bar. The error bars are estimated from the dispersion of the ˆ D simG , established fromauto- and cross- spectra of the 500 correlated MC Gaussian realizations performed in thesection 4.3. The solid black line denote the curve expected in the fiducial Planck Λ CDMmodel, D fidG ( z ) . As the function D fidG ( z ) is directly related to the cosmological parameters Ω m σ H , we consider the Planck chains to randomly draw 3000 points and calculate the lineargrowth function for each cosmology. The gray shaded region around the D fidG ( z ) indicatesthe σ scatter for the 3000 cosmologies. It is worth mention that for each cosmology i , wenormalize the curve by multiplying by the factor ( Ω m σ H ) i /( Ω m σ H ) fid .We can assess the amplitude of the linear growth function A D , with respect to the fiducialprediction, assuming a template shape of the D G ( z ) to be fixed by the D fidG ( z ) [1, 29, 32],such that D G ( z ) = A D D fidG ( z ) . (5.2)The result of the fit is A D = 1 . ± . , when we consider the ˆ D G of the six redshift bins.Due to the features of the highest redshift bins reported in section 5.1, as a sanity check werepeat the A D fit using only the redshift bins that have a robust bias fit, that is, only thethree lowest redshift bins, from which we obtain A D = 1 . ± . . In both cases, we foundconsistency with the fiducial value, only slightly higher than the expected value A Λ CDM D = 1 .Similar analyses using other galaxy samples, for nearer [29, 30] and for deeper [1, 32]redshifts than the considered in this work, also indicate agreement with the fiducial cosmologyestablished by Planck. In this sense, our analysis is complimentary, as we consider anothersurvey that covers a different region of the sky and therefore, extends to probing other possiblesystematics effects and redshifts intervals. Recent reports witness the increasing importance of measuring the growth of the cosmicstructures using the large deep surveys catalogues and CMB lensing data [1, 29, 30, 32], nowavailable. In fact, the linear structure growth factor as a function of redshift, D G ( z ) , havethe potential to discriminate between alternative models of cosmic acceleration. In this work,we present a tomographic estimate of the linear growth factor by combining the auto- andcross-correlation of the CMB Planck convergence map, κ , and a galaxy density fluctuationsmap, δ g , where the δ g map was constructed from the photometric catalogue based on multi-band data from SCUSS, SDSS, and WISE [47]. We perform detailed analyses in six redshiftbins of width ∆ z = 0 . , spanning the redshift interval . < z < . .We have studied the evolution of the linear galaxy bias, b , and the amplitude of the cross-correlation, A , using the galaxy-galaxy and galaxy-CMB lensing power spectra, respectively.We found a significant detection of the best-fit parameters, with SN spanning ∼ − σ for b and ∼ . − . σ for A . However, we have found a poor fit for b in the highest two redshiftbins, possibly due to a lack of high-z galaxies in the training set during the constructionof the photometric redshift catalogue [47], which consequently generate an underestimatedphotometric redshifts for z (cid:38) . , increasing the uncertainties of dn/dz in these redshifts bins.We have found the linear galaxy bias is in agreement with the cross-correlation amplitude and,therefore, with the lensing amplitude, in all redshift bins. When we consider the catalogue– 19 – .0 0.2 0.4 0.6 0.8 z D G ( z ) CDM
Figure 10 : The linear growth factor estimated from the ˆ D G estimator to the six redshiftbins. The solid black line denotes the theoretical growth function for the Planck fiducialcosmology. The σ scatter for cosmologies randomly drawn from the Planck chains isshown in the gray shaded region.in the redshift range . < z < . instead of a tomographic approach to fit the parameters,we found a trend of A < b , although the difference between them is reduced significantlywhen using galaxies only up to z < . , suggesting that this discrepancy is driven by the highredshift. The main results are summarized in Table 2 and section 5.1.In addition, we perform null tests to check if our measured signal is affected by artifactsfrom the survey’s systematics or other undesirable effects. To this end, we performed variousinvestigations of the robustness of the results, showing that the null test indicates that thecross-correlation is unlikely to be affected by such effects and a possible tSZ and extinctioncontamination are negligible effects. These main results are displayed in Figure 7, 8 and 9.By combining the auto and the cross-correlation estimates, we measure the linear growthfactor at different epochs of the Universe by using the bias-independent estimator ˆ D G intro-duced by [1]. The main result displayed in Figure 10, shows the measured linear structuregrowth factor in comparison with the expected in the fiducial Λ CDM scenario. Compilingthe measurements of growth in the six tomographic bins, we find the amplitude of the lineargrowth function A D = 1 . ± . , closely consistent with the expected by the fiducial model A Λ CDM D = 1 .The CMB lensing tomography is an efficient method to test the linear growth of cosmicstructures and, by extension, to test dark energy scenarios and/or alternative gravity models.In the near future, the CMB and galaxy surveys such as the Simons Observatory, CMB-S4,LSST, and WFIRST will produce comprehensive data and will enable us to reach a deepmapping of the galaxies and a high sensitivity in reconstructing the CMB lensing potential.Thus, we may expect that the CMB lensing tomography, through analysis as the one usedhere, will be fundamental to find shrunken bounds in the scenario that better explains thehistory of the cosmic structure growth. – 20 – cknowledgments We thank Carlos Bengaly and Kevin Huffenberger for many useful discussions. GAM issupported by the CAPES Foundation of the Ministry of Education of Brazil fellowships. ABacknowledges a CNPq fellowship. We acknowledge the use of public data [21, 47] and theuse of many python packages: Numpy [83], Astropy a community-developed core Pythonpackage for Astronomy [84, 85], Matplotlib [86], IPython [87] and Scipy [88]. References [1] T. Giannantonio, P. Fosalba, R. Cawthon, Y. Omori, M. Crocce, F. Elsner et al.,
CMB lensingtomography with the DES Science Verification galaxies , Monthly Notices of the RoyalAstronomical Society (2016) 3213.[2] The Dark Energy Survey Collaboration,
The Dark Energy Survey , arXiv e-prints (2005) astro[ astro-ph/0510346 ].[3] G. T. Richards, A. D. Myers, A. G. Gray, R. N. Riegel, R. C. Nichol, R. J. Brunner et al., Efficient photometric selection of quasars from the Sloan Digital Sky Survey. II. 1, 000, 000quasars from Data Release 6 , The Astrophysical Journal Supplement Series (2008) 67.[4] M. Bilicki, T. H. Jarrett, J. A. Peacock, M. E. Cluver and L. Steward,
Two micron all skysurvey photometric redshift catalog: A comprehensive three-dimensional census of the wholesky , The Astrophysical Journal Supplement Series (2013) 9.[5] M. Bilicki, J. A. Peacock, T. H. Jarrett, M. E. Cluver, N. Maddox, M. J. Brown et al.,
WISE × SuperCOSMOS photometric redshift catalog: 20 million galaxies over 3 π steradians , TheAstrophysical Journal Supplement Series (2016) 5.[6] R. Beck, L. Dobos, T. Budavári, A. S. Szalay and I. Csabai,
Photometric redshifts for the SDSSData Release 12 , Monthly Notices of the Royal Astronomical Society (2016) 1371.[7] LSST Science Collaboration,
LSST Science Book, Version 2.0 , ArXiv:0912.0201 (2009) .[8] A. Bernui, C. Tsallis and T. Villela,
Temperature fluctuations of the cosmic microwavebackground radiation: A case of non-extensivity? , Physics Letters A (2006) 426.[9] C. Novaes, A. Bernui, I. Ferreira and C. Wuensche,
Searching for primordial non-gaussianity inplanck cmb maps using a combined estimator , Journal of Cosmology and Astroparticle Physics (2014) 018.[10] R. K. Sachs and A. M. Wolfe,
Perturbations of a cosmological model and angular variations ofthe microwave background , Astrophys. J. (1967) 73.[11] R. A. Sunyaev and Ya. B. Zeldovich,
Microwave background radiation as a probe of thecontemporary structure and history of the universe , Ann. Rev. Astron. Astrophys. (1980)537.[12] M. J. Rees and D. W. Sciama, Large scale Density Inhomogeneities in the Universe , Nature (1968) 511.[13] C. M. Hirata, N. Padmanabhan, U. Seljak, D. Schlegel and J. Brinkmann,
Cross-correlation ofCMB with large-scale structure: weak gravitational lensing , Physical Review D (2004)103501.[14] K. M. Smith, O. Zahn and O. Dore, Detection of gravitational lensing in the cosmic microwavebackground , Physical Review D (2007) 043510. – 21 –
15] S. Das, B. D. Sherwin, P. Aguirre, J. W. Appel, J. R. Bond, C. S. Carvalho et al.,
Detection ofthe power spectrum of cosmic microwave background lensing by the atacama cosmologytelescope , Physical Review Letters (2011) 021301.[16] A. Van Engelen, R. Keisler, O. Zahn, K. Aird, B. Benson, L. Bleem et al.,
A measurement ofgravitational lensing of the microwave background using South Pole Telescope data , TheAstrophysical Journal (2012) 142.[17] S. Das, T. Louis, M. R. Nolta, G. E. Addison, E. S. Battistelli, J. R. Bond et al.,
The AtacamaCosmology Telescope: temperature and gravitational lensing power spectrum measurementsfrom three seasons of data , Journal of Cosmology and Astroparticle Physics (2014) 014.[18] P. A. R. Ade, Y. Akiba, A. E. Anthony, K. Arnold, M. Atlas, D. Barron et al.,
Measurement ofthe Cosmic Microwave Background Polarization Lensing Power Spectrum with thePOLARBEAR Experiment , Phys. Rev. Lett (2014) 021301 [ ].[19] B. D. Sherwin, A. van Engelen, N. Sehgal, M. Madhavacheril, G. E. Addison, S. Aiola et al.,
Two-season Atacama Cosmology Telescope polarimeter lensing power spectrum , Phys. Rev. D (2017) 123529 [ ].[20] Planck Collaboration, Ade, P. A. R., Aghanim, N., Armitage-Caplan, C., Arnaud, M.,Ashdown, M. et al., Planck 2013 results. XVII. Gravitational lensing by large-scale structure , A&A (2014) A17.[21] Planck Collaboration, Ade, P. A. R., Aghanim, N., Arnaud, M., Ashdown, M., Aumont, J.et al.,
Planck 2015 results - xv. gravitational lensing , A&A (2016) A15.[22] N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini et al.,
Planck2018 results. viii. gravitational lensing , arXiv preprint arXiv:1807.06210 (2018) .[23] A. Krolewski, S. Ferraro, E. F. Schlafly and M. White, unwise tomography of planck cmblensing , arXiv preprint arXiv:1909.07412 (2019) .[24] C. M. Hirata, S. Ho, N. Padmanabhan, U. Seljak and N. A. Bahcall, Correlation of CMB withlarge-scale structure. II. Weak lensing , Physical Review D (2008) 043520.[25] Y. Omori and G. Holder, Cross-Correlation of CFHTLenS Galaxy Number Density and PlanckCMB Lensing , arXiv preprint arXiv:1502.03405 (2015) .[26] S. Singh, R. Mandelbaum and J. R. Brownstein, Cross-correlating Planck CMB lensing withSDSS: lensing–lensing and galaxy–lensing cross-correlations , Monthly Notices of the RoyalAstronomical Society (2016) 2120.[27] E. Giusarma, S. Vagnozzi, S. Ho, S. Ferraro, K. Freese, R. Kamen-Rubio et al.,
Scale-dependentgalaxy bias, CMB lensing-galaxy cross-correlation, and neutrino masses , Physical Review D (2018) 123526.[28] S. Singh, S. Alam, R. Mandelbaum, U. Seljak, S. Rodriguez-Torres and S. Ho, Probing gravitywith a joint analysis of galaxy and CMB lensing and SDSS spectroscopy , Monthly Notices of theRoyal Astronomical Society (2018) 785.[29] F. Bianchini and C. L. Reichardt,
Constraining Gravity at Large Scales with the 2MASSPhotometric Redshift Catalog and Planck Lensing , The Astrophysical Journal (2018) 81.[30] J. Peacock and M. Bilicki,
Wide-area tomography of CMB lensing and the growth ofcosmological density fluctuations , Monthly Notices of the Royal Astronomical Society (2018) 1133.[31] E. Baxter, J. Clampitt, T. Giannantonio, S. Dodelson, B. Jain, D. Huterer et al.,
Jointmeasurement of lensing-galaxy correlations using SPT and DES SV data , MNRAS (2016)4099 [ ]. – 22 –
32] Y. Omori, T. Giannantonio, A. Porredon, E. Baxter, C. Chang, M. Crocce et al.,
Dark EnergySurvey Year 1 Results: Tomographic cross-correlations between Dark Energy Survey galaxiesand CMB lensing from South Pole Telescope+ Planck , Physical Review D (2019) 043501.[33] S. Raghunathan, F. Bianchini and C. L. Reichardt,
Imprints of gravitational lensing in thePlanck cosmic microwave background data at the location of WISE × SCOS galaxies , PhysicalReview D (2018) 043506.[34] J. Liu and J. C. Hill, Cross-correlation of Planck CMB lensing and CFHTLenS galaxy weaklensing maps , Physical Review D (2015) 063517.[35] Y. Omori, E. J. Baxter, C. Chang, D. Kirk, A. Alarcon, G. M. Bernstein et al., Dark EnergySurvey Year 1 Results: Cross-correlation between Dark Energy Survey Y1 galaxy weak lensingand South Pole Telescope+P l a n c k CMB weak lensing , Phys. Rev. D (2019) 043517[ ].[36] S. Singh, R. Mandelbaum, U. Seljak, S. Rodríguez-Torres and A. Slosar,
Cosmologicalconstraints from galaxy-lensing cross correlations using BOSS galaxies with SDSS and CMBlensing , arXiv e-prints (2018) arXiv:1811.06499 [ ].[37] T. Namikawa, Y. Chinone, H. Miyatake, M. Oguri, R. Takahashi, A. Kusaka et al., Evidencefor the Cross-correlation between Cosmic Microwave Background Polarization Lensing fromPolarbear and Cosmic Shear from Subaru Hyper Suprime-Cam , ApJ (2019) 62[ ].[38] B. D. Sherwin, S. Das, A. Hajian, G. Addison, J. R. Bond, D. Crichton et al.,
The AtacamaCosmology Telescope: Cross-correlation of cosmic microwave background lensing and quasars , Physical Review D (2012) 083006.[39] J. Geach, R. Hickox, L. Bleem, M. Brodwin, G. Holder, K. Aird et al., A direct measurement ofthe linear bias of mid-infrared-selected quasars at z ∼ , The Astrophysical Journal Letters (2013) L41.[40] M. DiPompeo, A. Myers, R. Hickox, J. Geach, G. Holder, K. Hainline et al.,
Weighing obscuredand unobscured quasar hosts with the cosmic microwave background , Monthly Notices of theRoyal Astronomical Society (2014) 3492.[41] F. Bianchini, P. Bielewicz, A. Lapi, J. Gonzalez-Nuevo, C. Baccigalupi, G. De Zotti et al.,
Cross-correlation between the CMB lensing potential measured by Planck and high-zsubmillimeter galaxies detected by the Herschel-ATLAS survey , The Astrophysical Journal (2015) 64.[42] M. Aguilar Faundez, K. Arnold, C. Baccigalupi, D. Barron, D. Beck, F. Bianchini et al.,
Cross-correlation of POLARBEAR CMB Polarization Lensing with High- z Sub-mmHerschel-ATLAS galaxies , arXiv e-prints (2019) arXiv:1903.07046 [ ].[43] P. Zhang, M. Liguori, R. Bean and S. Dodelson, Probing gravity at cosmological scales bymeasurements which test the relationship between gravitational lensing and matter overdensity , Physical Review Letters (2007) 141302.[44] R. Reyes, R. Mandelbaum, U. Seljak, T. Baldauf, J. E. Gunn, L. Lombriser et al., Confirmation of general relativity on large scales from weak lensing and galaxy velocities , Nature (2010) 256.[45] A. R. Pullen, S. Alam and S. Ho,
Probing gravity at large scales through CMB lensing , MonthlyNotices of the Royal Astronomical Society (2015) 4326.[46] N. Kaiser,
Clustering in real space and in redshift space , Monthly Notices of the RoyalAstronomical Society (1987) 1.[47] J. Gao, H. Zou, X. Zhou and X. Kong,
A Photometric Redshift Catalog Based on SCUSS,SDSS, and WISE Surveys , The Astrophysical Journal (2018) 12. – 23 –
48] M. Bartelmann and P. Schneider,
Weak gravitational lensing , Physics Reports (2001) 291.[49] J. N. Fry and E. Gaztanaga,
Biasing and hierarchical statistics in large-scale structure , Astrophys. J. (1993) 447.[50] D. N. Limber,
The Analysis of Counts of the Extragalactic Nebulae in Terms of a FluctuatingDensity Field. , The Astrophysical Journal (1953) 134.[51] A. Lewis and A. Challinor,
CAMB: Code for anisotropies in the microwave background , Astrophysics Source Code Library (2011) .[52] R. E. Smith, J. A. Peacock, A. Jenkins, S. White, C. Frenk, F. Pearce et al.,
Stable clustering,the halo model and non-linear cosmological power spectra , Monthly Notices of the RoyalAstronomical Society (2003) 1311.[53] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi et al.,
Planck 2018 results. VI. Cosmological parameters , arXiv e-prints (2018) arXiv:1807.06209[ ].[54] X. Zhou, X.-H. Fan, Z. Fan, B.-L. He, L.-H. Jiang, Z.-J. Jiang et al., South Galactic Capu-band Sky Survey (SCUSS): Project Overview , Research in Astronomy and Astrophysics (2016) 069.[55] D. G. York, J. Adelman, J. E. Anderson Jr, S. F. Anderson, J. Annis, N. A. Bahcall et al., TheSloan Digital Sky Survey: Technical summary , The Astronomical Journal (2000) 1579.[56] E. L. Wright, P. R. Eisenhardt, A. K. Mainzer, M. E. Ressler, R. M. Cutri, T. Jarrett et al.,
The Wide-field Infrared Survey Explorer (WISE): mission description and initial on-orbitperformance , The Astronomical Journal (2010) 1868.[57] H. Zou, X. Zhou, Z. Jiang, X. Peng, D. Fan, X. Fan et al.,
South Galactic Cap u-band SkySurvey (SCUSS): Data Release , The Astronomical Journal (2016) 37.[58] H. Zou, Z. Jiang, X. Zhou, Z. Wu, J. Ma, X. Fan et al.,
South galactic cap u-Band sky survey(SCUSS): Data reduction , The Astronomical Journal (2015) 104.[59] C. P. Ahn, R. Alexandroff, C. A. Prieto, F. Anders, S. F. Anderson, T. Anderton et al.,
Thetenth data release of the Sloan Digital Sky Survey: first spectroscopic data from the SDSS-IIIApache Point Observatory galactic evolution experiment , The Astrophysical JournalSupplement Series (2014) 17.[60] D. Lang, unwise: Unblurred coadds of the wise imaging , The Astronomical Journal (2014)108.[61] D. Lang, D. W. Hogg and D. J. Schlegel,
WISE photometry for 400 million SDSS sources , TheAstronomical Journal (2016) 36.[62] D. J. Schlegel, D. P. Finkbeiner and M. Davis,
Maps of Dust Infrared Emission for Use inEstimation of Reddening and Cosmic Microwave Background Radiation Foregrounds , ApJ (1998) 525 [ astro-ph/9710327 ].[63] K. M. Gorski, E. Hivon, A. Banday, B. D. Wandelt, F. K. Hansen, M. Reinecke et al.,
HEALPix: A framework for high-resolution discretization and fast analysis of data distributedon the sphere , The Astrophysical Journal (2005) 759.[64] T. Budavari, A. J. Connolly, A. S. Szalay, I. Szapudi, I. Csabai, R. Scranton et al.,
Angularclustering with photometric redshifts in the Sloan Digital Sky Survey: Bimodality in theclustering properties of galaxies , The Astrophysical Journal (2003) 59.[65] R. K. Sheth and G. Rossi,
Convolution-and deconvolution-based estimates of galaxy scalingrelations from photometric redshift surveys , Monthly Notices of the Royal Astronomical Society (2010) 2137. – 24 –
66] T. Okamoto and W. Hu,
Cosmic microwave background lensing reconstruction on the full sky , Physical Review D (2003) 083002.[67] N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini et al., Planck2018 results. iii. high frequency instrument data processing and frequency maps , arXiv preprintarXiv:1807.06207 (2018) .[68] M. Hauser and P. Peebles, Statistical analysis of catalogs of extragalactic objects. II. The Abellcatalog of rich clusters , The Astrophysical Journal (1973) 757.[69] E. Hivon, K. M. Górski, C. B. Netterfield, B. P. Crill, S. Prunet and F. Hansen,
Master of thecosmic microwave background anisotropy power spectrum: a fast method for statistical analysisof large and complex cosmic microwave background data sets , The Astrophysical Journal (2002) 2.[70] G. Hinshaw, D. Spergel, L. Verde, R. Hill, S. Meyer, C. Barnes et al.,
First-Year WilkinsonMicrowave Anisotropy Probe (WMAP) Observations: The Angular Power Spectrum , TheAstrophysical Journal Supplement Series (2003) 135.[71] A. Balaguera-Antolínez, M. Bilicki, E. Branchini and A. Postiglione,
Extracting cosmologicalinformation from the angular power spectrum of the 2MASS Photometric Redshift catalogue , Monthly Notices of the Royal Astronomical Society (2018) 1050.[72] F. Lacasa,
Covariance of the galaxy angular power spectrum with the halo model , Astronomy &Astrophysics (2018) A1.[73] C. J. Copi, M. O’Dwyer and G. D. Starkman,
The ISW effect and the lack of large-angle CMBtemperature correlations , Mon. Not. Roy. Astron. Soc. (2016) 3305 [ ].[74] M. Kamionkowski, A. Kosowsky and A. Stebbins,
Statistics of cosmic microwave backgroundpolarization , Phys. Rev.
D55 (1997) 7368 [ astro-ph/9611125 ].[75] P. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi et al.,
Planck 2015results-XXI. The integrated Sachs-Wolfe effect , Astronomy & Astrophysics (2016) A21.[76] J. Hartlap, P. Simon and P. Schneider,
Why your model parameter confidences might be toooptimistic. Unbiased estimation of the inverse covariance matrix , Astronomy & Astrophysics (2007) 399.[77] D. Foreman-Mackey, D. W. Hogg, D. Lang and J. Goodman, emcee: the MCMC hammer , Publications of the Astronomical Society of the Pacific (2013) 306.[78] A. Van Engelen, S. Bhattacharya, N. Sehgal, G. Holder, O. Zahn and D. Nagai,
Cmb lensingpower spectrum biases from galaxies and clusters using high-angular resolution temperaturemaps , The Astrophysical Journal (2014) 13.[79] M. S. Madhavacheril and J. C. Hill,
Mitigating foreground biases in CMB lensingreconstruction using cleaned gradients , Physical Review D (2018) 023534.[80] J. E. Geach and J. A. Peacock, Cluster richness–mass calibration with cosmic microwavebackground lensing , Nature Astronomy (2017) 795.[81] E. Schaan and S. Ferraro, Foreground-immune cosmic microwave background lensing withshear-only reconstruction , Physical review letters (2019) 181301.[82] A. Abergel, P. A. Ade, N. Aghanim, M. Alves, G. Aniano, C. Armitage-Caplan et al.,
Planck2013 results. xi. all-sky model of thermal dust emission , Astronomy & Astrophysics (2014)A11.[83] T. E. Oliphant,
Guide to numpy, 2nd , USA: CreateS-pace Independent Publishing Platform (2015) . – 25 –
84] Astropy Collaboration, T. P. Robitaille, E. J. Tollerud, P. Greenfield, M. Droettboom, E. Brayet al.,
Astropy: A community Python package for astronomy , A&A (2013) A33[ ].[85] A. M. Price-Whelan, B. M. Sipőcz, H. M. Günther, P. L. Lim, S. M. Crawford, S. Conseilet al.,
The Astropy Project: Building an Open-science Project and Status of the v2.0 CorePackage , AJ (2018) 123.[86] J. D. Hunter, Matplotlib: A 2D graphics environment , Computing in science & engineering (2007) 90.[87] F. Pérez and B. E. Granger, Ipython: a system for interactive scientific computing , Computingin Science & Engineering (2007) 21.[88] E. Jones, T. Oliphant, P. Peterson et al., SciPy: Open source scientific tools for Python , 2001–.
A Covariance matrix validation