An Updated Tomographic Analysis of the Integrated Sachs-Wolfe Effect and implications for Dark Energy
Benjamin Stölzner, Alessandro Cuoco, Julien Lesgourgues, Maciej Bilicki
AAn Updated Tomographic Analysis of the Integrated Sachs-Wolfe Effect andImplications for Dark Energy
Benjamin St¨olzner, ∗ Alessandro Cuoco, † Julien Lesgourgues, ‡ and Maciej Bilicki
2, 3, 4, § Institute for Theoretical Particle Physics and Cosmology (TTK),RWTH Aachen University, Otto-Blumenthal-Strasse, 52057, Aachen, Germany. Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA Leiden, The Netherlands National Centre for Nuclear Research, Astrophysics Division, P.O.Box 447, 90-950 (cid:32)L´od´z, Poland Janusz Gil Institute of Astronomy, University of Zielona G´ora, ul. Lubuska 2, 65-265 Zielona G´ora, Poland
We derive updated constraints on the Integrated Sachs-Wolfe (ISW) effect through cross-correlation of the cosmic microwave background with galaxy surveys. We improve with respectto similar previous analyses in several ways. First, we use the most recent versions of extragalacticobject catalogs: SDSS DR12 photometric redshift (photo- z ) and 2MASS Photo- z datasets, as wellas employed earlier for ISW, SDSS QSO photo- z and NVSS samples. Second, we use for the firsttime the WISE × SuperCOSMOS catalog, which allows us to perform an all-sky analysis of the ISWup to z ∼ .
4. Third, thanks to the use of photo- z s, we separate each dataset into different redshiftbins, deriving the cross-correlation in each bin. This last step leads to a significant improvementin sensitivity. We remove cross-correlation between catalogs using masks which mutually excludecommon regions of the sky. We use two methods to quantify the significance of the ISW effect. Inthe first one, we fix the cosmological model, derive linear galaxy biases of the catalogs, and thenevaluate the significance of the ISW using a single parameter. In the second approach we perform aglobal fit of the ISW and of the galaxy biases varying the cosmological model. We find significancesof the ISW in the range 4.7-5.0 σ thus reaching, for the first time in such an analysis, the thresholdof 5 σ . Without the redshift tomography we find a significance of ∼ σ , which shows the impor-tance of the binning method. Finally we use the ISW data to infer constraints on the Dark Energyredshift evolution and equation of state. We find that the redshift range covered by the catalogs isstill not optimal to derive strong constraints, although this goal will be likely reached using futuredatasets such as from Euclid, LSST, and SKA. Keywords: cosmology: theory – cosmology: observations – cosmology: large scale structure of the universe– cosmology: cosmic microwave background – cosmology: dark energy
I. INTRODUCTION
We have, at present, strong evidence for Dark Energy(DE) from the large amount of available cosmologicaldata [e.g., 1]. Nonetheless, this evidence is mostly basedon precise constraints from the Cosmic Microwave Back-ground (CMB) epoch extrapolated to the present time.Local, or present-day, constraints on DE are, instead,mostly given by SuperNovae (SN) data, which are notyet precise enough for accurately constraining the prop-erties and time evolution of DE [e.g., 2].Thus, it is important to look for alternative local DEprobes. In this respect such a DE-sensitive measurementis given by the late-time Integrated Sachs-Wolfe effect(ISW) on the CMB [3]. This effect is imprinted in theangular pattern of the CMB in the presence of a time-varying cosmological gravitational potential, which ap-pears in the case of a non-flat universe [4, 5], as well asfor a flat one in the presence of DE, but also for variousmodified gravity theories [e.g., 6, 7]. Thus, for standardGeneral Relativity (GR) and flat cosmology a non-zero ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ISW implies the presence of DE. The effect is very smalland cannot be well measured using the CMB alone sinceit peaks at large angular scales (small multipoles, (cid:96) (cid:46) z (cid:46)
1) redshift [8, 9], with most of the signal lying inthe range z ∈ [0 . , .
5] for a standard ΛCDM cosmologi-cal model [10].In the past, many ISW analyses were performed usinga large variety of tracers at different redshifts [11–28].In a few cases, global analyses were performed combin-ing different LSS tracers, giving the most stringent con-straints and evidence for the ISW effect at the level of ∼ σ [29–31]. Related methodology, which has been ex-plored more recently, consists in stacking CMB patchesoverlapping with locations of large-scale structures, suchas superclusters or voids [32–38]. A further idea, whichwas sometimes exploited, is to use the redshift informa-tion of a given catalog to divide it into different redshiftbins, compute the cross-correlation in each bin, and thencombine the information. This tomographic approachwas pursued, for example, in the study of 2MASS [39] orSDSS galaxies [40, 41]. Typically, the use of tomographydoes not provide strong improvement over the no-binningcase, either because the catalog does not contain a large a r X i v : . [ a s t r o - ph . C O ] A p r enough number of objects and splitting them increasesthe shot-noise, or because the redshift range is not wellsuited for ISW studies.Nonetheless, in the recent years, several catalogs withredshift information and with a very large number of ob-jects have become available thanks to the use of photo-metric redshifts (photo- z s) instead of spectroscopic ones.Although photo- z s are not as accurate as their spectro-scopic counterparts, the former are sufficient for perform-ing a tomographic analysis of the ISW with coarse z bins.Hence we can exploit these large catalogs, which have theadvantage of giving a low shot noise even when dividedinto sub-samples. In this work, we combine for the firsttime the two above approaches: we use several datasetscovering different redshifts ranges, and we bin them intoredshift sub-samples to perform a global tomography. Weshow that in this way we are able to improve the signif-icance of the ISW effect from ∼ σ without redshiftbinning to ∼ σ exploiting the full tomography infor-mation. When combining the various catalogs, we takespecial care to minimize their overlap both in terms ofcommon sources and the same LSS traced, in order notto use the same information many times. This is doneby appropriate data cleaning and masking. We then usethese improved measurement of the ISW effect to studydeviations of DE from the simplest assumption of a cos-mological constant.Finally, the correlation data derived in this work andthe associated likelihood will soon be made publicly avail-able, in the next release of the MontePython pack-age [42]. II. THEORY
The expression for the cross-correlation angular powerspectrum (CAPS) between two fields I and J is given by: C I,Jl = 2 π (cid:90) k P ( k )[ G I(cid:96) ( k )][ G J(cid:96) ( k )] dk, (1)where P ( k ) is the present-day power spectrum of mat-ter fluctuations. In the above expression we have as-sumed an underlying cosmological model, like ΛCDM,in which the evolution of density fluctuations is separa-ble in wavenumber k and redshift z on linear scales. Adifferent expression applies, for example, in the presenceof massive neutrinos [43], where the k and z evolutionis not separable. Moreover, in the following, we assumestandard GR and a flat ΛCDM model. For studies of theISW effect for non-zero curvature or modified gravity see[4–7]. See http://baudren.github.io/montepython.html and https://github.com/brinckmann/montepython_public
For the case I = c of the fluctuation field of a catalogof discrete objects, one has G c(cid:96) ( k ) = (cid:90) dN ( z ) dz b c ( z ) D ( z ) j (cid:96) [ kχ ( z )] dz, (2)where dN ( z ) /dz and b c ( z ) represent the redshift distri-bution and the galaxy bias factor of the sources, respec-tively, j (cid:96) [ kχ ( z )] are spherical Bessel functions, D ( z ) =( P ( k, z ) /P ( k )) / is the linear growth factor of densityfluctuations and χ ( z ) is the comoving distance to redshift z . For the case of cross-correlation with the temperaturefluctuation field obtained from the CMB maps ( J = T ),the ISW effect in real space is given by [e.g., 44]Θ(ˆ n ) = − (cid:90) d Φ(ˆ nχ, χ ) dχ dχ, (3)where Φ represents the gravitational potential. In theexpression, we neglect a factor of exp( − τ ), which intro-duces an error of the order of 10%, smaller than the typ-ical accuracy achieved in the determination of the ISWitself. Furthermore, using the Poisson and Friedmannequations, and considering scales sufficiently within thehorizon Φ( k, z ) = − c Ω m a ( z ) H k δ ( k, z ) (4)where c is the speed of light, a ( z ) is the cosmological scalefactor, H is the Hubble parameter today, Ω m = Ω b + Ω c is the fractional density of matter today, and δ ( k, z ) is thematter fluctuation field in Fourier space, we can write G T(cid:96) ( k ) = 3 Ω m c H k (cid:90) ddz (cid:18) D ( z ) a ( z ) (cid:19) j (cid:96) [ kχ ( z )] dz . (5)Finally, the equations above can be combined throughEq. (1) to give the CAPS expected for the ISW effectresulting from the correlation between a catalog of ex-tragalactic objects, tracing the underlying mass distribu-tion, and the CMB. Using the Limber approximation [45]the correlation becomes [31] C cT(cid:96) = 3Ω m H c (cid:0) l + (cid:1) (cid:90) dz b c ( z ) dNdz H ( z ) D ( z ) ddz (cid:18) D ( z ) a ( z ) (cid:19) × P (cid:18) k = l + χ ( z ) (cid:19) . (6)The Limber approximation is very accurate at (cid:96) > (cid:96) <
10 [45], which issufficient for the present analysis. Eqs. 3-4 are valid assuming GR. For modified gravity differentappropriate expressions would apply (see, e.g., [6, 7]).
In our study, we use the public code class [46] tocompute the linear power spectrum of density fluctua-tions. As an option, this code can compute internallythe spectra C cT(cid:96) and C cc(cid:96) , for arbitrary redshift distribu-tion functions, using either the Limber approximation ora full integral in ( k, z ) space. We prefer, nonetheless, touse the Limber approximation since CAPS calculationsare significantly faster. Also, to get better performancesand more flexibility, we choose to perform these calcula-tions directly inside our python likelihood, reading only P ( k, z ) from the class output. We checked on a few ex-amples that our spectra do agree with those computedinternally by class . III. CMB MAPS
We use CMB maps from the Planck 2015 data release [1] which have been produced using four different meth-ods of foreground subtraction: Commander , NILC , SEVEM ,and
SMICA . Each method provides a confidence maskwhich defines the region of the sky in which the CMBmaps can be used. We construct a combined mask asthe union of these four confidence masks. This mask isapplied on the CMB maps before calculating the cross-correlation. We will use the
SEVEM map as default forthe analysis. Nonetheless, we have also tested the othermaps to check the robustness of the results. The test isdescribed in more detail in Sec. VIII.As the ISW effect is achromatic, for further cross-checks we also use CMB maps at different frequencies.In particular we use maps at 100 GHz, 143 GHz, and 217GHz. The results using these maps are also described inSec. VIII.
IV. ADDITIONAL COSMOLOGICALDATASETS
In the following we will perform parameter fits us-ing the ISW data obtained with the cross-correlation.Beside this, in some setups, we will also use othercosmological datasets in conjunction. In particu-lar, we will employ the Planck 2015 public likeli-hoods [1] and the corresponding MontePython in-terfaces
Planck highl lite (for high- (cid:96) temperature),
Planck lowl (for low- (cid:96) temperature and polarization),and
Planck lensing (CMB lensing reconstruction). Theaccuracy of the
Planck highl lite likelihood (whichperforms an internal marginalization over all the nui-sance parameters except one) with respect to the fullPlanck likelihood (where the nuisance parameters are not See http://class-code.net See http://pla.esac.esa.int/pla/ See http://pla.esac.esa.int/pla/ n o r m a li z e d d N / d l n ( z ) SDSS DR12 photometricWISE x SuperCOSMOS2MPZSDSS DR6 QSONVSS
FIG. 1. Photometric redshift distributions for the five cat-alogs used for the cross-correlation. The dN/dz curves arenormalized to a unit integral. For the NVSS case the ana-lytical approximation described in the text is used, since noredshifts information is available for the single catalog objects. marginalized) has been tested in [47, 48] where the au-thors find that the difference in the inferred cosmologi-cal parameters is at the level of 0.1 σ . Finally we willuse BAO data from 6dF [49], SDSS DR7 [50] and BOSSDR10&11 [51], which are implemented as bao boss and bao boss aniso in MontePython . V. CATALOGS OF DISCRETE SOURCES
For the cross-correlation with the CMB, as tracers ofmatter distribution we use five catalogs of extragalacticsources. As the ISW is a wide-angle effect, they were cho-sen to cover as large angular scales as possible, and two ofthem are all-sky. Furthermore, our study does not requireexact, i.e. spectroscopic, redshift information, thus pho-tometric samples are sufficient. Except for one case, thedatasets employed here include individual photo- z s foreach source, which allows us to perform a tomographicapproach by splitting the datasets into redshift bins.The catalogs we use span a wide redshift range; seeFig. 1 for their individual redshift distributions. Ta-ble I quantifies their properties (sky coverage, numberof sources, mean projected density) as effectively usedfor the analysis, i.e., after applying both the catalog andCMB masks.For a plot of the sky maps and masks of the catalogs de-scribed below, and for their detailed description, see [52].Below we provide a short summary of the properties ofthe datasets. A. 2MPZ
As a tracer of the most local LSS in this study weuse the 2MASS Photometric Redshift catalog [2MPZ, Available from http://ssa.roe.ac.uk/TWOMPZ.html . source sky number mean surfacecatalog coverage of sources density [deg − ]NVSS 62.3% 431,724 67.22MPZ 64.2% 661,060 24.9WISE × SCOS 64.5% 17,695,635 665SDSS DR12 18.7% 23,907,634 3095SDSS DR6 QSO 15.6% 461,093 71.8TABLE I. Statistics of the catalogs used in the analysis. Thenumbers refer to the area of the sky effectively employed inthe analysis, i.e., applying both the catalog and CMB masks. z s were subsequentlyestimated for all the included sources, by calibrating onoverlapping spectroscopic datasets.2MPZ includes ∼ ,
000 galaxies over almost the fullsky. Part of this area is however undersampled due to theGalactic foreground and instrumental artifacts, we thusapplied a mask described in [57]. When combined withthe CMB mask, this leaves over 660,000 2MPZ galaxieson ∼
64% of the sky (Table I).2MPZ provides the best-constrained photo- z s amongthe catalogs used in this paper. They are practically un-biased ( (cid:104) δz (cid:105) ∼
0) and their random errors have RMSscatter σ δz (cid:39) . z ∼ .
06 while the mean (cid:104) z (cid:105) ∼ .
08. The overall surfacedensity of 2MPZ is ∼
25 sources per square degree.For the tomographic analysis we split the catalog inthree redshift bins: z ∈ [0 . , . . , . . , . dN/dz where most of the ISW signal is expected.A precursor of 2MPZ, based on 2MASS and Super-COSMOS only, was used in a tomographic ISW analysisby [39], while an early application of 2MPZ itself to ISWtomography is presented in [58]. In both cases no signifi-cant ISW signal was found, consistent with expectations.Another ISW-related application of 2MPZ is presented in[13], where it was applied to reconstruct ISW anisotropiescaused by the LSS. B. WISE × SuperCOSMOS
The WISE × SuperCOSMOS photo- z catalog [WI × SC, 59] is an all-sky extension of 2MPZ obtained Available from http://ssa.roe.ac.uk/WISExSCOS.html . by cross-matching WISE and SuperCOSMOS samples.WI × SC reaches roughly 3 times deeper than 2MPZ andhas almost 30 times larger surface density. However, itsuffers from more severe foreground contamination, andits useful area is ∼
70% of the sky after applying its de-fault mask. This is further reduced to ∼
65% once thePlanck mask is also used; the resulting WI × SC sampleincludes about 17.5 million galaxies.WI × SC photo- z s have overall mean error (cid:104) δz (cid:105) ∼ σ δz (cid:39) . z ). Theredshift distribution is shown in Fig. 1 with the dashedorange curve. The peak is at z ∼ .
2, and the majorityof the sources are within z < .
5. In the tomographicapproach, the WI × SC sample is divided into four red-shift bins: z ∈ [0 . , . . , . . , . . , . × SCdataset for an ISW analysis for the first time. Variousstudies based using WISE have been performed in thepast [23, 25, 60, 61]. However, the samples used therediffered significantly from WI × SC, and none included in-dividual redshift estimates which would allow for redshiftbinning.
C. SDSS DR12 photometric
Currently there are no all-sky photo- z catalogs avail-able reaching beyond WI × SC. Therefore, in order to lookfor the ISW signal at z > .
5, we used datasets of smallersky coverage. The first of them, with the largest num-ber density of all employed in this paper, is based on theSloan Digital Sky Survey Data Release 12 (SDSS-DR12)photo- z sample compiled by [62]; to our knowledge, ourstudy is its first application to an ISW analysis, althoughearlier versions (DR 6 and DR 8) were used in [29, 30](but without z binning).The parent SDSS-DR12 photo- z dataset includes over200 million galaxies. Here we however use a subsampledescribed in detail in [52], which was obtained via appro-priate cleaning as recommended by [62], together withour own subsequent purification of problematic sky ar-eas. In particular, as the SDSS galaxies are distributed intwo disconnected regions in the Galactic south and north,with most of the area in the northern part, and unevensampling in the south, we have excluded the latter re-gion from the analysis. After additionally employing thePlanck CMB mask, we were left with about 24 millionSDSS DR12 sources with mean (cid:104) z (cid:105) = 0 .
34 and mostlywithin z < .
6. The resulting sky coverage is ∼
19% andthe mean surface density is ∼ − . The redshiftdistribution is shown in Fig. 1 with the solid blue line.Thanks to the very large projected density of objects,we were able to split the SDSS-DR12 sample into severalredshift bins, keeping low shot-noise in each shell. For thetomographic analysis we divided the dataset into six bins: z ∈ [0 . , . . , . . , . . , . . , .
7] and[0 . , . z ∈ [0 . , .
3] is not subdivided fur-ther since this redshift range is best covered by WI × SC,where we already have sub-bins. The photo- z accuracyof SDSS-DR12 depends on the ‘photo- z class’ defined by[62], and each class has an associated error estimate. Ourspecific preselection detailed in [52] leads to an effectivephoto- z scatter of σ δz = 0 . z ) based on the overallerror estimates from [62]. D. SDSS DR6 QSO
As a tracer of high- z LSS, we use a catalog of photo-metric quasars (QSOs) compiled by [63] from the SDSSDR6 dataset (DR6-QSO in the following), used previ-ously in ISW studies by e.g. [22, 29] and [30]. We applythe same preselections as in [22], and the resulting sam-ple includes 6 × QSOs on ∼
25% of the sky. Weexclude from the analysis three narrow stripes present inthe south Galactic sky and use only the northern region.The DR6-QSO sources are provided with photo- z sspanning formally 0 < z < .
75 but with a relativelypeaked dN/dz and mean (cid:104) z (cid:105) (cid:39) . z ∈ [0 . , . . , . . , . z ∈ [0 . , .
5] in order to minimize the overlap with theother catalogs in this redshift range. Nonetheless, thereare very few DR6-QSO catalog objects at these redshifts,thus this choice has only a very minor impact on the re-sults. The typical photo- z accuracy of this dataset is σ δz ∼ .
24 as reported by [63], and we will use this num-ber for the extended modeling of underlying dN/dz s perredshift bin in Sec. VIII.
E. NVSS
The NRAO VLA Sky Survey [NVSS, 64] is a catalog ofradio sources, most of which are extragalactic. This sam-ple has already been used for multiple ISW studies [e.g.9, 20, 26–28]. The dataset covers the whole sky avail-able for the VLA instrument; after appropriate cleanupof likely Galactic entries and artifacts, the NVSS sampleincludes ∼ . × objects flux-limited to >
10 mJy,located at declinations δ (cid:38) − ◦ and Galactic latitudes | b | > ◦ . This is the only of the datasets considered inthis work which does not provide even crude redshift in-formation for the individual sources. We thus use it with-out tomographic binning and, where relevant, assume its dN/dz to follow the model of [65] (purple short-long-dashed line in Fig. 1). This sample spans the broad-est redshift range of all the considered catalogs, namely0 < z < F. Masks
In the correlation of the CMB with each catalog weuse the CMB mask, described in Sec. III, combined withthe specific mask of the given catalog. Beside this, wedefine specific masks which we use when combining thesignal from the different catalogs in order to circumventincluding the same information twice, and to avoid theneed to take into account the cross-correlations betweenvarious tracers of the same LSS. We proceeded in thefollowing way. • SDSS catalogs (i.e. SDSS DR6 QSOs and SDSSDR12 galaxies) are used without additional masks.When combining the information with other cata-logs we, however, exclude the first SDSS DR12 bin,since the region z ∈ [0 . , .
1] is best covered by2MPZ. • To avoid correlations with the SDSS catalogs, whenusing all the remaining ones (i.e. NVSS, 2MPZ,WI × SC) we apply a mask which is a complemen-tary of the joint mask of SDSS DR12 galaxies andSDSS DR6 QSOs (in short, SDSS mask in the fol-lowing). • For 2MPZ and WI × SC it is not possible to definemutually exclusive masks since both these datasetscover practically the same part of the sky. Nonethe-less, we use them together, since WI × SC was builtexcluding most of the objects already contained in2MPZ [59]. The two catalogs, thus, have practi-cally no common sources. In this way the corre-lation among the two datasets is significantly sup-pressed, although not totally, since both trace thesame underlying LSS in the overlapping redshiftranges. We will, however, not consider the first bin, z ∈ [0 . , . × SC in the combined analysissince in this redshift range 2MPZ has better red-shift determination and basically no stellar contam-ination. Nonetheless, as we will show in Sec. VII,the evidence for ISW in the range z ∈ [0 . , . × SC have most of the over-lap, is very small, so, in practice, this has only amarginal effect on the final ISW significance. • Similarly, also for NVSS, 2MPZ and WI × SC it isnot possible to define a mutually exclusive maskdue to the large common area of the sky. In thiscase, we note that 2MPZ and WI × SC cover onlythe low redshift tail of NVSS. Thus, the overlapand correlation among them is minimal.We will thus use the above setup when reporting com-bined significances of the ISW from the different catalogs.For simplicity, we will use the same setup also to de-rive auto-correlations of the single catalogs. In this casethe significances could be increased slightly for NVSS,2MPZ, and WI × SC if their proper masks were used, butwe checked that the improvement is only marginal. C gg SDSS DR12 z=0.1-0.3 10 C g T
1e 8
SDSS DR12 z=0.3-0.4
FIG. 2. Left: Example of measured source catalog auto-correlation and best-fit model with free galaxy bias, referring to thecase of SDSS-DR12 in the labeled z bin. Right: example of measured cross-correlation between sources and CMB temperatureand best-fit model, referring to the case of SDSS-DR12 in the labeled z bin. Dots refer to the measured single multipoles, whiledata points with error bars refer to binned measurements. VI. CROSS-CORRELATION ANALYSIS
In the previous section we have presented the cata-logs of extragalactic objects that we use in the analysis.Their input format is that of a 2D pixelized map of objectcounts n ( ˆΩ i ), where ˆΩ i specifies the angular coordinateof the i -th pixel. For the cross-correlation analysis weconsider maps of normalized counts n ( ˆΩ i ) / ¯ n , where ¯ n is the mean object density in the unmasked area, andCMB temperature maps, also pixelized with a matchingangular resolution.In our analysis we compute both the angular 2-pointcross-correlation function, CCF, w ( cT ) ( θ ), and its har-monic transform, the angular power spectrum ¯ C ( cT ) (cid:96) ,CAPS. However, we restrict the quantitative analysis tothe CAPS only. The reason for this choice is that theCAPS has the advantage that different multipoles are al-most uncorrelated, especially after binning. Their covari-ance matrix is therefore close to diagonal, which simpli-fies the comparison between models and data. Similarly,we compute also the auto-correlation power spectrum ofthe catalogs (APS) and the related auto-correlation func-tion (ACF).We use the PolSpice statistical toolkit [66–69] to es-timate the correlation functions and power spectra. Pol-Spice automatically corrects for the effect of the mask.In this respect, we point out that the effective geometryof the mask used for the correlation analysis is obtainedby combining that of the CMB maps with those of eachcatalog of astrophysical objects. The accuracy of the
Pol-Spice estimator has been assessed in [70] by comparingthe measured CCF with the one computed using the pop-ular Landy-Szalay method [71]. The two were found to See be in very good agreement.
PolSpice also provides thecovariance matrix for the angular power spectrum, ¯ V (cid:96)(cid:96) (cid:48) [72].For the case of source catalog APS a further step isrequired. Contrary to the CAPS, the APS contains shotnoise due to the discrete nature of the objects in themap. The shot noise is constant in multipole and canbe expressed as C N = 4 πf sky /N gal , where f sky is thefraction of sky covered by the catalog in the unmaskedarea and N gal is the number of catalog objects, againin the unmasked area. The above shot-noise has beensubtracted from our final estimated APS.The Planck Point Spread Function and the map pix-elization affect in principle the estimate of the CAPS.However, the CAPS contains information on the ISWonly up to (cid:96) ∼
100 where these effects are negligible. Wewill thus not consider them further.Finally, to reduce the correlation in nearby multipolesinduced by the angular mask, we use an (cid:96) − binned ver-sion of the measured CAPS. The number of bins andthe maximum and minimum (cid:96) used in the analysis willbe varied to assess the robustness of the results. We in-dicate the binned CAPS with the same symbol as theunbinned one, C ( cT ) (cid:96) . It should be clear from the contextwhich one is used. The C ( cT ) (cid:96) in each bin is given by thesimple unweighted average of the C ( cT ) (cid:96) within the bin.For the binned C ( cT ) (cid:96) we build the corresponding covari-ance matrix as a block average of the unbinned covariancematrix V (cid:96)(cid:96) (cid:48) , i.e., (cid:80) (cid:96)(cid:96) (cid:48) V (cid:96)(cid:96) (cid:48) / ∆ (cid:96)/ ∆ (cid:96) (cid:48) , where ∆ (cid:96), ∆ (cid:96) (cid:48) are thewidths of the two multipole bins, and (cid:96), (cid:96) (cid:48) run over themultipoles of the first and the second bin. The binningprocedure is very efficient in removing correlation amongnearby multipoles, resulting in a block covariance ma-trix that is, to a good approximation, diagonal. We willuse nonetheless the full block covariance matrix in thefollowing, although we have checked that using the diag- catalog z b χ min b Halofit χ min b Halofit+ σ δ z χ min SDSS 0-0.1 0 . ± .
02 3.59 0 . ± .
02 3.76 0 . ± .
02 4.110.1-0.3 1 . ± .
03 1.71 1 . ± .
03 1.68 1 . ± .
03 1.630.3-0.4 0 . ± .
03 0.64 0 . ± .
03 0.63 0 . ± .
03 0.610.4-0.5 0 . ± .
02 4.84 0 . ± .
02 4.65 0 . ± .
03 4.990.5-0.7 1 . ± .
04 6.16 1 . ± .
04 5.86 1 . ± .
04 6.350.7-1 1 . ± .
11 15.04 1 . ± .
11 14.99 1 . ± .
13 15.16WIxSC 0-0.09 0 . ± .
03 0.46 0 . ± .
03 0.38 0 . ± .
03 0.280.09-0.21 0 . ± .
03 2.38 0 . ± .
03 2.67 0 . ± .
03 2.620.21-0.3 0 . ± .
02 10.07 0 . ± .
02 10.14 0 . ± .
02 10.090.3-0.6 0 . ± .
03 5.62 1 . ± .
04 5.88 1 . ± .
04 5.55QSO 0-1 1 . ± .
16 5.9 1 . ± .
16 5.93 1 . ± .
15 4.970.5-1 1 . ± .
26 3.09 1 . ± .
26 3.07 1 . ± .
26 3.071-2 2 . ± .
27 3.61 2 . ± .
27 3.59 2 . ± .
27 3.62-3 3 . ± .
50 7.08 3 . ± .
51 7.05 3 . ± .
55 7.082MPZ 0-0.105 1 . ± .
03 4.41 1 . ± .
03 1.30 1 . ± .
03 1.260.105-0.195 1 . ± .
04 2.00 1 . ± .
04 2.07 1 . ± .
04 2.170.195-0.3 1 . ± .
09 6.54 1 . ± .
09 6.67 2 . ± .
09 6.34NVSS 0-6 2 . ± .
08 3.02 2 . ± .
08 0.64 — —catalog z b χ min b Halofit χ min SDSS 0-1 1 . ± .
04 1.25 1 . ± .
04 1.27 1 . ± .
04 1.59WIxSC 0-0.6 1 . ± .
03 3.15 1 . ± .
03 3.74 1 . ± .
03 3.94QSO 0-3 2 . ± .
23 2.77 2 . ± .
23 2.76 2 . ± .
23 2.42MPZ 0-0.3 1 . ± .
04 5.19 1 . ± .
04 2.04 1 . ± .
04 1.98TABLE II. Linear biases for the different redshift bins of the various catalogs fitted for a fixed cosmological model. The reportederrors on the bias are derived from the fit of Eq. (7); goodness of fit is quantified in the relevant χ columns. The χ refers tothe case of a fit with 4 bins in the multipole range 10-60. onal only gives minor differences. When showing CAPSplots, however, we use the diagonal terms to plot the er-rors on the C (cid:96) , (∆ C (cid:96) ) = (cid:80) (cid:96)(cid:96) (cid:48) V (cid:96)(cid:96) (cid:48) / ∆ (cid:96) , where the sumruns over the multipoles of the bin contributing to C (cid:96) . VII. DERIVATION OF THE ISWSIGNIFICANCE
In this section we illustrate the two methods we useto quantify the significance of the ISW. We will assumefor the first method a flat ΛCDM model with cosmo-logical parameters Ω b h = 0 . c h = 0 . τ = 0 . h = 0 . A s = 3 . k = 0 . − , and n s = 0 . A. Method 1
This is the usual method employed in previous publi-cations to study the significance of the ISW. In this casewe fix the cosmological model to the best-fit one mea-sured by Planck, and we derive with class the matterpower spectrum P ( k, z ), which is used to calculate theexpected auto-correlation C (cid:96) for each catalog for the ap- propriate redshift bin. The measured auto-correlation isthen used to fit the linear bias, as a proportionality con-stant in the predicted C (cid:96) . An example of this fit is shownin the left panel of Fig. 2. A simple χ over the bins ofthe auto-correlation is used for the fit: χ AC ≡ χ ( b ) = (cid:88) (cid:96) bins ( ˆ C c (cid:96) ( b ) − C c (cid:96) ) (∆ C c (cid:96) ) , (7)where ˆ C c (cid:96) and C c (cid:96) represent the model and the measuredCAPS, and the sum is over all (cid:96) bins.As mentioned in Sec. VI we tested that the use of thefull covariance matrix with respect to the diagonal ex-pression for the χ above does not give appreciable differ-ences. Table II summarizes the various measured biases,and the default binning used for the auto-correlations.We tested the robustness of the fitted biases changingthe number of bins from 4 to 6 and the maximum (cid:96) from40-80, and we found stable results, with variations of theorder of 10%. A maximum (cid:96) of 40-80 is chosen sinceabove this range typically non-linear effects become sig-nificant. As the default case, we use 4 bins in the range10-60.As a further test we checked the impact of usingnon-linear corrections to the matter power spectrum tomodel the auto-correlation of the catalogs. The non-linear corrections were implemented through the version catalog z A ISW Aσ A χ χ min ∆ χ SDSS 0-0.1 0 . ± .
35 0.07 1.224 1.219 0.0050.1-0.3 0 . ± .
03 0.87 3.89 3.12 0.760.3-0.4 1 . ± .
24 1.57 4.47 2.01 2.450.4-0.5 2 . ± .
36 2.03 6.57 2.45 4.120.5-0.7 2 . ± .
13 2.28 9.28 4.06 5.220.7-1 1 . ± .
72 0.37 6.76 6.62 0.13WIxSC 0-0.09 5 . ± .
86 1.08 2.84 1.68 1.160.09-0.21 0 . ± .
01 0.33 4.63 4.52 0.110.21-0.3 1 . ± .
94 1.1 3.62 2.4 1.210.3-0.6 1 . ± .
94 1.41 4.91 2.92 1.99QSO 0-1 2 . ± .
64 1.52 5.95 3.64 2.310.5-1 2 . ± .
65 1.45 7.46 5.34 2.111-2 2 . ± .
64 1.52 3.99 1.68 2.312-3 1 . ± .
80 0.38 3.11 2.96 0.142MPZ 0-0.105 1 . ± .
43 0.36 1.26 1.13 0.130.105-0.195 0 . ± .
77 0.3 1.12 1.03 0.090.195-0.3 1 . ± .
47 0.71 1.66 1.16 0.5NVSS 0-6 1 . ± .
57 2.97 14.9 6.11 8.79catalog A ISW Aσ A χ χ min ∆ χ SDSS 1 . ± .
57 3.29 30.96 20.11 8.46WIxSC 0 . ± .
56 1.67 13.16 10.39 2.76Quasars 2 . ± .
13 2.13 14.55 10.01 2.992MPZ 0 . ± .
07 0.81 4.04 3.38 0.65SDSS+WIxSC 1 . ± .
40 3.49 44.12 31.94 11.21SDSS+Quasars 1 . ± .
51 3.9 45.51 30.28 11.45SDSS+WIxSC+Quasars 1 . ± .
38 4 58.67 42.66 14.2SDSS+WIxSC+Quasars+NVSS+2MPZ 1 . ± .
30 5 77.61 52.61 22.16SDSS+WIxSC+Quasars+NVSS 1 . ± .
31 4.97 73.57 48.85 21.52SDSS+WIxSC+NVSS+2MPZ 1 . ± .
31 4.6 63.06 41.92 19.17SDSS+Quasars+NVSS+2MPZ 1 . ± .
36 4.88 64.45 40.67 19.41SDSS+WIxSC+Quasars+2MPZ 1 . ± .
36 4.04 62.71 46.35 14.85WIxSC+Quasars+NVSS+2MPZ 1 . ± .
35 3.84 46.65 31.9 13.71TABLE III. Summary of the measured ISW and related significances for the single redshift bins of each catalogs (top table)and for various combinations of the catalogs, where, in the latter case, also the individual redshift bins of each catalog werecombined (bottom table). The last five rows give the cases in which a single catalog is excluded from the fit each time. The χ refers to the case of a fit with 4 bins in the multipole range 4-100.catalog A ISW Aσ A χ χ min ∆ χ SDSS 0 . ± .
65 1.49 5.3 3.09 2.21WIxSC 0 . ± .
61 1.02 5.28 4.24 0.65Quasars 1 . ± .
63 2.03 5.55 1.41 3.942MPZ 0 . ± .
32 0.39 0.87 0.72 0.15NVSS 1 . ± .
57 2.97 14.9 6.11 8.79SDSS+WIxSC 0 . ± .
42 2.23 18.47 13.48 4.96SDSS+Quasars 1 . ± .
56 2.35 19.85 14.33 5.2SDSS+WIxSC+Quasars 1 . ± .
40 2.84 33.02 24.97 7.95SDSS+WIxSC+Quasars+NVSS 1 . ± .
33 4.02 47.91 31.76 15.27SDSS+WIxSC+Quasars+NVSS+2MPZ 1 . ± .
31 4.08 51.95 35.28 15.92TABLE IV. Summary of the measured ISW and related significances for the the case of no redshift binning of the catalogs.Various combinations of the catalogs are shown.
Parameter 68% limits − ω b . ± . ω cdm . ± . n s . ± . − A s . ± . h . ± . τ reio . ± . − A Planck . ± . Ω Λ . ± . MontePython fit to Planck +BAO data only. Here Ω Λ is a derived parameter and A Planck a Planck nuisance parameter. of Halofit [73] implemented in class v2.6.1. The last 2columns of Table II show the bias and the best-fit χ ob-tained using the non-linear model. It can be seen that thebiases obtained with and without non-linear correctionsare fully compatible. The only exception is the first red-shift bin of 2MPZ where the best-fit bias changes at the2 σ level. More importantly, the fit shows a visible im-provement from χ ∼ . χ ∼ .
3. This is expected,since at these low redshifts even the small (cid:96) s correspondmostly to small, non-linear, physical scales. As we showbelow, however, 2MPZ presents little or no imprint ofthe ISW effect, so we conclude that the use of the linear P ( k, z ) has a negligible impact on the study of the ISWeffect in this analysis.As an additional comment about the galaxy biases re-ported in Table II, we note that the ∼
10% variationquoted above is typically larger than the statistical er-rors given in that Table, the latter being sometimes onlya few %; this means that the bias errors are systematics-rather statistics-limited. Also, in some cases, for exam-ple most notably in the z ∈ [0 . , .
0] bin of SDSS DR12galaxies, the minimum χ is quite large, indicating a poorquality of the fit. This is also visible in some of the ACplots provided in Appendix A. This is likely related tonon-uniformities of the catalogs, which are more severein the tails of the redshift distribution, which in particu-lar leads to excessively large measured low- (cid:96) AC power insome cases. Therefore, in such instances, the small sta-tistical errors on b should be taken with care. In general,we stress that the precise determination of the bias erroris not crucial in this analysis, which is, instead, focusedon the determination of the significance of the ISW ef-fect. To this aim, the error, and even the value of thebias, have only a limited impact. See further discussionbelow.In the second step, all the galaxy biases are fixed tobest-fit values previously derived, and only the measuredcross-correlations are used. At this point only a single pa-rameter A ISW is fitted using as data either a single mea-sured cross-correlation or a combination of them, with the χ statistics: χ CC ≡ χ ( A ISW ) = (cid:88) z − bins (cid:88) cat . (cid:88) (cid:96) bins ( A ISW ˆ C Tc (cid:96) − C Tc (cid:96) ) (∆ C Tc (cid:96) ) , (8)where ˆ C Tc (cid:96) and C Tc (cid:96) represent the model (for the standardΛCDM cosmological model considered) and the measuredcatalog – CMB temperature cross-correlation for a givenredshift bin, respectively, the sum is over all the (cid:96) bins,and over different catalogs and different redshift bins.The linear parameter A ISW quantifies the agreement withthe above standard model expectation. In the denomi-nator we use the error provided by
Polspice discussed inthe previous section. In principle, however, one shoulduse an error where the model is taken into account. Forthe case of binned data, however, this is a small effect(see for example discussion in [74]).An example of measured cross-correlation and fit tothe model is shown in the right panel of Fig. 2. Table IIIsummarizes the results of the fit for each single z -binof each catalog, for each catalog combining the different z -bins, and for different combinations of the catalogs,where, again, for each catalog z -binning has been used.For the default case we use four multipole bins between (cid:96) of 4 and 100, but, again, we have verified that the resultsare stable when changing the number of bins from 4 to 6and the maximum (cid:96) from 60 to 100, which is expected,since the ISW effect is rapidly decreasing as a functionof (cid:96) , and not much signal is expected beyond (cid:96) ∼ χ (0) − χ , (9)where χ min is the minimum χ , and χ (0) is the χ of the null hypothesis of no ISW effect, i.e. of the case A ISW = 0. TS is expected to behave asymptotically asa χ distribution with a number of degrees of freedomequal to the number of fitted parameters, allowing us toderive the significance level of a measurement based onthe measured TS. In this case, since there is only onefitted parameter, the significance in sigma is just givenby √ T S . From Table III one can see that the maxi-mum significance achieved with Method 1 when usingall the catalogs in combination is √ .
16 = 4 . σ . Fromthe different results it can also be seen that the maincontribution is given by NVSS and SDSS DR12 galax-ies. We remind that the cross-correlation with NVSS iscalculated masking the area of the sky used to calculatethe correlation with SDSS. The two are, thus, completelyindependent. A smaller, and comparable, contribution,is given by WI × SC and SDSS-QSO. 2MPZ instead showbasically no sign of ISW, which is expected given the verylow z range. In the Table we also include a column withthe signal to noise (S/N= A/σ A ) of the ISW measurementfor comparison with other works since this quantity is of-ten reported in the literature. We can see that the globalfit reaches a S/N of 5.0We also show in Table IV the result of the fit when noredshift binning is used. It is clear that without such bin-ning the significance of the ISW is significantly reduced,especially for SDSS-DR12 and WI × SC , while the signif-icance of SDSS-QSO is almost unchanged. Overall, whenno redshift binning is used, the significance of the ISWeffect combining all the catalog is 4.0 σ , which is signif-icantly reduced with respect to the 4.7 σ achieved withthe redshift binning.As mentioned above, the derived significance is veryweakly dependent on the exact values of the biases used.For the case of a single catalog redshift bin, this is clearlooking at Eqs. (6) & (8), which show that the ISW sig-nal is linear in b . The fit to the cross-correlation thusconstraints the quantity bA ISW and the value of b is notimportant for the determination of the significance, al-though, clearly, is relevant in determining the value of A ISW . When several redshift bins and catalogs are used,the above argument is not exact anymore, but remainsapproximately valid. We checked, indeed, that using dif-ferent biases derived from the autocorrelation fits usingdifferent (cid:96) max and different number of (cid:96) bins, gives un-changed significances.We can see that the preferred A ISW value from thecombined fit is slightly larger than 1 at a bit more than1 σ . In the single catalog fits, both NVSS, QSOs andSDSS seem to drive the A ISW value above 1. This isconfirmed in the last 5 rows of Table III where differentfits are performed each time excluding only one catalogand combining the remaining four. All the fits give com-patible results with A ISW above 1 at around 1 σ or abit more. This result is further scrutinized in Section IXwhere we investigate if this slight difference of A ISW from1 can be interpreted as an indication of departure of DEfrom the simple case of a cosmological constant.
B. Method 2
The first method is, in principle, not fully self-consistent, because the auto-correlations, and hence thebiases, are sensitive to the underlying matter power spec-trum. We fixed the matter power spectrum to the PlanckΛCDM best-fitting model, but this may not be the bestfit to the auto-correlation data. The induced error shouldbe negligible when CMB and BAO are also used, sincethey impose P ( k ) to be very close to the fiducial model.But more importantly, the cross-correlation determinesa given amount of ISW, and this has in principle an ef-fect on cosmology, since a different ISW means a differentDark Energy model and thus also a different P ( k ). Forthese reasons it is more consistent to fit to the data atthe same time as the bias parameters, the cosmologicalparameters, and the A ISW parameter used to assess thedetection significance.We perform such a fit using the
MontePython en-vironment. The fit typically involves many parameters( >
15) which can present degeneracies which are not known in advance. To scan efficiently this parameterspace we run
MontePython in the Multinest mode [75].In this way we can robustly explore the posterior withtypically ∼ likelihood evaluations, and efficiencies ofthe order of 10%. We consider two cases.In the first case, we only use cross-correlation and auto-correlation measurements. We call this dataset AC+CC,and we fit a total of 22 parameters, i.e, 15 biases, A ISW ,and the six ΛCDM parameters ( ω b , ω cdm , n s , h , A s , τ reio ). When Planck data are used, we also include thenuisance parameter A Planck [1]. For all cosmological pa-rameters except ω cdm , we use Gaussian priors derivedfrom a fit of Planck+BAO summarized in Table V, whichare consistent with those published in [1]. The errorbars from Planck+BAO are so small that we find es-sentially the same result for A ISW when fixing these fiveparameters to their best fit values instead of marginal-izing over them with Gaussian priors. Our results forthis fit are shown in the first column of Table VI. As ex-pected, the constraint on ω cdm coming from the AC+CCdata is weaker than that from Planck+BAO data, byabout a factor 6. Also, the ω cdm best-fit of the AC+CCanalysis is lower than the Planck+BAO fit, by about 2 σ .The fitted galaxy biases are typically compatible withthose of Method 1, although in several cases they are 10-20% larger, which can be understood as a consequence ofthe lower ω cdm , resulting in a lower P ( k ) normalization.Indeed, the measured auto-correlations basically fix theproduct of the squared biases and of the overall P ( k ) am-plitude. Comparing the case with free A ISW to the onewith A ISW = 0, we find TS=∆ χ = 22, giving a signifi-cance of 4.7 σ , identical to the one found in Method 1.With the same setup we also perform a fit using CC dataonly. The results are shown in the second column of Ta-ble VI. In this case the biases are determined from thecross-correlation only, without relying on the autocor-relation. It is interesting to see that, even in this case,good constraints on the biases can be achieved, although,clearly, the errors are much larger (by a factor of ∼ σ , thusreaching the 5 σ threshold. The increase in significanceseems to be due to the larger freedom in the fit of thebiases which allows to reach an overall better best-fit ofthe CC data with respect to the case in which the biasesare constrained by the AC data.In the second case, we fit the same parameters to thedata, but we now include the full Planck+BAO likeli-hoods instead of Gaussian priors on five parameters. For-mally, we use the Planck and BAO likelihoods combinedwith the χ from the AC+CC data:log L = log L PL + log L BAO − χ AC / − χ CC / . (10)It should be noted that the use of other data besidesAC+CC does not affect the ability to derive the signifi-cance of the ISW detection, which is only encoded in theparameter A ISW entering the AC+CC likelihood. Resultsof this fit are shown in the third column of Table VI.1
Parameter AC+CC CC PL+AC+CC − ω b . ± .
014 2 . ± .
013 2 . ± . ω cdm . ± . . +0 . − . . ± . n s . ± . . ± . . ± . − A s . ± .
049 2 . ± .
044 2 . ± . h . ± . . ± . . ± . τ reio — — 0 . ± . − A Planck — — 100 . ± . A ISW . ± .
29 1 . ± .
29 1 . ± . b , . ± .
059 1 . +0 . − . . ± . b , . ± .
049 1 . +0 . − . . ± . b , . ± .
080 1 . ± .
27 1 . ± . b , SDSS . ± .
043 1 . +0 . − . . ± . b , SDSS . ± .
030 0 . +0 . − . . ± . b , SDSS . ± .
027 0 . +0 . − . . ± . b , SDSS . ± .
038 1 . +0 . − . . ± . b , SDSS . +0 . − . . +0 . − . . +0 . − . b , WISC . ± .
040 1 . +0 . − . . ± . b , WISC . ± .
032 0 . +0 . − . . ± . b , WISC . ± .
038 0 . +0 . − . . ± . b , QSO . +0 . − . . +0 . − . . +0 . − . b , QSO . +0 . − . . +0 . − . . +0 . − . b , QSO . +0 . − . . . − . . +0 . − . b NVSS . ± .
11 2 . ± .
39 2 . ± . Ω Λ . ± .
014 0 . +0 . − . . ± . . . . σ . . . ∆ log(ev) . . . MontePython fits in the ΛCDM model with using several combinations of Planck data, AC dataand CC data. When the Planck data is not used, Gaussian priors on the cosmological parameters except ω cdm are assumed.Here Ω Λ is a derived parameter. The third to last row gives the Test Statistics (TS) which is equal to ∆ χ for the fit in thefirst two column and − L for the fit in the third column. The second to last row gives the significance σ = √ T S . Finally,the last row gives the logarithm of the Bayes factor, representing the evidence for non-zero A ISW in Bayesian terms.
The main difference with respect to the previous fit isthe value of ω cdm , now driven back to the Planck best-fit. This upward shift in ω cdm results, again, in a globaldownward shift of the biases, by about 10-20%, givingnow a better compatibility with the results of Method 1.In general, apart from the small degeneracy with ω cdm resolved by the inclusion of Planck+BAO data, the biasesare well constrained by the fit. This means that the sub-space of biases is approximately orthogonal to the rest ofthe global parameter space, which simplifies the fit andspeeds up its convergence. To measure the significance,in this case we define the test statistic as TS= − L ,which shares the same properties of the TS defined interms of the χ . Comparing the case with free A ISW tothe one with A ISW = 0, we now get TS= − L =24.9, which gives a significance of 5.0 σ . Since the cos-mology is basically fixed by the Planck+BAO data to apoint in parameter space very close to the fiducial modelof Method 1, this improvement in significance comes, ap- parently, from fitting jointly the biases and A ISW (whilein Method 1 the biases were kept fixed using the results ofthe first step of the method). The joint fit explores thecorrelations which exist between the biases and A ISW .This results in a better global fit, and also in a slightlyenhanced A ISW significance, reaching the 5 σ threshold.Finally, since the fit performed with Multinest auto-matically provides also the evidence of the Posterior, inthe last row of Table VI we additionally report the loga-rithm of the Bayes factor, i.e., the logarithm of the ratioof the evidences for the two fits where A ISW is free andwhere it is fixed to A ISW = 0. We find in all cases valuesaround ∼
12. Logarithm of the Bayes factors larger than5 represents strong evidence according to Jeffreys’ scale[76].2 C
1e 8 commandernilcsevemsmica
FIG. 3. Measured cross-correlation of 2MPZ in one singleredshift bin with CMB maps from
Commander , NILC , SEVEM ,and
SMICA . VIII. ROBUSTNESS TESTS
In this section we describe some further tests per-formed to verify the robustness of the results.As mentioned in Sec. III, several CMB maps are avail-able from Planck, resulting from different foregroundcleaning methods. In Fig. 3 we show the results of thecross-correlation using four CMB maps cleaned with fourdifferent methods. We pick up as an example the cross-correlation with the full 2MPZ catalog, without subdivi-sion in redshift bins. It clearly appears that the use ofdifferent maps has no appreciable impact on the result.Another important aspect is the possible frequency de-pendence of the correlation. In particular, while the ISWeffect is expected to be achromatic, some secondary ef-fects, like a correlation due to a Sunyaev-Zel’dovich [77]or Rees-Sciama [78] imprint in the CMB map, are ex-pected to be frequency dependent. To test this possi-bility, we use available Planck CMB maps at 100 GHz,143 GHz and 217 GHz. Again the the full 2MPZ cata-log is used as example, since these effects are expectedto peak at low redshift. Fig. 4 shows the result of thecorrelation at different frequencies. We observe a verysmall trend of the CAPS with frequency, especially forthe first (cid:96) bin, but this effect is negligible with respect tothe error bars of the data points. Results are similar forthe other catalogs, showing no frequency dependence.Finally, we tested the effect of photo- z errors. Inthe basic setup, the theoretical predictions for the auto-and cross-correlation functions per redshift bin are mod-eled by assuming that the true redshift distribution iswell approximated by the photo- z one, i.e. dN/dz true (cid:39) dN/dz phot . In reality, sharp cuts in dN/dz phot will cor-respond to more extended tails in dN/dz true becausethe photo- z s are smeared out in the radial direction.However, we can easily take photo- z errors into ac- C
1e 8
100 GHz143 GHz217 GHz
FIG. 4. Measured cross-correlation of 2MPZ in one singleredshift bin with CMB at 100 GHz, 143 GHz and 217 GHz. count if we know their statistical properties. In thecase of 2MPZ, the photo- z error is basically constantin z and has roughly Gaussian scatter of σ δz (cid:39) . (cid:104) δz (cid:105) = 0, while for WI × SC the scatter is σ δz ( z ) = 0 . z ) with also approximately zero meanin δz . For SDSS QSOs it is also approximately constantin z and equal to 0.24. Finally, for SDSS DR12 the erroris σ δz ( z ) = 0 . z ) (see Sec. V). We thus derive theeffective true redshift distribution of a given bin by con-volving the measured photo- z selection function in thatbin with a z -dependent Gaussian of width σ δz ( z ). Theresulting true- z distribution is a smoothed version of thephoto- z distribution, presenting tails outside the edgesof the bin. We then use this distribution to fit again theauto- and cross-correlations data. The results are shownin the last column of Table II. We find that the effect ofphoto- z errors has some impact on the determination ofthe biases. The effect is most important in the high- z tails of various catalogs, and, in particular, WI × SC andSDSS DR12. This is not surprising since, in these cases,the photo- z errors increase with redshift and are largestat high- z . The effect is at the level of 10-20%. This cor-responds to a decrease in A ISW of the same amount inthese bins. Nonetheless, since the above bins only havea limited weight on the combined fit, the impact on thefinal A ISW determined from the global fit of all bins andcatalogs is basically negligible.
IX. DARK ENERGY FIT
In this section we investigate the power of the cross-correlation data to constrain DE, in a similar frameworkas presented in [79, 80] and [81]. For this purpose, we donot use the A ISW parameter employed in Sec. VII, sinceit is only an artificial quantity necessary to evaluate theISW significance from the cross-correlation data. How-3 w a w w a Planck+BAO+CCPlanck+BAOCC
FIG. 5. Marginalized posterior in the w − w a plane for thethree different fits, Planck+BAO, Planck+BAO+CC and CConly. w a w w a Planck+BAOACCC
FIG. 6. Marginalized posterior in the w − w a plane for thethree different fits, Planck+BAO, CC only, and AC only. ever, as shown in Sec. VII, there is indication that thebest-fit value of A ISW is above 1 at slightly more than 1 σ .This suggests (although with low statistical significance)that DE could differ from a simple cosmological constant.To investigate this more in detail, we perform a fit withMethod 2 of Sec. VII, but with A ISW = 1, and with extraparameters accounting for dynamical Dark Energy. Forsimplicity, we use the w − w a empirical parametrization Parameter 68% limits − ω b . ± . ω cdm . ± . n s . ± . − A s . ± . h . +0 . − . w − . +0 . − . w a − . ± . Ω ,fld . +0 . − . TABLE VII. Results of the
MontePython fit with usingPlanck + BAO data. [82, 83] and the parameterized post-Friedmann frame-work of [84] and [85], which are implemented in class ,to study models with w < −
1. We test several differentfit setups. In particular, since the AC dataset is a cosmo-logical probe with its own sensitivity to the cosmologicalparameters, we test various combinations in which theAC and CC data are used separately. A further reasonto study the AC data separately from the CC ones is thatthe APS of extragalactic objects are typically difficult tomodel accurately, even at small (cid:96) , due to the non-linearityand possible stochasticity of the galaxy bias with respectto matter. Separate fits to the AC and CC data couldthen reveal inconsistencies that might be associated toour minimal assumption that the bias is linear and scale-independent. A further reason to study separately theAC and CC data is the fact that the AC ones are moreprone to possible systematic effects present in the cat-alogs like, for example, non-uniform calibration acrossthe sky. These systematics would more severely bias theAC-based cosmological inference, while the CC measure-ments are more robust in this respect, since systematicoffsets or mis-calibrations across the sky do not generallycorrelate with the LSS nor the CMB.We perform the following fits: (a) Planck+BAO, (b)Planck+BAO+CC+AC, (c) Planck+BAO+CC, (d) CConly, (e) AC only, (f) AC+CC. Case (a) has the standard6 ΛCDM parameters, plus w a , w , and one Planck nui-sance parameter, A Planck , required for the evaluation ofthe Planck likelihood [1], thus 9 parameters in total. Theresults of this baseline fit are shown in Table VII. Case(b) includes CC and AC datasets, and uses additionally15 bias parameters (24 parameters in total). Case (c) issimilar to (b) but without AC data. Since the biases arestill needed for the CC fitting, they are still included inthe fit, but with a Gaussian prior coming from fit (b).We verified that just fixing the biases to the best fit (b),instead of including them in the fit with Gaussian priors,does not actually change the results. Similarly, the resultdoes not change if the biases are taken from another fitthan (b), like (e) or (f). For fit (d), featuring only CCdata, all cosmological parameters except ( w a , w , ω cdm )and all bias parameters are either fixed or marginalizedwith Gaussian priors. For fit (e), featuring only AC data,4all cosmological parameters except ( w a , w , ω cdm ) arefixed to best-fit values, while the biases are left free, sincethey are constrained by the AC data. Finally fit (f) com-bines AC and CC data, and uses the same setup as fit(e).Figs. 5-6 show the results for w and w a (marginalizedover all the remaining parameters) for some of these fits.Table VIII gives the confidence intervals on all the pa-rameters for all our fits. The most evident result is thatthe AC-only fit selects a region of parameter space sig-nificantly in tension with the Planck+BAO constraints,basically excluding the standard case ( w , w a ) = ( − , σ . This is either a consequence of thelinear bias model not being accurate enough to providereliable cosmological constraints, or an indication of somesystematic effects in some of the catalogs. Problems inthe modeling of the bias might be particularly relevantfor the auto-correlation of the catalogs in the highest red-shift bins, which are the most sensitive to deviations froma standard cosmological constant, but also the ones ly-ing in the tail of the redshift distribution of the catalog,where different population of galaxies are probably se-lected, which requires more accurate modeling. More so-phisticated approach to the modeling of the catalog auto-correlations might be thus required to address properlythis issue. Various bias models have been proposed be-yond linear bias, like for instance models based on thehalo occupation distribution of the catalog objects (seefor instance [86]). We leave a systematic study of thissubject for future work. Intrinsic artifacts in the catalog,like non-uniformity in the sky coverage, or large errors inthe photo- z determination, are also a likely issue. Theseproblems can become more evident especially in the tailsof the redshift distribution. Indeed, the largest χ forAC fits from Table II are for the z -bins in the tail of thedistribution, especially for SDSS DR12 and QSOs, indi-cating a poor match between the model and the data.This can be seen more explicitly also in the related plotsin Appendix A.Hence, in deriving DE constraints it is more conser-vative to discard information from AC and focus on CConly. We see that the constraints from the CC data arecompatible with Planck+BAO results. However, giventhe relatively low significance of the ISW effect, the for-mer are about three times weaker than the latter for eachparameter. The direction of the degeneracy between w and w a is approximately the same in the two fits, whichwas not obvious a priori, since the two data sets are sen-sitive to Dark Energy through different physical effects(the ISW effect in CMB temperature angular spectrumfor the CC fit, and the constraint on the BAO scale forthe Planck + BAO fit). It appears that the valley of well-fitting models with w > − w ( z )crossing − . < z < .
5, but with very dif-ferent derivatives w (cid:48) ( z ). Even when w is very large, allmodels in this valley do feature accelerated expansion ofthe Universe in the recent past, but not necessarily today.In fact, when w increases while w a decreases simultane- ously, the stage of accelerated expansion is preserved buttranslated backward in time.Since the CC data are less sensitive than Planck anddo not feature a different direction of degeneracy, thejoint constraints from Planck+BAO+CC are basicallyunchanged with respect to Planck+BAO only. X. DISCUSSION AND CONCLUSIONS
We derived an updated measurement of the ISW ef-fect through cross-correlations of the cosmic microwavebackground with several galaxy surveys, namely, 2MASSPhotometric Redshift catalog (2MPZ), NVSS, SDSSQSOs, SDSS DR12 photometric redshift dataset, andWISE × SuperCOSMOS; the two latter are here usedfor the first time for an ISW analysis. We also improvedwith respect to previous analyses performing tomogra-phy within each catalog, i.e., exploiting the photomet-ric redshifts and dividing each catalog into redshift bins.We found that the current cross-correlation data providestrong evidence for the ISW effect and thus for Dark En-ergy, at the 5 σ level.However, current catalogs are still not optimal to de-rive cosmological constraints from the ISW, for two mainreasons. First, the clustering of objects requires compli-cated modeling, probably beyond the simple linear biasassumption. On this last point, improvements are possi-ble using more sophisticated modeling, but at a price ofintroducing more nuisance parameters. Also, the tails ofthe redshift distributions of the objects might be morestrongly affected by catalog systematics such as unevensampling or large photo- z errors.Second, the data used in this paper are sensitive mostlyto the redshift range 0 < z < .
6, while the ISW effectis expected to be important for 0 . < z < .
5. Severalplanned or forthcoming wide-angle galaxy surveys willcover this redshift range and should thus bring (major)improvement for ISW detection via cross-correlation withCMB. For the Euclid satellite, the predicted significanceof such a signal is ∼ σ [87], and one should expect simi-lar figures from the Large Synoptic Survey Telescope [88],and the Square-Kilometer Array [89]. The very high S/Nof ISW from these deep and wide future catalogs will notonly allow for much stronger constraints on dark energythan we obtained here, but even on some modified gravitymodels which often predict very different ISW signaturesthan ΛCDM [e.g. 90]. ACKNOWLEDGMENTS
Simulations were performed with computing resourcesgranted by RWTH Aachen University under projectthes0263.MB is supported by the Netherlands Organizationfor Scientific Research, NWO, through grant number5
Parameter AC+CC CC+bias priors AC PL+AC+CC PL+CC+bias priors − ω b . ± .
021 2 . ± .
022 2 . ± .
021 2 . ± .
022 2 . ± . ω cdm . ± . . +0 . − . . ± .
011 0 . ± . . ± . n s . ± . . ± . . ± . . ± . . ± . − A s . ± .
076 2 . ± .
080 2 . ± .
077 2 . ± .
065 2 . ± . h . +0 . − . . ± .
031 0 . ± .
058 0 . +0 . − . . +0 . − . τ reio — — — 0 . ± .
017 0 . ± . Ω Λ . ± .
029 0 . +0 . − . . +0 . − . . ± .
038 0 . +0 . − . w . +0 . − . . +0 . − . . +0 . − . − . ± . − . +0 . − . w a − . +1 . − . − . +1 . − . − . +0 . − . − . +1 . − . − . +1 . − . − A Planck — — — 100 . ± .
25 100 . ± . b , . +0 . − . . +0 . − . . +0 . − . . ± .
040 1 . +0 . − . b , . ± .
11 1 . ± .
030 1 . +0 . − . . ± .
041 1 . ± . b , . ± .
15 1 . ± .
070 2 . +0 . − . . ± .
076 1 . ± . b , SDSS . ± .
090 1 . ± .
030 1 . +0 . − . . ± .
033 1 . ± . b , SDSS . +0 . − . . ± .
030 0 . +0 . − . . ± .
027 0 . ± . b , SDSS . +0 . − . . ± .
025 0 . +0 . − . . ± .
024 0 . ± . b , SDSS . +0 . − . . ± .
035 1 . +0 . − . . ± .
036 1 . ± . b , SDSS . +0 . − . . ± .
020 0 . +0 . − . . +0 . − . . ± . b , WISC . ± .
083 0 . ± .
030 1 . +0 . − . . ± .
035 0 . ± . b , WISC . +0 . − . . ± .
030 0 . +0 . − . . ± .
029 0 . ± . b , WISC . +0 . − . . ± .
041 1 . ± .
070 0 . ± .
036 0 . ± . b , QSO . ± .
22 1 . ± .
030 1 . ± .
20 1 . ± .
22 1 . ± . b , QSO . ± .
26 2 . ± .
030 1 . +0 . − . . +0 . − . . ± . b , QSO . +0 . − . . ± .
050 2 . +0 . − . . +0 . − . . ± . b NVSS . +0 . − . . ± .
10 2 . +0 . − . . ± .
099 2 . ± . MontePython fits in the ΛCDM + w + w a model with using several combinations of Planck +BAO (PL) data, AC data and CC data. When the Planck data is not used, Gaussian priors on all cosmological parametersexcept ( ω cdm , w , w a ) are assumed. [91].This research has made use of data obtained from theSuperCOSMOS Science Archive, prepared and hosted bythe Wide Field Astronomy Unit, Institute for Astronomy,University of Edinburgh, which is funded by the UK Sci-ence and Technology Facilities Council.Funding for SDSS-III has been provided by the Al-fred P. Sloan Foundation, the Participating Institutions,the National Science Foundation, and the U.S. Depart-ment of Energy Office of Science. The SDSS-III website is . SDSS-III is managed bythe Astrophysical Research Consortium for the Partici-pating Institutions of the SDSS-III Collaboration includ-ing the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, Carnegie Mel-lon University, University of Florida, the French Partici-pation Group, the German Participation Group, HarvardUniversity, the Instituto de Astrofisica de Canarias, theMichigan State/Notre Dame/JINA Participation Group,Johns Hopkins University, Lawrence Berkeley NationalLaboratory, Max Planck Institute for Astrophysics, MaxPlanck Institute for Extraterrestrial Physics, New Mex-ico State University, New York University, Ohio StateUniversity, Pennsylvania State University, University ofPortsmouth, Princeton University, the Spanish Partici-pation Group, University of Tokyo, University of Utah,Vanderbilt University, University of Virginia, Universityof Washington, and Yale University.Some of the results in this paper have been derivedusing the GetDist package . http://healpix.sourceforge.net/ [1] Planck Collaboration, P. 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In this appendix we show the measured APS andCAPS and the related best-fit model for all the cata-logs and z -bins considered in the analysis. Dots refer tothe measured single multipoles, while data points witherror bars refer to binned measurements.9 C gg C gg C gg C gg Wi×SC z=0-0.0910 C gg Wi×SC z=0.09-0.21 10 C gg Wi×SC z=0.21-0.3
FIG. 7. Measured auto-correlation for different catalogs and redshift bins. C gg Wi×SC z=0.3-0.6 10 C gg SDSS DR12 z=0-0.110 C gg SDSS DR12 z=0.1-0.3 10 C gg SDSS DR12 z=0.3-0.410 C gg SDSS DR12 z=0.4-0.5 10 C gg SDSS DR12 z=0.5-0.7
FIG. 8. Measured auto-correlation for different catalogs and redshift bins. C gg SDSS DR12 z=0.7-1.0 10 C gg SDSS DR6 QSO z=0.5-110 C gg SDSS DR6 QSO z=0-1 10 C gg SDSS DR6 QSO z=1-210 C gg SDSS DR6 QSO z=2-3 10 C gg NVSS
FIG. 9. Measured auto-correlation for different catalogs and redshift bins. C g T
1e 7 C g T
1e 8 C g T
1e 8 C g T
1e 7
Wi×SC z=0-0.0910 C g T
1e 8
Wi×SC z=0.09-0.21 10 C g T
1e 8
Wi×SC z=0.21-0.3
FIG. 10. Measured cross-correlation with the CMB for different catalogs and redshift bins. C g T
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Wi×SC z=0.3-0.6 10 C g T
1e 8
SDSS DR12 z=0-0.110 C g T
1e 8
SDSS DR12 z=0.1-0.3 10 C g T
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SDSS DR12 z=0.3-0.410 C g T
1e 8
SDSS DR12 z=0.4-0.5 10 C g T
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SDSS DR12 z=0.5-0.7
FIG. 11. Measured cross-correlation with the CMB for different catalogs and redshift bins. C g T
1e 8
SDSS DR12 z=0.7-1.0 10 C g T
1e 8
SDSS DR6 QSO z=0.5-110 C g T
1e 8
SDSS DR6 QSO z=0-1 10 C g T
1e 8
SDSS DR6 QSO z=1-210 C g T
1e 8
SDSS DR6 QSO z=2-3 10 C g T
1e 8