MEASUREMENT OF THE INTEGRATED SACHS–WOLFE EFFECT USING THE ALLWISE DATA RELEASE

We used the 9 year foreground reduced WMAP temperature data provided by the LAMBDA website² (Bennett et al. 2013). We only used the Q, V, and W bands (41, 61, and 94 GHz, respectively) as they have the smallest amount of galactic contamination. As we are only interested in $l\leqslant 100$, the maps were re-binned into HEALPix (Hierarchical Equal Area isoLatitude Pixelization; Górski et al. 2005) maps with the resolution parameter ${n}_{\mathrm{side}}=128$. We have used the KQ75y9 extended temperature analysis mask with ${f}_{\mathrm{sky}}=0.65$, which excludes point sources detected by WMAP. The final mask is the combination of the WMAP mask and a mask for the WISE data described in Section 3.3. This final mask was applied to both of the maps before taking the cross-correlation.

The WISE mission surveyed the whole sky in four bands: 3.4 (W1), 4.6 (W2), 12 (W4), and 22 μm (W4). In this study, we used the AllWISE data release, which combines the 4-band cryogenic phase with the NEOWISE post-cryo phase (Mainzer et al. 2011). This data release is deeper than the previous all-sky data release by roughly a factor of two in the W1 and W2 bands as the NEOWISE post-cryo phase only used these two bands. The AllWISE source catalog has over 747 million objects with signal-to-noise ratio (S/N) $\geqslant \,5$ for profile-fit flux measurement in at least one band. We only select sources from the catalog using W1 and W2 magnitudes with S/N $\geqslant \,5$ for the W1 band and S/N $\geqslant \,3$ for the W2 band.

The coverage of WISE is not uniform throughout the sky. The median number of exposures for the AllWISE data release is 30.17 ± 0.02 in W1 and 30.00 ± 0.03 in W2 with each exposure being 7.7 s long for both bands. According to the AllWISE Explanatory Supplement, the catalog is 95% complete for W1 < 17.1. Therefore, we applied this magnitude cut to ensure uniformity and completeness for our galaxy sample.

In this study, galaxies are defined as sources in the AllWISE catalog that are not classified as stars or AGNs. To remove stars from the object catalog, we used the following color cut: [W1–W2 $\lt \,0.4$ & W1 $\lt \,10.5{\rm{]}}$ (Jarrett et al. 2011). We also removed any object with W1–W2 $\lt \,0$ to effectively remove galactic stars (Goto et al. 2012; Ferraro et al. 2015). To select AGNs from the catalog, we used the color cut criterion (Assef et al. 2013)

$\begin{eqnarray}&&{\rm{W}}1-{\rm{W}}2\gt 0.662\exp [0.232{({\rm{W}}2-13.97)}^{2}].\end{eqnarray} \tag{ 6 }$

For some of the objects in the AllWISE catalog, the W1 source flux uncertainty could not be measured because of the presence of a large number of saturated pixels in the 3-band cryo frames containing the source. These sources lie along a narrow strip of ecliptic longitude and are marked by null values for w1msigmpro. These objects are removed from the sample. We also discarded any object with cc_flags $\ne$ 0 in W1 or W2, as a non-zero value for cc_flags indicates a spurious detection (diffraction spike, persistence, halo, or optical ghost). After applying the S/N and magnitude cuts, we are left with approximately 383 million objects. Of these objects, roughly 192 million (50.0%) are classified as galaxies, 189 million as stars (49.3%), and 2.6 million (0.7%) as AGNs according to the adopted color cut criteria.

We constructed the mask for the overdensity-CMB cross-correlation analysis with HEALPix resolution parameter ${n}_{\mathrm{side}}=128$. The moon_lev flag in the AllWISE catalog indicates the fraction of frames contaminated by moonlight among the number of frames where the flux from a source was measured. We added HEALPix pixels with more than 20% sources with moon_lev > 2 to the mask. HEALPix pixels with more than 10% sources with cc_flags $\ne$ 0 out of the total source count within the pixel are also added to the mask. As mentioned in Section 3.2, some objects in the AllWISE catalog with null values for w1msigmpro were removed from the sample, and we excluded regions with more than 1% of such sources. We also excluded regions with galactic latitude $| b| \lt 10^\circ$ to effectively remove areas of galactic contamination. For the AGN overdensity map, some HEALPix pixels (<0.2%) had an abnormally high source count and we added these pixels to the mask for the AGN overdensity map. After applying the combined final mask, the unmasked sky fraction becomes ${f}_{\mathrm{sky}}=0.46$ (Figure 2). This unmasked region contains approximately 106 million galaxies and 1.5 million AGNs.

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It is computationally difficult to evaluate the spherical Bessel integrals in Equations (4) and (5) through brute force. For efficient computation, we reformulated these integrals as logarithmically discretized Hankel transforms following Hamilton (2000). In this form, the integrals can be evaluated through fast-Fourier-transform (FFT) convolutions using the FFTLog algorithm (Talman 1978).

Finally, we used CAMB with HALOFIT (Lewis et al. 2000; Smith et al. 2003) to generate the nonlinear matter power spectra for our fiducial cosmology.

We performed source matching between the SDSS DR12 (Alam et al. 2015) galaxy sample and our AllWISE galaxy sample with a matching radius of $3^{\prime\prime}$. The matching radius was chosen based on the angular resolutions for the WISE W1 and W2 bands, which are $6\buildrel{\prime\prime}\over{.} 1$ and $6\buildrel{\prime\prime}\over{.} 4$, respectively. We only chose approximately 82 million galaxies with $r\gt 22.2$ (95% completeness limit; Abazajian et al. 2004) from the SDSS DR12 Photoz catalog.³ The common sky fraction for our mask and SDSS coverage region is ${f}_{\mathrm{sky}}=0.24$ and contains approximately 56 million AllWISE galaxies. We find matching pairs for roughly 29% of the AllWISE galaxy sample. The redshift distribution was then inferred from the SDSS photometric redshift of the matched galaxies (Figure 3). The low matching percentage of the AllWISE galaxies with SDSS is expected because high-redshift galaxies are optically fainter with redder r–W1 color and the majority of the unmatched AllWISE galaxies can be massive ellipticals at $z\gtrsim 1$ (Yan et al. 2013). As the 95% completeness magnitude limit for WISE, W1 $\lt \,17.1$, goes quite deep into redshift space, many high-redshift, WISE-selected galaxies fall beyond the SDSS 95% completeness limit of $r\lt 22.2$ (Figure 4).

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To obtain the redshift distribution of the AGN sample, we executed source matching with approximately 750,000 objects flagged as "QSO" in the SDSS DR12 SpecObjAll catalog, which has spectroscopic redshifts for roughly 4.4 million objects. The matching radius was also taken as $3^{\prime\prime}$. Out of roughly 848,000 WISE selected AGNs in the common coverage region, we found matching pairs for approximately 15% of them.

It should be noted that SDSS had an uneven target selection strategy over different redshifts leading to a bias in the redshift distribution of the SDSS objects. Therefore, the redshift distribution obtained by source matching with SDSS objects would also be similarly biased. However, the ISW effect sensitivity function is widespread over a broad range of redshift peaking at ${z}_{\mathrm{peak}}\approx 0.66$ (Figure 3) and the ISW effect measurement by cross-correlation is not very sensitive to errors in the estimation of redshift distribution (Afshordi 2004).

We used weak lensing of the CMB by our tracers of the matter overdensity to measure the bias. This method has two advantages over measuring the bias from the auto-correlation of the tracers: (1) it takes into account contamination by stars or artifacts, and (2) it is less prone to systematic errors giving a more robust estimation of the bias. The observed CMB temperature $T(\hat{{\boldsymbol{n}}})$ is the lensed remapping of the original CMB temperature field ${T}_{0}(\hat{{\boldsymbol{n}}}+{\boldsymbol{d}})=T(\hat{{\boldsymbol{n}}})$, where ${\boldsymbol{d}}$ is the deflection field. Then, CMB lensing convergence is defined as $\kappa \equiv -{\boldsymbol{\nabla }}\cdot {\boldsymbol{d}}/2=-{{\rm{\nabla }}}^{2}\phi /2$, where ϕ is the lensing potential. The lensing convergence can be expressed as the line-of-sight integral of matter fluctuation as

$\begin{eqnarray}&&\kappa (\hat{{\boldsymbol{n}}})=\int \delta (\chi \hat{{\boldsymbol{n}}},z(\chi ))\ {W}^{\kappa }(\chi )\ d\chi ,\end{eqnarray} \tag{ 7 }$

where ${W}^{\kappa }$ is the lensing window function given by (Cooray & Hu 2000):

$\begin{eqnarray}&&{W}^{\kappa }(\chi )=\displaystyle \frac{3{{\rm{\Omega }}}_{m}{H}_{0}^{2}}{2{c}^{2}}\displaystyle \frac{\chi }{a(\chi )}\displaystyle \frac{{\chi }_{\mathrm{ls}}-\chi }{{\chi }_{\mathrm{ls}}}.\end{eqnarray} \tag{ 8 }$

Here, $a(\chi )$ is the scale factor and ${\chi }_{\mathrm{ls}}\approx 14$ Gpc is the comoving distance to the last scattering surface.

The cross-correlation between the lensing convergence and matter overdensity field can be calculated using the Limber approximation (Limber 1953; Kaiser 1992), which works well for our angular scale of interest $l\gtrsim 100$, as

$\begin{eqnarray}&&{C}_{l}^{\kappa g}\approx \int \displaystyle \frac{1}{{\chi }^{2}}{W}^{\kappa }(\chi ){W}^{g}(\chi )P\left(k=\displaystyle \frac{l+1/2}{\chi },z\right)\displaystyle \frac{d\chi }{{dz}}\ {dz},\end{eqnarray} \tag{ 9 }$

where $P(k,z)$ is the nonlinear matter power spectrum at redshift z for our fiducial cosmology and W^g is the tracer distribution window function given by

$\begin{eqnarray}&&{W}^{g}(\chi )=\displaystyle \frac{{dz}}{d\chi }\displaystyle \frac{{dN}}{{dz}}\,b(\chi ).\end{eqnarray} \tag{ 10 }$

We used the lensing convergence map provided by the Planck data release⁴ 2 (Planck Collaboration 2015b) to cross-correlate it with the overdensity maps of our LSS tracers to measure their effective biases. The correlation between the WISE and Planck lensing convergence was investigated by Geach et al. (2013) and Planck Collaboration (2014), where these authors found $\sim 7\sigma$ detection for both galaxies and AGNs. Here, we repeated a similar analysis. We converted the lensing convergence and overdensity maps to HEALPix resolution ${n}_{\mathrm{side}}=512$. The mask for this analysis was taken to be a combination of the mask for the ISW effect analysis and the lensing convergence mask provided in the Planck lensing package. The unmasked sky fraction for this combined mask is ${f}_{\mathrm{sky}}=0.45$. We obtained the pseudo-power spectrum ${\tilde{C}}_{l}^{\kappa g}$ of lensing-overdensity cross-correlation using the anafast facility of the HEALPix package. We deconvolved the effect of masking and pixelization using the master approach (Hivon et al. 2002) as

$\begin{eqnarray}&&{C}_{l^{\prime} }^{\kappa g}=\displaystyle \frac{1}{{B}_{l^{\prime} }}\,\displaystyle \sum _{l}{M}_{{ll}^{\prime} }^{-1}{\tilde{C}}_{l}^{\kappa g},\end{eqnarray} \tag{ 11 }$

where M_ll' is the mode–mode coupling kernel for the applied mask and B_l is the pixel window function for ${n}_{\mathrm{side}}=512$. We binned the power spectra into six bins (band powers) in the multipole range $100\leqslant l\leqslant 400$ as ${{ \mathcal C }}_{b}^{\kappa g}={\sum }_{l}{P}_{{bl}}{C}_{l}^{\kappa g}$, where P_bl is the binning operator

$\begin{eqnarray}{P}_{{bl}}=\left\{\begin{array}{ll}\displaystyle \frac{l}{{l}_{\mathrm{low}}^{(b+1)}-{l}_{\mathrm{low}}^{(b)}}, & \mathrm{if}\quad {l}_{\mathrm{low}}^{(b)}\leqslant l\lt {l}_{\mathrm{low}}^{(b+1)},\\ 0, & \mathrm{otherwise}.\end{array}\right.\end{eqnarray} \tag{ 12 }$

Here, ${l}_{\mathrm{low}}^{(b)}$ is the lower boundary of the bth bin.

We used 100 simulated lensing convergence maps provided in the Planck lensing package to calculate the covariance matrix ${\rm{C}}$ as

$\begin{eqnarray}&&{{\rm{C}}}_{{bb}^{\prime} }={\langle ({{ \mathcal C }}_{b}^{\kappa g}-{\langle {{ \mathcal C }}_{b}^{\kappa g}\rangle }_{\mathrm{sim}})({{ \mathcal C }}_{b^{\prime} }^{\kappa g}-{\langle {{ \mathcal C }}_{b}^{\kappa g}\rangle }_{\mathrm{sim}})\rangle }_{\mathrm{sim}},\end{eqnarray} \tag{ 13 }$

where $\langle {\rangle }_{\mathrm{sim}}$ denotes an average over the simulated maps.

We fit the estimated cross-correlation band powers to different bias models for both galaxies and AGNs. Several bias models have been proposed in the literature, e.g., constant bias model $b(z)={b}_{0}$ (Peacock & Dodds 1994), linear redshift evolution model $b(z)={b}_{0}(1+z)$ (Ferraro et al. 2015), fitting function for AGNs $b(z)={b}_{0}(0.55+0.289{(1+z)}^{2})$ (Croom et al. 2005), etc. We fit for the constant and linear evolution bias models in the lensing-overdensity cross-correlation analysis for galaxies, and for the constant and fitting function bias models in the lensing-overdensity cross-correlation analysis for AGNs (Figure 5).

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We obtained the best fit for each model by maximizing the likelihood function

$\begin{eqnarray}&&{ \mathcal L }({\boldsymbol{d}};{\boldsymbol{t}},{\rm{C}})\propto \exp \left[-\displaystyle \frac{1}{2}{({\boldsymbol{d}}-{\boldsymbol{t}})}^{T}{{\rm{C}}}^{-1}({\boldsymbol{d}}-{\boldsymbol{t}})\right],\end{eqnarray} \tag{ 14 }$

where ${\boldsymbol{d}}$ is the vector containing the measured band powers, ${\boldsymbol{t}}$ is the vector containing the expected band powers of the cross-correlation for each bias model, which depend on the model parameters, and ${\rm{C}}$ is the covariance matrix. Here, we have assumed that individual data points are Gaussian distributed. Maximizing the likelihood function is equivalent to minimizing ${\chi }^{2}={({\boldsymbol{d}}-{\boldsymbol{t}})}^{T}{{\rm{C}}}^{-1}({\boldsymbol{d}}-{\boldsymbol{t}})$ and the likelihood ratio between two models are given by $-2\mathrm{ln}({{ \mathcal L }}_{1}/{{ \mathcal L }}_{2})={\rm{\Delta }}{\chi }^{2}$. The best-fit parameters for each model are given in Table 1. We used the best-fit bias models, i.e., the linear evolution model for galaxies and the constant bias model for AGNs, in the CMB temperature-overdensity cross-correlation analysis.

Table 1.  Best-fit Parameters for Different Bias Models

LSS Tracer	Bias Model b(z)	b0	${\chi }^{2}$
Galaxy sample	b0	1.17 ± 0.02a	10.6
	${b}_{0}(1+z)$	0.86 ± 0.01a	9.7
AGN sample	b0	2.90 ± 0.07a	36.3
	${b}_{0}(0.55+0.289{(1+z)}^{2})$	1.44 ± 0.04a	37.0

Note.

^aThe errors are computed by fitting the likelihood function ${ \mathcal L }({\boldsymbol{d}};{\boldsymbol{t}}({b}_{0})$, ${\rm{C}})\propto \exp [{({\boldsymbol{d}}-{\boldsymbol{t}})}^{T}{{\rm{C}}}^{-1}({\boldsymbol{d}}-{\boldsymbol{t}})]$ to a Gaussian distribution and taking the standard deviation σ of the fit as the error. Here, ${\boldsymbol{d}}$ is the vector containing the measured band powers, ${\boldsymbol{t}}$ is the vector containing the expected band powers for a given bias model, and ${\rm{C}}$ is the covariance matrix.

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We measured the cross-correlation of the WMAP CMB maps in the Q, V, and W bands and the AllWISE galaxy and AGN overdensity maps. The complex geometry of the mask induces off-diagonal correlations between the multipoles. We deconvolved the effect of masking and pixelization from the pseudo-power spectrum ${\tilde{C}}_{l}^{{Tg}}$, which is obtained through anafast, as

$\begin{eqnarray}&&{C}_{l^{\prime} }^{{Tg}}=\displaystyle \frac{1}{{B}_{l^{\prime} }{F}_{l^{\prime} }}\,\displaystyle \sum _{l}{M}_{{ll}^{\prime} }^{-1}{\tilde{C}}_{l}^{{Tg}},\end{eqnarray} \tag{ 15 }$

where ${M}_{{ll}^{\prime} }$ is the mode–mode coupling kernel for our adopted mask, B_l is the pixel window function for ${n}_{\mathrm{side}}=128$, and F_l is the WMAP beam transfer function. WMAP provides beam transfer functions for each differencing assembly in a band. We took an average of the beam transfer functions for all of the differencing assemblies in a given band to obtain the beam transfer function for each band as ${F}_{l}^{2}={\sum }_{i}^{N}{({F}_{l}^{(i)})}^{2}/N$, where N is the number of differencing assemblies in each WMAP band and the index i goes over all of the differencing assemblies. We binned the deconvolved power spectra into eight logarithmic bins (band powers) using a binning operator P_bl given by

$\begin{eqnarray}{P}_{{bl}}=\left\{\begin{array}{ll}\displaystyle \frac{1}{2\pi }\displaystyle \frac{l(l+1)}{{l}_{\mathrm{low}}^{(b+1)}-{l}_{\mathrm{low}}^{(b)}}, & \mathrm{if}\quad {l}_{\mathrm{low}}^{(b)}\leqslant l\lt {l}_{\mathrm{low}}^{(b+1)},\\ 0, & \mathrm{otherwise},\end{array}\right.\end{eqnarray} \tag{ 16 }$

where ${l}_{\mathrm{low}}^{(b)}$ denotes the lower boundary of the bth bin. We took the bin boundaries as l = 2, 5, 8, 12, 17, 26, 41, 64, and 100; thus, the first band includes l = 2, 3, 4, etc. We avoided $l\geqslant 100$ as the ISW effect is not sensitive at these small scales.

To estimate the Monte Carlo error bars and covariance matrices, we ran 5000 simulations for each WMAP band. We used our fiducial cosmological parameters and WMAP beam transfer function to obtain the simulated CMB maps using synfast included in the HEALPix package. Then, we added noise to each pixel by adding a random value from a Gaussian distribution with zero mean and standard deviation given by $\sigma ={\sigma }_{0}/\sqrt{{N}_{\mathrm{obs}}}$, where ${\sigma }_{0}$ is 2.188, 3.131, and 6.544 mK for the Q, V, and W bands, respectively, and N_obs is the effective number of observations for the corresponding pixel in the WMAP survey. Some examples of the simulated CMB maps for different WMAP bands are shown in Figure 6. We cross-correlated these simulated maps with AllWISE galaxy and AGN overdensity maps to obtain the covariance matrices according to Equation (13). The error bars are taken to be the square roots of the diagonal elements of the covariance matrix. The neighboring bins are anti-correlated by 3%–20% in the lower multipole range and correlated by roughly 20%–30% in the higher end of the multipole range (Figure 7).

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We find that the band powers are consistent across different WMAP bands (Figure 8). This indicates that the CMB maps are not likely to have significant foreground contamination.

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We obtained the amplitude A of the signal by minimizing ${\chi }^{2}={({\boldsymbol{d}}-A{\boldsymbol{t}})}^{T}{{\rm{C}}}^{-1}({\boldsymbol{d}}-A{\boldsymbol{t}})$, where ${\boldsymbol{d}}$ is the vector containing the measured band powers, ${\boldsymbol{t}}$ is the vector containing corresponding band powers of the theoretically predicted power spectra for the ΛCDM model, and ${\rm{C}}$ is the Monte Carlo covariance matrix. Then, the signal amplitude and its error are given by

$\begin{eqnarray}A & = & {{\boldsymbol{d}}}^{T}{{\rm{C}}}^{-1}{\boldsymbol{t}}{[{{\boldsymbol{t}}}^{T}{{\rm{C}}}^{-1}{\boldsymbol{t}}]}^{-1},\\ {\sigma }_{A} & = & {[{{\boldsymbol{t}}}^{T}{{\rm{C}}}^{-1}{\boldsymbol{t}}]}^{-1/2}.\end{eqnarray} \tag{ 17 }$

We calculated the significance of the detection from

$\begin{eqnarray}{\rm{S}}/{\rm{N}} & = & \sqrt{{\chi }_{\mathrm{null}}^{2}-{\chi }_{\mathrm{fit}}^{2}}\\ & = & {{\boldsymbol{d}}}^{T}{{\rm{C}}}^{-1}{\boldsymbol{t}}{[{{\boldsymbol{t}}}^{T}{{\rm{C}}}^{-1}{\boldsymbol{t}}]}^{-1/2}=\displaystyle \frac{A}{{\sigma }_{A}},\end{eqnarray} \tag{ 18 }$

where ${\chi }_{\mathrm{fit}}^{2}$ is for the best fit and ${\chi }_{\mathrm{null}}^{2}$ is for the null hypothesis with ${\boldsymbol{t}}=0$.

We detected the ISW effect signal for AllWISE galaxies with $3.3\sigma$ significance for all three WMAP bands. The combined ISW effect signal amplitude for the three WMAP bands is $A=1.18\pm 0.36$, which agrees very well with the ΛCDM prediction of A = 1. For AGNs, the ISW effect amplitude is $A=0.64\pm 0.74$ with $0.9\sigma$ significance, which is also in agreement with the ΛCDM model. The signal amplitude and some basic statistical properties for each WMAP band are given in Table 2.

Table 2.  Statistical Properties of WISE-CMB Cross-correlation Amplitudes

LSS Tracer	WMAP Band	A	S/N	${\chi }^{2}$	dof	${\rm{\Delta }}{\chi }_{{\rm{\Lambda }}\mathrm{CDM}}^{2}$	${\rm{\Delta }}{\chi }_{\mathrm{null}}^{2}$
	Q	1.18 ± 0.35	3.3	6.09	7	0.26	11.17
Galaxy sample	V	1.19 ± 0.36	3.3	6.40	7	0.32	11.31
	W	1.17 ± 0.36	3.3	6.52	7	0.21	10.58
	Q	0.65 ± 0.74	0.9	9.12	7	0.21	0.77
AGN sample	V	0.62 ± 0.74	0.8	9.86	7	0.28	0.70
	W	0.65 ± 0.74	0.9	9.32	7	0.22	0.79

Note. The "dof" column refers to the degrees of freedom of the ${\chi }^{2}$ distribution. ${\rm{\Delta }}{\chi }_{\mathrm{null}}^{2}$ shows ${\rm{\Delta }}{\chi }^{2}$ of the best fit from the null hypothesis ${\boldsymbol{t}}=0$ and ${\rm{\Delta }}{\chi }_{{\rm{\Lambda }}\mathrm{CDM}}^{2}$ shows ${\rm{\Delta }}{\chi }^{2}$ of the best fit from the ΛCDM prediction.

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