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Correlating the CMB with Luminous Red Galaxies : The Integrated Sachs-Wolfe Effect

We present a 2.5 sigma detection of the Integrated Sachs-Wolfe (ISW) effect and discuss the constraints it places on cosmological parameters. We cross-correlate microwave temperature maps from the WMAP satellite with a 4000 deg^2 luminous red galaxy (LRG) overdensity map measured by the Sloan Digital Sky Survey. Accurate photometric redshifts allow us to perform a reliable auto-correlation analysis of the LRGs, eliminating the uncertainty in the galaxy bias, and combined with cross correlation signal, constrains cosmological parameters -- in particular, the matter density. We find a 2.5 sigma signal in the Ka, Q, V, and W WMAP bands, after combining the information from multipoles 2<= l<400. This is consistent with the expected amplitude of the ISW effect, but requires a lower matter density than is usually assumed: the amplitude, parametrized by the galaxy bias assuming \Omega_M=0.3, \Omega_\Lambda=0.7 and \sigma_8=0.9, is b_g = 4.05 \pm 1.54 for V band, with similar results for the other bands. This should be compared to b_g = 1.82 \pm 0.02 from the auto-correlation analysis. These data provide only a weak confirmation (2.5 sigma) of dark energy, but provide a significant upper limit: \Omega_\Lambda=0.80_{-0.06}^{+0.03} (1 sigma)_{-0.19}^{+0.05} (2 sigma), assuming a cosmology with \Omega_M+\Omega_\Lambda=1, \Omega_b = 0.05, and \sigma_8=0.9, and w=-1. The weak cross-correlation signal rules out low matter density/high dark energy density universes and, in combination with other data, strongly constrains models with w<-1.3. We provide a simple prescription to incorporate these constraints into cosmological parameter estimation methods for (\Omega_M, \sigma_8,w). (abridged)


I. INTRODUCTION
The cosmic microwave background (CMB) observed by the WMAP satellite [1] has been a font of information for cosmology. The positions of the acoustic oscillations in the angular power spectrum of temperature fluctuations imprinted at the last scattering surface measures the curvature of the universe to unprecedented accuracy. In addition, a careful modelling of the power spectrum combined with other cosmological probes have allowed extremely precise measurements of the parameters of the ΛCDM model. Subsequent measurements of the power spectrum by WMAP , as well as future experiments such as the PLANCK satellite will further constrain the space of cosmological models, as well as test our understanding of the physics underlying recombination.
CMB temperature fluctuations are not just sourced by density fluctuations at the last scattering surface; * Electronic address: npadmana@princeton.edu anisotropies also arise due to the interaction of photons with hot electrons in galaxies and clusters (the Sunyaev-Zeldovich [SZ,2] and kinetic-SZ [3] effects), as well as with the gravitational potentials along their propagation path (the Integrated Sachs-Wolfe [ISW,4] effect and gravitational lensing).
The ISW effect results from the red-(or blue-)shifting of CMB photons as they propagate through gravitational potential wells. If these potentials did not evolve, then the blueshift gained falling into a potential well would exactly cancel the redshift emerging from the well; evolving potentials spoil this cancellation. This is significant only on the largest scales, both because the power in the fluctuations of the potential is largest on large scales, and because integrating along the line of sight cancels out the temperature fluctuations (a photon is as likely to be redshifted as blueshifted).
Although the above effects are too small to unambigously detect in current CMB data (for eg. [5]), one can attempt to isolate them by cross-correlating the temperature maps with suitable tracer populations [6,7,8,9,10,11,12,13]. This paper is a continuation of these efforts, attempting to detect the ISW effect, but also goes one step further in making the ISW signal useful for cosmological parameter analysis. We combine the cross-correlation analysis with a galaxy auto-correlation measurement to remove the uncertainty due to galaxy bias. The auto-correlation analysis is only reliable when the galaxy population has well determined redshifts: we argue below that this is indeed the case for the sample used here. This paper does not focus on the autocorrelation analysis, so we perform a restricted analysis where we assume the shape of power spectrum and only constrain the amplitude. This is sufficient for the current purposes, as the errors are dominated by the statistical uncertainties in the ISW effect. A detailed autocorrelation analysis will be presented in a future paper.
A galaxy catalog used for detecting the ISW effect must satisfy several criteria. Since the ISW signal is only detectable on the largest scales, the catalog must cover as large an area as possible. In addition, a large number of galaxies is necessary to remove noise from random Poisson fluctuations. Finally, to make theoretical interpretation possible, these galaxies must be drawn from a uniform population with a well characterized redshift distribution.
Photometrically selected luminous red galaxies (LRGs) [14] from the Sloan Digital Sky Survey (SDSS) [15] are a promising candidate. These galaxies are amongst the most luminous in the Universe and so probe cosmologically interesting volumes, including the transition from matter to dark energy domination that sources the ISW signal. These galaxies are old stellar populations with very uniform spectral energy distributions that make uniform selection and accurate photometric redshifts possible. Finally, the deep and wide-angle imaging of the sky by the SDSS allows these galaxies to be selected over a large area with high densities.
Our goal is to extract the maximal amount of information on the ISW effect from the data. To do so, we employ quadratic estimator methods, which combine the data in an optimal way in the presence of noise and incomplete sky coverage [16,17]. In addition, we weight the information on different scales according to the expected scale dependence of the signal: the ISW effect is expected to dominate on large scales and be absent on small scales, so we weight these different scales accordingly. We apply these techniques to the latest SDSS data reductions, ensuring the largest available SDSS sky coverage to date.
Detecting an ISW effect would provide evidence of either a cosmological constant, quintessence or curvature independent of supernovae observations of the luminosity distance [18]. We investigate the cosmological implications of an ISW detection within the assumption of a flat universe, and in particular, on its ability to constrain the matter density. The rapid increase of the ISW signal with decreasing matter density allows us to place strong lower bounds on the matter density and suggests an alternative probe of the properties of the dark energy.
The paper is organized as follows : Sec. II intro-duces the ISW effect, developing the formalism to compute the predicted ISW signal. Sec. III describes the WMAP and SDSS data used in this paper. Sec. IV and Sec. V then discuss the cross correlation of these data, and the results obtained. Systematic effects are discussed in Sec. VI, while cosmological implications are considered in Sec. VII. We conclude in Sec. VIII. Unless otherwise specified, we assume a ΛCDM cosmology with Ω M = 0.3, Ω Λ = 0.7, H 0 = 100h km/s/Mpc and σ 8 = 0.9. Where necessary, we use h = 0.7.

II. THEORY
We briefly review the ISW effect, and its crosscorrelation with the galaxy density (see also eg. [6,19,20,21]) We consider linear functions of the density field projected onto the sky, specializing to the cases of the ISW temperature perturbations and the galaxy overdensity field. Decomposing these projected fields into spherical harmonics allows us to compute the expected autoand cross-power spectra. For simplicity, we restrict our discussion to flat universes (Ω m + Ω Λ = 1).

A. Projections onto the sky
We start with the 3D matter overdensity field, δ 3D (y), assumed to be an isotropic random variable. This allows us to define the 3D matter power spectrum, P (k), (1) We project this density field onto the sky, where f (y) weights the density field as a function of comoving distance, y, and L is a linear operator (independent of y) operating on the 3D density field. Simultaneously expanding the projected field in spherical harmonics, Y lm (n), and Fourier transforming the 3D density field, we obtain where we use the orthogonality of the Y lm 's and the expansion of a Fourier wave in spherical coordinates,

Galaxy and Temperature Fluctuations
Given the above equation, we specialize to the galaxy overdensity and the ISW temperature fluctuations. We assume that the galaxy overdensity, δ g,3D , is related to the matter overdensity by a linear bias, b g , which implies that (Eq. 3), Although linear bias must break down at some scale, it appears to be well motivated both theoretically [22] and observationally [23] on the large scales that the ISW effect is sensitive to. The redshift weighting f (y) is simply given by the galaxy selection function, φ(y), appropriately normalized, Working on large scales allows us to use linear perturbation theory where the growth of density fluctuations with time is separable, where D(y) is the growth factor. Substituting this above allows us to write Eq. 6 in the following useful form, where the window function W g (k) is given by The temperature fluctuations due to the ISW effect are given by the line of sight integral of the change in the gravitational potential to the last scattering surface, where Φ is the gravitational potential, and the overdot is the derivative with respect to conformal distance (or equivalently, conformal lookback time) at constant y 3D . The gravitational potential can be related to the 3D density fluctuations by Poisson's equation, where a is the scale factor, implying that, Taking the Fourier transform and substituting into Eq. 3, we find that, Again restricting to linear theory allows us to write the above equation in the same form as Eq. 9, where the ISW window function is given by It is straightforward to compute the relevant power spectra. The galaxy power spectrum, C gg l ≡ δ g,lm δ * g,lm , is given by, where we have used the definition of the three dimensional power spectrum (Eq. 1) and ∆ 2 (k) is the variance per logarithmic wavenumber, The cross correlation between the ISW temperature fluctuations and galaxy overdensities is similarly given by,

B. Predictions
The above formalism allows us to compute the expected ISW power spectrum given the redshift distribution of a tracer population and a particular cosmology. Fig. 1 shows the ISW power spectrum assuming the LRG redshift distribution introduced below (Sec. III B) and a flat cosmology. The growth factor is given by where a ≡ 1/(1 + z) is the scale factor, and H(a) is determined by the Friedmann equation. We note that (within the context of flat cosmologies), D/a is independent of redshift if Ω M = 1, leading to the lore that the detection of the ISW effect is evidence for some form of dark energy, although we note that universes with curvature and no dark energy/cosmological constant also lead to an ISW effect. Finally, a computational note on performing the spherical bessel integrals of the previous section : we follow [24] and recast these as logarithmically discretized Hankel transforms. In this form, the integrals can be efficiently performed via FFT convolutions using the FFTLOG algorithm [25] as implemented in [24]. The WMAP mission [1] is designed to produce all-sky maps of the CMB at multipoles up to l ∼several hundred. This analysis uses the first public release of WMAP data, consisting of the first year of observations from the Sun-Earth L2 Lagrange point. WMAP carries ten differencing assemblies (DAs), each of which measures the difference in intensity of the CMB at two points on the sky; a CMB map is constructed from these temperature differences as the satellite rotates. WMAP observes in the K (1 DA), Ka (1 DA), Q (2 DAs), V (2 DAs), and W (4 DAs) bands corresponding to central frequencies of 23, 33, 41, 61, and 94 GHz, respectively. The WMAP team has pixelized the data from each DA in the HEALPix [61] pixelization system at resolution 9 [26,27]. This system has 3,145,728 pixels, each 47.2 sq. arcmin in area. These maps are not beam-deconvolved; this, combined with the WMAP scan strategy, results in nearly uncorrelated Gaussian uncertainties on the temperature in each pixel.
We limit this analysis to the Ka through W bands. The K band is heavily contaminated by Galactic emission, increasing the number of possible systematic effects and making any error analysis unreliable. We apply the Kp0 mask specified by the WMAP team to mask out regions where Galactic foregrounds dominate. In addition, we reject pixels in the WMAP point source mask, including a 0.6 degree exclusion radius around each source. Finally, since the SDSS area used only covers about a tenth of the sky, we eliminate all pixels greater than 10 degrees from the SDSS mask, allowing us to speed up our crosscorrelation analysis. This mask, denoted Kp0∩S10\ps , admits 756,078 HEALPix pixels (9915 sq. deg). The variance in each of these pixels is computed directly from the number of observations of each pixel, as specified by the WMAP team.
We chose not to use either the WMAP "Internal Linear Combination" (ILC) map [26], or the foreground cleaned map of [28], to avoid a number of practical difficulties. These maps lose the frequency dependence of the original maps (useful to identify contaminating signals) and have complicated pixel-pixel noise correlations (increasing the complexity of the error analysis).

B. SDSS Luminous Red Galaxies
The Sloan Digital Sky Survey [15] is an ongoing effort to image approximately π steradians of the sky, and obtain spectra of approximately one million of the detected objects [29,30]. The imaging is carried out by driftscanning the sky in photometric conditions [31], in five bands (ugriz) [32,33] using a specially designed widefield camera [34]. Using these data, objects are targeted for spectroscopy [35] and are observed with a 640-fiber spectrograph on the same telescope. All of these data are processed by completely automated pipelines that detect and measure photometric properties of objects, and astrometrically calibrate the data [36,37]. The SDSS is nearing completion, and has had four major data releases [38,39,40] [62]. This paper uses all data observed through Fall 2003, processed and calibrated as described in [41].
The usefulness of LRGs as a cosmological probe has been appreciated by a number of authors [30,42]. These are typically the most luminous galaxies in the universe, and therefore probe cosmologically interesting volumes. In addition, these galaxies are generically old stellar systems with uniform spectral energy distributions (SEDs) characterized principally by a strong discontinuity at 4000Å. This combination of a uniform SED and a strong 4000Å break make LRGs an ideal candidate for photometric redshift algorithms, with redshift accuracies of σ z ∼ 0.03 [14]. We briefly outline the construction of the photometric LRG sample used in this paper below; a detailed discussion of the selection criteria and properties of the sample is in [14,43].
Our selection criteria are based on the spectroscopic selection of LRGs described in [30], extended to lower apparent luminosities. We select LRGs by choosing galaxies that both have colors consistent with an old stellar population, as well as absolute luminosities greater than a chosen threshold. The first criterion is simple to implement since the uniform SEDs of LRGs imply that they lie on an extremely tight locus in the space of galaxy colors; we simply select all galaxies that lie close to that locus. More specifically, we can define three (not independent) colors that describe this locus, where g, r, and i are the SDSS model magnitudes [38] in these bands respectively. We now make the following color selections, Making two cuts (Cut I and Cut II) is convenient since the LRG color locus changes direction sharply as the 4000 A break redshifts from the g to the r band; this division divides the sample into low redshift (Cut I, z < 0.4) and high redshift (Cut II, z > 0.4) samples.
In order to implement the absolute magnitude cut, we follow [30] and impose a cut in the galaxy colormagnitude space. The specific cuts we use are where r P etro is the SDSS r band Petrosian magnitude [38]. Finally, we reject all objects that resemble the pointspread function, or if they have colors inconsistent with normal galaxies; these cuts attempt to remove interloping stars. Applying these selection criteria to the ∼ 5500 degress of photometric SDSS imaging in the Galactic North yields a catalog of approximately 900,000 galaxies. Applying the simple template fitting photometric redshift algorithm of [14], we restrict this catalog to galaxies with 0.2 < z photo < 0.6, leaving us with ∼ 650,000 galaxies. We use the regularized inversion method of [14] as well as the photometric redshift error distribution presented there, to estimate the true redshift distribution of the sample. The results, comparing the photometric and true redshift distributions are shown in Fig. 2. The LRGs are approximately volume limited from z ∼ 0.2 to z ∼ 0.55. Comparing this to the S/N estimates of [44], we see that the LRG redshift distribution is well suited to detecting the ISW effect. Therefore, despite the availability of accurate photometric redshifts, we do not weight the LRGs in redshift any differently than they already are in the sample we use here.
This catalog is pixelized as a number overdensity, δ g = δn/n, onto a HEALPix pixelization of the sphere, with 3,145,728 pixels. Since the photometric catalogs are incomplete around bright stars, we mask regions around the stars in the Tycho astrometric catalog [45]. We also exclude data from the three southern SDSS stripes due to difficulties in photometrically calibrating them relative to the data in the Northern Galactic Cap. The final catalog covers an solid angle of 3,893 square degrees (296,872 HEALPix resolution 9 pixels) and contains 503,944 galaxies at a mean density of 1.70 galaxies per pixel. The sky coverage is shown in Fig. 3.

IV. ESTIMATING THE CROSS CORRELATION
We start by organizing the temperature fluctuations and the galaxy overdensities into a single data vector, where x B,T is a vector with the measured CMB temperature (with the monopole and dipole subtracted) in band B at every HEALPix pixel; analogously, x g is the LRG number overdensity. We suppress the band subscript for simplicity, with the implicit understanding that we always refer to the cross correlation of a single WMAP band with the LRG overdensity. The covariance matrix of x is, where C diag is given by, where N xx is the pixel noise matrix. The submatrices C T T , C gg and C gT are defined by wheren a i is the position (on the sky) of the i th point of the vector x a . The temperature-temperature, galaxy-galaxy and galaxy-temperature angular power spectra are denoted by C T T l , C gg l and C gT l respectively. In our analysis below, we use the best fit ΛCDM prediction for the temperature angular power spectrum, multiplied by the appropriate WMAP instrumental response at each multipole. The galaxy power spectrum is estimated using a pseudo-C l estimator [43], and fit by the non-linear power spectrum of [46], multiplied by a constant linear bias. We project out the monopole and dipole of both these power spectra by setting the power in the l = 0, 1 modes to a value (10 −2 ) much greater than the true power spectrum. In addition, the measured galaxy-galaxy power spectrum has excess power on large scales (l < 20) due to calibration errors; we boost the prior power spectrum to account for this.
We parametrize C gT l as a sum of bandpowers,P i,l , with amplitudes c i to be estimated, We consider two bandpowers in this paper. The first are "flat" bandpowers given bỹ where B(l) is the WMAP instrumental response [47]. This parametrizes the power spectrum as a sum of step functions and is useful when the shape of the power spectrum is unknown. However, as discussed in Sec. II, the shape of the ISW component of the galaxy-temperature correlation is well determined by the cosmology and the redshift distribution of the galaxies with only its amplitude (equivalently, the galaxy bias) unknown. We define the "template" bandpowers by replacing the flat bandpowers for l < l max with the shape of the ISW correlation, i.e.P This definition implies that c 1 directly measures the galaxy bias by optimally combining information from multipoles < l max . We are not restricted to estimating C gT l , but can simultaneously measure C T T l and C gg l . Since doing so is not the goal of this paper, we simply measure the amplitude of the input C T T l and C gg l analogous to the template bandpower above. This provides a useful runtime check of our pipeline, as well as allowing us to estimate the galaxy bias.
We estimate the c i by forming quadratic combinations of the data [16,17], These are related to the estimatedĉ i by the Fisher matrix, F,ĉ where If C gT l ≪ C gg l C T T l , then theĉ i are a good approximation to the maximum likelihood estimates of the c i . The covariance matrix of theĉ i is the inverse of the Fisher matrix, if the fiducial power spectra and noise used to compute C −1 diag correctly describe the data. These assumptions must be tested and calibrated with simulations (Sec. VI A).
Implementing the above algorithm is complicated by the sizes of the datasets; the data vector has 1,052,950 elements making both storing and naively manipulating the covariance matrix impossible on presently available computers. Working in harmonic space is also not possible due to the complicated geometry of the SDSS and WMAP masks. We implement the methods of [48] extended to the sphere. All matrix-vector operations are performed using convolutions using the spherical harmonic transform code of [7], which scales as N 3/2 , compared to the N 2 scaling of direct matrix multiplications. Matrix inversions are performed with a preconditioned   conjugate gradient code [49], using the preconditioner in Appendix B of [7]; this typically converges to a fractional precision (with an L 2 norm) of 10 −8 in ∼ 100 iterations. Finally, the Fisher matrix is computed using a Z 2 stochastic trace algorithm [48]; approximately 25 random vectors achieves the necessary precision.

V. RESULTS
The results of cross-correlating the WMAP temperature maps with the the LRG density are shown in Fig. 4, where the power spectrum has been estimated in flat bandpowers. For l < 50 where the ISW signal is expected to be the strongest (Fig. 1), we see evidence for a signal, peaking around l ∼ 20. This signal is seen in all four bands and appears to be independent of frequency, consistent with the achromatic ISW effect.
We want to estimate the amplitude of this signal, parametrized by the LRG bias. Performing a χ 2 fit to the above cross power spectra is undesirable for two reasons -(i) the flat bandpowers do not optimally use the information from the individual multipoles, and (ii) inverting a noisy covariance matrix (due to the stochastic trace estimation) is likely to incorrectly estimate the significance of the detection. To avoid both these problems, we use the template bandpowers described in the previous section to estimate the amplitude of the expected ISW signal (assuming our fiducial cosmology). We consider two values of l max at which we truncate the ISW templates, l max = 50, and l max = 400. Truncating the template at large l ensures that we optimally use all the information present in the multipoles. However, we do see (Fig. 4) some large fluctuations at small scales, l > 50. As we discuss in the next section, these appear to be statistical fluctuations, but choosing l max = 50 allows us to be conservative and ignore these multipoles.
The results are in Table I, and the best fit templates are plotted in Fig. 4. The signal again appears to be achromatic as expected. The values of the bias are consistent with measurements of the bias from the galaxygalaxy power spectrum, b g = 1.82 ± 0.02, estimated as discussed in the previous section. We also note the values of the bias obtained from truncating the templates at l max = 50 and l max = 400 are consistent, although, not surprisingly, the error associated with l max = 50 is higher.
Table I also presents the amplitude of the crosscorrelation between the LRG density and flipped (north and south reversed) and inverted (antipodal points exchanged) V band WMAP maps. These maps should not be correlated with the LRG density, and provide a useful systematic test internal to the data. Both biases obtained are consistent with zero as expected.

A. Pipeline Simulations
End-to-end simulations are essential both to validate the pipeline, as well as to calibrate errors obtained from the Fisher matrix. Since realistically simulating the galaxy population would involve understanding the formation of LRGs and therefore, is not currently feasible, we simulate the microwave sky and cross-correlate these Band bg (l < 50) bg (l < 400) χ 2 (flat) χ 2 (lmax < 50) χ 2 (lmax < 400) (  Recall that the simulated microwave maps are uncorrelated with actual LRG map, but that the different WMAP frequencies are correlated. Also shown is the χ 2 of the average power spectrum compared with zero and (in parentheses) the probability that χ 2 would be greater than the measured value for a correct model. Note that we also show the χ 2 value for the flat bandpowers in Fig. 4.
realizations with the actual LRG density map. Since the ISW cross-correlation is much smaller than the individual auto-power spectra, we simply generate microwave sky maps that are uncorrelated with the LRGs. The primary microwave temperature fluctuations are simulated by generating Gaussian random fields with an angular power spectrum identical to the best-fit ΛCDM spectrum determined by WMAP [50]. These maps are then convolved with the appropriate instrument beams [47]; we assume circular beams for simplicity. Note that the temperature fluctuations are correlated between the different frequency bands, allowing us to estimate the frequency correlation of the power spectrum estimates. We add noise by using the 110 simulated noise maps provided by the WMAP team [27]. In addition to white noise, these maps also simulate the 1/f detector noise and inter-detector correlations [51]. We make no attempt to add either Galactic or extragalactic foregrounds to these maps; these are small (especially for the V and W bands) outside of the Kp0 mask [52]. All these maps are then masked with the Kp0∩S10\ps mask, and then correlated with the actual LRG density in an identical manner to the actual temperature data.
The results from the 110 simulations are summarized in Table II. The frequency correlations on large scales are evident from the identical values of the average bias obtained. Note that measured power spectra are consistent with zero, both for the flat and template bandpowers, with χ 2 /dof ∼ 1 in all cases. Finally, we observe that the error on the bias from the Fisher matrix is consistent with the run-to-run simulation error (within the variance of the simulations, ∆σ/σ ∼ 1/2 √ 55).

B. Galactic Foregrounds
The two sources of contamination in the galaxy catalog that could correlate with Galactic microwave emission are stellar contamination and incorrect Galactic extinction corrections. The Kp0∩S10\ps mask by construction excludes regions with the worst contamination; we test for any residual contamination by directly estimating the level of contamination (Sec. VI B 1) and cross-correlating foreground emission templates with the galaxy density (Sec. VI B 2).

Stellar density and Galactic extinction contamination
In order to estimate the level of stellar contamination and incorrect extinction corrections, we compute the zero-lag correlation between the LRG density maps and maps of the stellar density and Galactic extinction. To avoid discretization effects, we re-pixelize the LRG catalog onto a HEALPIX sphere with 49152 (resolution 6) pixels. We estimate the stellar density by selecting stars identified by the SDSS photometric pipeline, with r band PSF magnitudes between 18.0 and 19.5, and pixelizing these onto the same HEALPIX sphere. The extinction map is constructed directly from the E(B −V ) reddening map of [53].
Contour plots of the LRG density as a function of stellar density and Galactic extinction are in Fig. 5; the contours show conditional probabilities of 5%, 50% and 95%. The near-horizontal contours (even at the extremes) suggest negligible levels of contamination. The cross-correlation coefficients of r = 0.006 (stellar density) and r = 0.004 (extinction) further support this conclusion.

Foreground Templates
A second test of Galactic foreground contamination is to cross-correlate templates of Galactic emission with the LRG density maps. We construct these templates with three components: • Thermal dust emission : We use the 100 micron maps of dust emission [53], extrapolated to the WMAP frequencies using the two-component dust model (Model 8) of [54].
• Free-Free emission : We model the free-free component with the all sky Hα maps of [55] using the prescription in [52].
• Spinning Dust/Synchrotron Emission: WMAP shows evidence for an additional source of emission at low frequencies due either to synchrotron emission [52], or spinning dust [56]. Since the source of this emission is still under debate, we . The contours are the 5%, 50% and 95% contours, while the horizontal line is the mean galaxy density. Also shown are the correlation coefficients, r for both datasets. There is no evidence for significant stellar contamination or incorrect extinction corrections in these data. adopt the simple phenomenological prescription in [56] to model this component.
These maps are constructed in the same pixelization as the WMAP temperature maps, and are masked with the Kp0∩S10\ps mask. The resulting maps (identical in form to the temperature maps) are then analyzed to compute the cross-correlation power spectrum. The results are shown in Fig. 6. As expected, the contamination increases with decreasing frequency, with Ka showing the worst contamination and V and W with negligible contamination. However, Galactic foreground contamination is no more than 20% of the detected signal for Ka, and is less than 10% in the other bands. Therefore, while Galactic foregrounds may be responsible for the slightly higher value of the bias in the Ka band (Table I), they cannot explain most of the signal in the temperature-galaxy correlation. Unlike the ISW effect, extragalactic foregrounds, particularly point sources, are expected to show a dependence on frequency. Table I and Fig. 4 suggest that the any such frequency dependence is much less than the errors on the measurements.

C. Redshift distribution uncertainties
Predicting the ISW signal requires knowing the redshift distribution of the LRGs. As discussed in [14], estimating the redshift distribution from measured photometric redshifts involves deconvolving the effect of photometric redshift errors. This process is unstable and must be regularized; unfortunately the choice of the regularization parameter and therefore, the estimated redshift distribution, is not unique. Fig. 7 estimates the error that this uncertainty introduces into the measurement of the bias. The upper panel compares the redshift distribution used throughout this paper with a distribution estimated with a smaller regularization parameter. The principal effect of reducing the regularization is ringing in the inversion, caused by noise in the photometric redshift distribution. However, as is evident from the lower panel, changing the redshift distribution makes only subpercent changes to the ISW signal. This insensitivity derives from the fact that the greatest contribution to the ISW signal is coming from z ∼ 0.5, where the inversion is most stable. In addition, integrating along the line of sight makes the signal principally sensitive to broadband features in the redshift distribution. The measured cross correlation (Fig. 4) shows a marginally significant (∼ 2σ) excess at multipoles around l ∼ 50, l ∼ 100 and l ∼ 200. This could either be a statistical fluctuation, or it could represent foreground contamination. As discussed above, there are a number of possible systematics -we consider each of these in turn below.
• Galactic contamination : Given the size of this effect and the expected contamination on these scales  Fig. 4. Note that the power spectra are consistent within the estimated errors. Also plotted is the ISW template scaled by the measured V band bias.
( Fig. 6), this would either require an order of magnitude mis-estimate of the known foregrounds, or a new component. Combined with the fact that this would have to be significant at high Galactic latitudes only (since the known templates agree well with the measured microwave emission in all regions outside the Kp0 mask), Galactic contamination is unlikely to be the source of this power.
• Extragalactic foregrounds : Extragalactic foregrounds broadly divide into two classes -SZ contamination and microwave point sources. Known point sources have very strong frequency dependent spectra, not observed in Fig. 4. The SZ effect is characterized by a temperature decrement [57] at these frequencies, the opposite of what is observed. The kinetic SZ effect, although frequency independent, is significantly smaller than the thermal SZ effect and should be negligible.
• Underestimation of Errors : A third possibility is that the errors (derived from the Fisher matrix) are underestimated. However, comparing with simulations, we find that the Fisher errors agree to better than ∼ 10%.
Since known systematics are unlikely to cause this excess power, we divide the LRG catalog into various sub- −0.05 (2σ). Note that the likelihood for ΩM is extremely non-Gaussian (see Table III).
samples to attempt to localize this effect. Fig. 8 shows the cross-correlation with the two contiguous regions of the SDSS (see Fig. 3), while Fig. 9 cross-correlates different photometric redshift slices. Within the errors, these different subsamples are consistent with each other around l ∼ 50; the redshift bin from z = 0.3 to 0.4 shows an excess of power at l ∼ 100. While this excess might be caused by a failure of the LRG selection algorithm at the junction of Cut I and Cut II, a precise mechanism that would cause an excess in power when correlating with the CMB is not apparent.
At the current level of accuracy, it is impossible to determine if this excess is simply statistical, or if it indicates systematic effects. Greater sky coverage as the SDSS nears completion could help answer this question. We note that this excess signal adds to the best fitted value of ISW amplitude in our analysis (compare columns 2 and 3 of Table I), but at a small level since these small scale modes are downweighted in the theoretical templates (Fig. 1). This differs from other analyses where this procedure was not adopted [10]. In addition, most of previous work has been based on correlation function analyses, where correlations between the bins in the correlation function make a scale dependence of the signal difficult to identify.

VII. COSMOLOGICAL CONSTRAINTS
The fact that the ISW signal → ∞ as Ω M → 0 suggests a method to place a lower bound on the matter density within the context of models with Ω M + Ω Λ = 1. If one computes the bias both from the galaxy autocorrelation function (b g ) and the ISW effect (b ISW ), then one finds that b ISW → 0 as Ω M → 0 faster than b g , as does the error on b ISW . Comparing the two values of the bias allows one to constrain the value of Ω M . Note that this only puts a strong lower bound on Ω M . The upper bound is determined by how strongly one detects the ISW effect; Ω M = 1 implies no ISW effect. Therefore, one would only expect to be able to rule out Ω M = 1 at ∼ 2.5σ, our quoted detection significance.
The results of performing this exercise are summarized in Fig. 10 and Table III. The galaxy and ISW bias are estimated by using their predicted templates from Sec. II. We have limited ourselves to the WMAP V band both for computational convenience and because the effects of galactic foregrounds are minimum there. We continue to use the fits to the measured power spectra used in previous sections to compute C −1 as these correctly measure the cosmic variance of these data.
The best fit value for the matter density is Ω M = 0.20 with 1σ limits of Ω M = 0.17 and Ω M = 0.26. The strength of the lower bound is evident from Table III, where we see that Ω M = 0.15 is ruled out at ∼ 2σ, while Ω M = 0.10 is ruled out at > 8σ. Modest increases in the ISW detection significance will translate into stronger constraints. For instance, increasing the survey area by a factor of 2 (e.g., with the SDSS area on completion) could rule out Ω M = 0.15 at > 3σ while an all sky survey at moderate redshift could rule it out at > 6σ. While these bounds will not replace traditional parameter estimates (for eg. [58]), they will provide much needed independent tests of dark energy.
So far we have held all other cosmological parameters fixed in the above analysis. Marginalizing over these parameters, in general, will weaken these constraints; these however will only weakly change the strength of the lower bound. For example, reducing σ 8 by a factor of x increases the ISW bias by a factor of x 2 , but the galaxygalaxy bias only by a factor of x. However, decreasing σ 8 from 0.9 to 0.8 (a 10% reduction) reduces the significance of Ω M = 0.15 to 1.6σ from 2.0σ, while making σ 8 = 1 increases it to 2.5σ.
The above discussion assumes that the acceleration is due to a cosmological constant, i.e. due to a component with equation of state, P = −ρ. An important generalization are quintessence models with equation of state, P = wρ. For w > −1, the ISW signal is enhanced, increasing the best fit value of Ω M relative to the w = −1 case, while w < −1 reduces the signal, allowing for even lower values of Ω M ; the likelihoods for Ω M assuming different values of w are shown in Fig. 11. Unfortunately, the weak ISW detection does not, by itself, allow us to place useful constraints on Ω M and w jointly. However,  where σ b,ISW is the error on the bias inferred from the ISW effect, and σ b,g ≪ 1. This describes number of σ a particular value of ΩM is away from the best fit value and can be used in likelihood analyses of cosmological parameters (χ 2 = [σ] 2 ). We also show the effect of changing σ8 on these results; increasing σ8 increases the inferred value of ΩM . The ISW effect is seen to provide a strong lower bound on the value of ΩM . one can combine the ISW constraints with other data. This would constrain models with w < −1 that require a low Ω M , currently disfavored by other data. We present an simple algorithm for computing the likelihood for cosmological models parametrized by (Ω M , σ 8 , w) from the ISW signal in Appendix A.
Finally, we make a parenthetical observation about the size of the errors for templates truncated at different l max values. We observe that for low Ω M universes, truncating at different multipoles has little effect, while for higher Ω M , there is a significant difference. This is a direct result of the template weighting; at low Ω M , most of the weight is at low multipoles where the signal peaks and so, including higher multipoles has little effect. As Ω M increases, the higher multipoles gain importance, decreasing the error.

VIII. DISCUSSION
We have cross-correlated the microwave temperature maps observed by WMAP with LRG overdensities obtained from the SDSS. Our goal is to extract the maximum amount of information on ISW available from the data. The major differences between our analysis and previous ISW analyses are: • Power Spectrum: Most analyses (with the exception of [6]) have used the correlation function instead of the power spectrum. While it is true that theoretically the power spectrum and correlation function are just Legendre transforms of each other, this is no longer true in the presence of noise and sky cuts. In particular, the correlation function has extremely correlated errors, while the power spectrum errors are almost uncorrelated on large scales. The presence of correlated errors makes an accurate determination of the covariance matrix with either Monte Carlo or jack-knife techniques more difficult.
• Optimal C −1 Weighting: We weight our data with the inverse covariance, resulting in minimumvariance error bars even in the presence of sky cuts.
• Fisher Matrix errors: We estimate our errors using the Fisher information matrix, instead of jack-knife error estimates favoured by a number of other analyses. Jack-knife errors tend to underestimate ISW errors because of the small number of uncorrelated jack-knife patches on the sky. These small numbers of samples also results in noisy covariance matrices, potentially biasing χ 2 fitting. We also note that the Fisher matrix errors are the smallest possible errors, given the intrinsic sample variance. We verify that we saturate this bound with simulations, indicating an optimal measurement.
• χ 2 Fitting: All analyses previous to this one have estimated the significance of their detection by fitting power spectrum or correlation function bins to ISW templates. We avoid this step by directly estimating the amplitude of a fiducial template; the choice of this fiducial template affects our conclusions about the significance of the effect only marginally, as seen in Table III. This has two advantages -(i) it optimally uses the information from all multipoles, and (ii) it avoids systematics introduced by χ 2 fitting with a noisy covariance matrix. The noise in the covariance matrix is further exacerbated by strong correlations between bins; this results in badly conditioned covariance matrices where errors in the most singular directions (the ones with the smallest eigenvalues) will dominate the fits. Note that these problems are made worse with the use of jack-knife errors, due to correlations in the different jack-knife samples on large scales and the small number of samples available.
The two analyses that use galaxies samples closest to ours are [8] and [10]; we note that both these analyses yield results similar to ours within the quoted errors. However, in addition to the discussion above, our analyses differ in a number of areas. The most trivial is in the difference in the sky coverage, ∼ 4000 deg 2 compared to the ∼ 2000 and ∼ 3400 deg 2 respectively. Furthermore, we use accurately calibrated photometric redshift distributions, including a deconvolution of the redshift errors [14]. Most importantly, we use theoretical templates to optimally weight the different scales, immunizing us to contamination from statistical fluctuations on smaller scales.
Despite our attempt to be nearly optimal and our use of the latest and largest SDSS data sample we do not find a strong detection of ISW. We observe a correlation on large scales at 2.5 σ, consistent with the ISW effect. We fit to the amplitude of this effect, optimally combining the information from multipoles < 400 by using a predicted template, and, for Ω M = 0.3 and σ 8 = 0.9, obtain a bias for the LRGs of b g = 4.05 ± 1.56 for the V band correlation, and similar values for the other bands. Restricting to multipoles < 50 yields b g = 2.89 ± 1.65, consistent with the previous value although with a somewhat larger error. These are consistent with the value from the galaxy-galaxy auto-correlation, b g = 1.82 ± 0.02 and even better agreement is obtained for a smaller value of Ω M (∼ 0.2).
We explore systematic effects that could contaminate the signal. Known Galactic and extragalactic foregrounds are subdominant to the measured ISW signal. However, we do detect a marginally significant (∼ 2σ) correlation at scales of l ∼ 50 and l ∼ 100. The frequency dependence and sign of the correlation are inconsistent with extragalactic foregrounds, and Galactic foregrounds are approximately an order of magnitude too small to influence our results. Subdividing the LRG sample both on the sky, and in redshift slices, yields power spectra consistent with each other, further ruling out systematic effects. We tentatively conclude that this excess power is a statistical fluctuation, and wait for future data to confirm or disprove this conclusion. This small scale excess power slightly increases the statistical significance of ISW in our analysis, but less so than in previous analyses where theoretical template weighting was not used.
The ISW effect provides a useful and independent probe of the properties of the dark energy. Although the weakness of the current detections preclude detailed parameter estimations, we attempt to estimate the matter density, by comparing the ISW bias to the value obtained from the auto-power spectrum analysis. The rapidly increasing amplitude of the ISW signal with decreasing matter density provides a strong lower bound on the matter density, ruling out Ω M = 0.15 at 2σ. These represent fits to Ω M only and assume σ 8 = 0.9 and w = −1 [58]. The current ISW detection is not significant enough to jointly constrain Ω M and w. However, these constraints can be incorporated into parameter estimation efforts; we provide a simple likelihood prescription to do this. As an example, assuming w = −1.3 reduces the predicted ISW signal and the matter density must be decreased to match observations: we find the best fit value for matter density is reduced to Ω m = 0.14 with a 2σ upper limit of 0.29. Such a low value is in conflict with other determinations of the matter density (see e.g. [59]) and as a result w < −1 models are strongly constrained.
In the present analysis we have ignored any evolution of the bias with redshift, since the redshift distribution is relatively narrow. Furthermore, estimates of the bias from the galaxy-galaxy power spectrum for different redshift slices yields a bias of between 1.7 and 1.9; this variation is too small to affect our analysis given the current errors. However, this could be a more significant issue for heterogeneous catalogs like the X-ray background and the NVSS radio catalog.
ISW detection efforts are still in their infancy, and future observations should lead to a stronger detection. This will hopefully realize the full potential of using the ISW effect as a probe of dark energy.